esalq course on models for longitudinal and incomplete

ESALQ Course on Models forLongitudinal and Incomplete Data

Exercises

Geert Verbeke & Geert Molenberghs

I-BioStat

Katholieke Universiteit Leuven & Universiteit Hasselt

ESALQ Course, Piracicaba, November 2014

Contents

1 Growth curves 1

1.1 Description of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Elements of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.2 SAS Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Growth Data 19

2.1 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


2.3.1 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 SAS Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Mastitis in Dairy Cattle 30

3.1 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Missing Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.2 Intraclass Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


i

CONTENTS 2

3.3.1 Data Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.2 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.4 Intraclass Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Age-related Macular Degeneration Study 50


4.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


4.3.1 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.2 SAS Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5 Blood Pressure Data 171

5.1 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.2.1 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172


5.3.1 Basic Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.3.2 Raw Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5.3.3 Covariance Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6 Marital Satisfaction Data 184


6.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186


6.3.1 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.3.2 SAS Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

6.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Dataset 1

Growth curves

1.1 Description of the Data

Goldstein (1979, Table 4.3, p. 101) and Verbeke and Molenberghs (1997) have analyzed growth curves of 20pre-adolescent girls, measured on a yearly basis from age 6 to 10. The girls were classified according to the heightof their mother (small: < 155 cm, medium: 155-164 cm, tall: > 164 cm). The individual profiles are shownin Figure 1.1, for each group separately. The measurements are given at exact years of age, some having beenpreviously adjusted to these.

The aim of the analysis is to study growth in schoolgirls and to investigate the relation of this growth to theheigth of the mother.

growthgv.sas7bdat

height : response, measured in cm

child : child identification number

age : age of the child at the time of the measurement

group : factor defining the groups ( 1: small mother, 2: medium mother, 3: tall mother)

References

Goldstein, H. (1979) The Design and Analysis of Longitudinal Studies. London: Academic Press.

Verbeke, G. and Molenberghs, G. (2000) Linear Mixed Models for Longitudinal Data. New York: Springer.

1.2 Questions

1. Fit a full multivariate model (unstructured mean, unstructured covariance matrix). Parameterize the meanin the following two ways :

1

DATASET 1. GROWTH CURVES 2

Figure 1.1: Heights of Schoolgirls. Growth curves of 20 school girls from age 6 to 10, for girls with small,

medium, or tall mothers.

(a) age group age*group

(b) age*group, no intercept

Compare the results, and interpret the parameter estimates obtained under both parameterizations.

2. Simplify the mean structure, assuming height to increase linearly as a function of age. Parameterize themean such that the estimates for the intercept and the slope of each group separately is immediatelyobtained. Interpret the obtained results.

3. Use an F -test to test whether the three groups have equal average slopes. Can this also be tested using alikelihood ratio test ?

4. Estimate the three pairwise differences of average slopes.

5. Fit a linear mixed model, assuming linear average evolutions, and including random intercepts and randomslopes for time.

6. Compare the model with a model with the same mean structure but unstructured covariance matrix, usingthe LR test statistic as well as information criteria.

7. Compare robust inference for the mean structure to naive inference.

1.3 Elements of Solution

1.3.1 Programs

1. Full multivariate model, two different parameterizations for the mean:

proc mixed data = growthgv method=ml;

class child group age;

model height = group age group*age / solution;


repeated age / type=un subject=child r rcorr;

run;

proc mixed data = growthgv method=ml;

class child group age;

model height = group*age / noint solution;

repeated age / type=un subject=child r rcorr;

run;

2. Simplified mean structure:

data test;

set growthgv;

age = age-6;

ageclss = age;

run;

proc mixed data = test method=ml ;

class child group ageclss ;

model height = group group*age / solution noint;

repeated ageclss / type=un subject=child r rcorr;

run;

3. F -test for interaction:

proc mixed data = test method=ml;




contrast ’interactie’ group*age 1 -1 0, group*age 1 0 -1;

run;



model height = group age group*age / solution;


run;

Likelihood ratio test, using ML:



model height = group age / solution;


run;

4. Estimation of pairwise differences of average slopes:





estimate ’small-medium’ group*age 1 -1 0;

estimate ’small-tall’ group*age 1 0 -1;

estimate ’medium-tall’ group*age 0 1 -1;

run;


5. Fitting the linear mixed model:

proc mixed data = test;

class child group;

model height = group group*age / noint solution;

random intercept age / type = un subject=child g;

run;

6. Comparison with model with unstructured covariance matrix:

proc mixed data = test ic;

class child group;



run;

proc mixed data = test ic;




run;

7. Robust inference:

proc mixed data = test empirical;

class child group;



run;

1.3.2 SAS Output

1. Full multivariate model, first parameterization for the mean:

Class Level Information

Class Levels Values

child 20 1 2 3 4 5 6 7 8 9 10 11 12 13

14 15 16 17 18 19 20

group 3 1 2 3

age 5 6 7 8 9 10

Iteration History

Iteration Evaluations -2 Log Like Criterion

0 1 494.40648596

1 1 239.97938687 0.00000000

Convergence criteria met.


Estimated R Matrix for child 1

Row Col1 Col2 Col3 Col4 Col5

1 5.1910 5.8479 7.0247 6.3238 6.1008

2 5.8479 7.2479 8.3169 7.6600 7.3970

3 7.0247 8.3169 10.2337 9.3406 9.3120

4 6.3238 7.6600 9.3406 8.8596 8.9731

5 6.1008 7.3970 9.3120 8.9731 9.5521

Estimated R Correlation Matrix for child 1


1 1.0000 0.9534 0.9638 0.9325 0.8664

2 0.9534 1.0000 0.9657 0.9559 0.8890

3 0.9638 0.9657 1.0000 0.9810 0.9418

4 0.9325 0.9559 0.9810 1.0000 0.9754

5 0.8664 0.8890 0.9418 0.9754 1.0000

Covariance Parameter Estimates

Cov Parm Subject Estimate

UN(1,1) child 5.1910

UN(2,1) child 5.8479

UN(2,2) child 7.2479

UN(3,1) child 7.0247

UN(3,2) child 8.3169

UN(3,3) child 10.2337

UN(4,1) child 6.3238

UN(4,2) child 7.6600

UN(4,3) child 9.3406

UN(4,4) child 8.8596

UN(5,1) child 6.1008

UN(5,2) child 7.3970

UN(5,3) child 9.3120

UN(5,4) child 8.9731

UN(5,5) child 9.5521

Fit Statistics

-2 Log Likelihood 240.0

AIC (smaller is better) 300.0

AICC (smaller is better) 326.9

BIC (smaller is better) 329.9

Null Model Likelihood Ratio Test


DF Chi-Square Pr > ChiSq

14 254.43 <.0001

Solution for Fixed Effects

Standard

Effect group age Estimate Error DF t Value Pr > |t|

Intercept 144.94 1.1682 17 124.08 <.0001

group 1 -11.2595 1.7195 17 -6.55 <.0001

group 2 -6.4857 1.6520 17 -3.93 0.0011

group 3 0 . . . .

age 6 -24.6286 0.6025 17 -40.87 <.0001

age 7 -18.2286 0.5353 17 -34.05 <.0001

age 8 -11.3000 0.4074 17 -27.74 <.0001

age 9 -5.0000 0.2579 17 -19.39 <.0001

age 10 0 . . . .

group*age 1 6 3.4786 0.8869 17 3.92 0.0011

group*age 1 7 2.8952 0.7880 17 3.67 0.0019

group*age 1 8 1.6000 0.5997 17 2.67 0.0162

group*age 1 9 0.06667 0.3796 17 0.18 0.8627

group*age 1 10 0 . . . .

group*age 2 6 2.4286 0.8521 17 2.85 0.0111

group*age 2 7 1.7000 0.7571 17 2.25 0.0383

group*age 2 8 0.5571 0.5762 17 0.97 0.3471

group*age 2 9 -0.2571 0.3647 17 -0.71 0.4903

group*age 2 10 0 . . . .

group*age 3 6 0 . . . .

group*age 3 7 0 . . . .

group*age 3 8 0 . . . .

group*age 3 9 0 . . . .

group*age 3 10 0 . . . .

Type 3 Tests of Fixed Effects

Num Den

Effect DF DF F Value Pr > F

group 2 17 19.81 <.0001

age 4 17 1263.53 <.0001

group*age 8 17 9.35 <.0001

2. Full multivariate model, second parameterization for the mean:



1 5.1910 5.8479 7.0247 6.3238 6.1008


2 5.8479 7.2479 8.3169 7.6600 7.3970

3 7.0247 8.3169 10.2337 9.3406 9.3120

4 6.3238 7.6600 9.3406 8.8596 8.9731

5 6.1008 7.3970 9.3120 8.9731 9.5521



1 1.0000 0.9534 0.9638 0.9325 0.8664

2 0.9534 1.0000 0.9657 0.9559 0.8890

3 0.9638 0.9657 1.0000 0.9810 0.9418

4 0.9325 0.9559 0.9810 1.0000 0.9754

5 0.8664 0.8890 0.9418 0.9754 1.0000



UN(1,1) child 5.1910

UN(2,1) child 5.8479

UN(2,2) child 7.2479

UN(3,1) child 7.0247

UN(3,2) child 8.3169

UN(3,3) child 10.2337

UN(4,1) child 6.3238

UN(4,2) child 7.6600

UN(4,3) child 9.3406

UN(4,4) child 8.8596

UN(5,1) child 6.1008

UN(5,2) child 7.3970

UN(5,3) child 9.3120

UN(5,4) child 8.9731

UN(5,5) child 9.5521

Fit Statistics







14 254.43 <.0001



Standard

Effect group age Estimate Error DF t Value Pr > |t|

group*age 1 6 112.53 0.9301 20 120.99 <.0001

group*age 1 7 118.35 1.0991 20 107.68 <.0001

group*age 1 8 123.98 1.3060 20 94.93 <.0001

group*age 1 9 128.75 1.2152 20 105.95 <.0001

group*age 1 10 133.68 1.2618 20 105.95 <.0001

group*age 2 6 116.26 0.8611 20 135.00 <.0001

group*age 2 7 121.93 1.0176 20 119.83 <.0001

group*age 2 8 127.71 1.2091 20 105.63 <.0001

group*age 2 9 133.20 1.1250 20 118.40 <.0001

group*age 2 10 138.46 1.1682 20 118.53 <.0001

group*age 3 6 120.31 0.8611 20 139.71 <.0001

group*age 3 7 126.71 1.0176 20 124.53 <.0001

group*age 3 8 133.64 1.2091 20 110.53 <.0001

group*age 3 9 139.94 1.1250 20 124.39 <.0001

group*age 3 10 144.94 1.1682 20 124.08 <.0001


Num Den


group*age 15 20 8511.09 <.0001

3. Model with simplified mean structure:



1 8.3335 9.2121 10.9714 9.7768 8.0299

2 9.2121 10.9370 12.6383 11.4137 9.6365

3 10.9714 12.6383 15.2964 13.7401 11.9267

4 9.7768 11.4137 13.7401 12.6910 11.2065

5 8.0299 9.6365 11.9267 11.2065 11.0842



1 1.0000 0.9649 0.9717 0.9507 0.8355

2 0.9649 1.0000 0.9771 0.9688 0.8752

3 0.9717 0.9771 1.0000 0.9862 0.9160

4 0.9507 0.9688 0.9862 1.0000 0.9449

5 0.8355 0.8752 0.9160 0.9449 1.0000




UN(1,1) child 8.3335

UN(2,1) child 9.2121

UN(2,2) child 10.9370

UN(3,1) child 10.9714

UN(3,2) child 12.6383

UN(3,3) child 15.2964

UN(4,1) child 9.7768

UN(4,2) child 11.4137

UN(4,3) child 13.7401

UN(4,4) child 12.6910

UN(5,1) child 8.0299

UN(5,2) child 9.6365

UN(5,3) child 11.9267

UN(5,4) child 11.2065

UN(5,5) child 11.0842

Fit Statistics







14 220.49 <.0001


Standard

Effect group Estimate Error DF t Value Pr > |t|

group 1 110.69 0.5916 17 187.09 <.0001

group 2 115.46 0.5477 17 210.79 <.0001

group 3 117.99 0.5477 17 215.41 <.0001

age*group 1 5.2503 0.1492 17 35.19 <.0001

age*group 2 5.6202 0.1381 17 40.69 <.0001

age*group 3 6.5309 0.1381 17 47.29 <.0001


Num Den


group 3 17 41946.6 <.0001


age*group 3 17 1710.16 <.0001

4. F -test for interaction, using CONTRAST statement:

Fit Statistics







14 220.49 <.0001


Standard


group 1 110.69 0.5916 17 187.09 <.0001

group 2 115.46 0.5477 17 210.79 <.0001

group 3 117.99 0.5477 17 215.41 <.0001

age*group 1 5.2503 0.1492 17 35.19 <.0001

age*group 2 5.6202 0.1381 17 40.69 <.0001

age*group 3 6.5309 0.1381 17 47.29 <.0001


Num Den


group 3 17 41946.6 <.0001

age*group 3 17 1710.16 <.0001

Contrasts

Num Den

Label DF DF F Value Pr > F

interactie 2 17 21.58 <.0001

5. F -test for interaction, using reparameterized mean structure:


1 8.3335 9.2121 10.9714 9.7768 8.0299


2 9.2121 10.9370 12.6383 11.4137 9.6365

3 10.9714 12.6383 15.2964 13.7401 11.9267

4 9.7768 11.4137 13.7401 12.6910 11.2065

5 8.0299 9.6365 11.9267 11.2065 11.0842



1 1.0000 0.9649 0.9717 0.9507 0.8355

2 0.9649 1.0000 0.9771 0.9688 0.8752

3 0.9717 0.9771 1.0000 0.9862 0.9160

4 0.9507 0.9688 0.9862 1.0000 0.9449

5 0.8355 0.8752 0.9160 0.9449 1.0000



UN(1,1) child 8.3335

UN(2,1) child 9.2121

UN(2,2) child 10.9370

UN(3,1) child 10.9714

UN(3,2) child 12.6383

UN(3,3) child 15.2964

UN(4,1) child 9.7768

UN(4,2) child 11.4137

UN(4,3) child 13.7401

UN(4,4) child 12.6910

UN(5,1) child 8.0299

UN(5,2) child 9.6365

UN(5,3) child 11.9267

UN(5,4) child 11.2065

UN(5,5) child 11.0842

Fit Statistics







14 220.49 <.0001



Standard


Intercept 117.99 0.5477 17 215.41 <.0001

group 1 -7.3019 0.8062 17 -9.06 <.0001

group 2 -2.5309 0.7746 17 -3.27 0.0045

group 3 0 . . . .

age 6.5309 0.1381 17 47.29 <.0001

age*group 1 -1.2806 0.2033 17 -6.30 <.0001

age*group 2 -0.9107 0.1953 17 -4.66 0.0002

age*group 3 0 . . . .


Num Den


group 2 17 41.78 <.0001

age 1 17 5012.88 <.0001

age*group 2 17 21.58 <.0001

6. LR test for interaction:



1 10.7331 12.7128 15.8314 15.7841 14.5792

2 12.7128 15.7540 19.2380 19.4131 17.9419

3 15.8314 19.2380 24.3104 24.6143 23.0795

4 15.7841 19.4131 24.6143 25.7152 24.3106

5 14.5792 17.9419 23.0795 24.3106 23.5812



1 1.0000 0.9777 0.9801 0.9501 0.9164

2 0.9777 1.0000 0.9830 0.9645 0.9309

3 0.9801 0.9830 1.0000 0.9845 0.9639

4 0.9501 0.9645 0.9845 1.0000 0.9872

5 0.9164 0.9309 0.9639 0.9872 1.0000



UN(1,1) child 10.7331

UN(2,1) child 12.7128

UN(2,2) child 15.7540


UN(3,1) child 15.8314

UN(3,2) child 19.2380

UN(3,3) child 24.3104

UN(4,1) child 15.7841

UN(4,2) child 19.4131

UN(4,3) child 24.6143

UN(4,4) child 25.7152

UN(5,1) child 14.5792

UN(5,2) child 17.9419

UN(5,3) child 23.0795

UN(5,4) child 24.3106

UN(5,5) child 23.5812

Fit Statistics







14 211.76 <.0001


Standard


Intercept 116.72 0.5385 17 216.74 <.0001

group 1 -5.7142 0.7658 17 -7.46 <.0001

group 2 -1.4018 0.7358 17 -1.91 0.0738

group 3 0 . . . .

age 5.5060 0.1121 17 49.11 <.0001


Num Den


group 2 17 29.67 <.0001

age 1 17 2412.16 <.0001

7. Estimation of pairwise differences of average slopes:

Fit Statistics








14 220.49 <.0001


Standard


group 1 110.69 0.5916 17 187.09 <.0001

group 2 115.46 0.5477 17 210.79 <.0001

group 3 117.99 0.5477 17 215.41 <.0001

age*group 1 5.2503 0.1492 17 35.19 <.0001

age*group 2 5.6202 0.1381 17 40.69 <.0001

age*group 3 6.5309 0.1381 17 47.29 <.0001


Num Den


group 3 17 41946.6 <.0001

age*group 3 17 1710.16 <.0001

Estimates

Standard

Label Estimate Error DF t Value Pr > |t|

small-medium -0.3699 0.2033 17 -1.82 0.0865

small-tall -1.2806 0.2033 17 -6.30 <.0001

medium-tall -0.9107 0.1953 17 -4.66 0.0002

8. Linear mixed model with random intercepts and slopes:

Estimated G Matrix

Row Effect child Col1 Col2

1 Intercept 1 7.0684 0.3546

2 age 1 0.3546 0.1331




UN(1,1) child 7.0684

UN(2,1) child 0.3546

UN(2,2) child 0.1331

Residual 0.4758

Fit Statistics

-2 Res Log Likelihood 315.7






3 179.31 <.0001


Standard


group 1 112.92 1.1071 60 102.00 <.0001

group 2 116.38 1.0250 60 113.54 <.0001

group 3 120.61 1.0250 60 117.68 <.0001

age*group 1 5.2700 0.1735 60 30.37 <.0001

age*group 2 5.5671 0.1606 60 34.66 <.0001

age*group 3 6.2486 0.1606 60 38.90 <.0001


Num Den


group 3 60 12381.1 <.0001

age*group 3 60 1212.19 <.0001

9. Calculation of information criteria for random-effects model:

Fit Statistics








3 179.31 <.0001

Information Criteria

Neg2LogLike Parms AIC AICC HQIC BIC CAIC

315.7 4 323.7 324.2 324.5 327.7 331.7

10. Calculation of information criteria for model with unstructured covariance matrix:

Fit Statistics







14 215.67 <.0001

Information Criteria

Neg2LogLike Parms AIC AICC HQIC BIC CAIC

279.4 15 309.4 315.5 312.3 324.3 339.3

11. Robust inference:

Estimated G Matrix

Row Effect child Col1 Col2

1 Intercept 1 7.0684 0.3546

2 age 1 0.3546 0.1331



UN(1,1) child 7.0684

UN(2,1) child 0.3546

UN(2,2) child 0.1331


Residual 0.4758

Fit Statistics







3 179.31 <.0001


Standard


group 1 112.92 0.7545 60 149.66 <.0001

group 2 116.38 0.9886 60 117.72 <.0001

group 3 120.61 1.0751 60 112.19 <.0001

age*group 1 5.2700 0.1337 60 39.40 <.0001

age*group 2 5.5671 0.1059 60 52.55 <.0001

age*group 3 6.2486 0.1957 60 31.93 <.0001


Num Den


group 3 60 16280.8 <.0001

age*group 3 60 1777.78 <.0001

1.3.3 Discussion

1. Two different parameterizations of the full multivariate model: As is to be expected, the samemaximized likelihood value is obtained, as well as the same marginal covariance matrix. Under the firstparameterization, the parameters represent contrasts between specific group means. Under the first param-eterization, the parameters represent average response values at each combination of group with age.

2. Simplified mean structure: The value 6 is first subtracted from the variable age, such that intercepts canbe interpreted as averages at the start of the study, rather than averages at the age of 0 years. In order toimmediately obtain estimates for the three intercepts as well as three slopes (rather than contrasts betweenthose parameters), the mean is parameterized as group group*age.

3. F -test for interaction: The F -test can be obtained under the original parameterization, using a CON-TRAST statement, but also as a default F -test under the parameterization group age group*age.


4. LR test: Since the models we are comparing have different mean structures, a valid LR test can only beobtained under ML estimation. The difference in minus twice the maximized log-likelihood value equals288.0493− 275.3983 = 12.651 which is significant when compared to a chi-squared distribution with twodegrees of freedom (p = 0.00179).

5. Pairwise comparisons of slopes: There is no significant difference in average trend between the childrenwith small mothers and children with medium mothers (p = 0.0865), but the children with tall mothersgrow significantly faster than those with small mothers or those with medium mothers (p < 0.0001, andp = 0.0002, respectively).

6. Comparison with random-effects model:

• Using LR test: Since both models have the same mean structure, a valid LR test can be obtained,even under REML estimation. The difference in minus twice the maximized REML log-likelihood valueequals 315.7478−279.3868 = 36.361 which is significant when compared to a chi-squared distributionwith 11 degrees of freedom (p = 0.000147).

• Using Information Criteria: AIC and HQIC prefer the unstructured model, while BIC and CAICprefer the random-effects model. This illustrates the informal, exploratory, nature of the informationcriteria.

The LR test clearly suggests that the random-effects covariance matrix is not appropriate. This impliesthat inferences, under the random-effects model, for the fixed effects need correction for this, hence robustinference is required.

7. Robust inference: Note that the same maximized likelihood values are obtained, as well as the sameparameter estimates. The only difference is in the standard errors for the fixed effects. Some robust s.e.’sare smaller, others are larger than the original naive s.e.’s.

Dataset 2

Growth Data

2.1 The Data

These data, introduced by Potthof and Roy (1964), contain growth measurements for 11 girls and 16 boys. Foreach subject the distance from the center of the pituitary to the maxillary fissure was recorded at the ages 8,10, 12, and 14. The data were used by Jennrich and Schluchter (1986) to illustrate estimation methods forunbalanced data, where unbalancedness is now to be interpreted in the sense of an unequal number of boys andgirls.

Little and Rubin (1987) deleted 9 of the [(11 +16)× 4] measurements, rendering 9 incomplete subjects. Deletionis confined to the age 10 measurements. Little and Rubin (1987) describe the mechanism to be such that subjectswith a low value at age 8 are more likely to have a missing value at age 10.

These data are analyzed at length in Verbeke and Molenberghs (1997; Ch. 4.4 and throughout Ch. 5) and inVerbeke and Molenberghs (2000, Ch. 17). We refer to these texts for tabular and graphical representations ofthe data.

2.2 Questions

Consider the complete cases (GROWTHCC.SAS7BDAT). In particular, we focus on Model 6:

• separate linear profiles for boys and girls,

• random intercept and random slope with general 2 × 2 covariance matrix D.

Fit this model under the following 2 × 2 “factorial design”:

• untransformed age (representing time as well) versus

age2 =age− 11

3,

• with and without the ‘nobound’ option.

19

DATASET 2. GROWTH DATA 20

Upon fitting these models, formulate an answer to the following questions:

• In which statement is the ‘nobound’ option to be placed ?

• What is the use of the ‘nobound’ option ?

• Study the LOG screen of SAS with care.

• Study the covariance matrices of the random effects in all four models.

• Calculate the covariance matrix of the 4 repeated measures and discuss.

• Do the different models (substantially) alter the estimates of the mean response profiles ?

In addition, starting from Model 7, formulate two additional models, with the same fixed-effects structure andthe following variance-covariance structures:

• Model 9: random intercept and AR(1) serial structure;

• Model 10: random intercept, AR(1) serial structure, and measurement error.

(Note: the answer to this last question can be found in Verbeke and Molenberghs (1997, p. 261).)

The relevant variables in GROWTHCC.SAS7BDAT are:

MEASURE: response variable, growth measurement;

SEX: 1 for boys, 2 for girls;

AGE: age in years (8, 10, 12, 14).


2.3.1 Programs

libname m ’\bartsas\gent’;

proc mixed data=m.growthcc method=ml;

title ’Growth Data (Complete Cases), Model 6’;

title2 ’Untransformed age’;

class sex idnr;

model measure=sex age*sex / s;

random intercept age / type=un subject=idnr g v vcorr;

run;

proc mixed data=m.growthcc method=ml;


title2 ’Untransformed age - Nobound’;

class sex idnr;

model measure=sex age*sex / s;

parms / nobound;


random intercept age / type=un subject=idnr g v vcorr;

run;

data hulp;

set m.growthcc;

age2=age-11;

age2=age2/3;

run;

proc mixed data=hulp method=ml;


title2 ’Transformed age: (age-11)/3’;

class sex idnr;

model measure=sex age2*sex / s;

random intercept age2 / type=un subject=idnr g v vcorr;

run;

proc mixed data=hulp method=ml;


title2 ’Transformed age: (age-11)/3 - Nobound’;

class sex idnr;

model measure=sex age2*sex / s;

parms / nobound;

random intercept age2 / type=un subject=idnr g v vcorr;

run;

These programs include the options ‘g’, ‘v’, and ‘vcorr’ in the RANDOM statement. They produce the covariancematrix of the random effects (D in the notes), and the total variance-covariance matrix together with the derivedcorrelation matrix, respectively:

V = ZDZ′ +D = ZDZ′ +G.

Indeed, the design matrix for the random effects is common to all subjects:

Z =

1 8

1 10

1 12

1 14

.

2.3.2 SAS Output

Growth Data (Complete Cases), Model 6 11

Untransformed age 15:29 Monday, April 24, 2000

The MIXED Procedure


Class Levels Values

SEX 2 1 2

IDNR 18 1 2 4 5 7 8 11 12 14 15 17 18

19 20 21 22 25 26


ML Estimation Iteration History

Iteration Evaluations Objective Criterion

0 1 173.83597294

1 2 217.95942441 0.31464225

2 1 207.78119685 1.02096475

3 1 197.51841663 3.35649391

4 1 187.22267468 11.09067497

5 1 176.90118943 36.62110390

6 1 166.51223080 118.41214091

7 1 155.90700618 275.34606092

8 3 152.48908720 35.92986685

9 1 151.30481194 16.48576496

10 2 147.72234368 292.28142431

11 2 146.11148938 0.45023154

12 3 144.78806054 .

13 3 144.61828469 .

14 1 144.51081783 0.00004756

15 1 144.50729532 0.00000007

16 1 144.50729017 0.00000000


G Matrix

Effect IDNR Row COL1 COL2

INTERCEPT 1 1 -0.00000000 0.11027204

AGE 1 2 0.11027204 0.00000000

V Matrix for IDNR 1

Row COL1 COL2 COL3 COL4

1 3.49309330 1.98489673 2.20544081 2.42598489

2 1.98489673 3.93418146 2.42598489 2.64652897

3 2.20544081 2.42598489 4.37526962 2.86707306

4 2.42598489 2.64652897 2.86707306 4.81635778


Untransformed age 15:29 Monday, April 24, 2000

V Correlation Matrix for IDNR 1


1 1.00000000 0.53543361 0.56414131 0.59145762

2 0.53543361 1.00000000 0.58473419 0.60798111

3 0.56414131 0.58473419 1.00000000 0.62456404

4 0.59145762 0.60798111 0.62456404 1.00000000

Covariance Parameter Estimates (MLE)



UN(1,1) IDNR -0.00000000

UN(2,1) IDNR 0.11027204

UN(2,2) IDNR 0.00000000

Residual 1.72874065

Model Fitting Information for MEASURE

Description Value

Observations 72.0000

Log Likelihood -138.417

Akaike’s Information Criterion -142.417

Schwarz’s Bayesian Criterion -146.971


Null Model LRT Chi-Square 29.3287

Null Model LRT DF 3.0000

Null Model LRT P-Value 0.0000


Effect SEX Estimate Std Error DF t Pr > |t|

INTERCEPT 18.29285714 1.24734381 16 14.67 0.0001

SEX 1 -0.11558442 1.59560756 36 -0.07 0.9427

SEX 2 0.00000000 . . . .

AGE*SEX 1 0.67500000 0.08864486 36 7.61 0.0001

AGE*SEX 2 0.47500000 0.11112222 36 4.27 0.0001

Tests of Fixed Effects

Source NDF DDF Type III F Pr > F

SEX 1 36 0.01 0.9427

AGE*SEX 2 36 38.13 0.0001


Untransformed age - Nobound 15:29 Monday, April 24, 2000

The MIXED Procedure


Class Levels Values

SEX 2 1 2

IDNR 18 1 2 4 5 7 8 11 12 14 15 17 18

19 20 21 22 25 26




0 1 173.83597294

1 1 143.34782275 0.00000000


G Matrix


INTERCEPT 1 1 -4.40880772 0.49152778

AGE 1 2 0.49152778 -0.03388889

V Matrix for IDNR 1


1 3.26591450 1.72758117 2.16841450 2.60924784

2 1.72758117 4.01202561 2.33813672 2.64341450

3 2.16841450 2.33813672 4.48702561 2.67758117

4 2.60924784 2.64341450 2.67758117 4.69091450



1 1.00000000 0.47725909 0.56644875 0.66662895

2 0.47725909 1.00000000 0.55107255 0.60933295

3 0.56644875 0.55107255 1.00000000 0.58362597

4 0.66662895 0.60933295 0.58362597 1.00000000



UN(1,1) IDNR -4.40880772


Untransformed age - Nobound 15:29 Monday, April 24, 2000



UN(2,1) IDNR 0.49152778

UN(2,2) IDNR -0.03388889

Residual 1.97916667


Description Value







PARMS Model LRT Chi-Square 30.4882

PARMS Model LRT DF 3.0000

PARMS Model LRT P-Value 0.0000



INTERCEPT 18.29285714 1.07304256 16 17.05 0.0001

SEX 1 -0.11558442 1.37264065 36 -0.08 0.9334

SEX 2 0.00000000 . . . .

AGE*SEX 1 0.67500000 0.07691166 36 8.78 0.0001

AGE*SEX 2 0.47500000 0.09641387 36 4.93 0.0001



SEX 1 36 0.01 0.9334

AGE*SEX 2 36 50.65 0.0001


Transformed age: (age-11)/3 15:29 Monday, April 24, 2000

The MIXED Procedure


Class Levels Values

SEX 2 1 2

IDNR 18 1 2 4 5 7 8 11 12 14 15 17 18

19 20 21 22 25 26



0 1 173.83597294

1 4 144.51319717 0.00239701

2 1 144.35391719 0.00006774

3 1 144.34888414 0.00000014

4 1 144.34887385 0.00000000


G Matrix



INTERCEPT 1 1 2.43597060 0.47994377

AGE2 1 2 0.47994377 0.00000000

V Matrix for IDNR 1


1 3.22932491 1.79604558 2.11600809 2.43597060

2 1.79604558 3.86924994 2.43597060 2.75593312

3 2.11600809 2.43597060 4.50917496 3.07589563

4 2.43597060 2.75593312 3.07589563 5.14909999



1 1.00000000 0.50809871 0.55451450 0.59737977

2 0.50809871 1.00000000 0.58319005 0.61743269

3 0.55451450 0.58319005 1.00000000 0.63834783

4 0.59737977 0.61743269 0.63834783 1.00000000


Transformed age: (age-11)/3 15:29 Monday, April 24, 2000



UN(1,1) IDNR 2.43597060

UN(2,1) IDNR 0.47994377

UN(2,2) IDNR -0.00000000

Residual 1.75324185


Description Value











INTERCEPT 23.51785714 0.64078981 16 36.70 0.0001

SEX 1 2.08441558 0.81970108 36 2.54 0.0154

SEX 2 0.00000000 . . . .

AGE2*SEX 1 2.02500000 0.26781249 36 7.56 0.0001


AGE2*SEX 2 1.42500000 0.33572072 36 4.24 0.0001



SEX 1 36 6.47 0.0154

AGE2*SEX 2 36 37.59 0.0001


Transformed age: (age-11)/3 - Nobound

15:29 Monday, April 24, 2000

The MIXED Procedure


Class Levels Values

SEX 2 1 2

IDNR 18 1 2 4 5 7 8 11 12 14 15 17 18

19 20 21 22 25 26



0 1 173.83597294

1 1 143.34782275 0.00000000


G Matrix


INTERCEPT 1 1 2.30424784 0.35625000

AGE2 1 2 0.35625000 -0.30500000

V Matrix for IDNR 1


1 3.26591450 1.72758117 2.16841450 2.60924784

2 1.72758117 4.01202561 2.33813672 2.64341450

3 2.16841450 2.33813672 4.48702561 2.67758117

4 2.60924784 2.64341450 2.67758117 4.69091450



1 1.00000000 0.47725909 0.56644875 0.66662895


2 0.47725909 1.00000000 0.55107255 0.60933295

3 0.56644875 0.55107255 1.00000000 0.58362597

4 0.66662895 0.60933295 0.58362597 1.00000000


Transformed age: (age-11)/3 - Nobound

15:29 Monday, April 24, 2000



UN(1,1) IDNR 2.30424784

UN(2,1) IDNR 0.35625000

UN(2,2) IDNR -0.30500000

Residual 1.97916667


Description Value






PARMS Model LRT Chi-Square 30.4882

PARMS Model LRT DF 3.0000

PARMS Model LRT P-Value 0.0000



INTERCEPT 23.51785714 0.63234705 16 37.19 0.0001

SEX 1 2.08441558 0.80890106 36 2.58 0.0142

SEX 2 0.00000000 . . . .

AGE2*SEX 1 2.02500000 0.23073499 36 8.78 0.0001

AGE2*SEX 2 1.42500000 0.28924162 36 4.93 0.0001



SEX 1 36 6.64 0.0142

AGE2*SEX 2 36 50.65 0.0001

2.3.3 Discussion

Note that the omission of ‘nobound’ forces the G matrix to have nonnegative diagonal elements. However, evenin that case the G matrix is non-positive definite ! Hence, this solution makes little sense. This non-PSD propertyis listed in the LOG screen of SAS.


However, the resulting V matrix is positive definite in all cases and hence a valid covariance structure is obtained.

Both models with ‘nobound’ produce the same V matrix, which is different from the other two. The modelswithout ‘nobound’ have a different V matrix: they ‘hit’ the boundary of the parameter space considered at adifferent location.

Finally, there is no impact on the fixed effects.

Dataset 3

Mastitis in Dairy Cattle

3.1 The Data

This example, concerning the occurrence of the infectious disease mastitis in dairy cows, was introduced in Diggleand Kenward (1994) and reanalyzed in Kenward (1998). Data were available of the milk yields in thousands ofliters of 107 dairy cows from a single herd in 2 consecutive years: Yij (i = 1, . . . , 107; j = 1, 2). In the first year,all animals were supposedly free of mastitis; in the second year, 27 became infected. Mastitis typically reducesmilk yield, and the question of scientific interest is whether the probability of occurrence of mastitis is related tothe yield that would have been observed had mastitis not occurred. A graphical representation of the completedata is given in Figure 3.1.

The data are analyzed at length in Verbeke and Molenberghs (2000).

Figure 3.1: Mastitis in Dairy Cattle. The first panel shows a scatter plot of the second measurement versus

the first measurement. The second panel shows a scatter plot of the change versus the baseline measurement.

30

DATASET 3. MASTITIS IN DAIRY CATTLE 31

3.2 Questions

3.2.1 Missing Data Analysis

Construct an unstructured bivariate model for these data, under a variety of missing data assumptions:

• complete case analysis

• last observation carried forward

• unconditional mean imputation

• conditional mean imputation

• likelihood-based available case analysis

Is it possible to code the models such that both variances are produced, together with the correlation ?

3.2.2 Intraclass Correlation

Calculate the intraclass correlation, under the assumptions of heterogeneous and homogeneous variance.


3.3.1 Data Manipulation

The following GAUSS code was used to construct the datasets:

data1="\\bartsas\\gent\\mast01";

open handle=^data1 for read;

mast01=readr(handle,1000);

close(handle);

ind=mast01[.,1];

tijd=mast01[.,2];

resp=mast01[.,3];

n=rows(resp)/2;

resp=ind[seqa(1,2,n)]~reshape(resp,n,2);

respcc=packr(resp);

resplocf=resp;

test=missrv(resplocf[.,3],10000);

resplocf[.,3]=(test.*(test./=10000))+(resplocf[.,2].*(test.==10000));


hulp=meanc(respcc[.,3]);

respmean=resp;

respmean[.,3]=missrv(respmean[.,3],hulp);

respcond=resp;

hulp=6.4435+0.6479*inv(0.9115)*(resp[.,2]-6.4435+0.7359);

respcond[.,3]=(test.*(test./=10000))+(hulp.*(test.==10000));

h1=(respcc[.,1].*.ones(2,1));

h2=ones(80,1).*.(0|1);

h3=vec(respcc[.,2 3]’);

let naam=ident ti yy;

dataset="\\bartsas\\gent\\mastcc";

create handle=^dataset with ^naam,0,4;

writer(handle,h1~h2~h3);

close(handle);

h1=(resplocf[.,1].*.ones(2,1));

h2=ones(107,1).*.(0|1);

h3=vec(resplocf[.,2 3]’);


dataset="\\bartsas\\gent\\mastlocf";



close(handle);

h1=(respmean[.,1].*.ones(2,1));

h2=ones(107,1).*.(0|1);

h3=vec(respmean[.,2 3]’);


dataset="\\bartsas\\gent\\mastmean";



close(handle);

h1=(respcond[.,1].*.ones(2,1));

h2=ones(107,1).*.(0|1);

h3=vec(respcond[.,2 3]’);


dataset="\\bartsas\\gent\\mastcond";



close(handle);

3.3.2 Datasets

MAST01.SAS7BDAT: Original data.

MASTCC.SAS7BDAT: Complete cases only.

MASTLOCF.SAS7BDAT: Last observation carried forward.


MASTMEAN.SAS7BDAT: Unconditional mean imputation.

MASTCOND.SAS7BDAT: Conditional mean imputation.

3.3.3 Model Formulation

SAS Code

libname m ’c:\bartsas\gent’;

proc mixed data=m.mastcc method=ml covtest;

title ’Complete Case Analysis’;

class ti;

model yy=ti / s;

repeated ti / type=csh subject=ident r;

run;


title ’Complete Case Analysis - Random Effects Version’;

class ti;

model yy=ti / s;

repeated ti / type=un(1) subject=ident r;

random intercept / subject=ident v;

id ident ti;

run;

proc mixed data=m.mastlocf method=ml covtest;

title ’Last Observation Carried Forward’;

class ti;

model yy=ti / s;


run;

proc mixed data=m.mastmean method=ml covtest;

title ’Unconditional Mean Imputation’;

class ti;

model yy=ti / s;


run;

proc mixed data=m.mastcond method=ml covtest;

title ’Conditional Mean Imputation’;

class ti;

model yy=ti / s;


run;

proc mixed data=m.mast01 method=ml covtest;

title ’Available Case/Ignorable Analysis’;

class ti;

model yy=ti / s;



run;

SAS Output

Complete Case Analysis 12:17 Sunday, April 23, 2000 523

The MIXED Procedure


Class Levels Values

TI 2 0 1



0 1 177.40559384

1 1 140.31178413 0.00000000


R Matrix for Subject 1

Row COL1 COL2

1 0.91148521 0.64793800

2 0.64793800 1.31835993


Cov Parm Subject Estimate Std Error Z Pr > |Z|

Var(1) IDENT 0.91148521 0.14411847 6.32 0.0001

Var(2) IDENT 1.31835993 0.20845101 6.32 0.0001

CSH IDENT 0.59107371 0.07274285 8.13 0.0001

Model Fitting Information for YY

Description Value











Effect TI Estimate Std Error DF t Pr > |t|

INTERCEPT 6.44348574 0.12837250 79 50.19 0.0001




TI 0 -0.73588719 0.10804913 79 -6.81 0.0001

TI 1 0.00000000 . . . .



TI 1 79 46.39 0.0001

Complete Case Analysis - Random Effects Version 525

12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TI 2 0 1



0 1 177.40559384

1 1 140.31178413 0.00000000



Row COL1 COL2

1 0.26354721

2 0.67042193

V Matrix for Subject 1

Row COL1 COL2

1 0.91148521 0.64793800


2 0.64793800 1.31835993



INTERCEPT IDENT 0.64793800 0.14236789 4.55 0.0001

UN(1,1) IDENT 0.26354721 0.10728212 2.46 0.0140

UN(2,1) IDENT 0.00000000 . . .

UN(2,2) IDENT 0.67042193 0.14494710 4.63 0.0001


Description Value






Complete Case Analysis - Random Effects Version 526

12:17 Sunday, April 23, 2000



INTERCEPT 6.44348574 0.12837250 79 50.19 0.0001

TI 0 -0.73588719 0.10804913 79 -6.81 0.0001

TI 1 0.00000000 . . . .



TI 1 79 46.39 0.0001

Last Observation Carried Forward 527

12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TI 2 0 1




0 1 222.10149267

1 1 166.53560980 0.00000000



Row COL1 COL2

1 0.86701950 0.63835255

2 0.63835255 1.21014683



Var(1) IDENT 0.86701950 0.11853647 7.31 0.0001

Var(2) IDENT 1.21014683 0.16544787 7.31 0.0001

CSH IDENT 0.62319955 0.05912776 10.54 0.0001


Description Value










12:17 Sunday, April 23, 2000



INTERCEPT 6.31526919 0.10634747 106 59.38 0.0001

TI 0 -0.55019603 0.08649246 106 -6.36 0.0001

TI 1 0.00000000 . . . .



TI 1 106 40.46 0.0001

Unconditional Mean Imputation 529

12:17 Sunday, April 23, 2000


The MIXED Procedure


Class Levels Values

TI 2 0 1



0 1 197.62938561

1 1 162.83816167 0.00000000



Row COL1 COL2

1 0.86701950 0.48443961

2 0.48443961 0.98568967



Var(1) IDENT 0.86701950 0.11853647 7.31 0.0001

Var(2) IDENT 0.98568967 0.13476072 7.31 0.0001

CSH IDENT 0.52402875 0.07012647 7.47 0.0001


Description Value










12:17 Sunday, April 23, 2000



INTERCEPT 6.44348562 0.09597944 106 67.13 0.0001

TI 0 -0.67841246 0.09088505 106 -7.46 0.0001


TI 1 0.00000000 . . . .



TI 1 106 55.72 0.0001

Conditional Mean Imputation 531

12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TI 2 0 1



0 1 208.19212620

1 1 151.19968795 0.00000000



Row COL1 COL2

1 0.86701950 0.61631974

2 0.61631974 1.07943125



Var(1) IDENT 0.86701950 0.11853647 7.31 0.0001

Var(2) IDENT 1.07943125 0.14757680 7.31 0.0001

CSH IDENT 0.63708027 0.05743659 11.09 0.0001


Description Value











12:17 Sunday, April 23, 2000



INTERCEPT 6.48434231 0.10043975 106 64.56 0.0001

TI 0 -0.71926916 0.08167701 106 -8.81 0.0001

TI 1 0.00000000 . . . .



TI 1 106 77.55 0.0001

Available Case/Ignorable Analysis 533

12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TI 2 0 1



0 1 197.91504041

1 2 159.45922194 0.00001599

2 1 159.45793872 0.00000000



Row COL1 COL2

1 0.86702230 0.61635644

2 0.61635644 1.29589716



Var(1) IDENT 0.86702230 0.11853604 7.31 0.0001


Var(2) IDENT 1.29589716 0.19963832 6.49 0.0001

CSH IDENT 0.58147565 0.07117100 8.17 0.0001


Description Value










12:17 Sunday, April 23, 2000



INTERCEPT 6.48434379 0.12172372 106 53.27 0.0001

TI 0 -0.71927064 0.10676608 79 -6.74 0.0001

TI 1 0.00000000 . . . .



TI 1 79 45.39 0.0001

3.3.4 Intraclass Correlation

The programs considered sofar employed heterogeneous compound symmetry models which, in the case of twooutcomes, produce two variances, one for each time point, and a correlation coefficient. While not entirelyconsistent with the definition of intraclass correlation, it is a relevant concept in this case since there is a naturalordering of the measurements within a cow.

In case variances are equal (which they are in this case, as can be verified from, for example, likelihood ratiotests), then a common variance can be considered using a number of equivalent models such as:

• compound symmetry,

• random intercepts,

• AR(1) (this equivalence holds only in the case of two outcomes).

The advantage of the latter model is that it explicitly produces a correlation coefficient. For the other models,the correlation has to be calculated from the variance parameters.


Program



title ’Complete Case Analysis’;

class ti;

model yy=ti / s;

repeated ti / type=ar(1) subject=ident r;

run;

proc mixed data=m.mastlocf method=ml covtest;

title ’Last Observation Carried Forward’;

class ti;

model yy=ti / s;


run;

proc mixed data=m.mastmean method=ml covtest;

title ’Unconditional Mean Imputation’;

class ti;

model yy=ti / s;


run;

proc mixed data=m.mastcond method=ml covtest;

title ’Conditional Mean Imputation’;

class ti;

model yy=ti / s;


run;

proc mixed data=m.mast01 method=ml covtest;

title ’Available Case/Ignorable Analysis’;

class ti;

model yy=ti / s;


run;

SAS Output


The MIXED Procedure


Class Levels Values

TI 2 0 1




0 1 177.40559384

1 1 144.43830022 0.00000000



Row COL1 COL2

1 1.11492257 0.64793800

2 0.64793800 1.11492257



AR(1) IDENT 0.58115067 0.07404335 7.85 0.0001

Residual 1.11492257 0.14417333 7.73 0.0001


Description Value











INTERCEPT 6.44348574 0.11805309 79 54.58 0.0001

TI 0 -0.73588719 0.10804913 79 -6.81 0.0001




TI 1 0.00000000 . . . .




TI 1 79 46.39 0.0001


12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TI 2 0 1



0 1 222.10149267

1 1 171.33415449 0.00000000



Row COL1 COL2

1 1.03858316 0.63835255

2 0.63835255 1.03858316



AR(1) IDENT 0.61463788 0.06015230 10.22 0.0001

Residual 1.03858316 0.11785263 8.81 0.0001


Description Value












INTERCEPT 6.31526919 0.09852099 106 64.10 0.0001


12:17 Sunday, April 23, 2000



TI 0 -0.55019603 0.08649246 106 -6.36 0.0001

TI 1 0.00000000 . . . .



TI 1 106 40.46 0.0001


12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TI 2 0 1



0 1 197.62938561

1 1 163.44410735 0.00000000



Row COL1 COL2

1 0.92635458 0.48443961

2 0.48443961 0.92635458



AR(1) IDENT 0.52295268 0.07023539 7.45 0.0001

Residual 0.92635458 0.10106048 9.17 0.0001



Description Value











INTERCEPT 6.44348562 0.09304579 106 69.25 0.0001


12:17 Sunday, April 23, 2000



TI 0 -0.67841246 0.09088505 106 -7.46 0.0001

TI 1 0.00000000 . . . .



TI 1 106 55.72 0.0001


12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TI 2 0 1



0 1 208.19212620

1 1 153.34854853 0.00000000




Row COL1 COL2

1 0.97322537 0.61631974

2 0.61631974 0.97322537



AR(1) IDENT 0.63327546 0.05790386 10.94 0.0001

Residual 0.97322537 0.11136442 8.74 0.0001


Description Value











INTERCEPT 6.48434231 0.09537067 106 67.99 0.0001


12:17 Sunday, April 23, 2000



TI 0 -0.71926916 0.08167701 106 -8.81 0.0001

TI 1 0.00000000 . . . .



TI 1 106 77.55 0.0001



12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TI 2 0 1



0 1 197.91504041

1 2 164.89047899 0.00008734

2 1 164.88320081 0.00000002

3 1 164.88319889 0.00000000



Row COL1 COL2

1 1.04588712 0.58890927

2 0.58890927 1.04588712



AR(1) IDENT 0.56307154 0.07233631 7.78 0.0001

Residual 1.04588712 0.12061328 8.67 0.0001


Description Value










12:17 Sunday, April 23, 2000




INTERCEPT 6.47584805 0.10967069 106 59.05 0.0001

TI 0 -0.71077490 0.10389734 79 -6.84 0.0001

TI 1 0.00000000 . . . .



TI 1 79 46.80 0.0001

Summary of Intraclass Correlations

The combined evidence from both approaches (heterogeneous and homogeneous variance structure) and all 5missing data strategies is as follows (estimates of correlation with their standard errors in parenthesis):

Method Heterogeneous Homogeneous

Complete Cases 0.591 (0.073) 0.581 (0.074)

LOCF 0.623 (0.059) 0.615 (0.060)

Unconditional Mean 0.524 (0.070) 0.523 (0.070)

Conditional Mean 0.637 (0.057) 0.633 (0.058)

Ignorable Analysis 0.581 (0.071) 0.563 (0.072)

Dataset 4

Age-related Macular Degeneration Study


These data arise from a randomized multi-centric clinical trial comparing an experimental treatment (interferon-α) to a corresponding placebo in the treatment of patients with age-related macular degeneration. Throughoutthe analyses done, we focus on the comparison between placebo and the highest dose (6 million units daily) ofinterferon-α (Z), but the full results of this trial have been reported elsewhere (Pharmacological Therapy forMacular Degeneration Study Group 1997). Patients with macular degeneration progressively lose vision. In thetrial, the patients’ visual acuity was assessed at different time points (4 weeks, 12 weeks, 24 weeks, and 52 weeks)through their ability to read lines of letters on standardized vision charts. These charts display lines of 5 lettersof decreasing size, which the patient must read from top (largest letters) to bottom (smallest letters). Each linewith at least 4 letters correctly read is called one ‘line of vision.’ The patient’s visual acuity is the total numberof letters correctly read. The primary endpoint of the trial was the loss of at least 3 lines of vision at 1 year,compared to their baseline performance (a binary endpoint). The secondary endpoint of the trial was the visualacuity at 1 year (treated as a continuous endpoint).

Table 4.1: Age Related Macular Degeneration Trial. Loss of at least 3 lines of vision at 1 year according to loss ofat least 2 lines of vision at 6 months and according to randomized treatment group (placebo versus interferon-α).

12 months

Placebo Active

6 months 0 1 0 1

No event (0) 56 9 31 9

Event (1) 8 30 9 38

Table 4.2 shows the visual acuity (mean and standard error) by treatment group at baseline, at 6 months, andat 1 year. Visual acuity can be measured in several ways. First, one can record the number of letters read.Alternatively, dichotomized versions (at least 3 lines of vision lost, or at least 3 lines of vision lost) can be usedas well. Although there are 190 subjects with both month 6 and month 12 measurements available, the totalnumber of longitudinal profiles is 240, but only for 188 of these have the four follow-up measurements been made.

50

DATASET 4. AGE-RELATED MACULAR DEGENERATION STUDY 51

Table 4.2: Age Related Macular Degeneration Trial. Mean (standard error) of visual acuity at baseline, at 6months and at 1 year according to randomized treatment group (placebo versus interferon-α).

Time point Placebo Active Total

Baseline 55.3 (1.4) 54.6 (1.3) 55.0 (1.0)

6 months 49.3 (1.8) 45.5 (1.8) 47.5 (1.3)

1 year 44.4 (1.8) 39.1 (1.9) 42.0 (1.3)

allarmd.sas7bdat

1. crf: patient identification number

2. trt: treatment indicator (1: placebo, 4: interferon-α)

3. visual0, visual12, visual24, visual52: responses at baseline, 4, 12, 24, and 52 weeks

4. lesion: the number of lesions

References

Molenberghs, G. and Verbeke, G. (2005) Models for Discrete Longitudinal Data. New York: Springer.

4.2 Questions

1. Describe the various missingness patterns and discuss. Pay attention to the different status of completers,monotone sequences, and non-monotone sequences.

2. Formulate and fit a standard GEE and linearization-based GEE model. Apply CC, LOCF, and use the dataas they are. Pay attention to the choice of the working correlation structure and discuss the differencebetween model based and empirically corrected standard errors. The covariate effects of interest are time(unstructured) as well as treatment by time interaction.

3. Supplement the GEE analysis with a WGEE counterpart. Describe the differences and explain.

4. Discuss the relationship between the (absence of) evidence for MAR and the difference between the GEE–WGEE results.

5. Are these differences important for the conclusions drawn from the study?

6. Formulate and fit a random-intercepts model, with the same covariates included as for the GEE model. UseCC, LOCF, and direct likelihood. Discuss.

7. Supplement the WGEE analysis with an MI-GEE analysis, where the data are first multiply imputed andthen a standard GEE analysis is performed, before combining them into a single set of inferences.

8. Fit the random-intercepts logistic regression using the PQL and MQL methods, under ML and REML, andcompare the results.

9. Refit the model using numerical integration with an increasing number of quadrature points. Compareresults between Gaussian Quadrature, Laplace approximation, and adaptive Gaussian Quadrature. Extractthe Empirical Bayes estimates for the random-effects and compute summary statistics.


10. Test for random-slopes and treatment effect using a likelihood ratio test.

11. Plot the marginal average evolutions under the final model for the two treatment groups.

12. Formulate a GLMM in which the time variable is considered categorical and test for a treatment differenceat week 52.


4.3.1 Programs

1. Exploration of missing data mechanisms.

data armd14;

set allarmd;

diff4=visual4-visual0;




bindif4=0; if diff4 <= 0 then bindif4=1;

bindif12=0;if diff12 <= 0 then bindif12=1;



if diff4=. then bindif4=.;




if trt=1 then treat=1;


subject=_n_;

run;

proc sort data=armd14;

by treat;

run;

proc mi data=armd14 seed=675938 simple nimpute=0;

title ’standard EM’;

em itprint outem=growthem1;

var diff4 diff12 diff24 diff52;

by treat;

run;

proc mi data=armd14 seed=675938 simple nimpute=0;

title ’EM with CC initial values’;

em itprint outem=growthem1 initial=cc;

var diff4 diff12 diff24 diff52;

by treat;

run;

2. Preparation of CC, LOCF and observed data datasets. Preparation of WGEE analysis.


/*

** we create a longitudinal dataset with four binary outcomes - complete & incomplete

**

*/

proc print data=allarmd;

run;

data armd11;

set allarmd;















run;

proc print data=armd11;

run;

/* switch from horizontal to vertical dataset */

data armd111;

set armd11;

array x (4) bindif4 bindif12 bindif24 bindif52;

do j=1 to 4;

bindif=x(j);

time=j;

subject=_n_;

output;

end;

run;

proc print data=armd111;

var subject bindif4 bindif12 bindif24 bindif52 treat time bindif;

run;

%macro cc(data=,id=,time=,response=,out=);

%if %bquote(&data)= %then %let data=&syslast;

proc freq data=&data noprint;

tables &id /out=freqsub;

tables &time / out=freqtime;

run;


proc iml;

use freqsub;

read all var {&id,count};

nsub = nrow(&id);

use freqtime;

read all var {&time,count};

ntime = nrow(&time);

use &data;

read all var {&id,&time,&response};

n = nrow(&response);

complete = j(n,1,1);

ind = 1;

do while (ind <= nsub);

j = 1;

do while (j <= ntime);

if (&response[(ind-1)*ntime+j]=.) then

complete[(ind-1)*ntime+1:(ind-1)*ntime+ntime]=0;

j = j+1;

end;

ind = ind+1;

end;

create help var {&id &time &response complete};

append;

quit;

data &out;

merge &data help;

if complete=0 then delete;

drop complete;

run;

%mend;

%cc(data=armd111,id=subject,time=time,response=bindif,out=armdcc);

proc print data=armdcc;

run;

%macro locf(data=,id=,time=,response=,out=);



tables &id /out=freqsub;


run;

proc iml;

use freqsub;

read all var {&id,count};

nsub = nrow(&id);

use freqtime;

read all var {&time,count};


use &data;

read all var {&id,&time,&response};



locf = &response;

ind = 1;

print nsub;

print ntime;


j=2;


if (locf[(ind-1)*ntime+j]=.) then locf[(ind-1)*ntime+j]=locf[(ind-1)*ntime+j-1];

j= j+1;

end;

ind = ind+1;

end;

create help var {&id &time &response locf};

append;

quit;

data &out;

merge &data help;

run;

%mend;

%locf(data=armd111,id=subject,time=time,response=bindif,out=armdlocf);

proc print data=armdlocf;

var subject treat time bindif locf;

run;

* WGEE: macro for creating variables "dropout" and "prev" */

%macro dropout(data=,id=,time=,response=,out=);



tables &id /out=freqid;


run;

proc iml;

reset noprint;

use freqid;

read all var {&id};

nsub = nrow(&id);

use freqtime;

read all var {&time};


time = &time;

use &data;

read all var {&id &time &response};


dropout = j(n,1,0);

ind = 1;


j=1;

if (&response[(ind-1)*ntime+j]=.) then print "First Measurement is Missing";


if (&response[(ind-1)*ntime+j]^=.) then

do;

j = ntime;

do until (j=1);

if (&response[(ind-1)*ntime+j]=.) then

do;

dropout[(ind-1)*ntime+j]=1;

j = j-1;

end;

else j = 1;

end;

end;

ind = ind+1;

end;

prev = j(n,1,1);

prev[2:n] = &response[1:n-1];

i=1;

do while (i<=n);

if &time[i]=time[1] then prev[i]=.;

i = i+1;

end;

create help var {&id &time &response dropout prev};

append;

quit;

data &out;

merge &data help;

run;

%mend;

%dropout(data=armd111,id=subject,time=time,response=bindif,out=armdhlp);

proc genmod data=armdhlp descending;

class trt prev lesion time;

model dropout = prev trt lesion time / pred dist=b;

ods output obstats=pred;

ods listing exclude obstats;

run;

proc print data=pred;

run;

data pred;

set pred;

keep observation pred;

run;

data armdhlp;

merge pred armdhlp;

run;

proc print data=armdhlp;

run;


%macro dropwgt(data=,id=,time=,pred=,dropout=,out=);



tables &id /out=freqid;


run;

proc iml;

reset noprint;

use freqid;

read all var {&id};

nsub = nrow(&id);

use freqtime;

read all var {&time};


time = &time;

use &data;

read all var {&id &time &pred &dropout};

n = nrow(&pred);

wi = j(n,1,1);

ind = 1;


wihlp = 1;

stay = 1;

/* first measurement */

if (&dropout[(ind-1)*ntime+2]=1)

then do;

wihlp = pred[(ind-1)*ntime+2];

stay = 0;

end;

else if (&dropout[(ind-1)*ntime+2]=0)

then wihlp = 1-pred[(ind-1)*ntime+2];

/* second to penultimate measurement */

j=2;

do while ((j <= ntime-1) & stay);

if (&dropout[(ind-1)*ntime+j+1]=1)

then do;

wihlp = wihlp*pred[(ind-1)*ntime+j+1];

stay = 0;

end;

else if (&dropout[(ind-1)*ntime+j+1]=0)

then wihlp = wihlp*(1-pred[(ind-1)*ntime+j+1]);

j = j+1;

end;

j = 1;


wi[(ind-1)*ntime+j]=wihlp;

j = j+1;

end;

ind = ind+1;

end;

create help var {&id &time &pred &dropout wi};

append;

quit;

data &out;


merge &data help;

data &out;

set &out;

wi=1/wi;

run;

%mend;

%dropwgt(data=armdhlp,id=subject,time=time,pred=pred,dropout=dropout,out=armdwgee);

proc print data=armdwgee;

var subject time bindif dropout prev pred wi;

run;

3. CC, LOCF, and observed data analyses.

/* LOCF, CC, direct likelihood, (W)GEE analysis */

proc genmod data=armdcc;

title ’CC - GEE’;

class time treat subject;

model bindif = time treat*time / noint dist=binomial;

repeated subject=subject / withinsubject=time type=exch modelse;

run;

proc glimmix data=armdcc;

title ’CC - GEE - linearized version’;

nloptions maxiter=50 technique=newrap;


model bindif = time treat*time / noint solution dist=binary;

random _residual_ / subject=subject type=cs;

run;

proc glimmix data=armdcc empirical;

title ’CC - GEE - linearized version - empirical’;





run;

proc genmod data=armdlocf;

title ’LOCF - GEE’;


model locf = time treat*time / noint dist=binomial;


run;

proc glimmix data=armdlocf;

title ’LOCF - GEE - linearized version’;



model locf = time treat*time / noint solution dist=binary;


run;


proc glimmix data=armdlocf empirical;

title ’LOCF - GEE - linearized version - empirical’;



model locf = time treat*time / noint solution dist=binary ;


run;

proc genmod data=armdwgee;

title ’data as is - GEE’;




run;

proc glimmix data=armdwgee;

title ’data as is - GEE - linearized version’;





run;

proc glimmix data=armdwgee empirical;

title ’data as is - GEE - linearized version - empirical’;



model bindif = time treat*time / noint solution dist=binary ;


run;

proc genmod data=armdwgee;

title ’data as is - WGEE’;

scwgt wi;




run;

proc glimmix data=armdwgee;

title ’data as is - WGEE - linearized version’;


weight wi;




run;

proc glimmix data=armdwgee empirical;

title ’data as is - WGEE - linearized version - empirical’;

weight wi;




model bindif = time treat*time / noint solution dist=binary ;


run;

proc glimmix data=armdcc method=rspl;

title ’CC - mixed - PQL’;




random intercept / subject=subject type=un g gcorr;

run;

proc glimmix data=armdcc method=rspl;

title ’CC - mixed - PQL’;





run;

proc glimmix data=armdlocf method=rspl;

title ’LOCF - mixed - PQL’;



model locf = time treat*time / noint solution dist=binary;


run;

proc glimmix data=armdwgee method=rspl;

title ’as is - mixed - PQL’;





run;

data help;

set armdcc;

time1=0;

time2=0;

time3=0;

time4=0;

if time=1 then time1=1;




run;

proc nlmixed data=help qpoints=20 maxiter=100 technique=newrap;

title ’CC - mixed - numerical integration’;

eta = beta11*time1+beta12*time2+beta13*time3+beta14*time4

+b

+(beta21*time1+beta22*time2+beta23*time3+beta24*time4)*(2-treat);

p = exp(eta)/(1+exp(eta));


model bindif ~ binary(p);

random b ~ normal(0,tau*tau) subject=subject;

estimate ’tau^2’ tau*tau;

run;

data help;

set armdlocf;

time1=0;

time2=0;

time3=0;

time4=0;





run;


title ’LOCF - mixed - numerical integration’;


+b



model locf ~ binary(p);



run;

data help;

set armdwgee;

time1=0;

time2=0;

time3=0;

time4=0;





run;


title ’as is - mixed - numerical integration’;


+b






run;

4. Multiple imputation.


libname m ’\bartsas\mvboek’ ;

options nocenter;

data m.armd13;

set m.allarmd;







subject=_n_;

run;

proc sort data=m.armd13;

by treat;

run;

proc mi data=m.armd13 seed=486048 simple out=m.armd13a nimpute=10 round=0.1;

var lesion diff4 diff12 diff24 diff52;

by treat;

run;

data m.armd13a;

set m.armd13a;









run;

proc print data=m.armd13a;

var _imputation_ diff4 diff12 diff24 diff52 bindif4 bindif12 bindif24 bindif52;

where (subject=1);

run;

data m.armd13b;

set m.armd13a;

array x (4) bindif4 bindif12 bindif24 bindif52;

array y (4) diff4 diff12 diff24 diff52;

do j=1 to 4;

bindif=x(j);

diff=y(j);

time=j;

output;

end;

run;

proc print data=m.armd13b;

title ’Dataset after imputation’;


var _imputation_ subject time diff bindif;

run;

data m.armd13c;

set m.armd13b;

time1=0;

time2=0;

time3=0;

time4=0;

trttime1=0;

trttime2=0;

trttime3=0;

trttime4=0;





if (time=1 & treat=1) then trttime1=1;




run;

proc sort data=m.armd13c;

by _imputation_ subject time;

run;

proc genmod data=m.armd13c;

title ’GEE after multiple imputation’;

class time subject;

by _imputation_;

model bindif = time1 time2 time3 time4 trttime1 trttime2 trttime3 trttime4

/ noint dist=binomial covb;


ods output ParameterEstimates=gmparms parminfo=gmpinfo CovB=gmcovb;

run;

data gmpinfo;

set gmpinfo;

if parameter=’Prm1’ then delete;

run;

proc print data=gmparms;

run;

proc print data=gmcovb;

run;

proc print data=gmpinfo;

run;

proc mianalyze parms=gmparms covb=gmcovb parminfo=gmpinfo wcov bcov tcov;

modeleffects time1 time2 time3 time4 trttime1 trttime2 trttime3 trttime4;


run;

proc nlmixed data=m.armd13c qpoints=20 maxiter=100 technique=newrap cov ecov;

title ’NLMIXED after multiple imputation’;

by _imputation_;


+b

+beta21*trttime1+beta22*trttime2+beta23*trttime3+beta24*trttime4;




estimate ’tau2’ tau*tau;

ods output ParameterEstimates=nlparms CovMatParmEst=nlcovb

AdditionalEstimates=nlparmsa CovMatAddEst=nlcovba;

run;

proc mianalyze parms=nlparms covb=nlcovb;

title ’MIANALYZE for NLMIXED’;

modeleffects beta11 beta12 beta13 beta14 beta21 beta22 beta23 beta24 tau;

run;

5. Fit GLMMs using PQL and MQL.

proc glimmix data = m.armd method = RSPL;

title ’PQL REML’;

class subject;

model bindif (event = ’1’) = timec treat * timec / dist = binary solution;

random intercept / subject = subject;

run;

proc glimmix data = m.armd method = MSPL noclprint noitprint;

title ’PQL ML’;

class subject;



run;

proc glimmix data = m.armd method = RMPL noclprint noitprint;

title ’MQL REML’;

class subject;



run;

proc glimmix data = m.armd method = MMPL noclprint noitprint;

title ’MQL ML’;

class subject;



run;

6. Fit GLMMs using Gaussian Quadrature.


*Get initial values for GLMMs;

proc genmod data = m.armd descending;

title ’Initial Values’;

model bindif = timec treat * timec / dist = bin;

run;

* Gauss-Quadrature non-adaptive Q = 25;

proc nlmixed data = m.armd noad qpoints = 25;

title ’MML GQ 25’;

parms beta0 = 0.5670 beta1 = 0.0098 beta2 = 0.0133 sigmab = 2;

eta = beta0 + b + beta1 * timec + beta2 * treat * timec;

pr = exp(eta) / (1 + exp(eta));

model bindif ~ binary(pr);

random b ~ normal(0, sigmab**2) subject = subject;

estimate ’sigmab^2’ sigmab**2;

run;

* Gauss-Quadrature non-adaptive Q = 51;

proc nlmixed data = m.armd noad qpoints = 51;

title ’MML GQ 51’;







run;

* Laplace approximation;

proc nlmixed data = m.armd qpoints = 1;

title ’MML Laplace’;







run;

* Gauss-Quadrature adaptive Q = 5;


title ’MML AGQ 5’;







run;

* Gauss-Quadrature adaptive Q = 11;


proc nlmixed data = m.armd qpoints = 11 ebopt;






random b ~ normal(0, sigmab**2) subject = subject out = EB;


run;

proc univariate data = EB;

var estimate;

histogram estimate;

title ’Empirical Bayes Estimates’;

run;

7. Likelihood Ratio Tests for random-slopes and treatment.

* Test for random slopes;


title ’MML AGQ 11 / Slopes’;

parms beta0 = 0.5670 beta1 = 0.0098 beta2 = 0.0133

sigmab1 = 2 sigmab2 = 1 rho = -0.4;

eta = beta0 + b1 + beta1 * timec + b2 * timec + beta2 * treat * timec;



random b1 b2 ~ normal([0, 0], [sigmab1**2, rho * sigmab1 * sigmab2, sigmab2**2])

subject = subject;

run;

data LRT;

L01 = -2 * (-449.285125 - (-443.794458));

df = 2;

pval = 1 - probchi(L01, 2);

run;

proc print data = LRT;

run;

* Increase quadrature points and test again;








run;


title ’MML AGQ 21 / Slopes’;

parms beta0 = 0.5670 beta1 = 0.0098 beta2 = 0.0133


sigmab1 = 2 sigmab2 = 1 rho = -0.4;

eta = beta0 + b1 + beta1 * timec + b2 * timec + beta2 * treat * timec;




subject = subject;

run;

data LRT;

L01 = -2 * (-449.295733 - (-443.902654));

df = 2;


run;


run;

* Test for a treatment effect;


title ’MML AGQ 21 / No Treatment’;

parms beta0 = 0.5670 beta1 = 0.0098 sigmab1 = 2 sigmab2 = 1 rho = -0.4;

eta = beta0 + b1 + beta1 * timec + b2 * timec;




subject = subject;

run;

data LRT;

L01 = 889.7 - 887.8;

df = 2;


run;


run;

8. Calculate and plot the marginal average evolutions.

%inc ’C:\ARMD\MVN.sas’;

data mean_b;

input m1;

cards;

0

0

run;

data var_b;

input m1-m2;

cards;

5.354133 -0.006765938


-0.006765938 0.006379217

run;

%mvn(varcov = var_b, means = mean_b, n = 4000, sample = b, seed = -1);

data b;

set b;

do t = 1 to 4 by 1;

output;

end;

drop t;

run;

data SimulateValues;

do treat = 0 to 1 by 1;

do subject = 1 to 2000 by 1;

do t = 1 to 4 by 1;

output;

end;

end;

end;

run;

proc sort data = SimulateValues;

by subject;

run;


merge SimulateValues b;

run;

proc sort data = SimulateValues;

by t treat;

run;


set SimulateValues;

timec = 0;

if t = 1 then timec = 4;




if treat = 0 then

y = 1 / (1 + exp(-(0.7860 + col1 + (0.04966 + col2)*timec)));

else

y = 1 / (1 + exp(-(0.7860 + col1 + (0.07458 + col2)*timec)));

run;

proc means data = SimulateValues;

var y;

by timec treat;

output out = out;

run;

proc gplot data = out;


plot y * timec = treat / haxis = axis1 vaxis = axis2 legend = legend1;

axis1 label = (h = 2 ’Time’) value = (h = 1.5) order = (3 to 53 by 10)

minor = none;

axis2 label = (h = 2 A = 90 ’P(Y = 1)’) value = (h = 1.5)

order = (0.6 to 0.9 by 0.1) minor = none;

legend1 label = (h = 1.5 ’Treatment: ’) value = (h = 1.5 ’A’ ’B’);

title h = 2.5 ’Marginal average evolutions (GLMM)’;

symbol1 c = black i = join w = 5 l = 1 mode = include;

symbol2 c = black i = join w = 5 l = 2 mode = include;

where _stat_ = ’MEAN’;

run; quit; run;

9. Test at week 52.

data m.armd52;

set m.armd;

time12 = 0;

time24 = 0;

time52 = 0;

if timec = 12 then time12 = 1;



run;

proc genmod data = m.armd52 descending;

title ’Initial Values’;

model bindif = time12 time24 time52 time12*treat

time24*treat time52*treat / dist = bin;

run;

proc nlmixed data = m.armd52 qpoints = 21;

title ’Time as factor’;

parms beta0 = 0.7522 beta1 = -0.3538 beta2 = -0.1253 beta3 = 0.5190

beta4 = 0.6288 beta5 = 0.4457 beta6 = 0.4205 sigmab = 2;

eta = beta0 + b + beta1 * time12 + beta2 * time24 + beta3 * time52 +

(beta4 * time12 + beta5 * time24 + beta6 * time52) * treat;




estimate ’MrgTrEff’ beta6 / sqrt(0.345843 * sigmab**2 + 1);

run;

4.3.2 SAS Output

1. Exploration of missing data mechanism

standard EM

treat=1

The MI Procedure


Model Information

Data Set M.ARMD14

Method MCMC

Multiple Imputation Chain Single Chain

Initial Estimates for MCMC EM Posterior Mode

Start Starting Value

Prior Jeffreys

Number of Imputations 0

Number of Burn-in Iterations 200

Number of Iterations 100

Seed for random number generator 675938

Missing Data Patterns

Group diff4 diff12 diff24 diff52 Freq Percent

1 X X X X 102 85.71

2 X X X . 9 7.56

3 X X . X 2 1.68

4 X X . . 3 2.52

5 X . . . 1 0.84

6 . X X X 1 0.84

7 O O O O 1 0.84


-------------------------Group Means------------------------

Group diff4 diff12 diff24 diff52

1 -0.921569 -2.313725 -5.598039 -10.960784

2 -1.222222 2.111111 -7.666667 .

3 -12.500000 -19.000000 . -18.500000

4 -4.000000 -4.000000 . .

5 -10.000000 . . .

6 . 1.000000 1.000000 -19.000000

7 . . . .

Univariate Statistics

--Missing Values--

Variable N Mean Std Dev Minimum Maximum Count Percent

diff4 117 -1.29915 7.71839 -33.00000 30.00000 2 1.68

diff12 117 -2.27350 11.73458 -38.00000 31.00000 2 1.68

diff24 112 -5.70536 13.82819 -54.00000 26.00000 7 5.88

diff52 105 -11.18095 16.42921 -59.00000 23.00000 14 11.76

standard EM

treat=1

The MI Procedure


Pairwise Correlations

diff4 diff12 diff24 diff52

diff4 1.000000000 0.574952060 0.378557649 0.344100362

diff12 0.574952060 1.000000000 0.556745963 0.524513959

diff24 0.378557649 0.556745963 1.000000000 0.772469673

diff52 0.344100362 0.524513959 0.772469673 1.000000000

Initial Parameter Estimates for EM

_TYPE_ _NAME_ diff4 diff12 diff24 diff52

MEAN -1.299145 -2.273504 -5.705357 -11.180952

COV diff4 59.573534 0 0 0

COV diff12 0 137.700413 0 0

COV diff24 0 0 191.218710 0

COV diff52 0 0 0 269.918864

EM (MLE) Iteration History

_Iteration_ -2 Log L diff4 diff12 diff24 diff52

0 2677.623690 -1.299145 -2.273504 -5.705357 -11.180952

1 2499.142589 -1.299145 -2.273504 -5.705357 -11.180952

2 2490.124996 -1.288106 -2.337030 -5.920341 -11.287948

3 2489.821035 -1.284122 -2.338551 -5.968896 -11.326957

4 2489.809193 -1.282958 -2.338624 -5.981039 -11.336978

5 2489.808488 -1.282661 -2.338637 -5.984320 -11.339102

6 2489.808431 -1.282579 -2.338640 -5.985249 -11.339499

7 2489.808426 -1.282555 -2.338641 -5.985521 -11.339563

8 2489.808426 -1.282547 -2.338641 -5.985603 -11.339571

9 2489.808426 -1.282544 -2.338641 -5.985628 -11.339571

EM (MLE) Parameter Estimates


MEAN -1.282544 -2.338641 -5.985628 -11.339571

COV diff4 58.922448 51.511233 47.562350 47.117941

COV diff12 51.511233 136.639243 96.740432 106.108044

COV diff24 47.562350 96.740432 198.718732 184.115237

COV diff52 47.117941 106.108044 184.115237 279.107171

standard EM

treat=1

The MI Procedure


EM (Posterior Mode) Estimates


MEAN -1.282517 -2.338656 -5.985517 -11.339728

COV diff4 56.514446 49.417496 45.626747 45.202035

COV diff12 49.417496 131.054420 92.785565 101.774088

COV diff24 45.626747 92.785565 190.400964 176.476791

COV diff52 45.202035 101.774088 176.476791 267.124726

standard EM

treat=2

The MI Procedure

Model Information

Data Set M.ARMD14

Method MCMC




Prior Jeffreys







1 X X X X 86 71.07

2 X X X . 15 12.40

3 X X . X 2 1.65

4 X X . . 5 4.13

5 X . . X 1 0.83

6 X . . . 5 4.13

7 . X X X 1 0.83

8 . X . . 1 0.83

9 O O O O 5 4.13


-------------------------Group Means------------------------


1 -3.244186 -4.662791 -8.325581 -15.127907

2 -7.200000 -14.600000 -13.133333 .

3 5.000000 3.000000 . -30.000000

4 -1.400000 -6.200000 . .

5 4.000000 . . -14.000000


6 -4.000000 . . .

7 . -1.000000 -12.000000 -18.000000

8 . -1.000000 . .

9 . . . .

standard EM

treat=2

The MI Procedure


--Missing Values--


diff4 114 -3.50877 8.86002 -38.00000 23.00000 7 5.79

diff12 110 -5.88182 11.62387 -46.00000 14.00000 11 9.09

diff24 102 -9.06863 14.08479 -52.00000 26.00000 19 15.70

diff52 90 -15.47778 15.38511 -49.00000 17.00000 31 25.62



diff4 1.000000000 0.613307315 0.510262505 0.288867147

diff12 0.613307315 1.000000000 0.771255576 0.526190089

diff24 0.510262505 0.771255576 1.000000000 0.728372871

diff52 0.288867147 0.526190089 0.728372871 1.000000000



MEAN -3.508772 -5.881818 -9.068627 -15.477778

COV diff4 78.499922 0 0 0

COV diff12 0 135.114345 0 0

COV diff24 0 0 198.381382 0

COV diff52 0 0 0 236.701748



0 2480.678086 -3.508772 -5.881818 -9.068627 -15.477778

1 2300.270300 -3.508772 -5.881818 -9.068627 -15.477778

2 2278.786779 -3.477478 -5.852476 -8.941456 -15.797125

3 2274.350562 -3.472907 -5.850379 -8.961648 -15.957037

4 2273.402622 -3.472710 -5.851200 -8.988433 -16.028507

5 2273.203404 -3.472699 -5.851816 -9.003220 -16.062034


6 2273.160311 -3.472669 -5.852158 -9.010063 -16.078693

7 2273.150702 -3.472645 -5.852336 -9.013093 -16.087193

8 2273.148517 -3.472630 -5.852426 -9.014439 -16.091541

9 2273.148015 -3.472622 -5.852472 -9.015047 -16.093748

10 2273.147899 -3.472618 -5.852494 -9.015327 -16.094857

11 2273.147872 -3.472616 -5.852505 -9.015458 -16.095408

12 2273.147866 -3.472615 -5.852511 -9.015520 -16.095681

standard EM

treat=2

The MI Procedure



13 2273.147865 -3.472615 -5.852513 -9.015550 -16.095814

14 2273.147864 -3.472615 -5.852515 -9.015564 -16.095880

15 2273.147864 -3.472614 -5.852515 -9.015571 -16.095911



MEAN -3.472614 -5.852515 -9.015571 -16.095911

COV diff4 77.341311 65.693484 63.603263 53.184996

COV diff12 65.693484 139.657824 126.204014 116.793951

COV diff24 63.603263 126.204014 194.336855 181.980490

COV diff52 53.184996 116.793951 181.980490 283.052328



MEAN -3.472576 -5.852548 -9.016530 -16.095481

COV diff4 74.112751 62.983750 60.970726 51.001689

COV diff12 62.983750 133.719788 120.846899 111.863551

COV diff24 60.970726 120.846899 185.801334 174.099672

COV diff52 51.001689 111.863551 174.099672 269.756733

EM with CC initial values

treat=1

The MI Procedure

Model Information

Data Set M.ARMD14

Method MCMC





Prior Jeffreys







1 X X X X 102 85.71

2 X X X . 9 7.56

3 X X . X 2 1.68

4 X X . . 3 2.52

5 X . . . 1 0.84

6 . X X X 1 0.84

7 O O O O 1 0.84


-------------------------Group Means------------------------


1 -0.921569 -2.313725 -5.598039 -10.960784

2 -1.222222 2.111111 -7.666667 .

3 -12.500000 -19.000000 . -18.500000

4 -4.000000 -4.000000 . .

5 -10.000000 . . .

6 . 1.000000 1.000000 -19.000000

7 . . . .


--Missing Values--


diff4 117 -1.29915 7.71839 -33.00000 30.00000 2 1.68

diff12 117 -2.27350 11.73458 -38.00000 31.00000 2 1.68

diff24 112 -5.70536 13.82819 -54.00000 26.00000 7 5.88

diff52 105 -11.18095 16.42921 -59.00000 23.00000 14 11.76


treat=1

The MI Procedure




diff4 1.000000000 0.574952060 0.378557649 0.344100362

diff12 0.574952060 1.000000000 0.556745963 0.524513959

diff24 0.378557649 0.556745963 1.000000000 0.772469673

diff52 0.344100362 0.524513959 0.772469673 1.000000000



MEAN -0.921569 -2.313725 -5.598039 -10.960784

COV diff4 47.756164 35.628810 38.542419 38.838478

COV diff12 35.628810 122.791691 84.968938 97.408464

COV diff24 38.542419 84.968938 188.361580 176.993982

COV diff52 38.838478 97.408464 176.993982 275.820229



0 2494.806915 -0.921569 -2.313725 -5.598039 -10.960784

1 2489.822930 -1.282544 -2.331244 -5.984559 -11.332561

2 2489.809128 -1.282766 -2.338521 -5.987990 -11.341407

3 2489.808461 -1.282606 -2.338637 -5.986186 -11.340144

4 2489.808427 -1.282561 -2.338640 -5.985746 -11.339711

5 2489.808426 -1.282548 -2.338641 -5.985657 -11.339603

6 2489.808426 -1.282545 -2.338641 -5.985641 -11.339577



MEAN -1.282545 -2.338641 -5.985641 -11.339577

COV diff4 58.922445 51.511232 47.563032 47.118620

COV diff12 51.511232 136.639243 96.741241 106.108805

COV diff24 47.563032 96.741241 198.720008 184.116452

COV diff52 47.118620 106.108805 184.116452 279.108439


treat=1

The MI Procedure



MEAN -1.282517 -2.338656 -5.985517 -11.339728


COV diff4 56.514446 49.417496 45.626747 45.202035

COV diff12 49.417496 131.054420 92.785566 101.774088

COV diff24 45.626747 92.785566 190.400966 176.476791

COV diff52 45.202035 101.774088 176.476791 267.124727


treat=2

The MI Procedure

Model Information

Data Set M.ARMD14

Method MCMC




Prior Jeffreys







1 X X X X 86 71.07

2 X X X . 15 12.40

3 X X . X 2 1.65

4 X X . . 5 4.13

5 X . . X 1 0.83

6 X . . . 5 4.13

7 . X X X 1 0.83

8 . X . . 1 0.83

9 O O O O 5 4.13


-------------------------Group Means------------------------


1 -3.244186 -4.662791 -8.325581 -15.127907

2 -7.200000 -14.600000 -13.133333 .

3 5.000000 3.000000 . -30.000000

4 -1.400000 -6.200000 . .

5 4.000000 . . -14.000000

6 -4.000000 . . .

7 . -1.000000 -12.000000 -18.000000

8 . -1.000000 . .

9 . . . .



treat=2

The MI Procedure


--Missing Values--


diff4 114 -3.50877 8.86002 -38.00000 23.00000 7 5.79

diff12 110 -5.88182 11.62387 -46.00000 14.00000 11 9.09

diff24 102 -9.06863 14.08479 -52.00000 26.00000 19 15.70

diff52 90 -15.47778 15.38511 -49.00000 17.00000 31 25.62



diff4 1.000000000 0.613307315 0.510262505 0.288867147

diff12 0.613307315 1.000000000 0.771255576 0.526190089

diff24 0.510262505 0.771255576 1.000000000 0.728372871

diff52 0.288867147 0.526190089 0.728372871 1.000000000



MEAN -3.244186 -4.662791 -8.325581 -15.127907

COV diff4 57.127907 47.389193 42.143092 36.780164

COV diff12 47.389193 101.920246 90.275787 85.749521

COV diff24 42.143092 90.275787 141.986867 134.381395

COV diff52 36.780164 85.749521 134.381395 239.806977



0 2290.659556 -3.244186 -4.662791 -8.325581 -15.127907

1 2273.220666 -3.475149 -5.801777 -9.010677 -16.061625

2 2273.150112 -3.472286 -5.849916 -9.018991 -16.099338

3 2273.148152 -3.472508 -5.852453 -9.016772 -16.098936

4 2273.147922 -3.472587 -5.852546 -9.016004 -16.097430

5 2273.147877 -3.472606 -5.852533 -9.015752 -16.096626

6 2273.147867 -3.472611 -5.852524 -9.015655 -16.096251

7 2273.147865 -3.472613 -5.852520 -9.015614 -16.096082

8 2273.147864 -3.472614 -5.852518 -9.015595 -16.096006

9 2273.147864 -3.472614 -5.852517 -9.015586 -16.095971



treat=2

The MI Procedure



MEAN -3.472614 -5.852517 -9.015586 -16.095971

COV diff4 77.341315 65.693471 63.603097 53.188213

COV diff12 65.693471 139.657797 126.203835 116.797829

COV diff24 63.603097 126.203835 194.336656 181.986036

COV diff52 53.188213 116.797829 181.986036 283.062059



MEAN -3.472576 -5.852548 -9.016530 -16.095482

COV diff4 74.112751 62.983750 60.970724 51.001727

COV diff12 62.983750 133.719788 120.846896 111.863598

COV diff24 60.970724 120.846896 185.801331 174.099740

COV diff52 51.001727 111.863598 174.099740 269.756851

2. GEE and GLMM analyses

CC - GEE

The GENMOD Procedure

Model Information

Data Set M.ARMDCC

Distribution Binomial

Link Function Logit

Dependent Variable bindif

Number of Observations Read 752

Number of Observations Used 752

Number of Events 218

Number of Trials 752


Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 188 2 4 6 7 8 9 12 13 14 15 16 17 18 19 20 22 23 24 25

26 27 29 33 34 35 36 39 40 42 43 44 45 46 49 51 52


53 54 55 57 58 59 60 61 62 63 64 65 66 67 68 69 70

71 72 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

89 92 93 94 95 96 99 102 103 104 105 106 107 108

...

Response Profile

Ordered Total

Value bindif Frequency

1 0 218

2 1 534

PROC GENMOD is modeling the probability that bindif=’0’. One way to change this to model the

probability that bindif=’1’ is to specify the DESCENDING option in the PROC statement.

Parameter Information

Parameter Effect time treat

Prm1 Intercept

Prm2 time 1

Prm3 time 2

Prm4 time 3

Prm5 time 4

Prm6 time*treat 1 1

Prm7 time*treat 1 2

Prm8 time*treat 2 1

Prm9 time*treat 2 2

Prm10 time*treat 3 1




Criteria For Assessing Goodness Of Fit

Criterion DF Value Value/DF

Deviance 744 885.5608 1.1903

Scaled Deviance 744 885.5608 1.1903

Pearson Chi-Square 744 752.0000 1.0108

Scaled Pearson X2 744 752.0000 1.0108


Algorithm converged.

Analysis Of Initial Parameter Estimates

Standard Wald 95% Confidence Chi-


Parameter DF Estimate Error Limits Square Pr > ChiSq

Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

time 1 1 -1.0076 0.2436 -1.4851 -0.5302 17.11 <.0001

time 2 1 -0.8920 0.2375 -1.3574 -0.4266 14.11 0.0002

time 3 1 -1.1299 0.2510 -1.6218 -0.6379 20.26 <.0001

time 4 1 -1.6376 0.2921 -2.2101 -1.0651 31.43 <.0001

time*treat 1 1 1 0.4015 0.3198 -0.2253 1.0283 1.58 0.2093

time*treat 1 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 1 0.4947 0.3117 -0.1163 1.1057 2.52 0.1125

time*treat 2 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 1 0.4805 0.3263 -0.1591 1.1201 2.17 0.1409

time*treat 3 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 1 0.4037 0.3761 -0.3335 1.1408 1.15 0.2832

time*treat 4 2 0 0.0000 0.0000 0.0000 0.0000 . .

Scale 0 1.0000 0.0000 1.0000 1.0000

NOTE: The scale parameter was held fixed.


Lagrange Multiplier Statistics

Parameter Chi-Square Pr > ChiSq

Intercept . .

GEE Model Information

Correlation Structure Exchangeable

Within-Subject Effect time (4 levels)

Subject Effect subject (188 levels)

Number of Clusters 188

Correlation Matrix Dimension 4

Maximum Cluster Size 4

Minimum Cluster Size 4


Exchangeable Working

Correlation

Correlation 0.3878480087

Analysis Of GEE Parameter Estimates

Empirical Standard Error Estimates

Standard 95% Confidence

Parameter Estimate Error Limits Z Pr > |Z|


Intercept 0.0000 0.0000 0.0000 0.0000 . .

time 1 -1.0076 0.2436 -1.4851 -0.5302 -4.14 <.0001

time 2 -0.8920 0.2375 -1.3574 -0.4266 -3.76 0.0002

time 3 -1.1299 0.2510 -1.6218 -0.6379 -4.50 <.0001

time 4 -1.6376 0.2921 -2.2101 -1.0651 -5.61 <.0001

time*treat 1 1 0.4015 0.3198 -0.2253 1.0283 1.26 0.2093

time*treat 1 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 0.4947 0.3117 -0.1163 1.1057 1.59 0.1125

time*treat 2 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 0.4805 0.3263 -0.1591 1.1201 1.47 0.1409

time*treat 3 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 0.4037 0.3761 -0.3335 1.1408 1.07 0.2832

time*treat 4 2 0.0000 0.0000 0.0000 0.0000 . .



Model-Based Standard Error Estimates



Intercept 0.0000 0.0000 0.0000 0.0000 . .

time 1 -1.0076 0.2436 -1.4851 -0.5302 -4.14 <.0001

time 2 -0.8920 0.2375 -1.3574 -0.4266 -3.76 0.0002

time 3 -1.1299 0.2510 -1.6218 -0.6379 -4.50 <.0001

time 4 -1.6376 0.2921 -2.2101 -1.0651 -5.61 <.0001

time*treat 1 1 0.4015 0.3198 -0.2253 1.0283 1.26 0.2093

time*treat 1 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 0.4947 0.3117 -0.1163 1.1057 1.59 0.1125

time*treat 2 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 0.4805 0.3263 -0.1591 1.1201 1.47 0.1409

time*treat 3 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 0.4037 0.3761 -0.3335 1.1408 1.07 0.2832

time*treat 4 2 0.0000 0.0000 0.0000 0.0000 . .

Scale 1.0000 . . . . .


CC - GEE - linearized version 20:20 Monday, May 16, 2005 179

The GLIMMIX Procedure

Model Information

Data Set M.ARMDCC

Response Variable bindif

Response Distribution Binary

Link Function Logit

Variance Function Default

Variance Matrix Blocked By subject

Estimation Technique Residual PL


Degrees of Freedom Method Between-Within


Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 188 2 4 6 7 8 9 12 13 14 15 16 17 18 19 20 22 23

24 25 26 27 29 33 34 35 36 39 40 42 43 44 45

46 49 51 52 53 54 55 57 58 59 60 61 62 63 64

65 66 67 68 69 70 71 72 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 92 93 94 95 96 99

102 103 104 105 106 107 108 109 110 111 112

113 115 116 117 118 119 120 121 123 124 125

126 127 128 129 130 132 134 135 137 138 139

140 141 142 143 145 146 149 151 152 153 154

155 156 157 158 159 160 161 162 164 165 166

168 169 170 171 172 173 175 176 178 179 180

181 182 183 184 185 187 188 190 192 193 194

195 199 202 203 205 206 208 209 210 211 212

214 215 217 218 220 221 222 223 224 225 226

227 228 229 232 233 234 235 236 237 238 239

240



Response Profile

Ordered Total


1 0 218

2 1 534

The GLIMMIX procedure is modeling the probability that bindif=’0’.


Dimensions

R-side Cov. Parameters 2

Columns in X 12

Columns in Z per Subject 0

Subjects (Blocks in V) 188

Max Obs per Subject 4

Optimization Information


Optimization Technique Newton-Raphson

Parameters in Optimization 1

Lower Boundaries 0

Upper Boundaries 0

Fixed Effects Profiled

Residual Variance Profiled

Starting From Data

Iteration History

Objective Max

Iteration Restarts Subiterations Function Change Gradient

0 0 2 3290.4920765 0.28577085 1.98E-8

1 0 1 3228.4753932 0.00736502 0.000054

2 0 1 3229.9762787 0.00002566 9.344E-8

3 0 0 3229.9774035 0.00000000 3.758E-6

Convergence criterion (PCONV=1.11022E-8) satisfied.

Fit Statistics

-2 Res Log Pseudo-Likelihood 3229.98

Generalized Chi-Square 459.16

Gener. Chi-Square / DF 0.62


Standard

Cov Parm Subject Estimate Error

CS subject 0.3936 0.05756

Residual 0.6172 0.03695

Solutions for Fixed Effects

Standard

Effect time treat Estimate Error DF t Value Pr > |t|

time 1 -1.0076 0.2449 556 -4.11 <.0001

time 2 -0.8920 0.2387 556 -3.74 0.0002

time 3 -1.1299 0.2524 556 -4.48 <.0001

time 4 -1.6376 0.2937 556 -5.58 <.0001

time*treat 1 1 0.4015 0.3215 556 1.25 0.2123

time*treat 1 2 0 . . . .

time*treat 2 1 0.4947 0.3134 556 1.58 0.1150

time*treat 2 2 0 . . . .

time*treat 3 1 0.4805 0.3281 556 1.46 0.1436

time*treat 3 2 0 . . . .


time*treat 4 1 0.4037 0.3781 556 1.07 0.2862

time*treat 4 2 0 . . . .

Type III Tests of Fixed Effects

Num Den


time 4 556 16.82 <.0001

time*treat 4 556 0.89 0.4687

CC - GEE - linearized version - empirical


Standard


CS subject 0.3936 0.05756

Residual 0.6172 0.03695


Standard


time 1 -1.0076 0.2436 556 -4.14 <.0001

time 2 -0.8920 0.2375 556 -3.76 0.0002

time 3 -1.1299 0.2510 556 -4.50 <.0001

time 4 -1.6376 0.2921 556 -5.61 <.0001

time*treat 1 1 0.4015 0.3198 556 1.26 0.2098

time*treat 1 2 0 . . . .

time*treat 2 1 0.4947 0.3117 556 1.59 0.1131

time*treat 2 2 0 . . . .

time*treat 3 1 0.4805 0.3263 556 1.47 0.1415

time*treat 3 2 0 . . . .

time*treat 4 1 0.4037 0.3761 556 1.07 0.2836

time*treat 4 2 0 . . . .


Num Den


time 4 556 18.06 <.0001

time*treat 4 556 0.87 0.4797

LOCF - GEE



Model Information

Data Set M.ARMDLOCF


Link Function Logit

Dependent Variable LOCF





Missing Values 27


Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 240 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

...

Response Profile

Ordered Total

Value LOCF Frequency

1 0 273

2 1 660

PROC GENMOD is modeling the probability that LOCF=’0’. One way to change this to model the

probability that LOCF=’1’ is to specify the DESCENDING option in the PROC statement.



Prm1 Intercept

Prm2 time 1

Prm3 time 2

Prm4 time 3

Prm5 time 4

Prm6 time*treat 1 1

LOCF - GEE





Prm7 time*treat 1 2

Prm8 time*treat 2 1

Prm9 time*treat 2 2







Deviance 925 1108.6193 1.1985

Scaled Deviance 925 1108.6193 1.1985


Scaled Pearson X2 925 932.9997 1.0086






Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

time 1 1 -0.8557 0.2047 -1.2569 -0.4544 17.47 <.0001

time 2 1 -0.9651 0.2077 -1.3722 -0.5579 21.58 <.0001

time 3 1 -1.0531 0.2120 -1.4687 -0.6376 24.67 <.0001

time 4 1 -1.5094 0.2411 -1.9820 -1.0367 39.18 <.0001

time*treat 1 1 1 0.2007 0.2827 -0.3533 0.7548 0.50 0.4776

time*treat 1 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 1 0.5525 0.2802 0.0034 1.1017 3.89 0.0486

time*treat 2 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 1 0.4229 0.2869 -0.1395 0.9853 2.17 0.1405

time*treat 3 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 1 0.3417 0.3240 -0.2933 0.9768 1.11 0.2915

time*treat 4 2 0 0.0000 0.0000 0.0000 0.0000 . .

Scale 0 1.0000 0.0000 1.0000 1.0000





Intercept . .






Clusters With Missing Values 9






Correlation






Intercept 0.0000 0.0000 0.0000 0.0000 . .

time 1 -0.8707 0.2055 -1.2735 -0.4678 -4.24 <.0001

time 2 -0.9651 0.2077 -1.3722 -0.5579 -4.65 <.0001

time 3 -1.0531 0.2120 -1.4687 -0.6376 -4.97 <.0001

time 4 -1.5094 0.2411 -1.9820 -1.0367 -6.26 <.0001

time*treat 1 1 0.2243 0.2827 -0.3298 0.7785 0.79 0.4275

time*treat 1 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 0.5525 0.2802 0.0034 1.1017 1.97 0.0486

time*treat 2 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 0.4229 0.2869 -0.1395 0.9853 1.47 0.1405

time*treat 3 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 0.3417 0.3240 -0.2933 0.9768 1.05 0.2915

time*treat 4 2 0.0000 0.0000 0.0000 0.0000 . .





Intercept 0.0000 0.0000 0.0000 0.0000 . .


time 1 -0.8707 0.2048 -1.2721 -0.4693 -4.25 <.0001

time 2 -0.9651 0.2077 -1.3722 -0.5579 -4.65 <.0001

time 3 -1.0531 0.2120 -1.4687 -0.6376 -4.97 <.0001

time 4 -1.5094 0.2411 -1.9820 -1.0367 -6.26 <.0001

time*treat 1 1 0.2243 0.2824 -0.3291 0.7778 0.79 0.4269

time*treat 1 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 0.5525 0.2802 0.0034 1.1017 1.97 0.0486

time*treat 2 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 0.4229 0.2869 -0.1395 0.9853 1.47 0.1405

time*treat 3 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 0.3417 0.3240 -0.2933 0.9768 1.05 0.2915

time*treat 4 2 0.0000 0.0000 0.0000 0.0000 . .

Scale 1.0000 . . . . .


LOCF - GEE - linearized version


Model Information

Data Set M.ARMDLOCF

Response Variable LOCF


Link Function Logit






Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 234 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20 22 23 24 25 26 27 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 49 50 51 52

53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

98 99 100 101 102 103 104 105 106 107 108 109

110 111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130 131

132 133 134 135 136 137 138 139 140 141 142

143 145 146 147 148 149 150 151 152 153 154

155 156 157 158 159 160 161 162 163 164 165

166 167 168 169 170 171 172 173 174 175 176

177 178 179 180 181 182 183 184 185 186 187

188 190 191 192 193 194 195 196 197 198 199

200 201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220 221


222 223 224 225 226 227 228 229 230 231 232

233 234 235 236 237 238 239 240



Response Profile

Ordered Total


1 0 273

The GLIMMIX procedure is modeling the probability that LOCF=’0’.

Response Profile

Ordered Total


2 1 660


Dimensions


Columns in X 12







Lower Boundaries 0

Upper Boundaries 0



Starting From Data

Iteration History

Objective Max


0 0 2 4033.7303417 0.22962606 7.948E-7

1 0 1 3949.5889542 0.00826676 7.989E-6


2 0 1 3951.5825139 0.00000864 8.041E-9

3 0 0 3951.5841239 0.00000002 2.353E-6

4 0 0 3951.5841245 0.00000000 2.337E-6


Fit Statistics





Standard


CS subject 0.4415 0.05472

Residual 0.5672 0.03046


Standard


time 1 -0.8707 0.2057 691 -4.23 <.0001

time 2 -0.9651 0.2086 691 -4.63 <.0001

time 3 -1.0531 0.2130 691 -4.95 <.0001

time 4 -1.5094 0.2422 691 -6.23 <.0001

time*treat 1 1 0.2243 0.2836 691 0.79 0.4292

time*treat 1 2 0 . . . .

time*treat 2 1 0.5525 0.2814 691 1.96 0.0500

time*treat 2 2 0 . . . .

time*treat 3 1 0.4229 0.2882 691 1.47 0.1427

time*treat 3 2 0 . . . .

time*treat 4 1 0.3417 0.3254 691 1.05 0.2940

time*treat 4 2 0 . . . .


Num Den


time 4 691 18.92 <.0001

time*treat 4 691 1.10 0.3552

LOCF - GEE - linearized version - empirical


Standard



CS subject 0.4415 0.05472

Residual 0.5672 0.03046


Standard


time 1 -0.8707 0.2055 691 -4.24 <.0001

time 2 -0.9651 0.2077 691 -4.65 <.0001

time 3 -1.0531 0.2120 691 -4.97 <.0001

time 4 -1.5094 0.2411 691 -6.26 <.0001

time*treat 1 1 0.2243 0.2827 691 0.79 0.4278

time*treat 1 2 0 . . . .

time*treat 2 1 0.5525 0.2802 691 1.97 0.0490

time*treat 2 2 0 . . . .

time*treat 3 1 0.4229 0.2869 691 1.47 0.1410

time*treat 3 2 0 . . . .

time*treat 4 1 0.3417 0.3240 691 1.05 0.2919

time*treat 4 2 0 . . . .


Num Den


time 4 691 20.10 <.0001

time*treat 4 691 1.01 0.4028

data as is - GEE


Model Information

Data Set M.ARMDWGEE


Link Function Logit






Missing Values 93



Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 240 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

...

Response Profile

Ordered Total


1 0 252

2 1 615





Prm1 Intercept

Prm2 time 1

Prm3 time 2

Prm4 time 3

Prm5 time 4

Prm6 time*treat 1 1

Prm7 time*treat 1 2

Prm8 time*treat 2 1

Prm9 time*treat 2 2







Deviance 859 1022.5545 1.1904

Scaled Deviance 859 1022.5545 1.1904


Scaled Pearson X2 859 867.0000 1.0093







Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

time 1 1 -0.8557 0.2047 -1.2569 -0.4544 17.47 <.0001

time 2 1 -1.0272 0.2164 -1.4513 -0.6030 22.53 <.0001

time 3 1 -1.0726 0.2272 -1.5179 -0.6273 22.29 <.0001

time 4 1 -1.6917 0.2908 -2.2617 -1.1216 33.83 <.0001

time*treat 1 1 1 0.2007 0.2827 -0.3533 0.7548 0.50 0.4776

time*treat 1 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 1 0.6288 0.2870 0.0662 1.1914 4.80 0.0285

time*treat 2 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 1 0.4457 0.3016 -0.1454 1.0369 2.18 0.1394

time*treat 3 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 1 0.4205 0.3745 -0.3136 1.1545 1.26 0.2616

time*treat 4 2 0 0.0000 0.0000 0.0000 0.0000 . .

Scale 0 1.0000 0.0000 1.0000 1.0000




Intercept . .












Correlation







Intercept 0.0000 0.0000 0.0000 0.0000 . .

time 1 -0.8669 0.2054 -1.2695 -0.4644 -4.22 <.0001

time 2 -1.0116 0.2145 -1.4320 -0.5911 -4.72 <.0001

time 3 -1.0703 0.2232 -1.5077 -0.6329 -4.80 <.0001

time 4 -1.7091 0.2899 -2.2773 -1.1408 -5.89 <.0001

time*treat 1 1 0.2201 0.2827 -0.3339 0.7741 0.78 0.4361

time*treat 1 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 0.6083 0.2856 0.0486 1.1680 2.13 0.0332

time*treat 2 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 0.4404 0.2981 -0.1438 1.0247 1.48 0.1396

time*treat 3 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 0.4359 0.3726 -0.2944 1.1662 1.17 0.2421

time*treat 4 2 0.0000 0.0000 0.0000 0.0000 . .





Intercept 0.0000 0.0000 0.0000 0.0000 . .

time 1 -0.8669 0.2048 -1.2684 -0.4655 -4.23 <.0001

time 2 -1.0116 0.2147 -1.4324 -0.5908 -4.71 <.0001

time 3 -1.0703 0.2243 -1.5100 -0.6306 -4.77 <.0001

time 4 -1.7091 0.2850 -2.2677 -1.1505 -6.00 <.0001

time*treat 1 1 0.2201 0.2824 -0.3334 0.7737 0.78 0.4357

time*treat 1 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 0.6083 0.2857 0.0483 1.1684 2.13 0.0333

time*treat 2 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 0.4404 0.2988 -0.1452 1.0260 1.47 0.1404

time*treat 3 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 0.4359 0.3681 -0.2856 1.1573 1.18 0.2364

time*treat 4 2 0.0000 0.0000 0.0000 0.0000 . .

Scale 1.0000 . . . . .


data as is - GEE - linearized version

Model Information

Data Set M.ARMDWGEE



Link Function Logit







Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 234 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20 22 23 24 25 26 27 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 49 50 51 52

53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

98 99 100 101 102 103 104 105 106 107 108 109

110 111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130 131

132 133 134 135 136 137 138 139 140 141 142

143 145 146 147 148 149 150 151 152 153 154

155 156 157 158 159 160 161 162 163 164 165

166 167 168 169 170 171 172 173 174 175 176

177 178 179 180 181 182 183 184 185 186 187

188 190 191 192 193 194 195 196 197 198 199

200 201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220 221

222 223 224 225 226 227 228 229 230 231 232

233 234 235 236 237 238 239 240



Response Profile

Ordered Total


1 0 252


data as is - GEE - linearized version


Response Profile

Ordered Total


2 1 615



Dimensions


Columns in X 12







Lower Boundaries 0

Upper Boundaries 0



Starting From Data

Iteration History

Objective Max


0 0 2 3798.0526759 0.31166692 2.144E-6

1 0 1 3728.7297728 0.00742859 0.000041

2 0 1 3730.3894543 0.00004304 4.076E-8

3 0 1 3730.3935466 . 4.43E-11

4 0 7 3730.3935466 2.00000000 2.249E-7

5 0 0 3730.3935419 0.00000000 6.846E-7


Fit Statistics





Standard


CS subject 0.3916 0.05303

Residual 0.6178 0.03483



Standard


time 1 -0.8669 0.2058 625 -4.21 <.0001

time 2 -1.0116 0.2157 625 -4.69 <.0001

time 3 -1.0703 0.2254 625 -4.75 <.0001

time 4 -1.7091 0.2864 625 -5.97 <.0001

time*treat 1 1 0.2201 0.2838 625 0.78 0.4382

time*treat 1 2 0 . . . .

time*treat 2 1 0.6083 0.2871 625 2.12 0.0345

time*treat 2 2 0 . . . .

time*treat 3 1 0.4404 0.3002 625 1.47 0.1428

time*treat 3 2 0 . . . .

time*treat 4 1 0.4358 0.3698 625 1.18 0.2390

time*treat 4 2 0 . . . .


Num Den


time 4 625 19.58 <.0001

time*treat 4 625 1.29 0.2712

data as is - GEE - linearized version - empirical



Standard


CS subject 0.3916 0.05303

Residual 0.6178 0.03483


Standard


time 1 -0.8669 0.2054 625 -4.22 <.0001

time 2 -1.0116 0.2145 625 -4.72 <.0001

time 3 -1.0703 0.2232 625 -4.80 <.0001

time 4 -1.7091 0.2899 625 -5.89 <.0001

time*treat 1 1 0.2201 0.2827 625 0.78 0.4364

time*treat 1 2 0 . . . .

time*treat 2 1 0.6083 0.2856 625 2.13 0.0335

time*treat 2 2 0 . . . .

time*treat 3 1 0.4404 0.2981 625 1.48 0.1401

time*treat 3 2 0 . . . .

time*treat 4 1 0.4358 0.3726 625 1.17 0.2426

time*treat 4 2 0 . . . .



Num Den


time 4 625 20.66 <.0001

time*treat 4 625 1.21 0.3059

data as is - WGEE


Model Information

Data Set M.ARMDWGEE


Link Function Logit


Scale Weight Variable WI



Sum of Weights 2249.749



Missing Values 114


Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 240 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

...

Response Profile

Ordered Total


1 0 733.2344

2 1 1516.515






Prm1 Intercept

Prm2 time 1

Prm3 time 2

Prm4 time 3

Prm5 time 4

Prm6 time*treat 1 1

Prm7 time*treat 1 2

Prm8 time*treat 2 1

Prm9 time*treat 2 2







Deviance 838 2665.1553 3.1804

Scaled Deviance 838 2665.1553 3.1804


Scaled Pearson X2 838 2249.7494 2.6847






Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

time 1 1 -0.9781 0.1034 -1.1809 -0.7754 89.44 <.0001

time 2 1 -1.7889 0.1569 -2.0963 -1.4814 130.05 <.0001

time 3 1 -0.9542 0.1528 -1.2538 -0.6547 38.98 <.0001

time 4 1 -1.5243 0.2490 -2.0123 -1.0363 37.49 <.0001

time*treat 1 1 1 0.8013 0.1457 0.5158 1.0869 30.25 <.0001

time*treat 1 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 1 1.8954 0.1883 1.5264 2.2644 101.36 <.0001

time*treat 2 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 1 0.2103 0.2031 -0.1877 0.6083 1.07 0.3004

time*treat 3 2 0 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 1 0.2995 0.3337 -0.3545 0.9535 0.81 0.3694

time*treat 4 2 0 0.0000 0.0000 0.0000 0.0000 . .


Scale 0 1.0000 0.0000 1.0000 1.0000




Intercept . .












Correlation






Intercept 0.0000 0.0000 0.0000 0.0000 . .

time 1 -0.9781 0.4407 -1.8419 -0.1144 -2.22 0.0265

time 2 -1.7822 0.3751 -2.5174 -1.0470 -4.75 <.0001

time 3 -1.1125 0.3334 -1.7659 -0.4591 -3.34 0.0008

time 4 -1.7189 0.3913 -2.4859 -0.9519 -4.39 <.0001

time*treat 1 1 0.8013 0.6696 -0.5111 2.1138 1.20 0.2314

time*treat 1 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 1.8679 0.6103 0.6717 3.0640 3.06 0.0022

time*treat 2 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 0.7337 0.5322 -0.3094 1.7769 1.38 0.1680

time*treat 3 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 0.7383 0.5166 -0.2743 1.7508 1.43 0.1530

time*treat 4 2 0.0000 0.0000 0.0000 0.0000 . .






Intercept 0.0000 0.0000 0.0000 0.0000 . .

time 1 -0.9781 0.1034 -1.1809 -0.7754 -9.46 <.0001

time 2 -1.7822 0.1540 -2.0840 -1.4804 -11.57 <.0001

time 3 -1.1125 0.1524 -1.4111 -0.8139 -7.30 <.0001

time 4 -1.7189 0.2472 -2.2034 -1.2344 -6.95 <.0001

time*treat 1 1 0.8013 0.1457 0.5158 1.0869 5.50 <.0001

time*treat 1 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 2 1 1.8679 0.1857 1.5038 2.2319 10.06 <.0001

time*treat 2 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 3 1 0.7337 0.1963 0.3489 1.1185 3.74 0.0002

time*treat 3 2 0.0000 0.0000 0.0000 0.0000 . .

time*treat 4 1 0.7383 0.3148 0.1213 1.3552 2.35 0.0190

time*treat 4 2 0.0000 0.0000 0.0000 0.0000 . .

Scale 1.0000 . . . . .


data as is - WGEE - linearized version


Model Information

Data Set M.ARMDWGEE



Link Function Logit


Weight Variable WI





Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 234 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20 22 23 24 25 26 27 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 49 50 51 52

53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

98 99 100 101 102 103 104 105 106 107 108 109

110 111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130 131

132 133 134 135 136 137 138 139 140 141 142

143 145 146 147 148 149 150 151 152 153 154


155 156 157 158 159 160 161 162 163 164 165

166 167 168 169 170 171 172 173 174 175 176

177 178 179 180 181 182 183 184 185 186 187

188 190 191 192 193 194 195 196 197 198 199

200 201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220 221

222 223 224 225 226 227 228 229 230 231 232

233 234 235 236 237 238 239 240



Response Profile

Ordered Total


1 0 246

2 1 600


Dimensions


Columns in X 12







Lower Boundaries 0

Upper Boundaries 0



Starting From Data

Iteration History

Objective Max


0 0 3 4097.6696637 0.15208155 0.000032

1 0 2 4077.0629889 0.00967492 3.701E-7

2 0 1 4078.8642714 0.00056760 2.889E-6

3 0 1 4078.7916245 0.00003595 1.162E-8


4 0 1 4078.7960437 . 6.67E-11

5 0 8 4078.7960437 2.00000000 5.777E-9

6 0 1 4078.7954956 0.00000043 2.34E-12

7 0 0 4078.7955404 0.00000003 1.037E-6

8 0 0 4078.795535 0.00000001 8.981E-7


data as is - WGEE - linearized version


Fit Statistics





Standard


CS subject 1.8446 0.2339

Residual 1.2946 0.07679


Standard


time 1 -0.9781 0.1832 604 -5.34 <.0001

time 2 -1.7769 0.2622 604 -6.78 <.0001

time 3 -1.1893 0.2532 604 -4.70 <.0001

time 4 -1.8112 0.3938 604 -4.60 <.0001

time*treat 1 1 0.8013 0.2581 604 3.10 0.0020

time*treat 1 2 0 . . . .

time*treat 2 1 1.8460 0.3199 604 5.77 <.0001

time*treat 2 2 0 . . . .

time*treat 3 1 0.9829 0.3265 604 3.01 0.0027

time*treat 3 2 0 . . . .

time*treat 4 1 0.9664 0.4923 604 1.96 0.0501

time*treat 4 2 0 . . . .


Num Den


time 4 604 10.98 <.0001


time*treat 4 604 8.36 <.0001

data as is - WGEE - linearized version - empirical


Model Information

Data Set M.ARMDWGEE



Link Function Logit


Weight Variable WI




Fixed Effects SE Adjustment Sandwich - Classical


Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 234 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20 22 23 24 25 26 27 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 49 50 51 52

53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

98 99 100 101 102 103 104 105 106 107 108 109

110 111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130 131

132 133 134 135 136 137 138 139 140 141 142

143 145 146 147 148 149 150 151 152 153 154

155 156 157 158 159 160 161 162 163 164 165

166 167 168 169 170 171 172 173 174 175 176

177 178 179 180 181 182 183 184 185 186 187

188 190 191 192 193 194 195 196 197 198 199

200 201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220 221

222 223 224 225 226 227 228 229 230 231 232

233 234 235 236 237 238 239 240



Response Profile

Ordered Total



1 0 246

2 1 600


Dimensions


Columns in X 12







Lower Boundaries 0

Upper Boundaries 0



Starting From Data

Iteration History

Objective Max


0 0 3 4097.6696637 0.15208155 0.000032

1 0 2 4077.0629889 0.00967492 3.701E-7

2 0 1 4078.8642714 0.00056760 2.889E-6

3 0 1 4078.7916245 0.00003595 1.162E-8

4 0 1 4078.7960437 . 6.67E-11

5 0 8 4078.7960437 2.00000000 5.777E-9

6 0 1 4078.7954956 0.00000043 2.34E-12

7 0 0 4078.7955404 0.00000003 1.037E-6

8 0 0 4078.795535 0.00000001 8.981E-7


Fit Statistics





Standard



CS subject 1.8446 0.2339

Residual 1.2946 0.07679


Standard


time 1 -0.9781 0.4407 604 -2.22 0.0268

time 2 -1.7769 0.4188 604 -4.24 <.0001

time 3 -1.1893 0.3775 604 -3.15 0.0017

time 4 -1.8112 0.4814 604 -3.76 0.0002

time*treat 1 1 0.8013 0.6696 604 1.20 0.2319

time*treat 1 2 0 . . . .

time*treat 2 1 1.8460 0.6367 604 2.90 0.0039

time*treat 2 2 0 . . . .

time*treat 3 1 0.9829 0.5969 604 1.65 0.1001

time*treat 3 2 0 . . . .

time*treat 4 1 0.9664 0.6454 604 1.50 0.1348

time*treat 4 2 0 . . . .


Num Den


time 4 604 5.16 0.0004

time*treat 4 604 2.49 0.0422

CC - mixed - PQL


Model Information

Data Set M.ARMDCC



Link Function Logit




Degrees of Freedom Method Containment


Class Levels Values

time 4 1 2 3 4


treat 2 1 2

subject 188 2 4 6 7 8 9 12 13 14 15 16 17 18 19 20 22 23

24 25 26 27 29 33 34 35 36 39 40 42 43 44 45

46 49 51 52 53 54 55 57 58 59 60 61 62 63 64

65 66 67 68 69 70 71 72 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 92 93 94 95 96 99

102 103 104 105 106 107 108 109 110 111 112

113 115 116 117 118 119 120 121 123 124 125

126 127 128 129 130 132 134 135 137 138 139

140 141 142 143 145 146 149 151 152 153 154

155 156 157 158 159 160 161 162 164 165 166

168 169 170 171 172 173 175 176 178 179 180

181 182 183 184 185 187 188 190 192 193 194

195 199 202 203 205 206 208 209 210 211 212

214 215 217 218 220 221 222 223 224 225 226

227 228 229 232 233 234 235 236 237 238 239

240



Response Profile

Ordered Total


1 0 218

2 1 534



Dimensions

G-side Cov. Parameters 1

Columns in X 12







Lower Boundaries 1

Upper Boundaries 0


Starting From Data


Iteration History

Objective Max


0 0 4 3373.5642704 0.32012192 4.854E-8

1 0 3 3451.2809298 0.10686186 4.923E-7

2 0 3 3479.5329722 0.03013256 2.85E-11

3 0 2 3486.3804256 0.00698688 8.052E-8

4 0 2 3487.882408 0.00151187 1.79E-10

5 0 1 3488.2026488 0.00032112 2.989E-6

6 0 1 3488.2704392 0.00006806 1.342E-7

7 0 1 3488.284796 0.00001439 6.003E-9

8 0 1 3488.2878318 0.00000304 2.68E-10

9 0 1 3488.2884735 0.00000064 1.21E-11

10 0 0 3488.2886091 0.00000000 3.619E-6


Fit Statistics





Estimated G Matrix

Effect Row Col1

Intercept 1 2.0263

Estimated G Correlation

Matrix

Effect Row Col1

Intercept 1 1.0000


Cov Standard

Parm Subject Estimate Error

UN(1,1) subject 2.0263 0.3902



Standard


time 1 -1.1929 0.3149 558 -3.79 0.0002

time 2 -1.0484 0.3087 558 -3.40 0.0007

time 3 -1.3451 0.3221 558 -4.18 <.0001

time 4 -1.9670 0.3611 558 -5.45 <.0001

time*treat 1 1 0.4529 0.4181 558 1.08 0.2792

time*treat 1 2 0 . . . .

time*treat 2 1 0.5768 0.4101 558 1.41 0.1601

time*treat 2 2 0 . . . .

time*treat 3 1 0.5498 0.4245 558 1.30 0.1958

time*treat 3 2 0 . . . .

time*treat 4 1 0.4371 0.4709 558 0.93 0.3536

time*treat 4 2 0 . . . .


Num Den


time 4 558 18.12 <.0001

time*treat 4 558 0.83 0.5039

CC - mixed - PQL 20:20 Monday, May 16, 2005 218


Model Information

Data Set M.ARMDCC



Link Function Logit






Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 188 2 4 6 7 8 9 12 13 14 15 16 17 18 19 20 22 23

24 25 26 27 29 33 34 35 36 39 40 42 43 44 45

46 49 51 52 53 54 55 57 58 59 60 61 62 63 64

65 66 67 68 69 70 71 72 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 92 93 94 95 96 99

102 103 104 105 106 107 108 109 110 111 112


113 115 116 117 118 119 120 121 123 124 125

126 127 128 129 130 132 134 135 137 138 139

140 141 142 143 145 146 149 151 152 153 154

155 156 157 158 159 160 161 162 164 165 166

168 169 170 171 172 173 175 176 178 179 180

181 182 183 184 185 187 188 190 192 193 194

195 199 202 203 205 206 208 209 210 211 212

214 215 217 218 220 221 222 223 224 225 226

227 228 229 232 233 234 235 236 237 238 239

240



Response Profile

Ordered Total


1 0 218

2 1 534



Dimensions


Columns in X 12







Lower Boundaries 1

Upper Boundaries 0


Starting From Data

Iteration History

Objective Max


0 0 4 3373.5642704 0.32012192 4.854E-8


1 0 3 3451.2809298 0.10686186 4.923E-7

2 0 3 3479.5329722 0.03013256 2.85E-11

3 0 2 3486.3804256 0.00698688 8.052E-8

4 0 2 3487.882408 0.00151187 1.79E-10

5 0 1 3488.2026488 0.00032112 2.989E-6

6 0 1 3488.2704392 0.00006806 1.342E-7

7 0 1 3488.284796 0.00001439 6.003E-9

8 0 1 3488.2878318 0.00000304 2.68E-10

9 0 1 3488.2884735 0.00000064 1.21E-11

10 0 0 3488.2886091 0.00000000 3.619E-6


Fit Statistics




CC - mixed - PQL


Estimated G Matrix

Effect Row Col1

Intercept 1 2.0263


Matrix

Effect Row Col1

Intercept 1 1.0000


Cov Standard


UN(1,1) subject 2.0263 0.3902


Standard


time 1 -1.1929 0.3149 558 -3.79 0.0002


time 2 -1.0484 0.3087 558 -3.40 0.0007

time 3 -1.3451 0.3221 558 -4.18 <.0001

time 4 -1.9670 0.3611 558 -5.45 <.0001

time*treat 1 1 0.4529 0.4181 558 1.08 0.2792

time*treat 1 2 0 . . . .

time*treat 2 1 0.5768 0.4101 558 1.41 0.1601

time*treat 2 2 0 . . . .

time*treat 3 1 0.5498 0.4245 558 1.30 0.1958

time*treat 3 2 0 . . . .

time*treat 4 1 0.4371 0.4709 558 0.93 0.3536

time*treat 4 2 0 . . . .


Num Den


time 4 558 18.12 <.0001

time*treat 4 558 0.83 0.5039

LOCF - mixed - PQL


Model Information

Data Set M.ARMDLOCF

Response Variable LOCF


Link Function Logit






Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 234 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20 22 23 24 25 26 27 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 49 50 51 52

53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

98 99 100 101 102 103 104 105 106 107 108 109

110 111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130 131

132 133 134 135 136 137 138 139 140 141 142

143 145 146 147 148 149 150 151 152 153 154


155 156 157 158 159 160 161 162 163 164 165

166 167 168 169 170 171 172 173 174 175 176

177 178 179 180 181 182 183 184 185 186 187

188 190 191 192 193 194 195 196 197 198 199

200 201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220 221

222 223 224 225 226 227 228 229 230 231 232

233 234 235 236 237 238 239 240



Response Profile

Ordered Total


1 0 273


LOCF - mixed - PQL


Response Profile

Ordered Total


2 1 660


Dimensions


Columns in X 12







Lower Boundaries 1

Upper Boundaries 0


Starting From Data


Iteration History

Objective Max


0 0 4 4189.2824592 0.31278061 7.91E-8

1 0 3 4298.5207801 0.11492834 1.32E-6

2 0 3 4339.6266764 0.03393012 1.11E-10

3 0 2 4350.2238767 0.00824645 2.055E-7

4 0 2 4352.654268 0.00185764 5.37E-10

5 0 1 4353.1927427 0.00040953 5.923E-6

6 0 1 4353.3109852 0.00009006 2.863E-7

7 0 1 4353.3369649 0.00001975 1.377E-8

8 0 1 4353.3426617 0.00000433 6.62E-10

9 0 1 4353.3439104 0.00000095 3.17E-11

10 0 0 4353.3441841 0.00000000 6.482E-6


LOCF - mixed - PQL


Fit Statistics




Estimated G Matrix

Effect Row Col1

Intercept 1 2.3422


Matrix

Effect Row Col1

Intercept 1 1.0000


Cov Standard


UN(1,1) subject 2.3422 0.3877



Standard


time 1 -1.0529 0.2773 693 -3.80 0.0002

time 2 -1.1803 0.2800 693 -4.22 <.0001

time 3 -1.2961 0.2843 693 -4.56 <.0001

time 4 -1.8856 0.3122 693 -6.04 <.0001

time*treat 1 1 0.2370 0.3855 693 0.61 0.5389

time*treat 1 2 0 . . . .

time*treat 2 1 0.6788 0.3825 693 1.77 0.0764

time*treat 2 2 0 . . . .

time*treat 3 1 0.5026 0.3893 693 1.29 0.1971

time*treat 3 2 0 . . . .

time*treat 4 1 0.3874 0.4240 693 0.91 0.3612

time*treat 4 2 0 . . . .

LOCF - mixed - PQL



Num Den


time 4 693 20.29 <.0001

time*treat 4 693 0.98 0.4167

as is - mixed - PQL


Model Information

Data Set M.ARMDWGEE



Link Function Logit






Class Levels Values

time 4 1 2 3 4

treat 2 1 2

subject 234 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19


20 22 23 24 25 26 27 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 49 50 51 52

53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

98 99 100 101 102 103 104 105 106 107 108 109

110 111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130 131

132 133 134 135 136 137 138 139 140 141 142

143 145 146 147 148 149 150 151 152 153 154

155 156 157 158 159 160 161 162 163 164 165

166 167 168 169 170 171 172 173 174 175 176

177 178 179 180 181 182 183 184 185 186 187

188 190 191 192 193 194 195 196 197 198 199

200 201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220 221

222 223 224 225 226 227 228 229 230 231 232

233 234 235 236 237 238 239 240



Response Profile

Ordered Total


1 0 252


as is - mixed - PQL


Response Profile

Ordered Total


2 1 615


Dimensions


Columns in X 12








Lower Boundaries 1

Upper Boundaries 0


Starting From Data

Iteration History

Objective Max


0 0 4 3883.4998582 0.31708535 3.285E-8

1 0 3 3967.3330478 0.10156883 3.249E-7

2 0 3 3997.4203228 0.02877119 1.9E-11

3 0 2 4004.7088334 0.00677621 7.8E-8

4 0 2 4006.3408848 0.00150218 1.91E-10

5 0 1 4006.698031 0.00032765 3.624E-6

6 0 1 4006.7756985 0.00007134 1.718E-7

7 0 1 4006.7925978 0.00001550 8.11E-9

8 0 1 4006.7962695 0.00000337 3.83E-10

9 0 1 4006.797067 0.00000073 1.8E-11

10 0 0 4006.7972402 0.00000000 5.08E-6


as is - mixed - PQL


Fit Statistics




Estimated G Matrix

Effect Row Col1

Intercept 1 1.9455


Matrix

Effect Row Col1


Intercept 1 1.0000


Cov Standard


UN(1,1) subject 1.9455 0.3488


Standard


time 1 -1.0028 0.2626 627 -3.82 0.0001

time 2 -1.1886 0.2752 627 -4.32 <.0001

time 3 -1.2573 0.2881 627 -4.36 <.0001

time 4 -2.0235 0.3511 627 -5.76 <.0001

time*treat 1 1 0.2168 0.3666 627 0.59 0.5543

time*treat 1 2 0 . . . .

time*treat 2 1 0.7142 0.3716 627 1.92 0.0551

time*treat 2 2 0 . . . .

time*treat 3 1 0.4932 0.3880 627 1.27 0.2042

time*treat 3 2 0 . . . .

time*treat 4 1 0.4578 0.4592 627 1.00 0.3192

time*treat 4 2 0 . . . .

as is - mixed - PQL



Num Den


time 4 627 21.02 <.0001

time*treat 4 627 1.15 0.3312

CC - mixed - numerical integration

The NLMIXED Procedure

Specifications

Data Set WORK.HELP


Distribution for Dependent Variable Binary

Random Effects b

Distribution for Random Effects Normal

Subject Variable subject



Integration Method Adaptive Gaussian

Quadrature

Dimensions

Observations Used 752

Observations Not Used 0

Total Observations 752

Subjects 188

Max Obs Per Subject 4

Parameters 9

Quadrature Points 20

Parameters

beta11 beta12 beta13 beta14 beta21 beta22 beta23 beta24 tau

1 1 1 1 1 1 1 1 1

Parameters

NegLogLike

436.884377

Iteration History

Iter Calls NegLogLike Diff MaxGrad Slope

1 22 394.450701 42.43368 26.03214 -77.1839

2 33 385.186335 9.264366 6.747321 -15.7106

3 44 383.970148 1.216187 1.18193 -2.17069

4 55 383.91831 0.051839 0.060632 -0.10021

5 66 383.918161 0.000148 0.000184 -0.0003

6 77 383.918161 1.398E-9 1.456E-9 -2.8E-9

CC - mixed - numerical integration


NOTE: GCONV convergence criterion satisfied.

Fit Statistics






Parameter Estimates

Standard

Parameter Estimate Error DF t Value Pr > |t| Alpha Lower Upper Gradient

beta11 1.7261 0.4203 187 4.11 <.0001 0.05 0.8970 2.5553 -163E-13

beta12 1.5335 0.4118 187 3.72 0.0003 0.05 0.7212 2.3458 -747E-13

beta13 1.9277 0.4302 187 4.48 <.0001 0.05 1.0790 2.7765 1.53E-11

beta14 2.7405 0.4811 187 5.70 <.0001 0.05 1.7914 3.6896 1.58E-10

beta21 -0.6386 0.5399 187 -1.18 0.2384 0.05 -1.7037 0.4265 1.08E-10

beta22 -0.8086 0.5327 187 -1.52 0.1308 0.05 -1.8596 0.2424 7.43E-11

beta23 -0.7659 0.5467 187 -1.40 0.1629 0.05 -1.8444 0.3126 1.15E-10

beta24 -0.6039 0.5940 187 -1.02 0.3106 0.05 -1.7757 0.5679 1.75E-10

tau 2.1909 0.2670 187 8.21 <.0001 0.05 1.6642 2.7176 -1.46E-9

Additional Estimates

Standard

Label Estimate Error DF t Value Pr > |t| Alpha Lower Upper

tau^2 4.8001 1.1698 187 4.10 <.0001 0.05 2.4923 7.1078

LOCF - mixed - numerical integration


Specifications

Data Set WORK.HELP

Dependent Variable LOCF


Random Effects b





Quadrature

Dimensions




Subjects 234


Parameters 9


Parameters



1 1 1 1 1 1 1 1 1

Parameters

NegLogLike

534.360977

Iteration History


1 22 478.6083 55.75268 28.85649 -94.3375

2 33 467.330215 11.27808 8.671126 -18.6908

3 44 465.387462 1.942753 1.760915 -3.41454

4 55 465.274339 0.113123 0.124086 -0.2161

5 66 465.27373 0.000609 0.000733 -0.00121

6 77 465.27373 2.115E-8 2.469E-8 -4.23E-8

LOCF - mixed - numerical integration



Fit Statistics





Parameter Estimates

Standard


beta11 1.6327 0.3879 233 4.21 <.0001 0.05 0.8684 2.3969 5.82E-10

beta12 1.8036 0.3923 233 4.60 <.0001 0.05 1.0307 2.5766 3.9E-10

beta13 1.9626 0.3989 233 4.92 <.0001 0.05 1.1768 2.7485 1.117E-9

beta14 2.7632 0.4390 233 6.29 <.0001 0.05 1.8983 3.6280 2.909E-9

beta21 -0.3787 0.5180 233 -0.73 0.4654 0.05 -1.3993 0.6418 2.318E-9

beta22 -0.9835 0.5170 233 -1.90 0.0584 0.05 -2.0022 0.03517 1.895E-9

beta23 -0.7363 0.5226 233 -1.41 0.1602 0.05 -1.7659 0.2934 2.351E-9

beta24 -0.5723 0.5582 233 -1.03 0.3063 0.05 -1.6721 0.5275 3.018E-9

tau 2.4660 0.2680 233 9.20 <.0001 0.05 1.9380 2.9940 -2.47E-8



Standard


tau^2 6.0812 1.3217 233 4.60 <.0001 0.05 3.4771 8.6853

as is - mixed - numerical integration


Specifications

Data Set WORK.HELP



Random Effects b





Quadrature

Dimensions




Subjects 234


Parameters 9


Parameters


1 1 1 1 1 1 1 1 1

Parameters

NegLogLike

505.797259

Iteration History


1 22 457.311287 48.48597 27.93691 -87.3977

2 33 447.342964 9.968323 7.182104 -16.9176

3 44 446.067867 1.275097 1.217035 -2.28425

4 55 446.017704 0.050164 0.057482 -0.09723


5 66 446.017583 0.000121 0.000147 -0.00024

6 77 446.017583 8.06E-10 7.5E-10 -1.61E-9

as is - mixed - numerical integration



Fit Statistics





Parameter Estimates

Standard


beta11 1.4987 0.3584 233 4.18 <.0001 0.05 0.7926 2.2048 -268E-13

beta12 1.7347 0.3726 233 4.66 <.0001 0.05 1.0006 2.4689 -208E-13

beta13 1.8292 0.3877 233 4.72 <.0001 0.05 1.0654 2.5930 6.91E-12

beta14 2.8463 0.4661 233 6.11 <.0001 0.05 1.9280 3.7647 9.48E-11

beta21 -0.3366 0.4806 233 -0.70 0.4844 0.05 -1.2834 0.6103 7.4E-11

beta22 -0.9954 0.4878 233 -2.04 0.0424 0.05 -1.9564 -0.03438 5.81E-11

beta23 -0.6944 0.5036 233 -1.38 0.1693 0.05 -1.6867 0.2979 7.09E-11

beta24 -0.6385 0.5779 233 -1.10 0.2704 0.05 -1.7770 0.5000 9.63E-11

tau 2.1979 0.2514 233 8.74 <.0001 0.05 1.7026 2.6933 -75E-11


Standard


tau^2 4.8308 1.1052 233 4.37 <.0001 0.05 2.6533 7.0083


GEE after multiple imputation

Imputation Number=1


Model Information

Data Set M.ARMD13C



Link Function Logit






Missing Values 4


Class Levels Values

time 4 1 2 3 4

subject 240 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

...

Response Profile

Ordered Total


1 0 271

2 1 685




Parameter Effect

Prm1 Intercept

Prm2 time1

Prm3 time2

Prm4 time3

Prm5 time4

Prm6 trttime1

Prm7 trttime2

Prm8 trttime3

Prm9 trttime4




Deviance 948 1115.3857 1.1766

Scaled Deviance 948 1115.3857 1.1766


Scaled Pearson X2 948 956.0000 1.0084



Estimated Covariance Matrix

Prm2 Prm3 Prm4 Prm5 Prm6 Prm7 Prm8 Prm9

Prm2 0.03968 0 0 0 -0.03968 0 0 0

Prm3 0 0.04349 0 0 0 -0.04349 0 0

Prm4 0 0 0.04444 0 0 0 -0.04444 0

Prm5 0 0 0 0.06000 0 0 0 -0.06000

Prm6 -0.03968 0 0 0 0.07689 0 0 0

Prm7 0 -0.04349 0 0 0 0.07866 0 0

Prm8 0 0 -0.04444 0 0 0 0.08166 0

Prm9 0 0 0 -0.06000 0 0 0 0.11064




Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

time1 1 -0.8473 0.1992 -1.2377 -0.4569 18.09 <.0001

time2 1 -1.0546 0.2086 -1.4634 -0.6459 25.57 <.0001

time3 1 -1.0986 0.2108 -1.5118 -0.6854 27.16 <.0001

time4 1 -1.6094 0.2449 -2.0895 -1.1293 43.17 <.0001

trttime1 1 0.2042 0.2773 -0.3393 0.7477 0.54 0.4616




trttime2 1 0.6281 0.2805 0.0784 1.1778 5.02 0.0251

trttime3 1 0.4555 0.2858 -0.1046 1.0155 2.54 0.1109

trttime4 1 0.2850 0.3326 -0.3669 0.9369 0.73 0.3915

Scale 0 1.0000 0.0000 1.0000 1.0000





Intercept . .












Correlation






Intercept 0.0000 0.0000 0.0000 0.0000 . .

time1 -0.8473 0.1992 -1.2377 -0.4569 -4.25 <.0001

time2 -1.0546 0.2086 -1.4634 -0.6459 -5.06 <.0001

time3 -1.0986 0.2108 -1.5118 -0.6854 -5.21 <.0001

time4 -1.6094 0.2449 -2.0895 -1.1293 -6.57 <.0001

trttime1 0.2042 0.2773 -0.3393 0.7477 0.74 0.4616

trttime2 0.6281 0.2805 0.0784 1.1778 2.24 0.0251

trttime3 0.4555 0.2858 -0.1046 1.0155 1.59 0.1109

trttime4 0.2850 0.3326 -0.3669 0.9369 0.86 0.3915





Intercept 0.0000 0.0000 0.0000 0.0000 . .

time1 -0.8473 0.1992 -1.2377 -0.4569 -4.25 <.0001

time2 -1.0546 0.2086 -1.4634 -0.6459 -5.06 <.0001

time3 -1.0986 0.2108 -1.5118 -0.6854 -5.21 <.0001

time4 -1.6094 0.2449 -2.0895 -1.1293 -6.57 <.0001

trttime1 0.2042 0.2773 -0.3393 0.7477 0.74 0.4616


trttime2 0.6281 0.2805 0.0784 1.1778 2.24 0.0251

trttime3 0.4555 0.2858 -0.1046 1.0155 1.59 0.1109

trttime4 0.2850 0.3326 -0.3669 0.9369 0.86 0.3915

Scale 1.0000 . . . . .


GEE after multiple imputation

Lower Upper Prob

Obs _Imputation_ Parameter DF Estimate StdErr WaldCL WaldCL ChiSq ChiSq

1 1 Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

2 1 time1 1 -0.8473 0.1992 -1.2377 -0.4569 18.09 <.0001

3 1 time2 1 -1.0546 0.2086 -1.4634 -0.6459 25.57 <.0001

4 1 time3 1 -1.0986 0.2108 -1.5118 -0.6854 27.16 <.0001

5 1 time4 1 -1.6094 0.2449 -2.0895 -1.1293 43.17 <.0001

6 1 trttime1 1 0.2042 0.2773 -0.3393 0.7477 0.54 0.4616

7 1 trttime2 1 0.6281 0.2805 0.0784 1.1778 5.02 0.0251

8 1 trttime3 1 0.4555 0.2858 -0.1046 1.0155 2.54 0.1109

9 1 trttime4 1 0.2850 0.3326 -0.3669 0.9369 0.73 0.3915

10 1 Scale 0 1.0000 0.0000 1.0000 1.0000 _ _

11 2 Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

12 2 time1 1 -0.8079 0.1977 -1.1954 -0.4205 16.70 <.0001

13 2 time2 1 -1.0546 0.2086 -1.4634 -0.6459 25.57 <.0001

14 2 time3 1 -1.0986 0.2108 -1.5118 -0.6854 27.16 <.0001

15 2 time4 1 -1.6707 0.2501 -2.1608 -1.1806 44.64 <.0001

16 2 trttime1 1 0.1648 0.2762 -0.3766 0.7061 0.36 0.5508

17 2 trttime2 1 0.6632 0.2800 0.1143 1.2120 5.61 0.0179

18 2 trttime3 1 0.4555 0.2858 -0.1046 1.0155 2.54 0.1109

19 2 trttime4 1 0.3463 0.3364 -0.3131 1.0056 1.06 0.3033

20 2 Scale 0 1.0000 0.0000 1.0000 1.0000 _ _

21 3 Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

22 3 time1 1 -0.8079 0.1977 -1.1954 -0.4205 16.70 <.0001

23 3 time2 1 -1.0116 0.2064 -1.4162 -0.6070 24.01 <.0001

24 3 time3 1 -0.9694 0.2044 -1.3701 -0.5687 22.48 <.0001

25 3 time4 1 -1.6707 0.2501 -2.1608 -1.1806 44.64 <.0001

26 3 trttime1 1 0.1274 0.2770 -0.4156 0.6703 0.21 0.6457

27 3 trttime2 1 0.5851 0.2789 0.0385 1.1317 4.40 0.0359

28 3 trttime3 1 0.3263 0.2811 -0.2247 0.8772 1.35 0.2457

29 3 trttime4 1 0.4447 0.3323 -0.2066 1.0961 1.79 0.1808

30 3 Scale 0 1.0000 0.0000 1.0000 1.0000 _ _

31 4 Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

32 4 time1 1 -0.8873 0.2008 -1.2809 -0.4937 19.52 <.0001

33 4 time2 1 -1.0546 0.2086 -1.4634 -0.6459 25.57 <.0001

34 4 time3 1 -1.0546 0.2086 -1.4634 -0.6459 25.57 <.0001

35 4 time4 1 -1.5506 0.2402 -2.0215 -1.0797 41.66 <.0001

36 4 trttime1 1 0.2442 0.2785 -0.3016 0.7900 0.77 0.3806

37 4 trttime2 1 0.6281 0.2805 0.0784 1.1778 5.02 0.0251

38 4 trttime3 1 0.4115 0.2841 -0.1453 0.9683 2.10 0.1475

39 4 trttime4 1 0.3246 0.3250 -0.3124 0.9616 1.00 0.3179

40 4 Scale 0 1.0000 0.0000 1.0000 1.0000 _ _

41 5 Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

42 5 time1 1 -0.8473 0.1992 -1.2377 -0.4569 18.09 <.0001

43 5 time2 1 -0.9694 0.2044 -1.3701 -0.5687 22.48 <.0001


44 5 time3 1 -1.0986 0.2108 -1.5118 -0.6854 27.16 <.0001

45 5 time4 1 -1.4392 0.2319 -1.8938 -0.9847 38.51 <.0001

46 5 trttime1 1 0.2412 0.2765 -0.3009 0.7832 0.76 0.3832

47 5 trttime2 1 0.5779 0.2770 0.0351 1.1208 4.35 0.0369

48 5 trttime3 1 0.4555 0.2858 -0.1046 1.0155 2.54 0.1109

49 5 trttime4 1 0.2133 0.3189 -0.4118 0.8383 0.45 0.5036

50 5 Scale 0 1.0000 0.0000 1.0000 1.0000 _ _

51 6 Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

52 6 time1 1 -0.8473 0.1992 -1.2377 -0.4569 18.09 <.0001

53 6 time2 1 -1.0116 0.2064 -1.4162 -0.6070 24.01 <.0001

54 6 time3 1 -1.0116 0.2064 -1.4162 -0.6070 24.01 <.0001

55 6 time4 1 -1.8718 0.2685 -2.3981 -1.3455 48.58 <.0001

56 6 trttime1 1 0.2412 0.2765 -0.3009 0.7832 0.76 0.3832

57 6 trttime2 1 0.5851 0.2789 0.0385 1.1317 4.40 0.0359

58 6 trttime3 1 0.3685 0.2825 -0.1853 0.9222 1.70 0.1922

59 6 trttime4 1 0.6459 0.3464 -0.0332 1.3249 3.48 0.0623

60 6 Scale 0 1.0000 0.0000 1.0000 1.0000 _ _

61 7 Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

62 7 time1 1 -0.8079 0.1977 -1.1954 -0.4205 16.70 <.0001

63 7 time2 1 -0.9280 0.2026 -1.3250 -0.5309 20.98 <.0001

64 7 time3 1 -1.0546 0.2086 -1.4634 -0.6459 25.57 <.0001

65 7 time4 1 -1.6094 0.2449 -2.0895 -1.1293 43.17 <.0001

66 7 trttime1 1 0.1648 0.2762 -0.3766 0.7061 0.36 0.5508

67 7 trttime2 1 0.5015 0.2761 -0.0396 1.0425 3.30 0.0693

68 7 trttime3 1 0.3741 0.2849 -0.1843 0.9324 1.72 0.1891

69 7 trttime4 1 0.3835 0.3285 -0.2603 1.0273 1.36 0.2430

70 7 Scale 0 1.0000 0.0000 1.0000 1.0000 _ _

71 8 Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

72 8 time1 1 -0.8473 0.1992 -1.2377 -0.4569 18.09 <.0001

73 8 time2 1 -1.1436 0.2132 -1.5615 -0.7256 28.76 <.0001

74 8 time3 1 -1.2368 0.2186 -1.6652 -0.8083 32.01 <.0001

75 8 time4 1 -1.5506 0.2402 -2.0215 -1.0797 41.66 <.0001

76 8 trttime1 1 0.2412 0.2765 -0.3009 0.7832 0.76 0.3832

77 8 trttime2 1 0.7170 0.2840 0.1605 1.2736 6.38 0.0116

78 8 trttime3 1 0.5936 0.2915 0.0222 1.1651 4.15 0.0417

79 8 trttime4 1 0.3719 0.3231 -0.2614 1.0053 1.32 0.2497

80 8 Scale 0 1.0000 0.0000 1.0000 1.0000 _ _

81 9 Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

82 9 time1 1 -0.8873 0.2008 -1.2809 -0.4937 19.52 <.0001

83 9 time2 1 -1.0116 0.2064 -1.4162 -0.6070 24.01 <.0001

84 9 time3 1 -0.9280 0.2026 -1.3250 -0.5309 20.98 <.0001

85 9 time4 1 -1.4939 0.2359 -1.9563 -1.0315 40.10 <.0001

86 9 trttime1 1 0.2812 0.2777 -0.2632 0.8255 1.02 0.3114

87 9 trttime2 1 0.5851 0.2789 0.0385 1.1317 4.40 0.0359

88 9 trttime3 1 0.2849 0.2797 -0.2634 0.8331 1.04 0.3085

89 9 trttime4 1 0.2194 0.3238 -0.4153 0.8541 0.46 0.4981

90 9 Scale 0 1.0000 0.0000 1.0000 1.0000 _ _

91 10 Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

92 10 time1 1 -0.8473 0.1992 -1.2377 -0.4569 18.09 <.0001

93 10 time2 1 -0.9694 0.2044 -1.3701 -0.5687 22.48 <.0001

94 10 time3 1 -1.1436 0.2132 -1.5615 -0.7256 28.76 <.0001

95 10 time4 1 -1.6094 0.2449 -2.0895 -1.1293 43.17 <.0001

96 10 trttime1 1 0.2042 0.2773 -0.3393 0.7477 0.54 0.4616

97 10 trttime2 1 0.5779 0.2770 0.0351 1.1208 4.35 0.0369


98 10 trttime3 1 0.5740 0.2862 0.0132 1.1349 4.02 0.0449

99 10 trttime4 1 0.4308 0.3267 -0.2094 1.0710 1.74 0.1872

100 10 Scale 0 1.0000 0.0000 1.0000 1.0000 _ _

_

I

m

p

u

t R

a o

t w

i N P P P P P P P P

O o a r r r r r r r r

b n m m m m m m m m m

s _ e 2 3 4 5 6 7 8 9

1 1 Prm2 0.0396825 0 0 0 -0.039683 0 0 0

2 1 Prm3 0 0.043494 0 0 0 -0.043494 0 0

3 1 Prm4 0 0 0.0444444 0 0 0 -0.044444 0

4 1 Prm5 0 0 0 0.06 0 0 0 -0.06

5 1 Prm6 -0.039683 0 0 0 0.0768933 0 0 0

6 1 Prm7 0 -0.043494 0 0 0 0.0786595 0 0

7 1 Prm8 0 0 -0.044444 0 0 0 0.0816552 0

8 1 Prm9 0 0 0 -0.06 0 0 0 0.1106383

9 2 Prm2 0.0390752 0 0 0 -0.039075 0 0 0

10 2 Prm3 0 0.043494 0 0 0 -0.043494 0 0

11 2 Prm4 0 0 0.0444444 0 0 0 -0.044444 0

12 2 Prm5 0 0 0 0.0625326 0 0 0 -0.062533

13 2 Prm6 -0.039075 0 0 0 0.076286 0 0 0

14 2 Prm7 0 -0.043494 0 0 0 0.0784119 0 0

15 2 Prm8 0 0 -0.044444 0 0 0 0.0816552 0

16 2 Prm9 0 0 0 -0.062533 0 0 0 0.1131709

...

The MIANALYZE Procedure

Model Information

PARMS Data Set WORK.GMPARMS

PARMINFO Data Set WORK.GMPINFO

COVB Data Set WORK.GMCOVB


Multiple Imputation Variance Information

-----------------Variance-----------------

Parameter Between Within Total DF

time1 0.000856 0.039631 0.040573 16698

time2 0.003629 0.042843 0.046835 1238.9


time3 0.007760 0.043903 0.052440 339.63

time4 0.013950 0.060207 0.075551 218.18

trttime1 0.002235 0.076721 0.079179 9337.3

trttime2 0.003376 0.077934 0.081648 4350.7

trttime3 0.009839 0.081079 0.091902 648.95

trttime4 0.015836 0.108550 0.125969 470.66


Relative Fraction

Increase Missing Relative

Parameter in Variance Information Efficiency

time1 0.023768 0.023333 0.997672

time2 0.093173 0.086705 0.991404

time3 0.194438 0.167673 0.983509

time4 0.254868 0.210309 0.979402

trttime1 0.032041 0.031254 0.996884

trttime2 0.047649 0.045921 0.995429

trttime3 0.133485 0.120471 0.988096

trttime4 0.160474 0.141922 0.986006

Multiple Imputation Parameter Estimates

Parameter Estimate Std Error 95% Confidence Limits DF

time1 -0.843486 0.201427 -1.23831 -0.44867 16698

time2 -1.020910 0.216413 -1.44549 -0.59633 1238.9

time3 -1.069445 0.228997 -1.51988 -0.61901 339.63

time4 -1.607580 0.274866 -2.14931 -1.06585 218.18

trttime1 0.211407 0.281388 -0.34017 0.76299 9337.3

trttime2 0.604904 0.285740 0.04471 1.16510 4350.7

trttime3 0.429925 0.303153 -0.16535 1.02521 648.95

trttime4 0.366539 0.354921 -0.33089 1.06396 470.66


t for H0:

Parameter Minimum Maximum Theta0 Parameter=Theta0 Pr > |t|

time1 -0.887303 -0.807923 0 -4.19 <.0001

time2 -1.143564 -0.927987 0 -4.72 <.0001

time3 -1.236763 -0.927987 0 -4.67 <.0001

time4 -1.871802 -1.439215 0 -5.85 <.0001

trttime1 0.127354 0.281167 0 0.75 0.4525

trttime2 0.501468 0.717045 0 2.12 0.0343

trttime3 0.284850 0.593626 0 1.42 0.1566

trttime4 0.213264 0.645850 0 1.03 0.3023

Within-Imputation Covariance Matrix


time1 time2 time3 time4

time1 0.0396310626 0.0000000000 0.0000000000 0.0000000000

time2 0.0000000000 0.0428428975 0.0000000000 0.0000000000

time3 0.0000000000 0.0000000000 0.0439033458 0.0000000000

time4 0.0000000000 0.0000000000 0.0000000000 0.0602066141

trttime1 -.0396310626 0.0000000000 0.0000000000 0.0000000000

trttime2 0.0000000000 -.0428428975 0.0000000000 0.0000000000

trttime3 0.0000000000 0.0000000000 -.0439033458 0.0000000000

trttime4 0.0000000000 0.0000000000 0.0000000000 -.0602066141

Within-Imputation Covariance Matrix

trttime1 trttime2 trttime3 trttime4

time1 -.0396310626 0.0000000000 0.0000000000 0.0000000000

time2 0.0000000000 -.0428428975 0.0000000000 0.0000000000

time3 0.0000000000 0.0000000000 -.0439033458 0.0000000000

time4 0.0000000000 0.0000000000 0.0000000000 -.0602066141

trttime1 0.0767208784 0.0000000000 0.0000000000 0.0000000000

trttime2 0.0000000000 0.0779340889 0.0000000000 0.0000000000

trttime3 0.0000000000 0.0000000000 0.0810791448 0.0000000000

trttime4 0.0000000000 0.0000000000 0.0000000000 0.1085496567

Between-Imputation Covariance Matrix


time1 0.0008563099 0.0004082521 -.0003197344 -.0013188041

time2 0.0004082521 0.0036288964 0.0023147112 -.0000340066

time3 -.0003197344 0.0023147112 0.0077604498 -.0019018583

time4 -.0013188041 -.0000340066 -.0019018583 0.0139497428

trttime1 -.0012313718 -.0006629132 -.0000847749 0.0018887228

trttime2 -.0002994706 -.0033591680 -.0028303911 0.0004365905

trttime3 0.0001406515 -.0022799319 -.0084281421 0.0018943906

trttime4 0.0011677226 0.0003481757 0.0005081848 -.0135624411

Between-Imputation Covariance Matrix


time1 -.0012313718 -.0002994706 0.0001406515 0.0011677226

time2 -.0006629132 -.0033591680 -.0022799319 0.0003481757

time3 -.0000847749 -.0028303911 -.0084281421 0.0005081848

time4 0.0018887228 0.0004365905 0.0018943906 -.0135624411

trttime1 0.0022347438 0.0005690403 0.0002194178 -.0015574154

trttime2 0.0005690403 0.0033759208 0.0030399236 -.0007737411

trttime3 0.0002194178 0.0030399236 0.0098389196 -.0000532713

trttime4 -.0015574154 -.0007737411 -.0000532713 0.0158357980

Total Covariance Matrix



time1 0.0429818143 0.0000000000 0.0000000000 0.0000000000

time2 0.0000000000 0.0464652054 0.0000000000 0.0000000000

time3 0.0000000000 0.0000000000 0.0476153131 0.0000000000

time4 0.0000000000 0.0000000000 0.0000000000 0.0652970002

trttime1 -.0429818143 0.0000000000 0.0000000000 0.0000000000

trttime2 0.0000000000 -.0464652054 0.0000000000 0.0000000000

trttime3 0.0000000000 0.0000000000 -.0476153131 0.0000000000

trttime4 0.0000000000 0.0000000000 0.0000000000 -.0652970002

Total Covariance Matrix


time1 -.0429818143 0.0000000000 0.0000000000 0.0000000000

time2 0.0000000000 -.0464652054 0.0000000000 0.0000000000

time3 0.0000000000 0.0000000000 -.0476153131 0.0000000000

time4 0.0000000000 0.0000000000 0.0000000000 -.0652970002

trttime1 0.0832075228 0.0000000000 0.0000000000 0.0000000000

trttime2 0.0000000000 0.0845233085 0.0000000000 0.0000000000

trttime3 0.0000000000 0.0000000000 0.0879342745 0.0000000000

trttime4 0.0000000000 0.0000000000 0.0000000000 0.1177273803

NLMIXED after multiple imputation


Specifications

Data Set M.ARMD13C



Random Effects b





Quadrature

Imputation Number=1


Dimensions




Subjects 239


Parameters 9



Parameters


1 1 1 1 1 1 1 1 1

Parameters

NegLogLike

545.864244

Iteration History


1 22 493.592025 52.27222 27.84385 -91.265

2 33 483.920182 9.671844 7.473753 -16.282

3 44 482.586491 1.333691 1.280772 -2.38803

4 55 482.533795 0.052696 0.060668 -0.10216

5 66 482.53367 0.000125 0.000154 -0.00025

6 77 482.53367 8.03E-10 7.6E-10 -1.61E-9


Fit Statistics






Parameter Estimates

Standard


beta11 1.4664 0.3482 238 4.21 <.0001 0.05 0.7804 2.1524 -743E-14

beta12 1.8086 0.3611 238 5.01 <.0001 0.05 1.0972 2.5200 5.63E-12

beta13 1.8803 0.3641 238 5.16 <.0001 0.05 1.1630 2.5977 2.45E-11

beta14 2.6907 0.4060 238 6.63 <.0001 0.05 1.8908 3.4905 9.92E-11

beta21 -0.2967 0.4749 238 -0.62 0.5326 0.05 -1.2322 0.6387 8.4E-11

beta22 -1.0237 0.4799 238 -2.13 0.0339 0.05 -1.9692 -0.07828 7.25E-11

beta23 -0.7107 0.4838 238 -1.47 0.1432 0.05 -1.6637 0.2424 8.52E-11

beta24 -0.3622 0.5276 238 -0.69 0.4931 0.05 -1.4017 0.6772 1.07E-10

tau 2.2181 0.2409 238 9.21 <.0001 0.05 1.7435 2.6927 -76E-11


Covariance Matrix of Parameter Estimates

Row Parameter beta11 beta12 beta13 beta14 beta21 beta22 beta23 beta24

1 beta11 0.1213 0.05701 0.05737 0.06086 -0.1143 -0.05181 -0.05038 -0.04892

2 beta12 0.05701 0.1304 0.05976 0.06429 -0.04886 -0.1243 -0.05160 -0.05037

3 beta13 0.05737 0.05976 0.1326 0.06498 -0.04898 -0.05351 -0.1242 -0.05066

4 beta14 0.06086 0.06429 0.06498 0.1649 -0.04988 -0.05612 -0.05401 -0.1461

5 beta21 -0.1143 -0.04886 -0.04898 -0.04988 0.2255 0.09419 0.09468 0.09549

6 beta22 -0.05181 -0.1243 -0.05351 -0.05612 0.09419 0.2303 0.09590 0.09567

7 beta23 -0.05038 -0.05160 -0.1242 -0.05401 0.09468 0.09590 0.2340 0.09723

8 beta24 -0.04892 -0.05037 -0.05066 -0.1461 0.09549 0.09567 0.09723 0.2784

9 tau 0.02181 0.02544 0.02618 0.03422 -0.00320 -0.01159 -0.00757 -0.00246

Covariance

Matrix of

Parameter

Estimates

Row tau

1 0.02181

2 0.02544

3 0.02618

4 0.03422

5 -0.00320

6 -0.01159

7 -0.00757

8 -0.00246

9 0.05804


Standard


tau2 4.9200 1.0688 238 4.60 <.0001 0.05 2.8146 7.0255

Covariance Matrix of


Row Label Cov1

1 tau2 1.1422


Model Information

PARMS Data Set WORK.NLPARMS

COVB Data Set WORK.NLCOVB




-----------------Variance-----------------


beta11 0.006200 0.120558 0.127378 3139.1

beta12 0.012235 0.128374 0.141832 999.52

beta13 0.014881 0.130702 0.147071 726.52

beta14 0.030602 0.165001 0.198664 313.46

beta21 0.007052 0.223534 0.231291 8002.1

beta22 0.009883 0.227435 0.238307 4324.6

beta23 0.023720 0.231383 0.257475 876.41

beta24 0.040058 0.273874 0.317938 468.56


Relative Fraction



beta11 0.056574 0.054147 0.994614

beta12 0.104839 0.096697 0.990423

beta13 0.125240 0.113737 0.988754

beta14 0.204013 0.174693 0.982831

beta21 0.034700 0.033778 0.996634

beta22 0.047800 0.046061 0.995415

beta23 0.112764 0.103381 0.989768

beta24 0.160891 0.142246 0.985975



beta11 1.455346 0.356901 0.75556 2.15513 3139.1


t for H0:


beta11 1.338106 1.582843 0 4.08 <.0001

MIANALYZE for NLMIXED 16:58 Sunday, December 26, 2004 885





beta12 1.749244 0.376606 1.01021 2.48827 999.52

beta13 1.826672 0.383498 1.07377 2.57957 726.52

beta14 2.686402 0.445717 1.80943 3.56338 313.46

beta21 -0.315416 0.480927 -1.25816 0.62733 8002.1

beta22 -0.988679 0.488167 -1.94574 -0.03162 4324.6

beta23 -0.673317 0.507420 -1.66922 0.32258 876.41

beta24 -0.515931 0.563860 -1.62394 0.59208 468.56


t for H0:


beta12 1.529515 1.882212 0 4.64 <.0001

beta13 1.653527 2.028812 0 4.76 <.0001

beta14 2.481554 3.074676 0 6.03 <.0001

beta21 -0.445510 -0.192473 0 -0.66 0.5119

beta22 -1.126595 -0.769312 0 -2.03 0.0429

beta23 -0.907306 -0.449662 0 -1.33 0.1849

beta24 -0.954038 -0.276338 0 -0.91 0.3607


Model Information

PARMS Data Set WORK.NLPARMS

COVB Data Set WORK.NLCOVB



-----------------Variance-----------------


beta11 0.006200 0.120558 0.127378 3139.1

beta12 0.012235 0.128374 0.141832 999.52

beta13 0.014881 0.130702 0.147071 726.52

beta14 0.030602 0.165001 0.198664 313.46

beta21 0.007052 0.223534 0.231291 8002.1

beta22 0.009883 0.227435 0.238307 4324.6

beta23 0.023720 0.231383 0.257475 876.41

beta24 0.040058 0.273874 0.317938 468.56

tau 0.008134 0.056946 0.065893 488.15


Relative Fraction



beta11 0.056574 0.054147 0.994614

beta12 0.104839 0.096697 0.990423

beta13 0.125240 0.113737 0.988754


beta14 0.204013 0.174693 0.982831

beta21 0.034700 0.033778 0.996634

beta22 0.047800 0.046061 0.995415

beta23 0.112764 0.103381 0.989768

beta24 0.160891 0.142246 0.985975

tau 0.157117 0.139302 0.986261



beta11 1.455346 0.356901 0.75556 2.15513 3139.1

beta12 1.749244 0.376606 1.01021 2.48827 999.52

beta13 1.826672 0.383498 1.07377 2.57957 726.52

beta14 2.686402 0.445717 1.80943 3.56338 313.46

beta21 -0.315416 0.480927 -1.25816 0.62733 8002.1

beta22 -0.988679 0.488167 -1.94574 -0.03162 4324.6

beta23 -0.673317 0.507420 -1.66922 0.32258 876.41

beta24 -0.515931 0.563860 -1.62394 0.59208 468.56




tau 2.203316 0.256697 1.69895 2.70768 488.15


t for H0:


beta11 1.338106 1.582843 0 4.08 <.0001

beta12 1.529515 1.882212 0 4.64 <.0001

beta13 1.653527 2.028812 0 4.76 <.0001

beta14 2.481554 3.074676 0 6.03 <.0001

beta21 -0.445510 -0.192473 0 -0.66 0.5119

beta22 -1.126595 -0.769312 0 -2.03 0.0429

beta23 -0.907306 -0.449662 0 -1.33 0.1849

beta24 -0.954038 -0.276338 0 -0.91 0.3607

tau 2.081615 2.332432 0 8.58 <.0001

4. Results of PQL and MQL.

PQL REML


Model Information

Data Set M.ARMD




Link Function Logit






Class Levels Values

subject 234 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20 22 23 24 25 26 27 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 49 50 51 52

53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

98 99 100 ...



Response Profile

Ordered Total


1 0 252

2 1 615


Dimensions


Columns in X 3





Optimization Technique Dual Quasi-Newton


Lower Boundaries 1

Upper Boundaries 0


Starting From Data


Iteration History

Objective Max


0 0 4 3893.4790252 0.30055232 0.000151

1 0 3 3972.9263817 0.09539047 0.000236

2 0 3 4000.5480588 0.02662055 0.000017

3 0 2 4007.1652454 0.00622526 0.000016

4 0 3 4008.6430457 0.00137635 4.613E-8

5 0 2 4008.9659864 0.00030013 2.148E-6

6 0 2 4009.0362183 0.00006515 1.012E-7

7 0 1 4009.0514556 0.00002402 0.000321

8 0 1 4009.0570731 0.00000792 0.000106

9 0 1 4009.055221 0.00000261 0.000035

10 0 1 4009.0558317 0.00000086 0.000011

11 0 0 4009.0556303 0.00000000 5.426E-6


Fit Statistics





Standard


Intercept subject 1.9113 0.3432


Standard

Effect Estimate Error DF t Value Pr > |t|

Intercept 0.6637 0.1593 233 4.17 <.0001

timec 0.01491 0.005979 631 2.49 0.0129

timec*treat 0.01018 0.008135 631 1.25 0.2113


Num Den


timec 1 631 6.22 0.0129

timec*treat 1 631 1.57 0.2113


PQL ML


Model Information

Data Set M.ARMD



Link Function Logit



Estimation Technique PL




Response Profile

Ordered Total


1 0 252

2 1 615


Dimensions


Columns in X 3







Lower Boundaries 1

Upper Boundaries 0


Starting From Data



Fit Statistics

-2 Log Pseudo-Likelihood 3986.53




Standard




Standard


Intercept 0.6625 0.1587 233 4.17 <.0001

timec 0.01484 0.005967 631 2.49 0.0131

timec*treat 0.01021 0.008111 631 1.26 0.2088


Num Den


timec 1 631 6.19 0.0131

timec*treat 1 631 1.58 0.2088

MQL REML


Model Information

Data Set M.ARMD



Link Function Logit



Estimation Technique Residual MPL





Response Profile

Ordered Total


1 0 252

2 1 615


Dimensions


Columns in X 3







Lower Boundaries 1

Upper Boundaries 0


Starting From Data


Fit Statistics





Standard




Standard


Intercept 0.5682 0.1404 233 4.05 <.0001

timec 0.01146 0.005373 631 2.13 0.0334


timec*treat 0.009728 0.007188 631 1.35 0.1764


Num Den


timec 1 631 4.55 0.0334

timec*treat 1 631 1.83 0.1764

MQL ML


Model Information

Data Set M.ARMD



Link Function Logit



Estimation Technique MPL




Response Profile

Ordered Total


1 0 252

2 1 615


Dimensions


Columns in X 3








Lower Boundaries 1

Upper Boundaries 0


Starting From Data


Fit Statistics





Standard




Standard


Intercept 0.5682 0.1401 233 4.06 <.0001

timec 0.01144 0.005370 631 2.13 0.0335

timec*treat 0.009758 0.007179 631 1.36 0.1746


Num Den


timec 1 631 4.54 0.0335

timec*treat 1 631 1.85 0.1746

5. Results of Gaussian Quadrature

Initial Values


Model Information

Data Set M.ARMD



Link Function Logit






Missing Values 93

Response Profile

Ordered Total


1 1 615

2 0 252

PROC GENMOD is modeling the probability that bindif=’1’.



Deviance 864 1028.1346 1.1900

Scaled Deviance 864 1028.1346 1.1900


Scaled Pearson X2 864 867.9430 1.0046



Analysis Of Parameter Estimates



Intercept 1 0.5670 0.1167 0.3383 0.7956 23.62 <.0001

timec 1 0.0098 0.0050 0.0000 0.0196 3.87 0.0492

timec*treat 1 0.0133 0.0060 0.0015 0.0251 4.87 0.0273

Scale 0 1.0000 0.0000 1.0000 1.0000


MML GQ 25



Specifications

Data Set M.ARMD



Random Effects b




Integration Method Gaussian Quadrature

Dimensions




Subjects 234


Parameters 4


Parameters

beta0 beta1 beta2 sigmab NegLogLike

0.567 0.0098 0.0133 2 456.839932

Iteration History


1 4 451.981825 4.858107 125.1142 -5664.22

2 7 451.75583 0.225995 148.1353 -34.8464

3 9 450.232672 1.523158 148.7488 -1.52912

4 11 449.371915 0.860757 13.61276 -0.74256

5 13 449.327407 0.044507 13.61007 -0.04633

6 14 449.303743 0.023664 11.90489 -0.02619

7 16 449.292117 0.011626 0.156812 -0.02244

8 18 449.292115 2.304E-6 0.007391 -4.56E-6

9 20 449.292115 6.26E-9 0.000469 -1.15E-8


Fit Statistics






Parameter Estimates

Standard


beta0 1.0168 0.2271 233 4.48 <.0001 0.05 0.5694 1.4641 2.666E-6

beta1 0.02112 0.007315 233 2.89 0.0042 0.05 0.006712 0.03554 0.000469

beta2 0.01091 0.01019 233 1.07 0.2856 0.05 -0.00917 0.03099 0.000237

sigmab 2.1864 0.2510 233 8.71 <.0001 0.05 1.6918 2.6809 -6.68E-6


Standard


sigmab^2 4.7802 1.0977 233 4.35 <.0001 0.05 2.6175 6.9428

MML GQ 51


Specifications

Data Set M.ARMD



Random Effects b





Dimensions




Subjects 234


Parameters 4


Parameters


0.567 0.0098 0.0133 2 456.839837

Iteration History



1 4 451.983217 4.856621 124.87 -5660.5

2 7 451.757185 0.226032 147.9446 -34.8474

3 9 450.235463 1.521722 148.7502 -1.52714

4 11 449.374938 0.860525 13.66278 -0.74259

5 13 449.330649 0.044289 13.67193 -0.04632

6 14 449.307685 0.022965 12.29056 -0.02598

7 16 449.29571 0.011975 0.152137 -0.02313

8 18 449.295708 1.801E-6 0.007551 -3.54E-6


Fit Statistics



Fit Statistics



Parameter Estimates

Standard


beta0 1.0170 0.2273 233 4.47 <.0001 0.05 0.5692 1.4648 0.000547

beta1 0.02113 0.007314 233 2.89 0.0042 0.05 0.006719 0.03554 0.007551

beta2 0.01089 0.01019 233 1.07 0.2865 0.05 -0.00919 0.03097 0.002419

sigmab 2.1850 0.2500 233 8.74 <.0001 0.05 1.6924 2.6776 -0.0003


Standard


sigmab^2 4.7744 1.0927 233 4.37 <.0001 0.05 2.6216 6.9271

MML Laplace


Specifications

Data Set M.ARMD




Random Effects b





Quadrature

Dimensions




Subjects 234


Parameters 4

Quadrature Points 1

Parameters


0.567 0.0098 0.0133 2 462.458846

Iteration History


1 4 457.050292 5.408554 136.2014 -6291.71

2 7 456.814449 0.235842 160.1404 -35.5955

3 9 454.972757 1.841692 165.5178 -1.82297

4 11 454.144846 0.827912 18.57783 -0.86111

5 13 454.099807 0.045039 19.46722 -0.04876

6 14 454.061138 0.038669 11.38215 -0.03014

7 16 454.055794 0.005344 0.571366 -0.01023

8 18 454.055783 0.000011 0.004151 -0.00002

9 20 454.055783 1.592E-8 0.000108 -2.97E-8


Fit Statistics





Parameter Estimates


Standard


beta0 1.0382 0.2264 233 4.59 <.0001 0.05 0.5922 1.4842 -4.39E-6

beta1 0.02041 0.007220 233 2.83 0.0051 0.05 0.006188 0.03464 0.000108

beta2 0.01128 0.009975 233 1.13 0.2593 0.05 -0.00837 0.03093 -0.00005

sigmab 2.0543 0.2380 233 8.63 <.0001 0.05 1.5853 2.5233 1.078E-6


Standard


sigmab^2 4.2202 0.9779 233 4.32 <.0001 0.05 2.2934 6.1469

MML AGQ 5


Specifications

Data Set M.ARMD



Random Effects b





Quadrature

Dimensions




Subjects 234


Parameters 4

Quadrature Points 5

Parameters


0.567 0.0098 0.0133 2 457.266838

Iteration History



1 4 452.37829 4.888548 124.7635 -5686.5

2 7 452.152399 0.225891 147.905 -34.8432

3 9 450.700052 1.452347 143.1303 -1.4782

4 11 449.928621 0.771431 12.92075 -0.69273

5 13 449.898346 0.030275 13.53213 -0.03481

6 14 449.885732 0.012614 15.18799 -0.01832

7 15 449.87628 0.009452 5.89114 -0.02185

8 17 449.874431 0.001849 0.114113 -0.00374

9 19 449.87443 4.934E-7 0.000905 -9.67E-7


Fit Statistics





Parameter Estimates

Standard


beta0 0.9940 0.2210 233 4.50 <.0001 0.05 0.5586 1.4294 0.000057

beta1 0.02090 0.007262 233 2.88 0.0044 0.05 0.006592 0.03521 0.000726

beta2 0.01102 0.01009 233 1.09 0.2758 0.05 -0.00885 0.03089 0.000905

sigmab 2.1232 0.2328 233 9.12 <.0001 0.05 1.6645 2.5819 -0.00019


Standard


sigmab^2 4.5081 0.9887 233 4.56 <.0001 0.05 2.5602 6.4560

MML AGQ 11


Specifications

Data Set M.ARMD



Random Effects b






Quadrature

Dimensions




Subjects 234


Parameters 4


Parameters


0.567 0.0098 0.0133 2 456.835902

Iteration History


1 4 451.979375 4.856526 124.8787 -5660.55

2 7 451.75333 0.226045 147.9511 -34.8481

3 9 450.229448 1.523882 148.9323 -1.52855

4 11 449.366036 0.863412 13.69402 -0.74424

5 13 449.320871 0.045165 13.67323 -0.04692

6 14 449.296972 0.023898 11.8723 -0.02656

7 16 449.285127 0.011845 0.15199 -0.02286

8 18 449.285125 2.285E-6 0.007377 -4.52E-6

9 20 449.285125 6.276E-9 0.000473 -1.15E-8


Fit Statistics





Parameter Estimates

Standard


beta0 1.0181 0.2277 233 4.47 <.0001 0.05 0.5695 1.4666 2.801E-6


beta1 0.02114 0.007316 233 2.89 0.0042 0.05 0.006722 0.03555 0.000473

beta2 0.01088 0.01020 233 1.07 0.2869 0.05 -0.00920 0.03097 0.000243

sigmab 2.1874 0.2512 233 8.71 <.0001 0.05 1.6925 2.6823 -6.73E-6


Standard


sigmab^2 4.7846 1.0989 233 4.35 <.0001 0.05 2.6195 6.9496

Empirical Bayes Estimates

The UNIVARIATE Procedure

Variable: Estimate (Empirical Bayes Estimate)

Moments

N 240 Sum Weights 240

Mean -0.2015791 Sum Observations -48.378989

Std Deviation 1.46466961 Variance 2.14525706

Skewness -0.5771659 Kurtosis -0.9872685

Uncorrected SS 522.468632 Corrected SS 512.716438

Coeff Variation -726.59787 Std Error Mean 0.09454402

Basic Statistical Measures

Location Variability

Mean -0.20158 Std Deviation 1.46467

Median -0.14263 Variance 2.14526

Mode 1.13816 Range 4.62938

Interquartile Range 2.55898

NOTE: The mode displayed is the smallest of 2 modes with a count of 44.

Tests for Location: Mu0=0

Test -Statistic- -----p Value------

Student’s t t -2.13212 Pr > |t| 0.0340

Sign M -3 Pr >= |M| 0.7439

Signed Rank S -2757.5 Pr >= |S| 0.0074

Quantiles (Definition 5)

Quantile Estimate

100% Max 1.231996


99% 1.231996

95% 1.231996

90% 1.231996

75% Q3 1.138164

50% Median -0.142633

25% Q1 -1.420813

10% -2.316417

5% -3.170772

1% -3.397384

0% Min -3.397384

Extreme Observations

------Lowest----- ----Highest----

Value Obs Value Obs

-3.39738 232 1.232 209

-3.39738 183 1.232 215

-3.39738 126 1.232 225

-3.39738 116 1.232 228

-3.39738 80 1.232 233

6. Likelihood Ratio Test for Random-Slopes and Treatment

MML AGQ 11 / Slopes


Specifications

Data Set M.ARMD



Random Effects b1 b2





Quadrature

Dimensions




Subjects 234


Parameters 6



Parameters

beta0 beta1 beta2 sigmab1 sigmab2 rho NegLogLike

0.567 0.0098 0.0133 2 1 -0.4 544.043791

Iteration History


1 6 509.742775 34.30102 84.0385 -862.232

2 10 498.18797 11.55481 99.93202 -1148.82

3 11 492.660684 5.527286 258.3582 -87.4619

4 13 473.505389 19.1553 310.3252 -208.164

5 15 460.92047 12.58492 149.7404 -338.749

6 18 454.404707 6.515763 27.99769 -39.3351

7 22 452.922877 1.48183 93.24139 -13.0028

8 24 447.074267 5.84861 62.17724 -26.2715

9 26 444.280983 2.793284 39.7117 -7.89304

10 28 443.944389 0.336594 18.89171 -0.91955

11 30 443.865729 0.078661 7.777303 -0.25302

12 32 443.837887 0.027842 7.197167 -0.04727

13 33 443.8235 0.014387 11.41167 -0.02545

14 34 443.799149 0.024351 3.130796 -0.04482

15 36 443.794579 0.00457 0.580312 -0.00868

16 38 443.794463 0.000115 0.106616 -0.00016

17 40 443.794458 5.048E-6 0.017819 -0.00001

18 42 443.794458 9.262E-8 0.000285 -1.85E-7


Fit Statistics





Parameter Estimates

Standard


beta0 0.7705 0.2916 232 2.64 0.0088 0.05 0.1960 1.3450 1.239E-6

beta1 0.05198 0.01965 232 2.64 0.0087 0.05 0.01326 0.09070 -0.00023

beta2 0.02617 0.01986 232 1.32 0.1890 0.05 -0.01297 0.06530 0.000285

sigmab1 2.3409 0.4453 232 5.26 <.0001 0.05 1.4635 3.2183 -0.00001

sigmab2 0.08663 0.03030 232 2.86 0.0046 0.05 0.02694 0.1463 0.000176


rho -0.06296 0.2146 232 -0.29 0.7695 0.05 -0.4858 0.3599 9.422E-6

LRT for Random-Slopes

Obs L01 df pval

1 10.9813 2 0.00412509

MML AGQ 21


Specifications

Data Set M.ARMD



Random Effects b





Quadrature

Dimensions




Subjects 234


Parameters 4


Parameters


0.567 0.0098 0.0133 2 456.839858

Iteration History


1 4 451.983229 4.856629 124.8697 -5660.5

2 7 451.757196 0.226033 147.9445 -34.8476

3 9 450.235473 1.521723 148.7505 -1.52714

4 11 449.374956 0.860517 13.6623 -0.74259

5 13 449.33067 0.044286 13.67166 -0.04632


6 14 449.307713 0.022957 12.29118 -0.02598

7 16 449.295735 0.011978 0.151947 -0.02314

8 18 449.295733 1.798E-6 0.007548 -3.54E-6


Fit Statistics





Parameter Estimates

Standard


beta0 1.0170 0.2273 233 4.47 <.0001 0.05 0.5692 1.4648 0.000546

beta1 0.02113 0.007314 233 2.89 0.0042 0.05 0.006718 0.03554 0.007548

beta2 0.01089 0.01019 233 1.07 0.2865 0.05 -0.00919 0.03097 0.002414

sigmab 2.1850 0.2500 233 8.74 <.0001 0.05 1.6924 2.6776 -0.0003

MML AGQ 21 / Slopes


Specifications

Data Set M.ARMD








Quadrature

Dimensions




Subjects 234


Parameters 6



Parameters

beta0 beta1 beta2 sigmab1 sigmab2 rho NegLogLike

0.567 0.0098 0.0133 2 1 -0.4 544.645228

Iteration History


1 6 508.814127 35.8311 76.91253 -888.317

2 11 494.473049 14.34108 115.9135 -846.425

3 12 489.205057 5.267992 352.7759 -100.889

4 13 474.066284 15.13877 403.8682 -245.292

5 14 469.060912 5.005372 252.3568 -473.345

6 15 464.554016 4.506896 95.06775 -173.992

7 17 451.325728 13.22829 89.90595 -23.1047

8 20 445.896149 5.429579 97.98775 -10.7319

9 23 444.772811 1.123338 75.11573 -4.00958

10 25 444.224114 0.548697 46.71996 -1.83776

11 27 444.060575 0.163539 29.85323 -0.62587

12 29 443.975696 0.084879 10.21039 -0.18815

13 31 443.941786 0.03391 2.386618 -0.06049

14 32 443.926112 0.015674 7.71936 -0.02338

15 33 443.905028 0.021084 1.73063 -0.03699

Iteration History


16 35 443.902974 0.002054 0.296542 -0.00418

17 37 443.902656 0.000318 0.090368 -0.00065

18 39 443.902654 1.757E-6 0.004427 -3.63E-6


Fit Statistics





Parameter Estimates

Standard


beta0 0.7860 0.2863 232 2.75 0.0065 0.05 0.2218 1.3501 -0.00004


beta1 0.04966 0.01781 232 2.79 0.0057 0.05 0.01457 0.08475 0.004427

beta2 0.02492 0.01908 232 1.31 0.1929 0.05 -0.01268 0.06252 0.001834

sigmab1 2.3139 0.4379 232 5.28 <.0001 0.05 1.4512 3.1766 0.00009

sigmab2 0.07987 0.02266 232 3.52 0.0005 0.05 0.03522 0.1245 0.000881

rho -0.03661 0.2206 232 -0.17 0.8683 0.05 -0.4711 0.3979 0.00001

LRT for Random-Slopes with AGQ 21

Obs L01 df pval

1 10.7862 2 0.0045479

MML AGQ 21 / No Treatment


Specifications

Data Set M.ARMD








Quadrature

Dimensions




Subjects 234


Parameters 5


Parameters

beta0 beta1 sigmab1 sigmab2 rho NegLogLike

0.567 0.0098 2 1 -0.4 545.246331

Iteration History


1 2 518.676407 26.56992 211.711 -745.686

2 6 499.702865 18.97354 375.7346 -207.13


3 11 452.212061 47.4908 33.28212 -1135.49

4 14 451.778791 0.433269 13.0378 -28.5472

5 16 447.578188 4.200603 142.6708 -17.2894

6 18 445.641089 1.9371 34.58276 -7.15164

7 20 445.043715 0.597373 21.19192 -1.04116

8 22 444.929054 0.114661 8.070474 -0.3141

9 24 444.888969 0.040086 3.650612 -0.0484

10 25 444.843874 0.045095 8.507939 -0.03002

11 27 444.840645 0.003229 0.677189 -0.00692

12 29 444.840612 0.000033 0.023544 -0.00007

13 31 444.840611 8.09E-7 0.010103 -1.78E-6


Fit Statistics





Parameter Estimates

Standard


beta0 0.8148 0.2867 232 2.84 0.0049 0.05 0.2499 1.3798 0.000088

beta1 0.05914 0.01790 232 3.30 0.0011 0.05 0.02387 0.09440 -0.0101

sigmab1 2.3359 0.4384 232 5.33 <.0001 0.05 1.4721 3.1997 -0.00003

sigmab2 0.07712 0.02241 232 3.44 0.0007 0.05 0.03297 0.1213 0.004551

rho -0.02278 0.2275 232 -0.10 0.9203 0.05 -0.4709 0.4254 0.000635

LRT for Treatment

Obs L01 df pval

1 1.87591 1 0.17080

7. Marginal Average Evolutions


Figure 4.1: Fitted marginal average evolutions under the random-slopes model. Treatment A corresponds toplacebo and B to active treatment.

8. Test for treatment difference at week 52.

Initial Values


Model Information

Data Set M.ARMD52


Link Function Logit






Missing Values 93

Response Profile


Ordered Total


1 1 615

2 0 252

PROC GENMOD is modeling the probability that bindif=’1’.



Deviance 860 1023.0599 1.1896

Scaled Deviance 860 1023.0599 1.1896


Scaled Pearson X2 860 867.0000 1.0081






Intercept 1 0.7522 0.1410 0.4758 1.0285 28.46 <.0001

time12 1 -0.3538 0.2355 -0.8153 0.1077 2.26 0.1329

time24 1 -0.1253 0.2434 -0.6023 0.3517 0.27 0.6067

time52 1 0.5190 0.2749 -0.0197 1.0578 3.57 0.0590

time12*treat 1 0.6288 0.2870 0.0662 1.1914 4.80 0.0285

time24*treat 1 0.4457 0.3016 -0.1454 1.0369 2.18 0.1394

time52*treat 1 0.4205 0.3745 -0.3136 1.1545 1.26 0.2616

Scale 0 1.0000 0.0000 1.0000 1.0000


Time as factor


Specifications

Data Set M.ARMD52



Random Effects b






Quadrature

Dimensions




Subjects 234


Parameters 8


Iteration History


1 2 447.178432 6.839458 4.112587 -141.884

2 4 446.466303 0.712128 1.599455 -8.53051

3 6 446.366192 0.100112 1.21234 -0.78402

4 7 446.330371 0.035821 1.361423 -0.36477

5 8 446.290535 0.039835 0.403971 -0.31919

6 10 446.26413 0.026406 0.041269 -0.06805

7 12 446.263779 0.000351 0.012186 -0.0006

8 14 446.263727 0.000052 0.006959 -0.0001

9 16 446.263722 5.034E-6 0.002377 -0.00001

10 18 446.263721 5.587E-7 0.000314 -1.12E-6


Fit Statistics





Parameter Estimates

Standard


beta0 1.3271 0.2559 233 5.19 <.0001 0.05 0.8230 1.8313 0.000314

beta1 -0.5212 0.3333 233 -1.56 0.1193 0.05 -1.1779 0.1355 0.000149

beta2 -0.1247 0.3412 233 -0.37 0.7152 0.05 -0.7970 0.5476 -0.00015

beta3 0.9495 0.3803 233 2.50 0.0132 0.05 0.2002 1.6987 0.000232

beta4 0.8586 0.4451 233 1.93 0.0550 0.05 -0.01837 1.7356 0.000134

beta5 0.5561 0.4622 233 1.20 0.2302 0.05 -0.3546 1.4668 -0.00013

beta6 0.4982 0.5413 233 0.92 0.3583 0.05 -0.5682 1.5646 0.000099


sigmab 2.1981 0.2513 233 8.75 <.0001 0.05 1.7030 2.6931 -0.00003


Standard


MrgTrEff 0.3048 0.3316 233 0.92 0.3589 0.05 -0.3485 0.9582

4.3.3 Discussion

1. Exploring incomplete data.

The total number of subjects equals 240, meaning that a substantial portion of the data is subject tomissingness. Both intermittent missingness as well as dropout occurs. An overview is given in Table 4.3.Thus, 78.33% of the profiles are complete, while 18.33% exhibit monotone missingness. Out of the latter

Table 4.3: Age Related Macular Degeneration Trial. Overview of missingness patterns and the frequencies withwhich they occur. ‘O’ indicates observed and ‘M’ indicates missing.

Measurement occasion

4 wks 12 wks 24 wks 52 wks Number %

Completers

O O O O 188 78.33

Dropouts

O O O M 24 10.00

O O M M 8 3.33

O M M M 6 2.50

M M M M 6 2.50

Non-monotone missingness

O O M O 4 1.67

O M M O 1 0.42

M O O O 2 0.83

M O M M 1 0.42

group, 2.5% or 6 subjects have no follow-up measurements. The remaining 3.33%, representing 8 subjects,have intermittent missing values. Although the group of dropouts is of considerable magnitude, the ones withintermittent missingness is much smaller. Nevertheless, it is cautious to include all into the analyses. Thisis certainly possible for direct likelihood analyses and for standard GEE (generalized estimating equations),but WGEE is more complicated in this respect. One solution is to monotonize the missingness patterns bymeans of multiple imputation and then conduct WGEE.

2. GEE and WGEE analyses.

We perform analyses on the completers only (CC), on the LOCF imputed data, as well as on the observeddata. In all cases, standard GEE, and linearization-based GEE (an alternative to standard GEE; the detailsare not important at this stage) will be considered. For the observed, partially incomplete data, GEEis supplemented with WGEE. Further, a random-intercepts GLMM (generalized linear mixed model) isconsidered, based on both PQL (penalized quasi-likelihood) and numerical integration. The GEE analyses


are reported in Table 4.4 and the random-effects models in Table 4.6. In all cases, we use the logit link.For GEE, a working exchangeable correlation matrix is considered. The model has four intercepts andfour treatment effects. The advantage of having separate treatment effects at each time is that particularattention can be given at the treatment effect assessment at the last planned measurement occasion, i.e.,after one year.

Table 4.4: Age Related Macular Degeneration Trial. Parameter estimates (model-based standard errors; empir-ically corrected standard errors) for the marginal models: standard and linearization-based GEE on the CC andLOCF population, and on the observed data. In the latter case, also WGEE is used.

Effect Par. CC LOCF Observed data

Unweighted WGEE

Standard GEE

Int.4 β11 -1.01(0.24;0.24) -0.87(0.20;0.21) -0.87(0.21;0.21) -0.98(0.10;0.44)

Int.12 β21 -0.89(0.24;0.24) -0.97(0.21;0.21) -1.01(0.21;0.21) -1.78(0.15;0.38)

Int.24 β31 -1.13(0.25;0.25) -1.05(0.21;0.21) -1.07(0.22;0.22) -1.11(0.15;0.33)

Int.52 β41 -1.64(0.29;0.29) -1.51(0.24;0.24) -1.71(0.29;0.29) -1.72(0.25;0.39)

Tr.4 β12 0.40(0.32;0.32) 0.22(0.28;0.28) 0.22(0.28;0.28) 0.80(0.15;0.67)

Tr.12 β22 0.49(0.31;0.31) 0.55(0.28;0.28) 0.61(0.29;0.29) 1.87(0.19;0.61)

Tr.24 β32 0.48(0.33;0.33) 0.42(0.29;0.29) 0.44(0.30;0.30) 0.73(0.20;0.52)

Tr.52 β42 0.40(0.38;0.38) 0.34(0.32;0.32) 0.44(0.37;0.37) 0.74(0.31;0.52)

Corr. ρ 0.39 0.44 0.39 0.33

Linearization-based GEE

Int.4 β11 -1.01(0.24;0.24) -0.87(0.21;0.21) -0.87(0.21;0.21) -0.98(0.18;0.44)

Int.12 β21 -0.89(0.24;0.24) -0.97(0.21;0.21) -1.01(0.22;0.21) -1.78(0.26;0.42)

Int.24 β31 -1.13(0.25;0.25) -1.05(0.21;0.21) -1.07(0.23;0.22) -1.19(0.25;0.38)

Int.52 β41 -1.64(0.29;0.29) -1.51(0.24;0.24) -1.71(0.29;0.29) -1.81(0.39;0.48)

Tr.4 β12 0.40(0.32;0.32) 0.22(0.28;0.28) 0.22(0.29;0.29) 0.80(0.26;0.67)

Tr.12 β22 0.49(0.31;0.31) 0.55(0.28;0.28) 0.61(0.28;0.29) 1.85(0.32;0.64)

Tr.24 β32 0.48(0.33;0.33) 0.42(0.29;0.29) 0.44(0.30;0.30) 0.98(0.33;0.60)

Tr.52 β42 0.40(0.38;0.38) 0.34(0.32;0.32) 0.44(0.37;0.37) 0.97(0.49;0.65)

σ2 0.62 0.57 0.62 1.29

τ2 0.39 0.44 0.39 1.85

Corr. ρ 0.39 0.44 0.39 0.59

Note also that the treatment effect under LOCF, especially at 12 weeks and after 1 year, is biased downwardin comparison to the GEE analyses. To properly use the information in the missingness process, WGEE canbe used. To this end, a logistic regression for dropout, given covariates and previous outcomes, needs tobe fitted. Parameter estimates and standard errors are given in Table 4.5. Intermittent missingness will beignored. Covariates of importance are treatment assignment, the level of lesions at baseline (a four-pointcategorical variable, for which three dummies are needed), and time at which dropout occurs. For the lattercovariates, there are three levels, since dropout can occur at times 2, 3, or 4. Hence, two dummy variablesare included. Finally, the previous outcome does not have a significant impact, but will be kept in the modelnevertheless.

In spite of there being no strong evidence for MAR, the results between GEE and WGEE differ quite a bit.It is noteworthy that at 12 weeks, a treatment effect is observed with WGEE which goes unnoticed with theother marginal analyses. This finding is mildly confirmed by the random-intercept model, when the data asobserved are used.

3. Random-intercepts model.


Table 4.5: Age Related Macular Degeneration Trial. Parameter estimates (standard errors) for a logistic regressionmodel to describe dropout.

Effect Parameter Estimate (s.e.)

Intercept ψ0 0.14 (0.49)

Previous outcome ψ1 0.04 (0.38)

Treatment ψ2 -0.86 (0.37)

Lesion level 1 ψ31 -1.85 (0.49)

Lesion level 2 ψ32 -1.91 (0.52)

Lesion level 3 ψ33 -2.80 (0.72)

Time 2 ψ41 -1.75 (0.49)

Time 3 ψ42 -1.38 (0.44)

The results for the random-intercept models are given in Table 4.6. We observe the usual downward biasin the PQL versus numerical integration analysis, as well as the usual relationship between the marginalparameters of Table 4.4 and their random-effects counterparts. Note also that the random-interceptsvariance is largest under LOCF, underscoring again that this method artificially increases the associationbetween measurements on the same subject.


One complication with WGEE is that the calculation of the weights is difficult with non-monotone missing-ness. Standard GEE on the incomplete data is valid only when the missing data are MCAR. Precisely here,multiple imputation is an appealing alternative.

The binary indicators were created by dichotomizing the continuous visual acuity outcomes, as negativeversus non-negative. The continuous outcomes were defined as the change from baseline in number of lettersread. Therefore, multiple imputation could start from the continuous outcomes. Ten multiply-imputeddatasets were created. The imputation model included, apart from the four continuous outcomes variables,also the four-point categorical variable ‘lesions.’ For simplicity, the latter was treated as continuous. Separateimputations were conducted for each of the two treatment groups. These choices imply that the imputedvalues depend on lesions and treatment assignment, and hence analysis models that include one or both ofthese effects are proper in the sense of Rubin (1987). This means, broadly speaking, that the model usedfor imputation should include all relationships that later will be considered in the analysis and inferencetasks. The added advantage of including ‘lesions’ into the imputation model, is that even individuals forwhich none of the four follow-up measurements are available, are still imputed. The MCMC method wasused, with EM starting values, and a single chain for all imputations.

Upon imputation, the same marginal GEE and random-intercept models as before were fitted in the analysistask. Results from the inference task are reported in Table 4.7.

The parameter estimates and standard errors are very similar to their counterparts in Table 4.4 and 4.6. Ofcourse, in the GEE case, there is no direct counterpart, since the WGEE method is different from GEE aftermultiple imputation, even though both are valid under MAR. However, in particular the similarity betweenthe direct likelihood method (bottom right column of Table 4.6) is clear, with only a minor deviation inestimate for the treatment effect after 1 year.

5. PQL versus MQL

Small fluctuations in the fixed-effects estimates are observed between PQL and MQL methods. However,greater differences are observed for the variance component for which MQL and ML methods seem tounderestimate it.

6. Gaussian Quadrature


Table 4.6: Age Related Macular Degeneration Trial. Parameter estimates (standard errors) for the random-intercept models: PQL and numerical-integration based fits on the CC and LOCF population, and on the observeddata (direct-likelihood).

Effect Parameter CC LOCF Direct lik.

PQL

Int.4 β11 -1.19(0.31) -1.05(0.28) -1.00(0.26)

Int.12 β21 -1.05(0.31) -1.18(0.28) -1.19(0.28)

Int.24 β31 -1.35(0.32) -1.30(0.28) -1.26(0.29)

Int.52 β41 -1.97(0.36) -1.89(0.31) -2.02(0.35)

Trt.4 β12 0.45(0.42) 0.24(0.39) 0.22(0.37)

Trt.12 β22 0.58(0.41) 0.68(0.38) 0.71(0.37)

Trt.24 β32 0.55(0.42) 0.50(0.39) 0.49(0.39)

Trt.52 β42 0.44(0.47) 0.39(0.42) 0.46(0.46)

R.I. s.d. τ 1.42(0.14) 1.53(0.13) 1.40(0.13)

R.I. var. τ2 2.03(0.39) 2.34(0.39) 1.95(0.35)

Numerical integration

Int.4 β11 -1.73(0.42) -1.63(0.39) -1.50(0.36)

Int.12 β21 -1.53(0.41) -1.80(0.39) -1.73(0.37)

Int.24 β31 -1.93(0.43) -1.96(0.40) -1.83(0.39)

Int.52 β41 -2.74(0.48) -2.76(0.44) -2.85(0.47)

Trt.4 β12 0.64(0.54) 0.38(0.52) 0.34(0.48)

Trt.12 β22 0.81(0.53) 0.98(0.52) 1.00(0.49)

Trt.24 β32 0.77(0.55) 0.74(0.52) 0.69(0.50)

Trt.52 β42 0.60(0.59) 0.57(0.56) 0.64(0.58)

R.I. s.d. τ 2.19(0.27) 2.47(0.27) 2.20(0.25)

R.I. var. τ2 4.80(1.17) 6.08(1.32) 4.83(1.11)

The Gaussian quadrature parameter estimates seem stable except from the Laplace and AGQ 5 cases.Comparing these results with the PQL and MQL methods we observe some downwards bias for almost allthe parameters, and especially for σ2

b . This is expected due to the small number of repeated measurementper subject and the fact that the variance estimate for the random-effect is far from zero.

7. Hypothesis Testing

The likelihood ratio tests suggest that first the random-slopes model provide a better fit to the data, andsecond that evolutions in time are not statistically different for the two treatment groups. Note howeverthat the test for random-slopes is on the boundary of the parameter space for the variance term.

8. Marginal Average Evolutions

The marginal average evolutions under the final model are depicted in Figure 4.1. The probability of ‘bindif= 1’ seems greater for the active treatment even though this difference is not statistically significant.

9. Treatment effect at week 52

The test for ‘Time52*Treatment’ regression coefficient suggests a statistically nonsignificant differenceat week 52 for the two treatment groups. Note however, that this test refers to the conditional on therandom-effects treatment difference and not to the marginal one. In order to test for the marginal treatmentdifference at week 52 we use the approximate formula βM = βRE/

√

c2σ2b + 1, where c2 = [16

√3/(15π)]2 ≈

0.345843. This leads to the same conclusion with a similar p-value.


Table 4.7: Age Related Macular Degeneration Trial. Parameter estimates (standard errors) for the standard GEEand numerical-integration based random-intercept models, after generating 10 multiple imputations.

Effect Par. GEE GLMM

Int.4 β11 -0.84(0.20) -1.46(0.36)

Int.12 β21 -1.02(0.22) -1.75(0.38)

Int.24 β31 -1.07(0.23) -1.83(0.38)

Int.52 β41 -1.61(0.27) -2.69(0.45)

Trt.4 β12 0.21(0.28) 0.32(0.48)

Trt.12 β22 0.60(0.29) 0.99(0.49)

Trt.24 β32 0.43(0.30) 0.67(0.51)

Trt.52 β42 0.37(0.35) 0.52(0.56)

R.I. s.d. τ 2.20(0.26)

R.I. var. τ2 4.85(1.13)

Table 4.8: Parameter estimates (standard error) under PQL and MQL methods using either REML or ML.

PQL (REML) PQL (ML) MQL (REML) MQL (ML)

Intercept 0.664(0.159) 0.663(0.159) 0.568(0.140) 0.568(0.140)

Time 0.015(0.006) 0.015(0.006) 0.011(0.005) 0.011(0.005)

Time*Treatment 0.010(0.008) 0.010(0.008) 0.010(0.007) 0.010(0.007)

σ2b 1.911(0.342) 1.881(0.338) 1.412(0.255) 1.396(0.252)

Table 4.9: Parameter estimates (standard error) and log-likelihood values using Gaussian and adaptive GaussianQuadrature.

GQ 25 GQ 51 Laplace ≡ AGQ 1 AGQ 5 AGQ 11

Intercept 1.017(0.227) 1.017(0.227) 1.038(0.226) 0.994(0.221) 1.018(0.228)

Time 0.021(0.007) 0.021(0.007) 0.020(0.007) 0.021(0.007) 0.021(0.007)

Time*Treatment 0.011(0.010) 0.011(0.010) 0.011(0.010) 0.011(0.010) 0.011(0.010)

σ2b 4.780(1.098) 4.774(1.093) 4.220(0.978) 4.508(0.989) 4.785(1.099)

log-Lik −449.2921 −449.2957 −454.0558 −449.8744 −449.2851

Table 4.10: Likelihood ratio test for random-slopes and treatment effect.

LRT df p-value

random-slopes AGQ 11 10.98 2 0.0041

random-slopes AGQ 21 10.79 2 0.0045

treatment 1.88 1 0.1708


Table 4.11: Parameter estimates, standard errors and p-values under a GLMM treating time as factor.

Value Std. Error p-value

Intercept 1.327 0.256 < 0.0001

Time12 −0.521 0.333 0.1193

Time24 −0.125 0.341 0.7152

Time52 0.950 0.380 0.0132

Time12*Treatment 0.859 0.445 0.0550

Time24*Treatment 0.556 0.462 0.2302

Time52*Treatment 0.498 0.541 0.3583

σb 2.198 0.251 −Marginal Treat Effect W52 0.305 0.332 0.3589

Dataset 5

Blood Pressure Data

5.1 The Data

We consider data reported by Hand et al . (1994), data set #72. For 15 patients with moderate essential (unknowncause) hypertension, the supine (measured while patient is lying down) systolic and diastolic blood pressure wasmeasured immediately before and 2 hours after taking the drug captopril. The individual profiles are shown inFigure 5.1. The objective of the analysis is to investigate the effect of treatment on both responses. These datahave been analyzed in Verbeke and Molenberhs (1997, Ch. 4.3; 2000, Ch. 24.1). We refer to these text for anintroduction and an overview of the models that have been fitted. For example, the final model considered is

data blood;

set blood;

slope = (time = ’after’);

intsys = (meas = ’systolic’);

run;

proc mixed data = blood covtest;

class time meas id;

model bp = meas*time / noint s;

random intercept intsys slope / type = un(1) subject = id;

estimate ’trt_sys’ meas*time 0 -1 0 1 / cl alpha = 0.05;

estimate ’trt_dia’ meas*time -1 0 1 0 / cl alpha = 0.05;

Figure 5.1: Blood Pressure Data. Systolic and diastolic blood pressure in patients with moderate essential

hypertension, immediately before and 2 hours after taking captopril.

171

DATASET 5. BLOOD PRESSURE DATA 172

contrast ’trt_sys = 2xtrt_dia’ meas*time 2 -1 -2 1;

run;

5.2 Questions

While there are only 4 measurements per subject, there is a double structure to the data:

Bivariate structure: Both systolic and diastolic blood pressure are recorded.

Repeated measures: Both of these measures are taken at two well defined points in time: immediately beforeand 2 hours after taking captopril.

Some questions:

• Fit models which explicitly make use of PROC MIXED features for such multivariate repeated measures.

• Compare the covariance structures:

– unstructured; in the original model (Verbeke and Molenberghs 1997, 2000);

– in meaningful multivariate repeated measure covariance structures.

• What is the impact on the treatment effect assessments ?

5.2.1 Dataset

The dataset is BLOOD.SAS7BDAT.


5.3.1 Basic Program


data m.blood1;

set m.blood;

slope = (time = ’after’);

intsys = (meas = ’systolic’);

run;

proc print data=m.blood1;

title ’Blood Pressure Data’;

run;

proc mixed data = m.blood1 covtest;

title ’Model from Verbeke-Molenberghs’;

class time meas id;



random intercept intsys slope / type = un(1) subject = id v;



run;


title ’Model 1: 4x4 unstructured covariance matrix’;

class time meas id;


repeated / type=un subject = id r;



run;


title ’Model 2: unstructured-by-unstructured’;

class time meas id;


repeated meas time / type=un@un subject = id r;



run;


title ’Model 3: unstructured-by-compound symmetry’;

class time meas id;


repeated meas time / type=un@cs subject = id r;



run;

5.3.2 Raw Output

Blood Pressure Data 12:17 Sunday, April 23, 2000 434

OBS ID BP MEAS TIME SLOPE INTSYS

1 1 210 systolic before 0 1
















16 1 201 systolic after 1 1















31 1 130 diastolic before 0 0















46 1 125 diastolic after 1 0















Model from Verbeke-Molenberghs 436

12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TIME 2 after before

MEAS 2 diastolic systolic

ID 15 1 2 3 4 5 6 7 8 9 10 11 12 13


14 15

REML Estimation Iteration History


0 1 380.87904251

1 3 329.43913763 0.00019315

2 2 325.72780080 0.00008744

3 2 322.33035886 0.00772996

4 2 321.73878311 0.00106566

5 1 321.53994376 0.00009633

6 1 321.52346568 0.00000105

7 1 321.52329606 0.00000000


V Matrix for ID 1


1 323.99541234 311.13299395 95.40469590 95.40469590

2 311.13299395 376.04620370 95.40469590 147.45548726

3 95.40469590 95.40469590 108.26711430 95.40469590

4 95.40469590 147.45548726 95.40469590 160.31790566

Covariance Parameter Estimates (REML)


UN(1,1) ID 95.40469590 39.45381687 2.42 0.0156

UN(2,1) ID 0.00000000 . . .

UN(2,2) ID 215.72829804 86.31500453 2.50 0.0124

UN(3,1) ID 0.00000000 . . .

UN(3,2) ID 0.00000000 . . .

UN(3,3) ID 52.05079136 24.77124215 2.10 0.0356

Residual 12.86241840 4.69297016 2.74 0.0061

Model from Verbeke-Molenberghs 437

12:17 Sunday, April 23, 2000

Model Fitting Information for BP

Description Value


Res Log Likelihood -212.222









Effect TIME MEAS Estimate Std Error DF t Pr > |t|

TIME*MEAS after diastolic 103.06666667 3.26922932 14 31.53 0.0001

TIME*MEAS after systolic 158.00000000 5.00696983 14 31.56 0.0001

TIME*MEAS before diastolic 112.33333333 2.68659778 14 41.81 0.0001

TIME*MEAS before systolic 176.93333333 4.64754711 14 38.07 0.0001



TIME*MEAS 4 14 552.98 0.0001

ESTIMATE Statement Results

Parameter Estimate Std Error DF t Pr > |t| Alpha Lower

trt_sys 18.93333333 2.27706870 14 8.31 0.0001 0.05 14.0495

trt_dia 9.26666667 2.27706870 14 4.07 0.0011 0.05 4.3828


Upper

23.8172

14.1505

Model 1: 4x4 unstructured covariance matrix 438

12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TIME 2 after before


ID 15 1 2 3 4 5 6 7 8 9 10 11 12 13

14 15



0 1 380.87904251

1 1 317.17343397 0.00000000


R Matrix for ID 1



1 422.92380952 370.78571429 143.16666667 105.07619048

2 370.78571429 400.14285714 153.07142857 166.28571429

3 143.16666667 153.07142857 109.66666667 96.54761905

4 105.07619048 166.28571429 96.54761905 157.63809524



UN(1,1) ID 422.92380952 159.85017479 2.65 0.0082

UN(2,1) ID 370.78571429 148.01350986 2.51 0.0122

UN(2,2) ID 400.14285714 151.23978413 2.65 0.0082

UN(3,1) ID 143.16666667 69.11550317 2.07 0.0383

UN(3,2) ID 153.07142857 69.34035909 2.21 0.0273

UN(3,3) ID 109.66666667 41.45010387 2.65 0.0082

UN(4,1) ID 105.07619048 74.50307260 1.41 0.1584

UN(4,2) ID 166.28571429 80.50230547 2.07 0.0389

UN(4,3) ID 96.54761905 43.59643431 2.21 0.0268

UN(4,4) ID 157.63809524 59.58159959 2.65 0.0082


Description Value



Model 1: 4x4 unstructured covariance matrix 439

12:17 Sunday, April 23, 2000


Description Value
















TIME*MEAS 4 15 523.71 0.0001



trt_sys 18.93333333 2.33088307 15 8.12 0.0001 0.05 13.9652

trt_dia 9.26666667 2.22425304 15 4.17 0.0008 0.05 4.5258


Upper

23.9015

14.0075

Model 2: unstructured-by-unstructured 440

12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TIME 2 after before


ID 15 1 2 3 4 5 6 7 8 9 10 11 12 13

14 15



0 1 380.87904251

1 2 336.62036209 0.04601371

2 1 332.05734669 0.02325751

3 1 327.34729868 0.00540866

4 1 326.29877861 0.00078309

5 1 326.15606451 0.00003573

6 1 326.15003498 0.00000011

7 1 326.15001728 0.00000000


R Matrix for ID 1


1 322.80926185 278.31214857 145.86670785 125.75995073

2 278.31214857 348.89346798 125.75995073 157.65328811


3 145.86670785 125.75995073 162.50540747 140.10511609

4 125.75995073 157.65328811 140.10511609 175.63645743



MEAS UN(1,1) ID 175.63645743 61.35722200 2.86 0.0042

UN(2,1) ID 157.65328811 62.46750542 2.52 0.0116

UN(2,2) ID 348.89346798 110.45201199 3.16 0.0016

TIME UN(1,1) ID 1.00000000 . . .

UN(2,1) ID 0.79769951 0.11005599 7.25 0.0001

UN(2,2) ID 0.92523733 0.20140740 4.59 0.0001

Model 2: unstructured-by-unstructured 441

12:17 Sunday, April 23, 2000


Description Value

















TIME*MEAS 4 41 407.02 0.0001



trt_sys 18.93333333 2.76981868 41 6.84 0.0001 0.05 13.3396

trt_dia 9.26666667 1.96522488 41 4.72 0.0001 0.05 5.2978



Upper

24.5271

13.2355

Model 3: unstructured-by-compound symmetry 442

12:17 Sunday, April 23, 2000

The MIXED Procedure


Class Levels Values

TIME 2 after before


ID 15 1 2 3 4 5 6 7 8 9 10 11 12 13

14 15



0 1 380.87904251

1 2 1096.3346546 0.50771777

2 1 732.26557531 0.41020333

3 1 536.29626092 0.29326409

4 1 434.08848104 0.18179072

5 1 383.01870378 0.10026872

6 1 358.14570277 0.05382129

7 1 345.64310292 0.03255113

8 1 338.09161836 0.02500254

9 1 332.61388737 0.01656220

10 1 328.99303806 0.00915332

11 1 327.09144038 0.00345036

12 1 326.40478781 0.00066174

13 1 326.28365539 0.00003074

14 1 326.27848046 0.00000008

15 1 326.27846792 0.00000000


R Matrix for ID 1


1 333.87161924 277.27041515 153.17839611 127.21008626

2 277.27041515 333.87161924 127.21008626 153.17839611

3 153.17839611 127.21008626 172.31420485 143.10180429

4 127.21008626 153.17839611 143.10180429 172.31420485




MEAS UN(1,1) ID 172.31420485 59.94888923 2.87 0.0040

UN(2,1) ID 153.17839611 59.46490001 2.58 0.0100

UN(2,2) ID 333.87161924 96.66914257 3.45 0.0006

TIME Corr ID 0.83047015 0.06279775 13.22 0.0001

Model 3: unstructured-by-compound symmetry 443

12:17 Sunday, April 23, 2000


Description Value

















TIME*MEAS 4 41 394.34 0.0001



trt_sys 18.93333333 2.74714892 41 6.89 0.0001 0.05 13.3854

trt_dia 9.26666667 1.97357208 41 4.70 0.0001 0.05 5.2810


Upper

24.4813

13.2524


5.3.3 Covariance Structures

For the original model, the covariance structure is found by adding the ‘v’ option to the RANDOM statement.For the others, the option ‘r’ is added to the REPEATED statement.

For Model 1, the unstructured covariance matrix is self-evident. For Model 2, the covariance matrix is:

322.80926185 278.31214857 145.86670785 125.75995073

278.31214857 348.89346798 125.75995073 157.65328811

145.86670785 125.75995073 162.50540747 140.10511609

125.75995073 157.65328811 140.10511609 175.63645743

.

Manipulating the estimated covariane parameters yields:

(

175.636 157.653

157.653 348.893

)

⊗

(

1.000 0.7977

0.7977 0.9252

)

=

175.63600 140.10484 157.65300 125.75980

140.10484 162.49843 125.75980 145.86056

157.65300 125.75980 348.89300 278.31195

125.75980 145.86056 278.31195 322.79580

,

which, up to reversal, is the structure provided by the ‘r’ matrix.

For Model 3, we obtain

333.87161924 277.27041515 153.17839611 127.21008626

277.27041515 333.87161924 127.21008626 153.17839611

153.17839611 127.21008626 172.31420485 143.10180429

127.21008626 153.17839611 143.10180429 172.31420485

.

Manipulating the estimated covariane parameters yields:

(

172.314 153.178

153.178 333.872

)

⊗

(

1.0000 0.8305

0.8305 1.0000

)

=

172.31400 143.10678 153.17800 127.21433

143.10678 172.31400 127.21433 153.17800

153.17800 127.21433 333.87200 277.28070

127.21433 153.17800 277.28070 333.87200

.

In contrast, the orignal random-effects model produced:

323.99541234 311.13299395 95.40469590 95.40469590

311.13299395 376.04620370 95.40469590 147.45548726

95.40469590 95.40469590 108.26711430 95.40469590

95.40469590 147.45548726 95.40469590 160.31790566

.

A fully unstructured model yields:

422.92380952 370.78571429 143.16666667 105.07619048

370.78571429 400.14285714 153.07142857 166.28571429

143.16666667 153.07142857 109.66666667 96.54761905

105.07619048 166.28571429 96.54761905 157.63809524

.

The model fitting information can be summarized as follows:


Model −2`REML df

RI + RIsys + RS, uncorrelated 424.444 4

Model 1: Unstructured 420.095 10

Model 2: Unstructured-by-unstructured 429.071 5

Model 3: Unstructured-by-compound symmetry 429.200 4

Thus, the fit of the original model is still the best one.

The effect on the assessment of treatment effect:

Model Systolic Diastolic

Original Model 18.933 (2.777) 9.267 (2.277)

Model 1 18.933 (2.331) 9.267 (2.224)

Model 2 18.933 (2.770) 9.267 (1.965)

Model 3 18.933 (2.747) 9.267 (1.974)

Dataset 6

Marital Satisfaction Data


The research sample consists of married men and women participating in the longitudinal research project “Child-rearing and family in the Netherlands”. In 1990, 1995 and 2000, the same family members (wife, husband andtarget child) provided information about similar sets of measures. Families were recruited using a multi-stagesampling method. In a first stage, a sample was taken of all Dutch municipalities distinguished by regional zoneand degree of urbanization. In a second stage, a sample of children aged 9 to 16 years old was taken in theselected municipalities. The children were selected in such a way that in each city as much boys as girls and asmuch children aged 9 to 12 as aged 13 to 16 were chosen.

The data were gathered by means of structured interviews and questionnaires, completed by both the child andthe parents. In order to establish a homogeneous research group for a study on marital quality, only first marriagesin which both men and women have a Dutch nationality were selected. This selection resulted in a research groupof 647 couples in 1990, 386 couples in 1995 and 182 couples in 2000. 17 couples that didn’t participate in 1995,did participate in 2000. Data of a couple are considered when one of the two parents participated, not necessarilyboth parents.

Outcomes of Interest

Marital satisfaction is measured by the Marital Satisfaction scale of Gerris et al. (1992, 1993, 1998). For for-mulating the items, satisfaction with the relationship and/or the partner was used as the guiding principle (e.g.,“Generally, I’m dissatisfied with the relationship with my partner” or “If I could choose again, I would choosethe same partner”). The scale consists of seven 7-point Likert items, ranging from 1 = “do not agree” to 7 =“completely agree”.

The open communication scale of Gerris et al. (1992, 1993) maps out the open communicational styles of thecouple. Respondents were asked to indicate to what degree personal feelings and experiences were shared (e.g.,“I often talk to my partner about things we are both interested in” or “I often talk to my partner about personalproblems”). The open communication scale also consists of three 7-point Likert items.

The negative communication scale of Gerris et al. (1992, 1993) maps out the negative communication styles ofthe couple. Respondents were asked to indicate to what degree certain forms of negative communication arecharacteristic to their marital relationship (e.g., “My partner often blames me when we are quarrelling” or “My

184

DATASET 6. MARITAL SATISFACTION DATA 185

partner and I interrupt each other a lot when we are talking together”). The scale consists of six 7-point Likertitems.

The uniqueness and stability of these concepts were demonstrated in Van den Troost et al. (2001).

Covariates of Interest

Education was measured in response to the question “What is your highest educational level?”, using a nine-levelscale ranging from 1 = “elementary school” to 9 = “university education”. Family income was measured inresponse to the question “What is the monthly net family income (in guilders, approx. 0.45 euro)?” Seven incomegroups were distinguished: 1 = “1100–1600”, then in groups of 500 guilders, until 7 = “more than 4500”. Furthercovariates are year of birth, year of marriage, number of children and marital status of the parents of the couple.For the latter, to both husband and wife it was asked “The time you were living with your parents, they weremarried all the time?”. Response categories were 1 = “yes”, 2 = “no” and 3 = “not applicable”.

marsatCC.xls and marsatIC.xls

FAMNR : family identification number

IDNR : parent identification number

YEAR : year of interview (1990, 1995, 2000)

PARENT : parent (0=father, 1=mother)

SAT : marital satisfaction score

PCOM : open communication score

DCOM : negative communication score

CHILD : number of children

BIRTH : year of birth

EDUC : educational level

MARR : year of marriage

INCOME : income category

STATUS : marital status of the parents of the couple (1=yes, 2=no)

References

Gerris, J. R. M., Houtmans, M. J. M., Kwaaitaal-Roosen, E. M. G., Schipper, J. C., Vermulst, A. A. andJanssens, J. M. A. M. (1998) Parents, adolescents and young adults in Dutch families. A longitudinalstudy. Nijmegen: University of Nijmegen, Institute of Family Studies.

Gerris, J. R. M., Van Boxtel, D. A. A. M., Vermulst, A. A., Janssens, J. M. A. M., Van Zutphen, R. A. H. andFelling, A. J. A. (1992) Child-rearing, family relations and family processes in 1990. Nijmegen: Universityof Nijmegen, Institute of Family Studies.


Gerris, J. R. M., Vermulst, A. A., Van Boxtel, D. A. A. M., Janssens, J. M. A. M., Van Zutphen, R. A. H. andFelling, A. J. A. (1993) Parenting in Dutch families. Nijmegen: University of Nijmegen, Institute of FamilyStudies.

Van den Troost, A., Vermulst, A. A., Gerris, J. R. M. and Matthijs, K. (2001) Meetinvariantie van huwelijk-skwaliteit en satisfactie. Leuven/Nijmegen. Onderzoeksverslag van het Departement Sociologie - afdelingGezin, Bevolking & Gezondheidszorg en Orthopedagogiek. - Gezin en Gedrag, GB/2001-13.

6.2 Questions

Part 1

1. Import the Excel datafile marsatCC.xls in SAS. Then split this dataset into two separate files, one containingonly the information for the fathers, the other containing only the information for the mothers.

2. Fit a fixed effects model, including all covariates in the dataset (be careful, some of them are continuous,others categorical). Use the unstructured covariance structure. Perform this analysis for both fathers andmothers.

3. Reduce the mean structure of the above models, until you end up with only the significant covariates.

4. Now keep this best mean structure, and reduce the covariance structure as much as possible for both models.Is it necessary to consider the heterogeneous covariance structures?

5. From now on, use the full dataset marsatCC again (father and mother together). Consider the fixed effectsyear, parent and their interaction. Simplify the covariance structure. Be careful, there are two clusteringvariables now. Therefore, the covariance structure will be more complicated.

Part 2

Consider the following binary transformation BSAT of the marital satisfaction score which equals 0 when SAT≤ 4, and 1 otherwise. In this question we are interested in the impact of the number of years married at themoment of the interview, the year of the interview, the age, education and gender of the subjects. Use the GEEmethodology to study the impact of these variables on the marital satisfaction. Consider the following points inyour analysis

• choose an appropriate covariance structure

• compare the results for males and females separately

• does the correlation between responses of members of the same family influence these results?

Part 3

1. Consider the dichotomized version of the marital satisfaction score (BSAT) as the longitudinal outcome ofinterest. Discuss a suitable random effects structure for this score.

2. Fit a mixed effects logistic regression model for BSAT, using Penalized quasi-likelihood (PQL) and Marginalquasi-likelihood (MQL) implemented by PROC GLIMMIX. Formulate the random effects structure properly,accounting for the hierarchical data structure. Assume that the mean structure consists of the main effects


of the variables ’parent’, ’year’ (treated as categorical), ’child’, years of marriage and the interaction between’parent’ and ’year’. Observe the estimated random effects variance components. What is implied for theassumed random effects structure?

3. For the same random effects structure fit the model using adaptive Gaussian quadrature with 20 quadraturepoints implemented by PROC NLMIXED. Test whether the lower level classification factor is needed.

4. Investigate the single level random effects structure by fitting the mixed effects logistic regression modelusing PQL, MQL under both REML and ML. Compare the resulting parameter estimates.

5. For the same random effects structure, fit the same model using Gaussian quadrature.

• Fit the model using both non adaptive and adaptive Gaussian quadrature with varying number ofnodes i.e., 3, 5, 10, 20 and 50. Compare the parameter estimates and standard errors.

• Fit the model using the Laplace approximation.

6. Compare the results from the fitted models under PQL, MQL, adaptive Gauss-Hermite with 20 nodes andLaplace approximation.

7. Plot the fitted marginal evolutions for the 3 time points, i.e., 1990, 1995 and 2000, for each parent adjustingfor the covariates at their median value. Compare these marginal evolutions with the fitted evolutions forthe median parent, i.e., for the parent with bi = 0.

8. Test for the parent effect in the final model using the likelihood ratio test.

Part 4

1. Analyze the continuous response ‘sat’ using a mixed model where the mean structure includes covariates‘year’(class), ‘parent’(class) and their interaction and the covariance structure is defined as unstructured(parent) and compound symmetry (year).

(a) Perform a complete case analysis (CC)

(b) Use all available data (Direct Likelihood)

(c) Complete the data using multiple imputation (5 imputations) and analyze the completed datasets

(d) Compare and discuss the results of all analyses

2. Analyze the binary response ‘bsat’ using GEE with mean structure ‘parent year birth parent*year parent*birthbirth*year parent*birth*year’ where ‘year’and ‘parent’ are again class-variables.


(b) Use a weighted GEE approach where you determine the weights based on a model with ‘covariates‘parent’(class), ‘year’(class) and ‘birth’

(c) Perform an available case GEE

(d) Complete the data using multiple imputation (5 imputations) and analyze the completed datasetsusing GEE

(e) Compare and discuss the results of all analyses

3. Analyze the binary response ‘bsat’ using GLMM with mean structure ‘parent year birth parent*year par-ent*birth birth*year parent*birth*year’ where ‘year’and ‘parent’ are again class-variables


(b) Perform an available case analysis (Direct Likelihood)



6.3.1 Programs

Part 1

1. Splitting in separate datasets for father and mother:

data father;

set marital;

where parent=0;

run;

data mother;

set marital;

where parent=1;

run;

2. Full model with unstructured mean and covariance structure (for both datasets):

proc mixed data=father method=ml;

class year status;

model sat = year child birth educ marr income status / solution noint;

repeated year / type=un subject=idnr;

run;

proc mixed data=mother method=ml;

class year status;

model sat = year child birth educ marr income status / solution noint;


run;

3. Final model after reducing mean structure (for both datasets):


class year status;

model sat = year / solution noint;


run;


class year status;



run;

4. Reducing the covariance structure (for both datasets):


class year status;


repeated year / type=un(2) subject=idnr;

run;



class year status;


repeated year / type=arh(1) subject=idnr;

run;


class year status;


repeated year / type=csh subject=idnr;

run;


class year status;


repeated year / type=toeph subject=idnr;

run;


class year status;


repeated year / type=cs subject=idnr;

run;


class year status;


repeated year / type=un(2) subject=idnr;

run;


class year status;


repeated year / type=arh(1) subject=idnr;

run;


class year status;


repeated year / type=csh subject=idnr;

run;


class year status;


repeated year / type=toeph subject=idnr;

run;


class year status;


repeated year / type=cs subject=idnr;

run;



class year status;


repeated year / type=ar(1) subject=idnr;

run;


class year status;


repeated year / type=simple subject=idnr;

run;

5. Modelling longitudinal data of both parents simultaneously:

proc mixed data=marital method=ml;

class year parent;

model sat = parent year parent*year / solution noint;

repeated parent*year / type=un subject=famnr r;

run;


class year parent;


repeated parent year / type=un@un subject=famnr r;

run;


class year parent;


repeated parent year / type=un@ar(1) subject=famnr r;

run;


class year parent;


repeated parent year / type=un@cs subject=famnr r;

run;

Part 2

1. Data Management

libname m ’c:\data’;

data maritcc;

set m.maritcc;

YEARCLS = year;

PARENTCLS = parent;

YEARSMAR90 = 1990 - (1900+marr);

AGEBASE = 1990 - (1900+birth);

run;

2. Separate Analysis for Males & Females


proc sort data=maritcc; by parent; run;

proc genmod data=maritcc descending;

title ’Full Model by Parent: CC - GEE: Type=UN’;

class yearcls idnr;

model bsat = yearsmar90 yearsmar90*year yearsmar90*agebase yearsmar90*educ

yearsmar90*agebase*year yearsmar90*educ*year yearsmar90*educ*agebase /

dist=binomial;

repeated subject=idnr / within=yearcls type=un modelse;

by parent;

run;

* males;

data males;

set maritcc;

where parent=0;

run;

proc genmod data=males descending;

title ’Reduced Model for Males: CC - GEE: Type=UN’;

class yearcls idnr;

model bsat = / dist=binomial;


run;

* females;

data females;

set maritcc;

where parent=1;

run;

proc genmod data=females descending;

title ’Reduced Model for Females: CC - GEE: Type=UN’;

class yearcls idnr;

model bsat = yearsmar90 yearsmar90*year yearsmar90*agebase yearsmar90*educ

yearsmar90*agebase*year yearsmar90*educ*agebase / noint dist=binomial ;


run;

3. Joint Analysis

* full model;


title ’CC - GEE: Full Model: Type=UN’;

class famnr yearcls parentcls;

model bsat = parent*yearsmar90 parent*yearsmar90*year

parent*yearsmar90*agebase parent*yearsmar90*educ

parent*yearsmar90*agebase*year parent*yearsmar90*educ*year

parent*yearsmar90*educ*agebase / dist=binomial;

repeated subject=famnr / within=yearcls(parentcls) type=un modelse corrw;

run;

* reduced model;



title ’CC - GEE: Reduced Model: Type=UN’;

class famnr yearcls parentcls;

model bsat = parent*yearsmar90 parent*yearsmar90*year

parent*yearsmar90*agebase parent*yearsmar90*educ

parent*yearsmar90*agebase*year / dist=binomial ;

repeated subject=famnr / within=yearcls(parentcls) type=un modelse corrw;

run;

Part 3

1. Data management:

libname MarSat ’C:\QMSS\GLMMs\Data’;

data dat;

set MarSat.marsat (drop = SAT PCOM BPCOM DCOM BDCOM birth educ status income);

yearsmarr = year - (1900 + marr);

drop marr;

run;

proc sort data = dat;

by idnr year famnr;

run;

2. Procedure GLIMMIX - nested random effects:

proc glimmix data = dat method = RSPL NOCLPRINT NOITPRINT;

title ’Nested RE - PQL’;

class famnr parent year;

model bsat(event = ’1’) = year parent parent*year child yearsmarr/ dist = binary solution;

random intercept / subject = famnr;

random intercept / subject = parent(famnr);

run;

proc glimmix data = dat method = RMPL NOCLPRINT NOITPRINT;

title ’Nested RE - MQL’;


model bsat(event = ’1’) = year parent parent*year child yearsmarr/ dist = binary solution;


random intercept / subject = parent(famnr);

run;

3. Procedure NLMIXED - nested random effects:

proc nlmixed data = dat points = 10;

title ’Nested RE - Adaptive GH (q = 10)’;

parms b0 = 2.8528 b1 = -0.5461 b2 = 0.3003 b3 = 0.1212 b4 = -0.02857

b5 = -0.2657 b6 = 0.06638 b7 = -0.03084

V1 = 2.3957;

eta = b0 + b1 * (year = 1990) + b2 * (year = 1995) +

b3 * (parent = 0) + b4 * (parent = 0) * (year = 1990) +

b5 * (parent = 0) * (year = 1995) + b6 * child +

b7 * yearsmarr +


g1 * (parent = 0) + g2 * (parent = 1);

expeta = exp(eta);

p = expeta / (1 + expeta);

model bsat ~ binary(p);

random g1 g2 ~ normal([0, 0], [V1 + V2,

V2, V1 + V2]) subject = famnr;

run;

4. Test for the parent random effect:


title ’Adaptive GH (q = 10) - No parent RE’;

parms b0 = 2.8528 b1 = -0.5461 b2 = 0.3003 b3 = 0.1212 b4 = -0.02857

b5 = -0.2657 b6 = 0.06638 b7 = -0.03084

V1 = 2.3957;

eta = b0 + b1 * (year = 1990) + b2 * (year = 1995) +



b7 * yearsmarr + g1;

expeta = exp(eta);



random g1 ~ normal(0, V1**2) subject = famnr;

run;

data LRT;

logL0 = -1203.2/2;

logL1 = -1199.6/2;

LRT = -2 * (logL0 - logL1);

df = 1;

pval = 1 - probchi(LRT, df);

run;


run;

5. Procedure GLIMMIX for PQL and MQL - random intercepts:

proc glimmix data = dat method = RSPL NOCLPRINT NOITPRINT IC=PQ or;

title ’PQL (REML)’;


nloptions maxit = 50 technique = newrap;

model bsat(event = ’1’) = year parent parent*year child

yearsmarr/ dist = binary solution;


run;

proc glimmix data = dat method = MSPL NOCLPRINT NOITPRINT IC=PQ or;

title ’PQL (ML)’;







run;

proc glimmix data = dat method = RMPL NOCLPRINT NOITPRINT;

title ’MQL (REML)’;






run;

proc glimmix data = dat method = MMPL NOCLPRINT NOITPRINT;

title ’MQL (ML)’;






run;

6. Procedure NLMIXED for Gaussian quadrature - random intercepts:

proc nlmixed data = dat noad points = 20;

title ’GH (q = 20)’;

parms b0 = 2.8528 b1 = -0.5465 b2 = 0.3001 b3 = 0.1213 b4 = -0.02867

b5 = -0.2656 b6 = 0.06617 b7 = -0.03079

V1 = 2.3662;

eta = b0 + b1 * (year = 1990) + b2 * (year = 1995) +




expeta = exp(eta);




run;

7. Procedure NLMIXED for adaptive Gaussian quadrature - random intercepts:


title ’Adaptive GH (q = 20)’;

parms b0 = 2.8528 b1 = -0.5465 b2 = 0.3001 b3 = 0.1213 b4 = -0.02867

b5 = -0.2656 b6 = 0.06617 b7 = -0.03079

V1 = 2.3662;

eta = b0 + b1 * (year = 1990) + b2 * (year = 1995) +




expeta = exp(eta);




estimate ’V1^2’ V1*V1;

run;


8. Procedure NLMIXED for Laplace approximation:


title ’Laplace approximation’;

parms b0 = 2.8528 b1 = -0.5465 b2 = 0.3001 b3 = 0.1213 b4 = -0.02867

b5 = -0.2656 b6 = 0.06617 b7 = -0.03079

V1 = 2.3662;

eta = b0 + b1 * (year = 1990) + b2 * (year = 1995) +




expeta = exp(eta);




estimate ’V1^2’ V1*V1;

run;

9. Marginal evolutions:

proc univariate data=dat;

var child yearsmarr;

run;

data Simulate;

do parent=0 to 1 by 1;

do subject=1 to 1000 by 1;

b1=rannor(-1);

b1=2.1366*b1;

do t=1 to 3 by 1;

output;

end;

end;

end;

run;

proc sort data=Simulate;

by t parent;

run;

data Simulate;

set Simulate;

child = 2;

marr = 22;

year90 = 0;

year95 = 0;

parent0 = 0;

if t=1 then year90=1;


if parent=0 then parent0=1;

if parent=0 then

y=1/(1+exp(-(4.3383+b1-0.7549*year90+0.4583*year95+0.1511*parent0

-0.02017*parent0*year90-0.3720*parent0*year95+

0.1531*child-0.04538*marr)));

else

y=1/(1+exp(-(4.3383+b1-0.7549*year90+0.4583*year95+


0.1531*child-0.04538*marr)));

run;

proc means data=Simulate;

var y;

by t parent;

output out=out;

run;

proc gplot data=out;

plot y*t=parent / haxis=axis1 vaxis=axis2 legend=legend1;

axis1 label=(h=2 ’Time Points’) value=(h=1.5) minor=none;

axis2 label=(h=2 A=90 ’P(Y=1)’) value=(h=1.5) order=(0.84 to 0.95 by 0.02) minor=none;

legend1 label=(h=1.5 ’Parent: ’) value=(h=1.5 ’Husband’ ’Wife’);

title h=2.5 ’Marginal average evolutions (GLMM)’;

symbol1 c=black i=join w=5 l=1 mode=include;


where _stat_=’MEAN’;

run;

quit;

run;

10. Evolutions for the median individual:

proc univariate data=dat;

var child yearsmarr;

run;

data MedianPlot;

do parent=0 to 1 by 1;

b1=0;

b1=2.1366*b1;

do t=1 to 3 by 1;

output;

end;

end;

run;

proc sort data=MedianPlot;

by t parent;

run;

data MedianPlot;

set MedianPlot;

child = 2;

marr = 22;

year90 = 0;

year95 = 0;

parent0 = 0;



if parent=0 then parent0=1;

if parent=0 then

y=1/(1+exp(-(4.3383+b1-0.7549*year90+0.4583*year95+0.1511*parent0

-0.02017*parent0*year90-0.3720*parent0*year95+

0.1531*child-0.04538*marr)));

else


y=1/(1+exp(-(4.3383+b1-0.7549*year90+0.4583*year95+

0.1531*child-0.04538*marr)));

run;

proc gplot data=MedianPlot;

plot y*t=parent / haxis=axis1 vaxis=axis2 legend=legend1;

axis1 label=(h=2 ’Time Points’) value=(h=1.5) minor=none;

axis2 label=(h=2 A=90 ’P(Y=1)’) value=(h=1.5) order=(0.92 to 1.0 by 0.02) minor=none;

legend1 label=(h=1.5 ’Parent: ’) value=(h=1.5 ’Husband’ ’Wife’);

title h=2.5 ’Evolutions for the median parent’;



run;

quit;

run;

11. Test for the parent effect:


title ’Adaptive GH (q = 20) - No parent effect’;

parms b0 = 2.8528 b1 = -0.5465 b2 = 0.3001 b3 = 0.1213 b4 = -0.02867

V1 = 2.3662;

eta = b0 + b1 * (year = 1990) + b2 * (year = 1995) + b3 * child +


expeta = exp(eta);




run;

data LRT;

logL0 = -1203.8/2;

logL1 = -1203.0/2;

LRT = -2 * (logL0 - logL1);

df = 3;

pval = 1 - probchi(LRT, df);

run;


run;

Part 4

1. Data management:

/* IC */

PROC IMPORT OUT= WORK.MSDIC

DATAFILE= "C:\QMSS Workshop\Data\MSD\data3_IC.xls"

DBMS=EXCEL REPLACE;

SHEET="Sheet1$";

GETNAMES=YES;

MIXED=NO;

SCANTEXT=YES;

USEDATE=YES;


SCANTIME=YES;

RUN;

data msdic;

set msdic;

yearcls=year;

parentcls=parent;

drop pcom bpcom dcom bdcom;

run;

proc print data=msdic;

run;

/* CC */

PROC IMPORT OUT= WORK.MSDCC

DATAFILE= "C:\QMSS Workshop\Data\MSD\data3_CC.xls"

DBMS=EXCEL REPLACE;

SHEET="Sheet1$";

GETNAMES=YES;

MIXED=NO;

SCANTEXT=YES;

USEDATE=YES;

SCANTIME=YES;

RUN;

data msdcc;

set msdcc;

drop pcom bpcom dcom bdcom;

run;

proc print data=msdcc;

run;

2. Exploring Missing Data Patterns

data msd1;

set msdic;

if year=1990 then sat1=sat;

if year>1990 then delete;

keep famnr parent sat1;

run;

data msd2;

set msdic;


if year>1995|year<1995 then delete;


run;

data msd3;

set msdic;


if year<2000 then delete;


run;

data misexp;

merge msd1 msd2 msd3;


by famnr parent;

run;

proc mi data=misexp;

var sat1 sat2 sat3;

run;

3. Continuous Response: Procedure MIXED: CC

proc mixed data=msdcc method=ml;

title ’CONT MIXED, CC’;

class famnr idnr year status parent;

model sat = year parent year*parent / s noint;

repeated parent year / type=un@CS subject=famnr r rcorr;

run;

4. Continuous Response: Procedure MIXED: Direct Likelihood

proc mixed data=msdic method=ml;

title ’CONT MIXED Direct Likelihood’;

class famnr idnr year status parent;

model sat = year parent year*parent / s noint;

repeated parent year / type=un@cs subject=famnr r rcorr;

run;

5. Continuous Response: Procedure MIXED: Multiple Imputation

proc MI data = msdic out = msdiccomp NOPRINT;

mcmc initial=em (bootstrap = 0.85)

prior = JEFFREYS

chain = multiple;

run;

/* Transforming the data for Mixed model (Creating dummies) */

data msdiccomp;

title ’CONT MIXED MULTIPLE IMPUTATION’;

set msdiccomp;

year1990 = 0;

year1995 = 0;

year2000 = 0;

if year = 1990 then year1990 = 1;



father = 0;

mother = 0;

if Parent = 0 then father = 1;

if Parent = 1 then mother = 1;

father1990 = 0;

father1995 = 0;

father2000 = 0;

mother1990 = 0;

mother1995 = 0;

mother2000 = 0;

if (father = 1) and (year = 1990) then father1990 = 1;



if (mother = 1) and (year = 1990) then mother1990 = 1;




run;

/* Analysing 5 Completed Datasets */

proc mixed data=msdiccomp method = ml asycov covtest;

title2 ’MIXED MODEL ANALYSIS PER IMPUTATION’;

class famnr year parent ;

model sat = year1990 year1995 year2000 father father1990 father1995 / s noint covb;

repeated year parent / subject=famnr type=un@cs rcorr;

ods output solutionf = solution covb = covb covparms = covparms asycov = asycov;

by _imputation_;

data solution0;

set solution;

data covb0;

set covb;

data covparms0;

set covparms;

if CovParm=’YEAR UN(1,1)’ then effect = ’YEARUN11’;

if CovParm=’ UN(2,1)’ then effect = ’YEARUN21’;





if CovParm=’PARENT Corr’ then effect = ’PARENTCORR’ ;

drop covparm;

data asycov0;

set asycov;

Col1=CovP1;

Col2=CovP2;

Col3=CovP3;

Col4=CovP4;

Col5=CovP5;

Col6=CovP6;

Col7=CovP7;

if CovParm=’YEAR UN(1,1)’ then effect = ’YEARUN11’;






if CovParm=’PARENT Corr’ then effect = ’PARENTCORR’ ;

drop CovP1 CovP2 CovP3 CovP4 CovP5 CovP6 CovP7 covparm;

run;

/* Combining 5 Separate Analyses (mean structure) */

proc mianalyze parms=solution0 covb(effectvar=rowcol)=covb0;

title2 ’COMBINING 5 MIXED MODEL ANALYSES (MEAN STRUCTURE)’;

modeleffects year1990 year1995 year2000 father father1990 father1995;

run;

/* Combining 5 Separate Analyses (covariance structure) */

proc mianalyze parms=covparms0 covb(effectvar=rowcol)=asycov0;

title2 ’COMBINING 5 MIXED MODEL ANALYSES (COVARIANCE STRUCTURE)’;

modeleffects YEARUN11 YEARUN21 YEARUN22 YEARUN31 YEARUN32 YEARUN33 PARENTCO;

run;

6. Binary Response: Procedure GEE: CC

proc genmod data=msdcc descending;

title ’BIN GEE CC’;


class famnr year parent;

model bsat = parent year birth parent*year parent*birth birth*year parent*birth*year

/dist=binomial type3;

repeated subject=famnr / within=year(parent) type=un modelse corrw;

run;

7. Binary Response: Procedure GEE: WGEE

/* Use the WGEE MACRO FOR CREATING VARIABLES "DROPOUT" AND "PREV" */

%dropout(data=msdic,id=famnr,time=year,response=bsat,out=msdicwgee);

proc print data=msdicwgee;

title ’BIN WGEE’;

run;

proc genmod data=msdicwgee descending;


title2 ’Dropout Model’;

class famnr year status prev;

model dropout = prev parent year birth/ pred dist=b;

ods output obstats=pred;

ods listing exclude obstats;

run;

data pred;

set pred;

keep observation pred;

run;

data msdicwgee1;

merge pred msdicwgee;

run;

/* Use the WGEE MACRO TO CREATE THE WEIGHTING VARIABLE */

%dropwgt(data=msdicwgee1,id=famnr,time=year,pred=pred,dropout=dropout,out=msdicwgee2);

proc print data=msdicwgee2;

var famnr year bsat dropout prev pred wi;

run;

/* WGEE model */

proc genmod data=msdicwgee2 descending;


scwgt wi;


model bsat = parent year birth parent*year parent*birth birth*year parent*birth*year

/dist=binomial type3;


run;

8. Binary Response: Procedure GEE: Available Cases

proc genmod data=msdic descending;

title ’BIN GEE Available Cases’;


model bsat = parent year birth parent*year parent*birth year*birth parent*year*birth

/ dist=binomial type3;


run;


9. Binary Response: Procedure GEE: Multiple Imputation

proc MI data = msdic out = msdiccomp NOPRINT;

mcmc initial=em (bootstrap = 0.85)

prior = JEFFREYS

chain = multiple;

run;

/* Transforming the data for MI GEE (Creating dummies) */

data msdiccomp;

title1 ’BIN GEE MULTIPLE IMPUTATION’;

set msdiccomp;

if SAT <= 4 then BSAT = 0;

if SAT > 4 then BSAT = 1;

year1990 = 0;

year1995 = 0;

year2000 = 0;




father = 0;

mother = 0;

if Parent = 0 then father = 1;

if Parent = 1 then mother = 1;

father1990 = 0;

father1995 = 0;

father2000 = 0;

mother1990 = 0;

mother1995 = 0;

mother2000 = 0;







fatherbirth = 0;

motherbirth = 0;

if father = 1 then fatherbirth = birth;

if mother = 1 then motherbirth = birth;

birth1990 = 0;

birth1995 = 0;

birth2000 = 0;

if year = 1990 then birth1990 = birth;



father1990birth = 0;



mother1990birth = 0;



if (father = 1) and (year = 1990) then father1990birth = birth;



if (mother = 1) and (year = 1990) then mother1990birth = birth;



run;


/* Analysing 5 Completed Datasets */

proc genmod data=msdiccomp descending;

title2 ’GEE ANALYSIS PER IMPUTATION’;


model bsat = father year1990 year1995 BIRTH father1990 father1995 fatherbirth

birth1990 birth1995 father1990birth father1995birth

/ dist=binomial type3 covb;


ods output parameterestimates = gmparms parminfo = gmpinfo covb = gmcovb;

by _imputation_;

data gmparms;

set gmparms;

if parameter in (’Scale’) then delete;

data gmcovb;

set gmcovb;

data gmpinfo;

set gmpinfo;

run;

/* Combining 5 Separate Analyses */

proc mianalyze parms = gmparms covb = gmcovb parminfo = gmpinfo;

title2 ’COMBINING 5 GEE ANALYSES’;

modeleffects intercept father year1990 year1995 BIRTH father1990 father1995 fatherbirth

birth1990 birth1995 father1990birth father1995birth;

run;

10. Binary Response: Procedure NLMIXED: CC

proc logistic data=msdcc;

title ’Determining the starting values for the CC GLMM’;

class year parent;

model bsat=parent year birth parent*year parent*birth birth*year parent*birth*year;

run;

proc nlmixed data = msdcc points = 10;

title ’CC GLMM’;

parms b0 = -3 b1 = -1.5 b2 = -0.9 b3 = 0.1 b4 = 0.008

b5 = 1.1 b6 = 3.1 b7 = 0.002 b8 = 0.04 b9 = 0.01 b10 = -0.02 b11 = -0.065

V1 = 2.3662;

eta = b0 + b1 * (year = 1990) + b2 * (year = 1995) + b3 * (parent = 0) + b4 * birth +

b5 * (parent = 0) * (year = 1990) + b6 * (parent = 0) * (year = 1995) +

b7 * (parent = 0) * birth + b8 * birth * (year = 1990) +

b9 * birth * (year = 1995) + b10 * birth * (year = 1990) * (parent = 0) +

b11 * birth * (year = 1995) * (parent = 0) + g1;

expeta = exp(eta);




run;

11. Binary Response: Procedure NLMIXED: Direct Likelihood

proc nlmixed data = msdic points = 10;

title ’Direct Likelihood GLMM’;

parms b0 = -3 b1 = -1.5 b2 = -0.9 b3 = 0.1 b4 = 0.008

b5 = 1.1 b6 = 3.1 b7 = 0.002 b8 = 0.04 b9 = 0.01 b10 = -0.02 b11 = -0.065

V1 = 2.3662;

eta = b0 + b1 * (year = 1990) + b2 * (year = 1995) + b3 * (parent = 0) + b4 * birth +

b5 * (parent = 0) * (year = 1990) + b6 * (parent = 0) * (year = 1995) +

b7 * (parent = 0) * birth + b8 * birth * (year = 1990) +

b9 * birth * (year = 1995) + b10 * birth * (year = 1990) * (parent = 0) +


b11 * birth * (year = 1995) * (parent = 0) + g1;

expeta = exp(eta);




run;

6.3.2 SAS Output

Part 1

2. Full model with unstructured mean and covariance structure

(a) Fathers:



UN(1,1) IDNR 0.5494

UN(2,1) IDNR 0.3786

UN(2,2) IDNR 0.8045

UN(3,1) IDNR 0.4058

UN(3,2) IDNR 0.5957

UN(3,3) IDNR 0.9409

Fit Statistics







5 142.39 <.0001


Standard

Effect YEAR STATUS Estimate Error DF t Value Pr > |t|

YEAR 1990 4.2185 1.6399 118 2.57 0.0113

YEAR 1995 4.9972 1.6461 118 3.04 0.0030

YEAR 2000 4.7205 1.6464 118 2.87 0.0049

CHILD 0.04839 0.04897 118 0.99 0.3251

BIRTH 0.008740 0.01883 118 0.46 0.6435

EDUC -0.03250 0.03742 118 -0.87 0.3869


MARR 0.008137 0.02700 118 0.30 0.7637

INCOME 0.02918 0.05456 118 0.53 0.5937

STATUS 1 -0.1992 0.1857 118 -1.07 0.2856

STATUS 2 0 . . . .


Num Den


YEAR 2 118 53.77 <.0001

CHILD 1 118 0.98 0.3251

BIRTH 1 118 0.22 0.6435

EDUC 1 118 0.75 0.3869

MARR 1 118 0.09 0.7637

INCOME 1 118 0.29 0.5937

STATUS 1 118 1.15 0.2856

(b) Mothers:



UN(1,1) IDNR 0.6559

UN(2,1) IDNR 0.3990

UN(2,2) IDNR 0.9592

UN(3,1) IDNR 0.3583

UN(3,2) IDNR 0.5691

UN(3,3) IDNR 0.9002

Fit Statistics







5 103.99 <.0001


Standard

Effect YEAR STATUS Estimate Error DF t Value Pr > |t|

YEAR 1990 7.1128 1.7919 118 3.97 0.0001

YEAR 1995 7.7785 1.7988 118 4.32 <.0001


YEAR 2000 7.5638 1.7987 118 4.21 <.0001

CHILD -0.00229 0.05124 118 -0.04 0.9645

BIRTH 0.01256 0.02824 118 0.44 0.6574

EDUC -0.02235 0.04469 118 -0.50 0.6180

MARR -0.02964 0.03577 118 -0.83 0.4090

INCOME -0.01205 0.05385 118 -0.22 0.8234

STATUS 1 -0.2059 0.1894 118 -1.09 0.2792

STATUS 2 0 . . . .


Num Den


YEAR 2 118 30.03 <.0001

CHILD 1 118 0.00 0.9645

BIRTH 1 118 0.20 0.6574

EDUC 1 118 0.25 0.6180

MARR 1 118 0.69 0.4090

INCOME 1 118 0.05 0.8234

STATUS 1 118 1.18 0.2792

3. Final model after reducing mean structure

(a) Fathers:



UN(1,1) IDNR 0.5596

UN(2,1) IDNR 0.3881

UN(2,2) IDNR 0.8222

UN(3,1) IDNR 0.4049

UN(3,2) IDNR 0.5991

UN(3,3) IDNR 0.9300

Fit Statistics







5 144.10 <.0001



Standard

Effect YEAR Estimate Error DF t Value Pr > |t|

YEAR 1990 5.2266 0.06773 122 77.17 <.0001

YEAR 1995 6.0236 0.08209 122 73.38 <.0001

YEAR 2000 5.7469 0.08731 122 65.82 <.0001


Num Den


YEAR 3 122 2438.85 <.0001

(b) Mothers:

Fit Statistics







5 104.13 <.0001


Standard

Effect YEAR Estimate Error DF t Value Pr > |t|

YEAR 1990 5.2588 0.07476 122 70.34 <.0001

YEAR 1995 5.9184 0.08878 122 66.66 <.0001

YEAR 2000 5.7037 0.08520 122 66.95 <.0001


Num Den


YEAR 3 122 2268.91 <.0001

4. Reducing the covariance structure

(a) Fathers:

• Unstructured - UN




UN(1,1) IDNR 0.5596

UN(2,1) IDNR 0.3881

UN(2,2) IDNR 0.8222

UN(3,1) IDNR 0.4049

UN(3,2) IDNR 0.5991

UN(3,3) IDNR 0.9300

Fit Statistics







5 144.10 <.0001

• Banded - UN(2)



UN(1,1) IDNR 0.5596

UN(2,1) IDNR 0.1859

UN(2,2) IDNR 0.6878

UN(3,1) IDNR 0

UN(3,2) IDNR 0.4646

UN(3,3) IDNR 0.9300

Fit Statistics







4 97.95 <.0001

• Heterogeneous Autoregressive - ARH(1)




Var(1) IDNR 0.5856

Var(2) IDNR 0.8222

Var(3) IDNR 0.8906

ARH(1) IDNR 0.6298

Fit Statistics







3 131.22 <.0001

• Heterogeneous Compound Symmetry - CSH



Var(1) IDNR 0.5793

Var(2) IDNR 0.8066

Var(3) IDNR 0.9165

CSH IDNR 0.6067

Fit Statistics







3 138.87 <.0001

• Heterogeneous Toeplitz - TOEPH



Var(1) IDNR 0.5770

Var(2) IDNR 0.8222

Var(3) IDNR 0.9028

TOEPH(1) IDNR 0.6295



Fit Statistics







4 140.62 <.0001

• Compound Symmetry - CS



CS IDNR 0.4640

Residual 0.3066

Fit Statistics







1 128.48 <.0001

(b) Mothers:

• Unstructured - UN



UN(1,1) IDNR 0.6819

UN(2,1) IDNR 0.4139

UN(2,2) IDNR 0.9616

UN(3,1) IDNR 0.3637

UN(3,2) IDNR 0.5629

UN(3,3) IDNR 0.8855

Fit Statistics








5 104.13 <.0001

• Banded - UN(2)



UN(1,1) IDNR 0.6819

UN(2,1) IDNR 0.2340

UN(2,2) IDNR 0.8381

UN(3,1) IDNR 0

UN(3,2) IDNR 0.4381

UN(3,3) IDNR 0.8855

Fit Statistics







4 73.98 <.0001

• Heterogeneous Autoregressive - ARH(1)



Var(1) IDNR 0.7051

Var(2) IDNR 0.9616

Var(3) IDNR 0.8573

ARH(1) IDNR 0.5614

Fit Statistics








3 96.03 <.0001

• Heterogeneous Compound Symmetry - CSH



Var(1) IDNR 0.7037

Var(2) IDNR 0.9395

Var(3) IDNR 0.8791

CSH IDNR 0.5303

Fit Statistics







3 99.89 <.0001

• Heterogeneous Toeplitz - TOEPH



Var(1) IDNR 0.6991

Var(2) IDNR 0.9616

Var(3) IDNR 0.8643



Fit Statistics








4 102.26 <.0001

• Compound Symmetry - CS



CS IDNR 0.4468

Residual 0.3962

Fit Statistics







1 96.07 <.0001

• Simple Structure

Covariance Parameter

Estimates


YEAR IDNR 0.8430

Fit Statistics





5. Modelling longitudinal data of both parents simultaneously:

• Unstructured 6 × 6 covariance matrix

Estimated R Matrix for Subject 1

Row Col1 Col2 Col3 Col4 Col5 Col6

1 0.5596 0.2454 0.3881 0.2620 0.4049 0.2285

2 0.2454 0.6819 0.3069 0.4139 0.2871 0.3637

3 0.3881 0.3069 0.8222 0.4211 0.5991 0.3913

4 0.2620 0.4139 0.4211 0.9616 0.4414 0.5629

5 0.4049 0.2871 0.5991 0.4414 0.9300 0.5107

6 0.2285 0.3637 0.3913 0.5629 0.5107 0.8855




UN(1,1) FAMNR 0.5596

UN(2,1) FAMNR 0.2454

UN(2,2) FAMNR 0.6819

UN(3,1) FAMNR 0.3881

UN(3,2) FAMNR 0.3069

UN(3,3) FAMNR 0.8222

UN(4,1) FAMNR 0.2620

UN(4,2) FAMNR 0.4139

UN(4,3) FAMNR 0.4211

UN(4,4) FAMNR 0.9616

UN(5,1) FAMNR 0.4049

UN(5,2) FAMNR 0.2871

UN(5,3) FAMNR 0.5991

UN(5,4) FAMNR 0.4414

UN(5,5) FAMNR 0.9300

UN(6,1) FAMNR 0.2285

UN(6,2) FAMNR 0.3637

UN(6,3) FAMNR 0.3913

UN(6,4) FAMNR 0.5629

UN(6,5) FAMNR 0.5107

UN(6,6) FAMNR 0.8855

Fit Statistics







20 316.76 <.0001


Standard

Effect YEAR PARENT Estimate Error DF t Value Pr > |t|

PARENT 0 5.7469 0.08731 122 65.82 <.0001

PARENT 1 5.7037 0.08520 122 66.95 <.0001

YEAR 1990 -0.4450 0.08298 122 -5.36 <.0001

YEAR 1995 0.2147 0.07689 122 2.79 0.0061

YEAR 2000 0 . . . .

YEAR*PARENT 1990 0 -0.07533 0.09228 122 -0.82 0.4159


YEAR*PARENT 1990 1 0 . . . .

YEAR*PARENT 1995 0 0.06206 0.09396 122 0.66 0.5102

YEAR*PARENT 1995 1 0 . . . .

YEAR*PARENT 2000 0 0 . . . .

YEAR*PARENT 2000 1 0 . . . .


Num Den


PARENT 1 122 0.40 0.5285

YEAR 2 122 80.08 <.0001

YEAR*PARENT 2 122 0.95 0.3913

• Unstructured @ Unstructured



1 0.5490 0.2008 0.3143 0.1149 0.3094 0.1131

2 0.2008 0.6620 0.1149 0.3790 0.1131 0.3730

3 0.3143 0.1149 0.7672 0.2805 0.4574 0.1673

4 0.1149 0.3790 0.2805 0.9251 0.1673 0.5516

5 0.3094 0.1131 0.4574 0.1673 0.7618 0.2786

6 0.1131 0.3730 0.1673 0.5516 0.2786 0.9186



PARENT UN(1,1) FAMNR 0.5490

UN(2,1) FAMNR 0.2008

UN(2,2) FAMNR 0.6620

YEAR UN(1,1) FAMNR 1.0000

UN(2,1) FAMNR 0.5725

UN(2,2) FAMNR 1.3973

UN(3,1) FAMNR 0.5634

UN(3,2) FAMNR 0.8331

UN(3,3) FAMNR 1.3875

Fit Statistics








7 286.42 <.0001


Standard


PARENT 0 5.7469 0.07902 121 72.73 <.0001

PARENT 1 5.7037 0.08677 121 65.73 <.0001

YEAR 1990 -0.4450 0.08271 242 -5.38 <.0001

YEAR 1995 0.2147 0.07791 242 2.76 0.0063

YEAR 2000 0 . . . .

YEAR*PARENT 1990 0 -0.07533 0.09146 242 -0.82 0.4109

YEAR*PARENT 1990 1 0 . . . .

YEAR*PARENT 1995 0 0.06206 0.08615 242 0.72 0.4720

YEAR*PARENT 1995 1 0 . . . .

YEAR*PARENT 2000 0 0 . . . .

YEAR*PARENT 2000 1 0 . . . .


Num Den


PARENT 1 121 0.26 0.6089

YEAR 2 242 65.13 <.0001

YEAR*PARENT 2 242 1.14 0.3228

• Unstructured @ Autoregressive



1 0.6933 0.2562 0.3671 0.1357 0.1944 0.07183

2 0.2562 0.8171 0.1357 0.4327 0.07183 0.2291

3 0.3671 0.1357 0.6933 0.2562 0.3671 0.1357

4 0.1357 0.4327 0.2562 0.8171 0.1357 0.4327

5 0.1944 0.07183 0.3671 0.1357 0.6933 0.2562

6 0.07183 0.2291 0.1357 0.4327 0.2562 0.8171




UN(2,1) FAMNR 0.2562

UN(2,2) FAMNR 0.8171

YEAR AR(1) FAMNR 0.5295


Fit Statistics







3 256.45 <.0001


Standard


PARENT 0 5.7469 0.07539 121 76.23 <.0001

PARENT 1 5.7037 0.08184 121 69.69 <.0001

YEAR 1990 -0.4450 0.09818 242 -4.53 <.0001

YEAR 1995 0.2147 0.07939 242 2.70 0.0073

YEAR 2000 0 . . . .

YEAR*PARENT 1990 0 -0.07533 0.1085 242 -0.69 0.4882

YEAR*PARENT 1990 1 0 . . . .

YEAR*PARENT 1995 0 0.06206 0.08774 242 0.71 0.4800

YEAR*PARENT 1995 1 0 . . . .

YEAR*PARENT 2000 0 0 . . . .

YEAR*PARENT 2000 1 0 . . . .


Num Den


PARENT 1 121 0.29 0.5910

YEAR 2 242 68.74 <.0001

YEAR*PARENT 2 242 1.29 0.2780

• Unstructured @ Compound Symmetry



1 0.7011 0.2696 0.3620 0.1392 0.3620 0.1392

2 0.2696 0.8309 0.1392 0.4290 0.1392 0.4290

3 0.3620 0.1392 0.7011 0.2696 0.3620 0.1392

4 0.1392 0.4290 0.2696 0.8309 0.1392 0.4290

5 0.3620 0.1392 0.3620 0.1392 0.7011 0.2696

6 0.1392 0.4290 0.1392 0.4290 0.2696 0.8309





UN(2,1) FAMNR 0.2696

UN(2,2) FAMNR 0.8309

YEAR Corr FAMNR 0.5164

Fit Statistics







3 270.80 <.0001


Standard


PARENT 0 5.7469 0.07581 121 75.81 <.0001

PARENT 1 5.7037 0.08253 121 69.11 <.0001

YEAR 1990 -0.4450 0.08116 242 -5.48 <.0001

YEAR 1995 0.2147 0.08116 242 2.64 0.0087

YEAR 2000 0 . . . .

YEAR*PARENT 1990 0 -0.07533 0.08872 242 -0.85 0.3967

YEAR*PARENT 1990 1 0 . . . .

YEAR*PARENT 1995 0 0.06206 0.08872 242 0.70 0.4849

YEAR*PARENT 1995 1 0 . . . .

YEAR*PARENT 2000 0 0 . . . .

YEAR*PARENT 2000 1 0 . . . .


Num Den


PARENT 1 121 0.27 0.6031

YEAR 2 242 66.89 <.0001

YEAR*PARENT 2 242 1.20 0.3021


Part 2

1. Separate Analysis for Males & Females

Full Model by Parent: CC - GEE: Type=UN

PARENT=0

Model Information

Data Set WORK.MARITCC


Link Function Logit

Dependent Variable BSAT






Class Levels Values

YEARCLS 3 1990 1995 2000

IDNR 122 23 25 39 47 77 83 91 93 101 111 119 129 133 135

137 139 145 159 195 247 249 277 285 287 313 331

339 341 345 361 363 373 385 411 415 423 425 433

439 443 447 453 455 469 479 505 509 519 547 551

557 565 577 597 621 625 629 639 643 645 661 663

673 681 695 ...

Response Profile

Ordered Total

Value BSAT Frequency

1 1 339

2 0 27

PROC GENMOD is modeling the probability that BSAT=’1’.


Parameter Effect

Prm1 Intercept

Prm2 YEARSMAR90

Prm3 YEARSMAR90*YEAR

Prm4 YEARSMAR90*AGEBASE


Prm5 YEARSMAR90*EDUC

Prm6 YEARSMA*YEAR*AGEBASE

Prm7 YEARSMAR90*YEAR*EDUC

Prm8 YEARSMA*AGEBASE*EDUC



Deviance 358 191.3000 0.5344



Scaled Pearson X2 358 364.3476 1.0177




Standard Wald 95% Chi-

Parameter DF Estimate Error Confidence Limits Square Pr > ChiSq

Intercept 1 3.3960 1.6584 0.1455 6.6465 4.19 0.0406

YEARSMAR90 1 20.3422 58.0975 -93.5269 134.2112 0.12 0.7262

YEARSMAR90*YEAR 1 -0.0103 0.0291 -0.0673 0.0468 0.12 0.7245

YEARSMAR90*AGEBASE 1 -0.3881 1.3206 -2.9765 2.2003 0.09 0.7689

YEARSMAR90*EDUC 1 -1.1199 2.6958 -6.4036 4.1638 0.17 0.6778

YEARSMA*YEAR*AGEBASE 1 0.0002 0.0007 -0.0011 0.0015 0.09 0.7675

YEARSMAR90*YEAR*EDUC 1 0.0006 0.0013 -0.0021 0.0032 0.17 0.6795

YEARSMA*AGEBASE*EDUC 1 0.0001 0.0014 -0.0026 0.0028 0.00 0.9486

Scale 0 1.0000 0.0000 1.0000 1.0000



Correlation Structure Unstructured

Within-Subject Effect YEARCLS (3 levels)

Subject Effect IDNR (122 levels)











Intercept 3.1348 2.2989 -1.3709 7.6405 1.36 0.1727

YEARSMAR90 30.4460 28.6533 -25.7135 86.6055 1.06 0.2880

YEARSMAR90*YEAR -0.0153 0.0144 -0.0435 0.0128 -1.07 0.2856

YEARSMAR90*AGEBASE -0.6467 0.6056 -1.8338 0.5403 -1.07 0.2856

YEARSMAR90*EDUC -0.6050 2.1619 -4.8422 3.6322 -0.28 0.7796

YEARSMA*YEAR*AGEBASE 0.0003 0.0003 -0.0003 0.0009 1.07 0.2837

YEARSMAR90*YEAR*EDUC 0.0003 0.0011 -0.0018 0.0024 0.28 0.7766

YEARSMA*AGEBASE*EDUC -0.0003 0.0019 -0.0040 0.0035 -0.14 0.8857





Intercept 3.1348 2.3166 -1.4056 7.6751 1.35 0.1760

YEARSMAR90 30.4460 38.7133 -45.4308 106.3227 0.79 0.4316

YEARSMAR90*YEAR -0.0153 0.0194 -0.0534 0.0227 -0.79 0.4293

YEARSMAR90*AGEBASE -0.6467 0.8869 -2.3851 1.0916 -0.73 0.4659

YEARSMAR90*EDUC -0.6050 1.8193 -4.1708 2.9608 -0.33 0.7395

YEARSMA*YEAR*AGEBASE 0.0003 0.0004 -0.0005 0.0012 0.73 0.4638


YEARSMA*AGEBASE*EDUC -0.0003 0.0019 -0.0040 0.0035 -0.14 0.8857

Scale 1.0000 . . . . .


Full Model by Parent: CC - GEE: Type=UN

PARENT=1

Model Information



Link Function Logit







Class Levels Values


YEARCLS 3 1990 1995 2000

IDNR 122 24 26 40 48 78 84 92 94 102 112 120 130 134 136

138 140 146 160 196 248 250 278 286 288 314 332

340 342 346 362 364 374 386 412 416 424 426 434

440 444 448 454 456 470 480 506 510 520 548 552

558 566 578 598 622 626 630 640 644 646 662 664

674 682 696 ...

Response Profile

Ordered Total


1 1 340

2 0 26



Parameter Effect

Prm1 Intercept

Prm2 YEARSMAR90





Prm7 YEARSMAR90*YEAR*EDUC




Deviance 358 178.1734 0.4977



Scaled Pearson X2 358 358.6836 1.0019







Intercept 1 1.6766 1.9589 -2.1628 5.5160 0.73 0.3921

YEARSMAR90 1 -91.5897 63.2847 -215.625 32.4460 2.09 0.1478

YEARSMAR90*YEAR 1 0.0461 0.0317 -0.0160 0.1082 2.12 0.1457

YEARSMAR90*AGEBASE 1 2.4072 1.4827 -0.4989 5.3132 2.64 0.1045

YEARSMAR90*EDUC 1 -3.0702 3.6579 -10.2395 4.0991 0.70 0.4013

YEARSMA*YEAR*AGEBASE 1 -0.0012 0.0007 -0.0027 0.0002 2.66 0.1030

YEARSMAR90*YEAR*EDUC 1 0.0015 0.0018 -0.0021 0.0051 0.65 0.4193


Scale 0 1.0000 0.0000 1.0000 1.0000















Intercept 1.5132 2.0866 -2.5766 5.6029 0.73 0.4684

YEARSMAR90 -105.064 45.7407 -194.714 -15.4143 -2.30 0.0216

YEARSMAR90*YEAR 0.0528 0.0229 0.0080 0.0976 2.31 0.0209

YEARSMAR90*AGEBASE 2.7683 1.1125 0.5878 4.9488 2.49 0.0128

YEARSMAR90*EDUC -3.0568 3.5830 -10.0793 3.9656 -0.85 0.3936

YEARSMA*YEAR*AGEBASE -0.0014 0.0006 -0.0025 -0.0003 -2.50 0.0125


YEARSMA*AGEBASE*EDUC 0.0021 0.0014 -0.0006 0.0048 1.53 0.1264





Intercept 1.5132 2.3965 -3.1838 6.2101 0.63 0.5278

YEARSMAR90 -105.064 56.6281 -216.054 5.9247 -1.86 0.0635

YEARSMAR90*YEAR 0.0528 0.0283 -0.0027 0.1084 1.86 0.0623


YEARSMAR90*EDUC -3.0568 3.2344 -9.3961 3.2824 -0.95 0.3446





Scale 1.0000 . . . . .


Reduced Model for Males: CC - GEE: Type=UN

Model Information

Data Set WORK.MALES


Link Function Logit







Class Levels Values

YEARCLS 3 1990 1995 2000

IDNR 122 23 25 39 47 77 83 91 93 101 111 119 129 133 135

137 139 145 159 195 247 249 277 285 287 313 331

339 341 345 361 363 373 385 411 415 423 425 433

439 443 447 453 455 469 479 505 509 519 547 551

557 565 577 597 621 625 629 639 643 645 661 663

673 681 695 ...

Response Profile

Ordered Total


1 1 339

2 0 27



Parameter Effect

Prm1 Intercept




Deviance 365 192.7243 0.5280



Scaled Pearson X2 365 366.0000 1.0027






Intercept 1 2.5302 0.2000 2.1382 2.9221 160.10 <.0001

Scale 0 1.0000 0.0000 1.0000 1.0000















Intercept 2.5333 0.2800 1.9845 3.0822 9.05 <.0001






Intercept 2.5333 0.2803 1.9840 3.0827 9.04 <.0001

Scale 1.0000 . . . . .


Reduced Model for Females: CC - GEE: Type=UN

Model Information

Data Set WORK.FEMALES


Link Function Logit







Class Levels Values

YEARCLS 3 1990 1995 2000

IDNR 122 24 26 40 48 78 84 92 94 102 112 120 130 134 136

138 140 146 160 196 248 250 278 286 288 314 332

340 342 346 362 364 374 386 412 416 424 426 434

440 444 448 454 456 470 480 506 510 520 548 552

558 566 578 598 622 626 630 640 644 646 662 664

674 682 696 ...

Response Profile

Ordered Total


1 1 340

2 0 26



Parameter Effect

Prm1 Intercept

Prm2 YEARSMAR90









Deviance 360 179.5951 0.4989



Scaled Pearson X2 360 358.5169 0.9959






Intercept 0 0.0000 0.0000 0.0000 0.0000 . .

YEARSMAR90 1 -106.408 58.0239 -220.133 7.3170 3.36 0.0667

YEARSMAR90*YEAR 1 0.0537 0.0291 -0.0033 0.1106 3.41 0.0649

YEARSMAR90*AGEBASE 1 2.5085 1.4119 -0.2587 5.2758 3.16 0.0756

YEARSMAR90*EDUC 1 -0.1109 0.0761 -0.2600 0.0383 2.12 0.1452

YEARSMA*YEAR*AGEBASE 1 -0.0013 0.0007 -0.0026 0.0001 3.19 0.0741


Scale 0 1.0000 0.0000 1.0000 1.0000




Intercept 0.7457 0.3878















Intercept 0.0000 0.0000 0.0000 0.0000 . .

YEARSMAR90 -118.617 42.4646 -201.846 -35.3874 -2.79 0.0052

YEARSMAR90*YEAR 0.0598 0.0213 0.0180 0.1015 2.81 0.0050


YEARSMAR90*EDUC -0.1069 0.0525 -0.2098 -0.0040 -2.04 0.0417







Intercept 0.0000 0.0000 0.0000 0.0000 . .

YEARSMAR90 -118.617 52.3178 -221.158 -16.0755 -2.27 0.0234

YEARSMAR90*YEAR 0.0598 0.0262 0.0084 0.1111 2.28 0.0225


YEARSMAR90*EDUC -0.1069 0.0890 -0.2813 0.0675 -1.20 0.2296



Scale 1.0000 . . . . .


2. Joint Analysis

CC - GEE: Full Model: Type=UN

Model Information



Link Function Logit








Class Levels Values

FAMNR 122 95 96 116 129 281 299 324 331 349 364 380 402 405

415 421 423 438 547 627 1076 1087 1151 1181 1182

1253 1314 1329 1332 1339 1375 1382 1422 1476 1523

1557 1566 1571 1606 1611 1614 1617 1637 1640 1680

1700 1761 1767 1783 1939 1950 1962 2008 2023 2079

...

YEARCLS 3 1990 1995 2000

PARENTCLS 2 0 1

Response Profile

Ordered Total


1 1 679

2 0 53



Parameter Effect

Prm1 Intercept

Prm2 PARENT*YEARSMAR90

Prm3 PARENT*YEARSMAR*YEAR

Prm4 PARENT*YEARSM*AGEBAS

Prm5 PARENT*YEARSMAR*EDUC

Prm6 PARE*YEAR*YEAR*AGEBA

Prm7 PARE*YEARS*YEAR*EDUC

Prm8 PARE*YEAR*AGEBA*EDUC



Deviance 724 371.0826 0.5125



Scaled Pearson X2 724 722.3190 0.9977







Intercept 1 2.5216 0.1983 2.1330 2.9102 161.75 <.0001

PARENT*YEARSMAR90 1 -92.5206 63.4805 -216.940 31.8988 2.12 0.1450

PARENT*YEARSMAR*YEAR 1 0.0465 0.0318 -0.0158 0.1088 2.14 0.1434

PARENT*YEARSM*AGEBAS 1 2.4290 1.4893 -0.4900 5.3480 2.66 0.1029

PARENT*YEARSMAR*EDUC 1 -3.0624 3.6483 -10.2129 4.0880 0.70 0.4012

PARE*YEAR*YEAR*AGEBA 1 -0.0012 0.0007 -0.0027 0.0002 2.68 0.1016

PARE*YEARS*YEAR*EDUC 1 0.0015 0.0018 -0.0021 0.0051 0.65 0.4201

PARE*YEAR*AGEBA*EDUC 1 0.0025 0.0021 -0.0016 0.0065 1.42 0.2332

Scale 0 1.0000 0.0000 1.0000 1.0000




Within-Subject Effect YEARCLS(PARENTCLS) (6 levels)

Subject Effect FAMNR (122 levels)






Working Correlation Matrix

Col1 Col2 Col3 Col4 Col5 Col6

Row1 1.0000 0.4392 0.5290 0.1111 0.1343 0.0526

Row2 0.4392 1.0000 0.5777 0.1495 0.2695 0.2196

Row3 0.5290 0.5777 1.0000 0.2880 0.2219 0.3311

Row4 0.1111 0.1495 0.2880 1.0000 0.2834 0.2212

Row5 0.1343 0.2695 0.2219 0.2834 1.0000 0.4382

Row6 0.0526 0.2196 0.3311 0.2212 0.4382 1.0000





Intercept 2.5021 0.2724 1.9683 3.0360 9.19 <.0001

PARENT*YEARSMAR90 -112.543 41.2309 -193.354 -31.7318 -2.73 0.0063

PARENT*YEARSMAR*YEAR 0.0565 0.0206 0.0160 0.0969 2.73 0.0063

PARENT*YEARSM*AGEBAS 2.9083 1.0653 0.8203 4.9963 2.73 0.0063

PARENT*YEARSMAR*EDUC -2.2856 3.1255 -8.4114 3.8402 -0.73 0.4646

PARE*YEAR*YEAR*AGEBA -0.0015 0.0005 -0.0025 -0.0004 -2.74 0.0062


PARE*YEARS*YEAR*EDUC 0.0011 0.0016 -0.0020 0.0042 0.71 0.4757

PARE*YEAR*AGEBA*EDUC 0.0007 0.0012 -0.0017 0.0031 0.59 0.5526





Intercept 2.5021 0.2797 1.9540 3.0503 8.95 <.0001

PARENT*YEARSMAR90 -112.543 52.3981 -215.241 -9.8443 -2.15 0.0317





PARE*YEARS*YEAR*EDUC 0.0011 0.0015 -0.0018 0.0040 0.76 0.4499

PARE*YEAR*AGEBA*EDUC 0.0007 0.0018 -0.0029 0.0043 0.40 0.6925

Scale 1.0000 . . . . .


CC - GEE: Reduced Model: Type=UN

Model Information



Link Function Logit







Class Levels Values

FAMNR 122 95 96 116 129 281 299 324 331 349 364 380 402 405

415 421 423 438 547 627 1076 1087 1151 1181 1182

1253 1314 1329 1332 1339 1375 1382 1422 1476 1523

1557 1566 1571 1606 1611 1614 1617 1637 1640 1680

1700 1761 1767 1783 1939 1950 1962 2008 2023 2079

...

YEARCLS 3 1990 1995 2000

PARENTCLS 2 0 1

Response Profile


Ordered Total


1 1 679

2 0 53



Parameter Effect

Prm1 Intercept

Prm2 PARENT*YEARSMAR90

Prm3 PARENT*YEARSMAR*YEAR

Prm4 PARENT*YEARSM*AGEBAS

Prm5 PARENT*YEARSMAR*EDUC

Prm6 PARE*YEAR*YEAR*AGEBA



Deviance 726 374.1287 0.5153



Scaled Pearson X2 726 722.5451 0.9952






Intercept 1 2.5189 0.1980 2.1309 2.9070 161.88 <.0001

PARENT*YEARSMAR90 1 -118.313 58.4154 -232.805 -3.8211 4.10 0.0428

PARENT*YEARSMAR*YEAR 1 0.0593 0.0293 0.0019 0.1167 4.10 0.0428

PARENT*YEARSM*AGEBAS 1 2.8050 1.4207 0.0205 5.5895 3.90 0.0483

PARENT*YEARSMAR*EDUC 1 -0.0106 0.0068 -0.0238 0.0027 2.44 0.1183

PARE*YEAR*YEAR*AGEBA 1 -0.0014 0.0007 -0.0028 -0.0000 3.90 0.0484

Scale 0 1.0000 0.0000 1.0000 1.0000





Within-Subject Effect YEARCLS(PARENTCLS) (6 levels)









Row1 1.0000 0.4364 0.5257 0.1450 0.1390 0.0496

Row2 0.4364 1.0000 0.5740 0.1604 0.2665 0.2274

Row3 0.5257 0.5740 1.0000 0.2991 0.2201 0.3330

Row4 0.1450 0.1604 0.2991 1.0000 0.2805 0.2140

Row5 0.1390 0.2665 0.2201 0.2805 1.0000 0.4443

Row6 0.0496 0.2274 0.3330 0.2140 0.4443 1.0000





Intercept 2.5045 0.2734 1.9687 3.0404 9.16 <.0001

PARENT*YEARSMAR90 -123.410 41.0496 -203.866 -42.9542 -3.01 0.0026









Intercept 2.5045 0.2796 1.9565 3.0526 8.96 <.0001

PARENT*YEARSMAR90 -123.410 50.4685 -222.326 -24.4934 -2.45 0.0145





Scale 1.0000 . . . . .



Part 3

1. Nested random effects - PQL under REML:

Model Information

Data Set WORK.DAT

Response Variable bsat


Link Function Logit


Variance Matrix Blocked By FAMNR





Response Profile

Ordered Total

Value bsat Frequency

1 0 209

2 1 1893

The GLIMMIX procedure is modeling the probability that bsat=’1’.

Dimensions


Columns in X 14







Lower Boundaries 2

Upper Boundaries 0


Starting From Data


Estimated G matrix is not positive definite.

Fit Statistics






Standard


Intercept FAMNR 1.6640 0.2396

Intercept PARENT(FAMNR) 6.34E-18 .


Standard

Effect PARENT YEAR Estimate Error DF t Value Pr > |t|

Intercept 3.0527 0.8803 558 3.47 0.0006

YEAR 1990 -0.6198 0.4397 964 -1.41 0.1590

YEAR 1995 0.3469 0.4058 964 0.85 0.3928

YEAR 2000 0 . . . .

PARENT 0 0.1369 0.4310 558 0.32 0.7509

PARENT 1 0 . . . .

PARENT*YEAR 0 1990 -0.03129 0.4723 964 -0.07 0.9472

PARENT*YEAR 0 1995 -0.2938 0.5409 964 -0.54 0.5871

PARENT*YEAR 0 2000 0 . . . .

PARENT*YEAR 1 1990 0 . . . .

PARENT*YEAR 1 1995 0 . . . .

PARENT*YEAR 1 2000 0 . . . .

CHILD 0.08389 0.09584 964 0.88 0.3816

yearsmarr -0.02938 0.02956 964 -0.99 0.3205


Num Den


YEAR 2 964 5.91 0.0028

PARENT 1 558 0.02 0.8818

PARENT*YEAR 2 964 0.26 0.7695

CHILD 1 964 0.77 0.3816

yearsmarr 1 964 0.99 0.3205

2. Nested random effects - MQL under REML:

Model Information

Data Set WORK.DAT



Link Function Logit








Response Profile

Ordered Total


1 0 209

2 1 1893


Dimensions


Columns in X 14







Lower Boundaries 2

Upper Boundaries 0


Starting From Data


Estimated G matrix is not positive definite.

Fit Statistics





Standard



Intercept PARENT(FAMNR) 2.36E-18 .


Standard


Intercept 2.8528 0.8596 558 3.32 0.0010

YEAR 1990 -0.5461 0.4179 964 -1.31 0.1916

YEAR 1995 0.3003 0.3751 964 0.80 0.4236


YEAR 2000 0 . . . .

PARENT 0 0.1212 0.3880 558 0.31 0.7549

PARENT 1 0 . . . .

PARENT*YEAR 0 1990 -0.02857 0.4288 964 -0.07 0.9469

PARENT*YEAR 0 1995 -0.2657 0.4914 964 -0.54 0.5889

PARENT*YEAR 0 2000 0 . . . .

PARENT*YEAR 1 1990 0 . . . .

PARENT*YEAR 1 1995 0 . . . .

PARENT*YEAR 1 2000 0 . . . .

CHILD 0.06638 0.09408 964 0.71 0.4806

yearsmarr -0.03084 0.02935 964 -1.05 0.2936


Num Den


YEAR 2 964 4.99 0.0070

PARENT 1 558 0.02 0.8950

PARENT*YEAR 2 964 0.25 0.7786

CHILD 1 964 0.50 0.4806

yearsmarr 1 964 1.10 0.2936

3. Nested random effects - Adaptive Gauss Hermite:

Specifications

Data Set WORK.DAT

Dependent Variable bsat


Random Effects g1 g2


Subject Variable FAMNR



Quadrature

Dimensions




Subjects 573


Parameters 10


Parameters

b0 b1 b2 b3 b4 b5 b6 b7 V1 V2

2.8528 -0.5461 0.3003 0.1212 -0.02857 -0.2657 0.06638 -0.03084 2.3957 1

NegLogLike


668.813909

Iteration History


1 4 623.175172 45.63874 347.5408 -36275.4

2 5 618.372171 4.803001 24.9271 -7.84723

3 7 614.795292 3.576879 321.8137 -2.30974

4 9 605.846345 8.948948 22.89988 -3.58245

5 11 603.694606 2.151738 89.56562 -1.464

6 12 602.628206 1.066401 32.77747 -0.83238

7 13 601.97919 0.649016 16.91893 -1.43408

8 14 601.665098 0.314092 33.32883 -0.72667

9 15 601.208274 0.456824 9.661278 -0.61331

10 16 600.792377 0.415896 9.710633 -0.26597

11 18 600.615128 0.17725 1.659908 -0.26523

12 20 600.543975 0.071153 3.595091 -0.03407

13 22 600.175591 0.368385 20.66025 -0.07906

14 24 600.05314 0.122451 16.58005 -0.13738

15 26 599.99982 0.05332 2.320657 -0.07341

16 28 599.976771 0.023049 13.13735 -0.01359

17 30 599.924682 0.052088 13.35125 -0.02512

18 32 599.911162 0.01352 1.259838 -0.0205

19 34 599.908848 0.002315 0.994878 -0.00189

20 36 599.888726 0.020122 5.835654 -0.00316

21 38 599.845863 0.042863 2.997486 -0.02204

22 40 599.822665 0.023198 0.502865 -0.02128

23 42 599.81846 0.004205 0.280123 -0.00608

24 44 599.818405 0.000055 0.088641 -0.00009

25 46 599.818035 0.000371 0.179494 -0.00002

26 47 599.817587 0.000447 0.048878 -0.0004

27 49 599.817586 1.342E-6 0.002676 -2.7E-6


Fit Statistics





Parameter Estimates

Standard


b0 4.6847 1.3678 571 3.42 0.0007 0.05 1.9981 7.3712 0.000124

b1 -0.7946 0.6029 571 -1.32 0.1880 0.05 -1.9788 0.3895 0.000121

b2 0.4935 0.5058 571 0.98 0.3297 0.05 -0.5000 1.4869 -0.00008

b3 0.2228 0.5265 571 0.42 0.6722 0.05 -0.8112 1.2569 0.000025

b4 -0.06166 0.5684 571 -0.11 0.9136 0.05 -1.1780 1.0547 7.725E-6

b5 -0.4096 0.6436 571 -0.64 0.5248 0.05 -1.6738 0.8546 -0.00003


b6 0.1700 0.1487 571 1.14 0.2536 0.05 -0.1221 0.4620 0.000303

b7 -0.04985 0.04586 571 -1.09 0.2775 0.05 -0.1399 0.04023 0.002676

V1 0.9007 0.6175 571 1.46 0.1452 0.05 -0.3122 2.1135 -0.00006

V2 5.0277 1.1089 571 4.53 <.0001 0.05 2.8498 7.2057 -0.00002

4. Test for the parent RE:

Specifications

Data Set WORK.DAT



Random Effects g1





Quadrature

Dimensions




Subjects 573


Parameters 9


Parameters

b0 b1 b2 b3 b4 b5 b6 b7 V1 NegLogLike

2.8528 -0.5461 0.3003 0.1212 -0.02857 -0.2657 0.06638 -0.03084 2.3957 656.61496

Iteration History


1 4 607.195256 49.4197 54.36621 -20610.7

2 5 603.970472 3.224784 44.41349 -5.25338

3 7 603.145232 0.82524 19.90867 -0.9741

4 9 602.464818 0.680414 22.54237 -0.59021

5 11 602.283024 0.181794 21.38663 -0.21632

6 13 602.186664 0.09636 7.665615 -0.09384

7 15 602.124647 0.062017 17.10094 -0.04093

8 16 602.038507 0.08614 5.669192 -0.05049

9 18 602.005751 0.032756 1.03675 -0.02963

10 19 601.949704 0.056046 6.688754 -0.02942

11 20 601.881805 0.0679 3.206596 -0.0567

12 21 601.763532 0.118273 0.73193 -0.06214

13 23 601.731248 0.032283 1.728933 -0.05459

14 24 601.676946 0.054303 2.192901 -0.00995

15 26 601.665972 0.010974 0.21346 -0.02043

16 28 601.631547 0.034425 5.698429 -0.00125


17 29 601.606951 0.024595 0.439586 -0.03079

18 31 601.601369 0.005582 0.075129 -0.01109

19 33 601.601368 5.824E-7 0.000131 -1.16E-6


Fit Statistics





Parameter Estimates

Standard


b0 4.3240 1.2469 572 3.47 0.0006 0.05 1.8750 6.7730 5.133E-6

b1 -0.7547 0.5636 572 -1.34 0.1811 0.05 -1.8617 0.3523 -0.00002

b2 0.4557 0.4792 572 0.95 0.3420 0.05 -0.4855 1.3969 0.000053

b3 0.1510 0.4817 572 0.31 0.7540 0.05 -0.7952 1.0972 0.000014

b4 -0.02026 0.5307 572 -0.04 0.9696 0.05 -1.0627 1.0221 -0.00003

b5 -0.3706 0.6054 572 -0.61 0.5407 0.05 -1.5596 0.8184 0.000059

b6 0.1519 0.1371 572 1.11 0.2684 0.05 -0.1174 0.4211 -1E-5

b7 -0.04526 0.04224 572 -1.07 0.2844 0.05 -0.1282 0.03770 -0.00013

V1 2.1158 0.2082 572 10.16 <.0001 0.05 1.7069 2.5247 0.000019

Obs logL0 logL1 LRT df pval

1 -601.6 -599.8 3.6 1 0.057780

5. PQL under REML:

Model Information

Data Set WORK.DAT



Link Function Logit







Response Profile

Ordered Total



1 0 209

2 1 1893


Dimensions


Columns in X 14







Lower Boundaries 1

Upper Boundaries 0


Starting From Data


Fit Statistics





Standard




Standard


Intercept 3.0527 0.8803 572 3.47 0.0006

YEAR 1990 -0.6198 0.4397 1522 -1.41 0.1588

YEAR 1995 0.3469 0.4058 1522 0.85 0.3927

YEAR 2000 0 . . . .

PARENT 0 0.1369 0.4310 1522 0.32 0.7509

PARENT 1 0 . . . .

PARENT*YEAR 0 1990 -0.03129 0.4723 1522 -0.07 0.9472

PARENT*YEAR 0 1995 -0.2938 0.5409 1522 -0.54 0.5870

PARENT*YEAR 0 2000 0 . . . .

PARENT*YEAR 1 1990 0 . . . .

PARENT*YEAR 1 1995 0 . . . .

PARENT*YEAR 1 2000 0 . . . .


CHILD 0.08389 0.09584 1522 0.88 0.3816

yearsmarr -0.02938 0.02956 1522 -0.99 0.3205


Num Den


YEAR 2 1522 5.91 0.0028

PARENT 1 1522 0.02 0.8818

PARENT*YEAR 2 1522 0.26 0.7695

CHILD 1 1522 0.77 0.3816

yearsmarr 1 1522 0.99 0.3205

6. PQL under ML:

Model Information

Data Set WORK.DAT



Link Function Logit



Estimation Technique PL




Response Profile

Ordered Total


1 0 209

2 1 1893


Dimensions


Columns in X 14







Lower Boundaries 1


Upper Boundaries 0


Starting From Data


Fit Statistics





Standard




Standard


Intercept 3.0495 0.8773 572 3.48 0.0005

YEAR 1990 -0.6196 0.4387 1522 -1.41 0.1580

YEAR 1995 0.3456 0.4052 1522 0.85 0.3937

YEAR 2000 0 . . . .

PARENT 0 0.1369 0.4306 1522 0.32 0.7506

PARENT 1 0 . . . .

PARENT*YEAR 0 1990 -0.03138 0.4718 1522 -0.07 0.9470

PARENT*YEAR 0 1995 -0.2932 0.5403 1522 -0.54 0.5874

PARENT*YEAR 0 2000 0 . . . .

PARENT*YEAR 1 1990 0 . . . .

PARENT*YEAR 1 1995 0 . . . .

PARENT*YEAR 1 2000 0 . . . .

CHILD 0.08341 0.09547 1522 0.87 0.3824

yearsmarr -0.02932 0.02946 1522 -1.00 0.3196


Num Den


YEAR 2 1522 5.92 0.0028

PARENT 1 1522 0.02 0.8809

PARENT*YEAR 2 1522 0.26 0.7701

CHILD 1 1522 0.76 0.3824

yearsmarr 1 1522 0.99 0.3196

7. MQL under REML:

Model Information

Data Set WORK.DAT




Link Function Logit







Response Profile

Ordered Total


1 0 209

2 1 1893


Dimensions


Columns in X 14







Lower Boundaries 1

Upper Boundaries 0


Starting From Data


Fit Statistics





Standard





Standard


Intercept 2.8528 0.8596 572 3.32 0.0010

YEAR 1990 -0.5461 0.4179 1522 -1.31 0.1915

YEAR 1995 0.3003 0.3751 1522 0.80 0.4235

YEAR 2000 0 . . . .

PARENT 0 0.1212 0.3880 1522 0.31 0.7549

PARENT 1 0 . . . .

PARENT*YEAR 0 1990 -0.02857 0.4288 1522 -0.07 0.9469

PARENT*YEAR 0 1995 -0.2657 0.4914 1522 -0.54 0.5888

PARENT*YEAR 0 2000 0 . . . .

PARENT*YEAR 1 1990 0 . . . .

PARENT*YEAR 1 1995 0 . . . .

PARENT*YEAR 1 2000 0 . . . .

CHILD 0.06638 0.09408 1522 0.71 0.4806

yearsmarr -0.03084 0.02935 1522 -1.05 0.2935


Num Den


YEAR 2 1522 4.99 0.0069

PARENT 1 1522 0.02 0.8950

PARENT*YEAR 2 1522 0.25 0.7785

CHILD 1 1522 0.50 0.4806

yearsmarr 1 1522 1.10 0.2935

8. MQL under ML:

Model Information

Data Set WORK.DAT



Link Function Logit



Estimation Technique MPL




Response Profile

Ordered Total



1 0 209

2 1 1893


Dimensions


Columns in X 14







Lower Boundaries 1

Upper Boundaries 0


Starting From Data


Fit Statistics





Standard




Standard

Effect PARE YEAR Estimate Error DF t Value Pr > |t|

Intercept 2.8528 0.8575 572 3.33 0.0009

YEAR 1990 -0.5465 0.4174 1522 -1.31 0.1906

YEAR 1995 0.3001 0.3750 1522 0.80 0.4236

YEAR 2000 0 . . . .

PARENT 0 0.1213 0.3881 1522 0.31 0.7546

PARENT 1 0 . . . .

PARENT*YEAR 0 1990 -0.02867 0.4289 1522 -0.07 0.9467

PARENT*YEAR 0 1995 -0.2656 0.4915 1522 -0.54 0.5890

PARENT*YEAR 0 2000 0 . . . .


PARENT*YEAR 1 1990 0 . . . .

PARENT*YEAR 1 1995 0 . . . .

PARENT*YEAR 1 2000 0 . . . .

CHILD 0.06617 0.09384 1522 0.71 0.4808

yearsmarr -0.03079 0.02927 1522 -1.05 0.2930


Num Den


YEAR 2 1522 5.00 0.0069

PARENT 1 1522 0.02 0.8942

PARENT*YEAR 2 1522 0.25 0.7788

CHILD 1 1522 0.50 0.4808

yearsmarr 1 1522 1.11 0.2930

9. Gauss-Hermite quadrature method:Quadrature points 20

Specifications

Data Set WORK.DAT



Random Effects g1





Dimensions




Subjects 573


Parameters 9


Parameters


2.8528 -0.5465 0.3001 0.1213 -0.02867 -0.2656 0.06617 -0.03079 2.3662 655.301943

Iteration History


1 4 606.617711 48.68423 53.29391 -20839.3

2 5 603.642061 2.975649 36.4491 -4.99921

3 7 602.894991 0.747071 13.5725 -0.74237


4 9 602.269373 0.625618 12.21121 -0.47682

5 11 602.090493 0.17888 29.34825 -0.12393

6 13 602.042601 0.047892 11.49444 -0.05899

7 15 601.935339 0.107262 5.266845 -0.0396

8 17 601.883304 0.052035 9.171713 -0.03022

9 18 601.845179 0.038125 1.341823 -0.03418

10 19 601.789902 0.055277 2.722201 -0.04462

11 21 601.642531 0.147371 1.384209 -0.05951

12 23 601.599932 0.042599 1.510803 -0.05877

13 24 601.539909 0.060023 0.47715 -0.01021

14 26 601.530733 0.009176 0.24404 -0.01563

15 27 601.521776 0.008957 0.361106 -0.00141

16 29 601.481188 0.040588 0.585063 -0.01702

17 31 601.463088 0.0181 0.038528 -0.02317

18 33 601.463083 4.828E-6 0.006037 -0.00001

19 35 601.463083 2.319E-8 0.000212 -4.63E-8


Fit Statistics





Parameter Estimates

Standard


b0 4.3539 1.2535 572 3.47 0.0006 0.05 1.8920 6.8159 0.00002

b1 -0.7591 0.5643 572 -1.35 0.1791 0.05 -1.8674 0.3492 0.000031

b2 0.4569 0.4796 572 0.95 0.3412 0.05 -0.4852 1.3989 -7.42E-6

b3 0.1508 0.4824 572 0.31 0.7548 0.05 -0.7968 1.0983 4.646E-6

b4 -0.01974 0.5316 572 -0.04 0.9704 0.05 -1.0638 1.0243 0.000014

b5 -0.3720 0.6064 572 -0.61 0.5398 0.05 -1.5630 0.8189 -3.85E-6

b6 0.1530 0.1364 572 1.12 0.2625 0.05 -0.1149 0.4209 0.000035

b7 -0.04562 0.04241 572 -1.08 0.2825 0.05 -0.1289 0.03768 0.000212

V1 2.1419 0.2178 572 9.84 <.0001 0.05 1.7142 2.5697 -9.55E-6

10. Adaptive Gauss-Hermite quadrature method:Quadrature points 20

Specifications

Data Set WORK.DAT



Random Effects g1






Quadrature

Dimensions




Subjects 573


Parameters 9


Parameters


2.8528 -0.5465 0.3001 0.1213 -0.02867 -0.2656 0.06617 -0.03079 2.3662 655.478125

Iteration History


1 4 606.687032 48.79109 59.31878 -20769.3

2 5 603.687591 2.99944 37.88727 -4.97779

3 7 602.974451 0.71314 20.75368 -0.77795

4 9 602.339803 0.634648 20.77356 -0.48552

5 11 602.170863 0.16894 22.67167 -0.16091

6 13 602.090426 0.080437 2.310172 -0.08521

7 14 601.991847 0.098579 18.31438 -0.02939

8 16 601.948716 0.043131 12.91512 -0.04565

9 17 601.891331 0.057385 3.359225 -0.0295

10 18 601.791255 0.100076 7.200366 -0.02416

11 19 601.680281 0.110974 9.308333 -0.13945

12 21 601.654022 0.026259 2.466996 -0.03148

13 22 601.610985 0.043037 0.596406 -0.01201

14 24 601.588483 0.022502 0.609329 -0.03361

15 26 601.585733 0.00275 0.747981 -0.00243

16 29 601.524155 0.061578 4.385252 -0.0024

17 31 601.521304 0.002852 0.051081 -0.00561

18 33 601.5213 3.258E-6 0.003112 -6.43E-6

19 35 601.5213 1.78E-9 0.000086 -3.64E-9


Fit Statistics





Parameter Estimates

Standard



b0 4.3383 1.2543 572 3.46 0.0006 0.05 1.8747 6.8018 -2.31E-6

b1 -0.7549 0.5657 572 -1.33 0.1826 0.05 -1.8661 0.3562 -1.59E-6

b2 0.4583 0.4805 572 0.95 0.3405 0.05 -0.4853 1.4020 1.889E-6

b3 0.1511 0.4824 572 0.31 0.7543 0.05 -0.7964 1.0985 -3.57E-6

b4 -0.02017 0.5315 572 -0.04 0.9697 0.05 -1.0640 1.0237 -3.31E-6

b5 -0.3720 0.6063 572 -0.61 0.5398 0.05 -1.5629 0.8189 1.336E-6

b6 0.1531 0.1381 572 1.11 0.2680 0.05 -0.1181 0.4242 -0.00002

b7 -0.04538 0.04247 572 -1.07 0.2858 0.05 -0.1288 0.03804 -0.00009

V1 2.1366 0.2189 572 9.76 <.0001 0.05 1.7066 2.5665 -4.03E-6


Standard


V1^2 4.5649 0.9354 572 4.88 <.0001 0.05 2.7276 6.4021

11. Laplace approximation:

Specifications

Data Set WORK.DAT



Random Effects g1





Quadrature

Dimensions




Subjects 573


Parameters 9

Quadrature Points 1

Parameters


2.8528 -0.5465 0.3001 0.1213 -0.02867 -0.2656 0.06617 -0.03079 2.3662 657.172467

Iteration History



1 4 595.845906 61.32656 68.97154 -26737.2

2 5 592.559211 3.286695 4.377269 -5.83548

3 7 591.491572 1.067638 85.99581 -0.61735

4 9 589.77634 1.715232 10.01569 -0.98064

5 11 589.038678 0.737662 10.38077 -0.3982

6 13 588.806169 0.232509 7.468499 -0.1661

7 15 588.305644 0.500526 12.86172 -0.14436

8 17 588.09221 0.213434 2.394789 -0.15723

9 19 588.001329 0.090881 2.080519 -0.08249

10 21 587.949209 0.052119 5.456902 -0.02362

11 23 587.783881 0.165328 15.11569 -0.04455

12 25 587.700576 0.083305 10.9808 -0.0695

13 27 587.654848 0.045728 13.06041 -0.02491

14 29 587.487859 0.166989 12.27121 -0.03259

15 30 587.297756 0.190103 8.331522 -0.13393

16 32 587.2794 0.018356 0.325252 -0.03008

17 34 587.271866 0.007534 4.250006 -0.00337

18 36 587.160245 0.111622 4.205993 -0.01103

19 38 587.116511 0.043733 0.250661 -0.06816

20 40 587.115935 0.000576 0.017436 -0.00105

21 42 587.115934 1.021E-6 0.002356 -2E-6


Fit Statistics





Parameter Estimates

Standard


b0 6.0577 1.7343 572 3.49 0.0005 0.05 2.6514 9.4641 0.000116

b1 -0.7566 0.6993 572 -1.08 0.2798 0.05 -2.1302 0.6170 0.00005

b2 0.6263 0.5567 572 1.13 0.2610 0.05 -0.4671 1.7197 0.000159

b3 0.1576 0.5220 572 0.30 0.7628 0.05 -0.8676 1.1829 0.000195

b4 -0.01072 0.5792 572 -0.02 0.9852 0.05 -1.1483 1.1269 0.000082

b5 -0.4580 0.6614 572 -0.69 0.4889 0.05 -1.7572 0.8412 0.000107

b6 0.2188 0.1843 572 1.19 0.2356 0.05 -0.1432 0.5809 0.000238

b7 -0.04728 0.05652 572 -0.84 0.4033 0.05 -0.1583 0.06375 0.002356

V1 4.0665 0.8547 572 4.76 <.0001 0.05 2.3878 5.7452 -0.00011



Standard


V1^2 16.5364 6.9512 572 2.38 0.0177 0.05 2.8834 30.1894

12. Fitted marginal evolutions and evolutions for the median parent:

Figure 6.1: Fitted marginal average evolution per parent

Figure 6.2: Fitted evolutions for the median parent

13. Test for the parent effect:

Specifications


Data Set WORK.DAT



Random Effects g1





Quadrature

Dimensions




Subjects 573


Parameters 6


Parameters

b0 b1 b2 b3 b4 V1 NegLogLike

2.8528 -0.5465 0.3001 0.1213 -0.02867 2.3662 644.510467

Iteration History


1 4 606.325514 38.18495 45.12185 -15955.8

2 5 603.822625 2.502889 33.51163 -3.92549

3 6 603.329753 0.492872 9.345013 -0.73621

4 8 602.834818 0.494935 17.2674 -0.46348

5 10 602.54832 0.286498 1.79628 -0.31385

6 12 602.497864 0.050457 6.899228 -0.02872

7 13 602.416047 0.081816 1.318331 -0.05246

8 15 602.394229 0.021818 3.097594 -0.02785

9 17 602.254877 0.139352 2.300388 -0.01376

10 18 602.090257 0.16462 2.49476 -0.08792

11 19 601.936675 0.153583 5.918115 -0.12149

12 21 601.923015 0.013659 0.745047 -0.02845

13 23 601.922945 0.000071 0.036981 -0.00014

14 25 601.922944 2.162E-7 0.005385 -4.5E-7


Fit Statistics






Parameter Estimates

Standard


b0 4.4049 1.2345 572 3.57 0.0004 0.05 1.9802 6.8295 -0.00023

b1 -0.7615 0.5108 572 -1.49 0.1366 0.05 -1.7648 0.2419 -0.00014

b2 0.2789 0.3846 572 0.73 0.4686 0.05 -0.4765 1.0343 -0.00009

b3 0.1534 0.1380 572 1.11 0.2666 0.05 -0.1175 0.4244 -0.00036

b4 -0.04539 0.04242 572 -1.07 0.2852 0.05 -0.1287 0.03794 -0.00538

V1 2.1334 0.2185 572 9.76 <.0001 0.05 1.7041 2.5626 0.000179

Obs logL0 logL1 LRT df pval

1 -601.9 -601.5 0.8 3 0.84947

Part 4

1. Missing Data Patterns


Group sat1 sat2 sat3 Freq Percent

1 X X X 268 23.39

2 X X . 392 34.21

3 X . X 41 3.58

4 X . . 428 37.35

5 . X X 1 0.09

6 . X . 2 0.17

7 O O O 14 1.22

2. Continuous Response: Procedure MIXED: CC

Model Information

Data Set WORK.MSDCC

Dependent Variable SAT

Covariance Structure Unstructured @ Compound

Symmetry

Subject Effect FAMNR

Estimation Method ML

Residual Variance Method None

Fixed Effects SE Method Model-Based



Class Levels Values

FAMNR 122 95 96 116 129 281 299 324 ...

IDNR 244 23 24 25 26 39 40 47 48 77 ...

YEAR 3 1990 1995 2000


STATUS 2 1 2

PARENT 2 0 1

Dimensions

Covariance Parameters 4

Columns in X 11

Columns in Z 0

Subjects 122


Number of Observations



Number of Observations Not Used 0

Iteration History


0 1 1920.18682894

1 2 1650.94511948 0.00935836

2 1 1649.41391865 0.00014849

3 1 1649.39099699 0.00000009

4 1 1649.39098381 0.00000000


Estimated R Matrix for FAMNR 95


1 0.7011 0.2696 0.3620 0.1392 0.3620 0.1392

2 0.2696 0.8309 0.1392 0.4290 0.1392 0.4290

3 0.3620 0.1392 0.7011 0.2696 0.3620 0.1392

4 0.1392 0.4290 0.2696 0.8309 0.1392 0.4290

5 0.3620 0.1392 0.3620 0.1392 0.7011 0.2696

6 0.1392 0.4290 0.1392 0.4290 0.2696 0.8309

Estimated R Correlation Matrix for FAMNR 95


1 1.0000 0.3532 0.5164 0.1824 0.5164 0.1824

2 0.3532 1.0000 0.1824 0.5164 0.1824 0.5164

3 0.5164 0.1824 1.0000 0.3532 0.5164 0.1824

4 0.1824 0.5164 0.3532 1.0000 0.1824 0.5164

5 0.5164 0.1824 0.5164 0.1824 1.0000 0.3532

6 0.1824 0.5164 0.1824 0.5164 0.3532 1.0000




UN(2,1) FAMNR 0.2696

UN(2,2) FAMNR 0.8309


Fit Statistics







3 270.80 <.0001



Standard


YEAR 1990 5.2588 0.08253 242 63.72 <.0001

YEAR 1995 5.9184 0.08253 242 71.72 <.0001

YEAR 2000 5.7037 0.08253 242 69.11 <.0001

PARENT 0 0.04313 0.09021 121 0.48 0.6334

PARENT 1 0 . . . .

YEAR*PARENT 1990 0 -0.07533 0.08872 242 -0.85 0.3967

YEAR*PARENT 1990 1 0 . . . .

YEAR*PARENT 1995 0 0.06206 0.08872 242 0.70 0.4849

YEAR*PARENT 1995 1 0 . . . .

YEAR*PARENT 2000 0 0 . . . .

YEAR*PARENT 2000 1 0 . . . .


Num Den


YEAR 2 242 66.89 <.0001

PARENT 1 121 0.27 0.6031

YEAR*PARENT 2 242 1.20 0.3021

3. Continuous Response: Procedure MIXED: Direct Likelihood

Model Information

Data Set WORK.MSDIC



Symmetry







Class Levels Values

FAMNR 573 15 47 54 60 62 64 72 90 91 ...

IDNR 1146 1 2 3 4 5 6 7 8 9 10 11 12 ...

YEAR 3 1990 1995 2000

STATUS 2 1 2

PARENT 2 0 1

Dimensions


Columns in X 11

Columns in Z 0

Subjects 573






Iteration History


0 1 5748.17979945

1 2 5127.95955233 0.00038577


2 1 5127.71116168 0.00000032

3 1 5127.71095688 0.00000000


Estimated R Matrix

for FAMNR 15

Row Col1 Col2

1 0.8061 0.3822

2 0.3822 0.9431

Estimated R Correlation

Matrix for FAMNR 15

Row Col1 Col2

1 1.0000 0.4383

2 0.4383 1.0000




UN(2,1) FAMNR 0.3822

UN(2,2) FAMNR 0.9431


Fit Statistics







3 620.47 <.0001


Standard


YEAR 1990 5.1867 0.04071 511 127.42 <.0001

YEAR 1995 5.9319 0.04945 511 119.96 <.0001

YEAR 2000 5.6027 0.06653 511 84.21 <.0001

PARENT 0 0.06192 0.07052 558 0.88 0.3803

PARENT 1 0 . . . .

YEAR*PARENT 1990 0 -0.06323 0.07013 455 -0.90 0.3677

YEAR*PARENT 1990 1 0 . . . .

YEAR*PARENT 1995 0 -0.01516 0.07339 455 -0.21 0.8365

YEAR*PARENT 1995 1 0 . . . .

YEAR*PARENT 2000 0 0 . . . .

YEAR*PARENT 2000 1 0 . . . .


Num Den


YEAR 2 511 186.80 <.0001

PARENT 1 558 0.76 0.3852

YEAR*PARENT 2 455 0.65 0.5206

4. Continuous Response: Procedure MIXED: Multiple Imputation

------------------------------------- Imputation Number=1 --------------------------------------


Model Information

Data Set WORK.MSDICCOMP



Symmetry







Class Levels Values

FAMNR 573 15 47 54 60 62 64 72 90 91 ...

YEAR 3 1990 1995 2000

PARENT 2 0 1

Dimensions


Columns in X 6

Columns in Z 0

Subjects 573






Iteration History


0 1 9683.98628048

1 2 9360.46272795 0.00004508

2 1 9360.39341474 0.00000001

3 1 9360.39339499 0.00000000


Estimated R Correlation Matrix for FAMNR 15


1 1.0000 0.2731 0.2906 0.07937 0.1150 0.03140

2 0.2731 1.0000 0.07937 0.2906 0.03140 0.1150

3 0.2906 0.07937 1.0000 0.2731 0.1045 0.02853

4 0.07937 0.2906 0.2731 1.0000 0.02853 0.1045

5 0.1150 0.03140 0.1045 0.02853 1.0000 0.2731

6 0.03140 0.1150 0.02853 0.1045 0.2731 1.0000


Standard Z

Cov Parm Subject Estimate Error Value Pr Z

YEAR UN(1,1) FAMNR 0.7375 0.03084 23.92 <.0001

UN(2,1) FAMNR 0.2621 0.02780 9.43 <.0001

UN(2,2) FAMNR 1.1031 0.04650 23.72 <.0001

UN(3,1) FAMNR 0.1030 0.02663 3.87 0.0001

UN(3,2) FAMNR 0.1144 0.03255 3.52 0.0004

UN(3,3) FAMNR 1.0876 0.04681 23.24 <.0001

PARENT Corr FAMNR 0.2731 0.02268 12.04 <.0001

Asymptotic Covariance Matrix of Estimates


Row Cov Parm CovP1 CovP2 CovP3 CovP4 CovP5 CovP6 CovP7

1 YEAR UN(1,1) 0.000951 0.000335 0.000128 0.000132 0.000046 0.000033 0.000030

2 UN(2,1) 0.000335 0.000773 0.000494 0.000098 0.000127 1.964E-6 -0.00004

3 UN(2,2) 0.000128 0.000494 0.002162 0.000050 0.000214 0.000093 0.000141

4 UN(3,1) 0.000132 0.000098 0.000050 0.000709 0.000260 0.000191 -9.98E-6

5 UN(3,2) 0.000046 0.000127 0.000214 0.000260 0.001059 0.000206 -0.00002

6 UN(3,3) 0.000033 1.964E-6 0.000093 0.000191 0.000206 0.002191 0.000255

7 PARENT Corr 0.000030 -0.00004 0.000141 -9.98E-6 -0.00002 0.000255 0.000514

Fit Statistics







6 323.59 <.0001


Standard


year1990 5.1915 0.03588 2859 144.70 <.0001

year1995 5.7965 0.04388 2859 132.11 <.0001

year2000 5.9290 0.04357 2859 136.09 <.0001

father -0.03170 0.05253 2859 -0.60 0.5462

father1990 0.02522 0.06410 2859 0.39 0.6940

father1995 0.1374 0.07055 2859 1.95 0.0516

Covariance Matrix for Fixed Effects

Row Effect Col1 Col2 Col3 Col4 Col5 Col6

1 year1990 0.001287 0.000457 0.000180 -0.00013 -0.00080 -0.00020

2 year1995 0.000457 0.001925 0.000200 -0.00015 -0.00019 -0.00125

3 year2000 0.000180 0.000200 0.001898 -0.00138 0.001249 0.001235

4 father -0.00013 -0.00015 -0.00138 0.002760 -0.00250 -0.00247

5 father1990 -0.00080 -0.00019 0.001249 -0.00250 0.004108 0.002873

6 father1995 -0.00020 -0.00125 0.001235 -0.00247 0.002873 0.004978


Num Den


year1990 1 2859 20939.5 <.0001

year1995 1 2859 17452.9 <.0001

year2000 1 2859 18520.0 <.0001

father 1 2859 0.36 0.5462

father1990 1 2859 0.15 0.6940

father1995 1 2859 3.79 0.0516

...

COMBINING 5 MIXED MODEL ANALYSES: The MIANALYZE Procedure

Model Information

PARMS Data Set WORK.SOLUTION0

COVB Data Set WORK.COVB0



-----------------Variance-----------------



year1990 0.000016438 0.001298 0.001318 17846

year1995 0.000899 0.001851 0.002930 29.48

year2000 0.001198 0.001805 0.003242 20.352

father 0.004480 0.002638 0.008014 8.8887

father1990 0.003539 0.003991 0.008238 15.052

father1995 0.005888 0.004726 0.011792 11.142


Relative Fraction



year1990 0.015199 0.015082 0.996993

year1995 0.583167 0.407249 0.924685

year2000 0.796381 0.491001 0.910581

father 2.037909 0.726202 0.873179

father1990 1.064006 0.569183 0.897798

father1995 1.494889 0.655867 0.884038



year1990 5.188727 0.036298 5.11758 5.259875 17846

year1995 5.820415 0.054129 5.70979 5.931044 29.48

year2000 5.899974 0.056936 5.78134 6.018608 20.352

father 0.037407 0.089520 -0.16549 0.240302 8.8887

father1990 -0.033850 0.090766 -0.22725 0.159554 15.052

father1995 0.014804 0.108590 -0.22383 0.253440 11.142

year1990 5.11758 5.259875 17846

year1995 5.70979 5.931044 29.48

year2000 5.78134 6.018608 20.352

father -0.16549 0.240302 8.8887

father1990 -0.22725 0.159554 15.052

father1995 -0.22383 0.253440 11.142


t for H0:


year1990 5.182633 5.193179 0 142.95 <.0001

year1995 5.796500 5.863764 0 107.53 <.0001

year2000 5.847270 5.933238 0 103.63 <.0001

father -0.031703 0.110921 0 0.42 0.6860

father1990 -0.096769 0.025219 0 -0.37 0.7144

father1995 -0.059910 0.137388 0 0.14 0.8940

Model Information

PARMS Data Set WORK.COVPARMS0

COVB Data Set WORK.ASYCOV0



-----------------Variance-----------------


YEARUN11 0.000062945 0.000966 0.001042 761.21

YEARUN21 0.000138 0.000754 0.000920 123.71

YEARUN22 0.002220 0.002002 0.004666 12.276

YEARUN31 0.000107 0.000681 0.000810 158.78

YEARUN32 0.000197 0.000971 0.001207 104.61

YEARUN33 0.002091 0.001978 0.004487 12.797


PARENTCO 0.000148 0.000519 0.000697 61.598


Relative Fraction



YEARUN11 0.078155 0.074917 0.985238

YEARUN21 0.219243 0.192765 0.962878

YEARUN22 1.330063 0.627017 0.888570

YEARUN31 0.188663 0.169119 0.967283

YEARUN32 0.243076 0.210496 0.959602

YEARUN33 1.268001 0.614906 0.890487

PARENTCO 0.341971 0.277898 0.947347



YEARUN11 0.743651 0.032280 0.680282 0.807019 761.21

YEARUN21 0.266324 0.030328 0.206296 0.326353 123.71

YEARUN22 1.060462 0.068308 0.912003 1.208921 12.276

YEARUN31 0.106555 0.028458 0.050349 0.162760 158.78

YEARUN32 0.121139 0.034740 0.052252 0.190025 104.61

YEARUN33 1.034015 0.066986 0.889066 1.178965 12.797

PARENTCO 0.269105 0.026393 0.216340 0.321870 61.598


t for H0:


YEARUN11 0.736704 0.755575 0 23.04 <.0001

YEARUN21 0.258364 0.287125 0 8.78 <.0001

YEARUN22 1.016246 1.117173 0 15.52 <.0001

YEARUN31 0.091112 0.118612 0 3.74 0.0003

YEARUN32 0.108231 0.144682 0 3.49 0.0007

YEARUN33 0.973560 1.087629 0 15.44 <.0001

PARENTCO 0.254140 0.280608 0 10.20 <.0001

5. Binary Response: Procedure GEE: CC

Model Information

Data Set WORK.MSDCC


Link Function Logit

Dependent Variable BSAT BSAT






Class Levels Values

FAMNR 122 95 96 116 ...

YEAR 3 1990 1995 2000

PARENT 2 0 1

Response Profile

Ordered Total


1 1 679

2 0 53




Parameter Effect YEAR PARENT

Prm1 Intercept

Prm2 PARENT 0

Prm3 PARENT 1

Prm4 YEAR 1990

Prm5 YEAR 1995

Prm6 YEAR 2000

Prm7 BIRTH

Prm8 YEAR*PARENT 1990 0






Prm14 BIRTH*PARENT 0


Prm16 BIRTH*YEAR 1990



Prm19 BIRTH*YEAR*PARENT 1990 0








Deviance 720 372.2591 0.5170



Scaled Pearson X2 720 717.8982 0.9971






Intercept 1 -3.7378 4.5822 -12.7188 5.2432 0.67 0.4147

PARENT 0 1 8.5801 5.8816 -2.9475 20.1078 2.13 0.1446

PARENT 1 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR 1990 1 9.4072 6.0691 -2.4881 21.3025 2.40 0.1211

YEAR 1995 1 10.8278 7.1164 -3.1201 24.7756 2.32 0.1281

YEAR 2000 0 0.0000 0.0000 0.0000 0.0000 . .

BIRTH 1 0.1305 0.0948 -0.0552 0.3163 1.90 0.1684

YEAR*PARENT 1990 0 1 -10.8144 7.9104 -26.3186 4.6897 1.87 0.1716

YEAR*PARENT 1990 1 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 1995 0 1 -14.8140 9.1629 -32.7728 3.1449 2.61 0.1059

YEAR*PARENT 1995 1 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 0 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 1 0 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*PARENT 0 1 -0.1814 0.1217 -0.4199 0.0571 2.22 0.1361

BIRTH*PARENT 1 0 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*YEAR 1990 1 -0.1993 0.1230 -0.4403 0.0417 2.63 0.1051


BIRTH*YEAR 1995 1 -0.2125 0.1426 -0.4920 0.0671 2.22 0.1363

BIRTH*YEAR 2000 0 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*YEAR*PARENT 1990 0 1 0.2265 0.1620 -0.0910 0.5440 1.95 0.1621

BIRTH*YEAR*PARENT 1990 1 0 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*YEAR*PARENT 1995 0 1 0.3082 0.1876 -0.0595 0.6759 2.70 0.1004




Scale 0 1.0000 0.0000 1.0000 1.0000




Within-Subject Effect YEAR(PARENT) (6 levels)









Row1 1.0000 0.5610 0.4617 0.1900 0.2071 0.0649

Row2 0.5610 1.0000 0.7391 0.1685 0.3022 0.1989

Row3 0.4617 0.7391 1.0000 0.3001 0.2321 0.3163

Row4 0.1900 0.1685 0.3001 1.0000 0.3622 0.1770

Row5 0.2071 0.3022 0.2321 0.3622 1.0000 0.4760

Row6 0.0649 0.1989 0.3163 0.1770 0.4760 1.0000





Intercept -3.9428 3.4927 -10.7884 2.9029 -1.13 0.2590

PARENT 0 8.2868 3.6374 1.1576 15.4159 2.28 0.0227

PARENT 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR 1990 10.5390 4.1980 2.3111 18.7668 2.51 0.0121

YEAR 1995 11.1059 4.9853 1.3349 20.8768 2.23 0.0259

YEAR 2000 0.0000 0.0000 0.0000 0.0000 . .

BIRTH 0.1347 0.0718 -0.0061 0.2755 1.88 0.0607

YEAR*PARENT 1990 0 -12.4878 4.7397 -21.7775 -3.1980 -2.63 0.0084

YEAR*PARENT 1990 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 1995 0 -17.3831 5.2857 -27.7428 -7.0234 -3.29 0.0010

YEAR*PARENT 1995 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 1 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*PARENT 0 -0.1750 0.0748 -0.3215 -0.0285 -2.34 0.0193

BIRTH*PARENT 1 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*YEAR 1990 -0.2219 0.0846 -0.3878 -0.0561 -2.62 0.0087

BIRTH*YEAR 1995 -0.2182 0.1001 -0.4144 -0.0219 -2.18 0.0293

BIRTH*YEAR 2000 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*YEAR*PARENT 1990 0 0.2606 0.0967 0.0710 0.4502 2.69 0.0071

BIRTH*YEAR*PARENT 1990 1 0.0000 0.0000 0.0000 0.0000 . .










Intercept -3.9428 4.2124 -12.1989 4.3134 -0.94 0.3493

PARENT 0 8.2868 4.8847 -1.2871 17.8606 1.70 0.0898

PARENT 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR 1990 10.5390 5.4407 -0.1247 21.2026 1.94 0.0527

YEAR 1995 11.1059 5.3650 0.5907 21.6211 2.07 0.0384

YEAR 2000 0.0000 0.0000 0.0000 0.0000 . .

BIRTH 0.1347 0.0874 -0.0365 0.3060 1.54 0.1231

YEAR*PARENT 1990 0 -12.4878 6.3517 -24.9369 -0.0386 -1.97 0.0493

YEAR*PARENT 1990 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 1995 0 -17.3831 5.8132 -28.7768 -5.9894 -2.99 0.0028

YEAR*PARENT 1995 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 1 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*PARENT 0 -0.1750 0.1011 -0.3732 0.0232 -1.73 0.0835

BIRTH*PARENT 1 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*YEAR 1990 -0.2219 0.1101 -0.4378 -0.0061 -2.02 0.0439

BIRTH*YEAR 1995 -0.2182 0.1073 -0.4286 -0.0078 -2.03 0.0421

BIRTH*YEAR 2000 0.0000 0.0000 0.0000 0.0000 . .







Scale 1.0000 . . . . .


Score Statistics For Type 3 GEE Analysis

Chi-

Source DF Square Pr > ChiSq

PARENT 1 0.24 0.6242

YEAR 2 2.63 0.2685

BIRTH 1 0.01 0.9219

YEAR*PARENT 2 7.09 0.0289

BIRTH*PARENT 1 0.19 0.6596

BIRTH*YEAR 2 2.89 0.2355

BIRTH*YEAR*PARENT 2 7.63 0.0220

6. Binary Response: Procedure GEE: WGEE (Dropout Model)

Model Information

Data Set WORK.MSDICWGEE


Link Function Logit

Dependent Variable DROPOUT





Missing Values 2059



Class Levels Values

FAMNR 569 15 47 54 ...

YEAR 2 1995 2000

STATUS 2 1 2

PREV 2 0 1

Response Profile

Ordered Total

Value DROPOUT Frequency

1 1 225

2 0 1154

PROC GENMOD is modeling the probability that DROPOUT=’1’.



Deviance 1374 1034.2961 0.7528

Scaled Deviance 1374 1034.2961 0.7528


Scaled Pearson X2 1374 1298.8603 0.9453






Intercept 1 -1.3186 0.7591 -2.8064 0.1691 3.02 0.0824

PREV 0 1 0.2316 0.2462 -0.2509 0.7142 0.89 0.3468

PREV 1 0 0.0000 0.0000 0.0000 0.0000 . .

PARENT 1 -4.3282 0.7140 -5.7276 -2.9287 36.75 <.0001

YEAR 1995 1 -0.3261 0.1570 -0.6339 -0.0183 4.31 0.0379

YEAR 2000 0 0.0000 0.0000 0.0000 0.0000 . .

BIRTH 1 0.0077 0.0158 -0.0232 0.0386 0.24 0.6252

Scale 0 1.0000 0.0000 1.0000 1.0000


7. Binary Response: Procedure GEE: WGEE(Analysis)

Model Information

Data Set WORK.MSDICWGEE2


Link Function Logit


Scale Weight Variable WI



Sum of Weights 2730.865



Missing Values 2091


Class Levels Values

FAMNR 573 15 47 54 60 ...

YEAR 3 1990 1995 2000

PARENT 2 0 1


Response Profile

Ordered Total


1 1 2586.492

2 0 144.3727




Prm1 Intercept

Prm2 PARENT 0

Prm3 PARENT 1

Prm4 YEAR 1990

Prm5 YEAR 1995



Prm6 YEAR 2000

Prm7 BIRTH




















Deviance 1335 1037.2206 0.7769

Scaled Deviance 1335 1037.2206 0.7769


Scaled Pearson X2 1335 2673.0973 2.0023






Intercept 1 -3.7376 3.6758 -10.9420 3.4668 1.03 0.3092

PARENT 0 1 16.5267 5.6271 5.4979 27.5556 8.63 0.0033

PARENT 1 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR 1990 1 1.8725 4.1690 -6.2986 10.0436 0.20 0.6533

YEAR 1995 1 10.8424 4.5339 1.9562 19.7287 5.72 0.0168

YEAR 2000 0 0.0000 0.0000 0.0000 0.0000 . .

BIRTH 1 0.1318 0.0763 -0.0177 0.2813 2.99 0.0840





YEAR*PARENT 1990 0 1 -13.5968 6.2095 -25.7673 -1.4264 4.79 0.0285

YEAR*PARENT 1990 1 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 1995 0 1 -19.1413 6.8943 -32.6539 -5.6286 7.71 0.0055

YEAR*PARENT 1995 1 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 0 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 1 0 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*PARENT 0 1 -0.3181 0.1170 -0.5475 -0.0887 7.38 0.0066

BIRTH*PARENT 1 0 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*YEAR 1990 1 -0.0590 0.0860 -0.2276 0.1096 0.47 0.4926

BIRTH*YEAR 1995 1 -0.2065 0.0925 -0.3879 -0.0252 4.98 0.0256

BIRTH*YEAR 2000 0 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*YEAR*PARENT 1990 0 1 0.2675 0.1289 0.0148 0.5202 4.30 0.0380


BIRTH*YEAR*PARENT 1995 0 1 0.3549 0.1420 0.0767 0.6332 6.25 0.0124




Scale 0 1.0000 0.0000 1.0000 1.0000














Row1 1.0000 0.1993 0.2405 0.1980 0.2368 0.1082

Row2 0.1993 1.0000 0.6525 0.2134 0.4173 0.4624

Row3 0.2405 0.6525 1.0000 0.2492 0.5165 0.4306

Row4 0.1980 0.2134 0.2492 1.0000 0.2289 0.1564

Row5 0.2368 0.4173 0.5165 0.2289 1.0000 0.4599

Row6 0.1082 0.4624 0.4306 0.1564 0.4599 1.0000





Intercept -3.0799 3.6293 -10.1931 4.0333 -0.85 0.3961

PARENT 0 14.0251 4.8978 4.4256 23.6245 2.86 0.0042

PARENT 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR 1990 2.2806 3.9665 -5.4936 10.0547 0.57 0.5653

YEAR 1995 10.7817 4.8456 1.2844 20.2789 2.23 0.0261

YEAR 2000 0.0000 0.0000 0.0000 0.0000 . .

BIRTH 0.1257 0.0762 -0.0237 0.2750 1.65 0.0990

YEAR*PARENT 1990 0 -11.6462 5.1287 -21.6984 -1.5941 -2.27 0.0232

YEAR*PARENT 1990 1 0.0000 0.0000 0.0000 0.0000 . .


YEAR*PARENT 1995 0 -14.3987 8.0971 -30.2686 1.4713 -1.78 0.0754

YEAR*PARENT 1995 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 1 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*PARENT 0 -0.2854 0.1003 -0.4819 -0.0889 -2.85 0.0044

BIRTH*PARENT 1 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*YEAR 1990 -0.0716 0.0824 -0.2332 0.0900 -0.87 0.3851

BIRTH*YEAR 1995 -0.2130 0.0983 -0.4056 -0.0203 -2.17 0.0303

BIRTH*YEAR 2000 0.0000 0.0000 0.0000 0.0000 . .



BIRTH*YEAR*PARENT 1995 0 0.2897 0.1598 -0.0236 0.6030 1.81 0.0699








Intercept -3.0799 3.6647 -10.2625 4.1027 -0.84 0.4007

PARENT 0 14.0251 4.3825 5.4356 22.6145 3.20 0.0014

PARENT 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR 1990 2.2806 4.1313 -5.8166 10.3777 0.55 0.5809

YEAR 1995 10.7817 4.1139 2.7186 18.8447 2.62 0.0088

YEAR 2000 0.0000 0.0000 0.0000 0.0000 . .

BIRTH 0.1257 0.0762 -0.0237 0.2750 1.65 0.0990

YEAR*PARENT 1990 0 -11.6462 5.0433 -21.5309 -1.7616 -2.31 0.0209

YEAR*PARENT 1990 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 1995 0 -14.3987 5.7453 -25.6593 -3.1381 -2.51 0.0122

YEAR*PARENT 1995 1 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 0 0.0000 0.0000 0.0000 0.0000 . .

YEAR*PARENT 2000 1 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*PARENT 0 -0.2854 0.0911 -0.4639 -0.1069 -3.13 0.0017

BIRTH*PARENT 1 0.0000 0.0000 0.0000 0.0000 . .

BIRTH*YEAR 1990 -0.0716 0.0852 -0.2387 0.0955 -0.84 0.4009

BIRTH*YEAR 1995 -0.2130 0.0843 -0.3782 -0.0477 -2.53 0.0115

BIRTH*YEAR 2000 0.0000 0.0000 0.0000 0.0000 . .







Scale 1.0000 . . . . .



Chi-


PARENT 1 3.75 0.0528

YEAR 2 4.88 0.0871

BIRTH 1 0.38 0.5376

YEAR*PARENT 0 . .

BIRTH*PARENT 1 3.59 0.0580

BIRTH*YEAR 2 2.93 0.2308

BIRTH*YEAR*PARENT 2 3.71 0.1565

8. Binary Response: Procedure GEE: MULTIPLE IMPUTATION


------------------------------------- Imputation Number=1 --------------------------------------

Model Information

Data Set WORK.MSDICCOMP


Link Function Logit







Class Levels Values

FAMNR 573 15 47 54 60 ...

YEAR 3 1990 1995 2000

PARENT 2 0 1

Response Profile

Ordered Total


1 1 3195

2 0 243



Parameter Effect

Prm1 Intercept

Prm2 father

Prm3 year1990

Prm4 year1995

Prm5 BIRTH

Prm6 father1990

Prm7 father1995

Prm8 fatherbirth

Prm9 birth1990

Prm10 birth1995

Prm11 father1990birth

Prm12 father1995birth



Deviance 3426 1681.1648 0.4907

Scaled Deviance 3426 1681.1648 0.4907


Scaled Pearson X2 3426 3430.9140 1.0014




Prm1 Prm2 Prm3 Prm4 Prm5 Prm6

Prm1 5.60685 -5.60685 -5.60685 -5.60685 -0.11298 5.60685

Prm2 -5.60685 11.77476 5.60685 5.60685 0.11298 -11.77476

Prm3 -5.60685 5.60685 7.85402 5.60685 0.11298 -7.85402

Prm4 -5.60685 5.60685 5.60685 10.84247 0.11298 -5.60685

Prm5 -0.11298 0.11298 0.11298 0.11298 0.002293 -0.11298

Prm6 5.60685 -11.77476 -7.85402 -5.60685 -0.11298 15.62355


Prm7 5.60685 -11.77476 -5.60685 -10.84247 -0.11298 11.77476

Prm8 0.11298 -0.24114 -0.11298 -0.11298 -0.002293 0.24114

Prm9 0.11298 -0.11298 -0.15827 -0.11298 -0.002293 0.15827

Prm10 0.11298 -0.11298 -0.11298 -0.21666 -0.002293 0.11298

Prm11 -0.11298 0.24114 0.15827 0.11298 0.002293 -0.31988

Prm12 -0.11298 0.24114 0.11298 0.21666 0.002293 -0.24114


Prm7 Prm8 Prm9 Prm10 Prm11 Prm12

Prm1 5.60685 0.11298 0.11298 0.11298 -0.11298 -0.11298

Prm2 -11.77476 -0.24114 -0.11298 -0.11298 0.24114 0.24114

Prm3 -5.60685 -0.11298 -0.15827 -0.11298 0.15827 0.11298

Prm4 -10.84247 -0.11298 -0.11298 -0.21666 0.11298 0.21666

Prm5 -0.11298 -0.002293 -0.002293 -0.002293 0.002293 0.002293

Prm6 11.77476 0.24114 0.15827 0.11298 -0.31988 -0.24114

Prm7 20.73424 0.24114 0.11298 0.21666 -0.24114 -0.42018

Prm8 0.24114 0.004984 0.002293 0.002293 -0.004984 -0.004984

Prm9 0.11298 0.002293 0.003212 0.002293 -0.003212 -0.002293

Prm10 0.21666 0.002293 0.002293 0.004360 -0.002293 -0.004360

Prm11 -0.24114 -0.004984 -0.003212 -0.002293 0.006610 0.004984

Prm12 -0.42018 -0.004984 -0.002293 -0.004360 0.004984 0.008589




Intercept 1 0.7404 2.3679 -3.9005 5.3814 0.10 0.7545

father 1 2.8850 3.4314 -3.8405 9.6105 0.71 0.4005

year1990 1 -1.6792 2.8025 -7.1720 3.8136 0.36 0.5490

year1995 1 2.6716 3.2928 -3.7822 9.1253 0.66 0.4172

BIRTH 1 0.0465 0.0479 -0.0474 0.1403 0.94 0.3320

father1990 1 -0.4406 3.9527 -8.1877 7.3065 0.01 0.9112

father1995 1 -0.0413 4.5535 -8.9660 8.8834 0.00 0.9928

fatherbirth 1 -0.0480 0.0706 -0.1864 0.0903 0.46 0.4962

birth1990 1 0.0115 0.0567 -0.0995 0.1226 0.04 0.8385

birth1995 1 -0.0568 0.0660 -0.1862 0.0727 0.74 0.3900

father1990birth 1 0.0016 0.0813 -0.1577 0.1610 0.00 0.9840

father1995birth 1 -0.0139 0.0927 -0.1956 0.1677 0.02 0.8804

Scale 0 1.0000 0.0000 1.0000 1.0000













Row1 1.0000 0.2508 0.1414 0.3160 0.1403 0.0303

Row2 0.2508 1.0000 0.2823 0.1630 0.2344 0.1535

Row3 0.1414 0.2823 1.0000 0.1709 0.1551 0.2162

Row4 0.3160 0.1630 0.1709 1.0000 0.2242 0.0382

Row5 0.1403 0.2344 0.1551 0.2242 1.0000 0.1361


Row6 0.0303 0.1535 0.2162 0.0382 0.1361 1.0000





Intercept 0.6317 1.9889 -3.2664 4.5298 0.32 0.7508

father 3.6371 2.6703 -1.5966 8.8708 1.36 0.1732

year1990 -1.2479 2.3786 -5.9098 3.4140 -0.52 0.5998

year1995 3.0881 2.5941 -1.9963 8.1725 1.19 0.2339

BIRTH 0.0487 0.0403 -0.0304 0.1277 1.21 0.2275

father1990 -1.2229 3.1118 -7.3218 4.8761 -0.39 0.6943

father1995 -0.4566 3.8246 -7.9526 7.0394 -0.12 0.9050

fatherbirth -0.0637 0.0544 -0.1704 0.0429 -1.17 0.2415

birth1990 0.0029 0.0482 -0.0917 0.0974 0.06 0.9528

birth1995 -0.0651 0.0524 -0.1679 0.0377 -1.24 0.2145

father1990birth 0.0177 0.0639 -0.1076 0.1430 0.28 0.7824

father1995birth -0.0053 0.0773 -0.1568 0.1461 -0.07 0.9448





Intercept 0.6317 2.3328 -3.9406 5.2040 0.27 0.7866

father 3.6371 3.1326 -2.5028 9.7769 1.16 0.2456

year1990 -1.2479 2.7294 -6.5975 4.1017 -0.46 0.6475

year1995 3.0881 3.0585 -2.9064 9.0826 1.01 0.3126

BIRTH 0.0487 0.0472 -0.0439 0.1412 1.03 0.3026

father1990 -1.2229 3.5778 -8.2353 5.7896 -0.34 0.7325

father1995 -0.4566 3.9644 -8.2267 7.3136 -0.12 0.9083

fatherbirth -0.0637 0.0643 -0.1897 0.0622 -0.99 0.3213

birth1990 0.0029 0.0552 -0.1054 0.1111 0.05 0.9588

birth1995 -0.0651 0.0613 -0.1853 0.0550 -1.06 0.2882

father1990birth 0.0177 0.0733 -0.1261 0.1614 0.24 0.8097

father1995birth -0.0053 0.0804 -0.1630 0.1523 -0.07 0.9470

Scale 1.0000 . . . . .



Chi-


father 1 1.56 0.2113

year1990 1 0.26 0.6134

year1995 1 1.22 0.2687

BIRTH 1 1.28 0.2587

father1990 1 0.14 0.7110

father1995 1 0.01 0.9123

fatherbirth 1 1.16 0.2810

birth1990 1 0.00 0.9546

birth1995 1 1.34 0.2475

father1990birth 1 0.07 0.7951

father1995birth 1 0.00 0.9491

...

COMBINING 5 GEE ANALYSES: The MIANALYZE Procedure

Model Information


PARMS Data Set WORK.GMPARMS

PARMINFO Data Set WORK.GMPINFO

COVB Data Set WORK.GMCOVB



-----------------Variance-----------------


intercept 1.788537 4.805738 6.951982 41.968

father 4.173570 8.063189 13.071473 27.248

year1990 1.699900 7.032767 9.072646 79.126

year1995 2.487329 8.335452 11.320247 57.536

BIRTH 0.000976 0.001992 0.003163 29.186

father1990 4.315186 11.839142 17.017365 43.2

father1995 2.981649 13.968385 17.546364 96.196

fatherbirth 0.001995 0.003403 0.005796 23.456

birth1990 0.001008 0.002903 0.004113 46.236

birth1995 0.001451 0.003389 0.005131 34.712

father1990birth 0.002074 0.004995 0.007483 36.169

father1995birth 0.001529 0.005805 0.007640 69.369


Relative Fraction



intercept 0.446600 0.339469 0.936423

father 0.621130 0.423933 0.921840

year1990 0.290054 0.243716 0.953522

year1995 0.358084 0.287996 0.945538

BIRTH 0.587815 0.409338 0.924328

father1990 0.437382 0.334408 0.937311

father1995 0.256148 0.219966 0.957861

fatherbirth 0.703448 0.457334 0.916198

birth1990 0.416694 0.322804 0.939354

birth1995 0.513919 0.374494 0.930320

father1990birth 0.498249 0.366634 0.931683

father1995birth 0.316017 0.261131 0.950366



intercept -0.338938 2.636661 -5.6601 4.98218 41.968

father 2.879104 3.615449 -4.5360 10.29424 27.248

year1990 -0.639482 3.012083 -6.6347 5.35577 79.126

year1995 2.791960 3.364557 -3.9441 9.52801 57.536

BIRTH 0.059730 0.056242 -0.0553 0.17473 29.186

father1990 -0.287554 4.125211 -8.6057 8.03061 43.2

father1995 -1.685472 4.188838 -10.0000 6.62909 96.196

fatherbirth -0.059170 0.076132 -0.2165 0.09815 23.456

birth1990 -0.001177 0.064135 -0.1303 0.12790 46.236

birth1995 -0.062871 0.071631 -0.2083 0.08259 34.712

father1990birth 0.008734 0.086507 -0.1667 0.18415 36.169


father1995birth 0.032184 0.087407 -0.1422 0.20654 69.369


t for H0:


intercept -1.883722 1.128103 0 -0.13 0.8983

father 0.466879 5.716576 0 0.80 0.4327

year1990 -2.049814 0.944915 0 -0.21 0.8324

year1995 1.105505 4.815376 0 0.83 0.4101

BIRTH 0.033510 0.109385 0 1.06 0.2969

father1990 -3.200129 2.258947 0 -0.07 0.9447

father1995 -2.984348 0.744977 0 -0.40 0.6883

fatherbirth -0.121201 -0.008756 0 -0.78 0.4448

birth1990 -0.051380 0.023826 0 -0.02 0.9854

birth1995 -0.109385 -0.017590 0 -0.88 0.3861

father1990birth -0.045218 0.072547 0 0.10 0.9201

father1995birth -0.017459 0.061564 0 0.37 0.7138

9. Binary Response: Procedure NLMIXED: CC (LOGISTIC Start Par)

Analysis of Maximum Likelihood Estimates

Standard Wald

Parameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 1 -3.0258 1.7634 2.9445 0.0862

PARENT 0 1 -0.0187 1.7634 0.0001 0.9915

YEAR 1990 1 -1.5265 2.3326 0.4283 0.5128

YEAR 1995 1 -0.9470 2.6876 0.1242 0.7246

BIRTH 1 0.00830 0.0360 0.0532 0.8176

YEAR*PARENT 1990 0 1 1.1360 2.3326 0.2372 0.6263

YEAR*PARENT 1995 0 1 3.1353 2.6876 1.3609 0.2434

BIRTH*PARENT 0 1 0.00158 0.0360 0.0019 0.9651

BIRTH*YEAR 1990 1 0.0379 0.0474 0.6391 0.4240

BIRTH*YEAR 1995 1 0.0102 0.0547 0.0350 0.8516

BIRTH*YEAR*PARENT 1990 0 1 -0.0241 0.0474 0.2590 0.6108

BIRTH*YEAR*PARENT 1995 0 1 -0.0650 0.0547 1.4103 0.2350

10. Binary Response: Procedure NLMIXED: CC

Specifications

Data Set WORK.MSDCC



Random Effects g1





Quadrature

Dimensions




Subjects 122


Parameters 13



Parameters

b0 b1 b2 b3 b4 b5 b6 b7 b8

-3 -1.5 -0.9 0.1 0.008 1.1 3.1 0.002 0.04

Parameters

b9 b10 b11 V1 NegLogLike

0.01 -0.02 -0.065 2.3662 467.173863

Iteration History


1 4 251.192538 215.9813 2160.479 -329682

2 5 186.808564 64.38397 727.3123 -99.2999

3 6 173.65153 13.15703 218.1587 -20.2009

...

29 54 162.032463 0.000014 0.176163 -1.23E-6


Fit Statistics





Parameter Estimates

Standard


b0 1.7898 6.8875 121 0.26 0.7954 0.05 -11.8458 15.4255 -0.04705

b1 1.9448 8.1297 121 0.24 0.8113 0.05 -14.1501 18.0398 -0.07148

b2 -0.06616 9.3099 121 -0.01 0.9943 0.05 -18.4976 18.3653 -0.03293

b3 3.1077 7.9848 121 0.39 0.6978 0.05 -12.7004 18.9158 0.127968

b4 0.05051 0.1388 121 0.36 0.7165 0.05 -0.2243 0.3253 0.060808

b5 2.5654 10.4361 121 0.25 0.8062 0.05 -18.0955 23.2263 0.143268

b6 2.7611 11.9853 121 0.23 0.8182 0.05 -20.9669 26.4891 0.16376

b7 -0.07193 0.1635 121 -0.44 0.6608 0.05 -0.3957 0.2518 0.112962

b8 -0.05337 0.1636 121 -0.33 0.7448 0.05 -0.3773 0.2705 0.169062

b9 0.007904 0.1880 121 0.04 0.9665 0.05 -0.3643 0.3801 -0.08336

b10 -0.04318 0.2131 121 -0.20 0.8397 0.05 -0.4650 0.3786 0.176163

b11 -0.04736 0.2456 121 -0.19 0.8474 0.05 -0.5336 0.4389 0.00573

V1 2.1570 0.3846 121 5.61 <.0001 0.05 1.3956 2.9183 -0.00899

11. Binary Response: Procedure NLMIXED: Direct Likelihood

Specifications

Data Set WORK.MSDIC



Random Effects g1





Quadrature

Dimensions





Subjects 573


Parameters 13


Parameters

b0 b1 b2 b3 b4 b5 b6 b7 b8

-3 -1.5 -0.9 0.1 0.008 1.1 3.1 0.002 0.04

Parameters

b9 b10 b11 V1 NegLogLike

0.01 -0.02 -0.065 2.3662 1620.89446

Iteration History


1 5 765.515534 855.3789 2996.607 -5010959

2 6 647.589176 117.9264 1287.777 -166.939

3 7 632.241291 15.34789 860.1845 -44.1928

...

31 66 597.464496 2.254E-6 0.128487 -4.32E-6


Fit Statistics





Parameter Estimates

Standard


b0 0.8072 4.7726 572 0.17 0.8658 0.05 -8.5668 10.1811 -0.00608

b1 -1.6750 5.0301 572 -0.33 0.7393 0.05 -11.5548 8.2048 -0.07512

b2 3.2998 5.6827 572 0.58 0.5617 0.05 -7.8617 14.4613 -0.09102

b3 2.9449 6.2556 572 0.47 0.6380 0.05 -9.3418 15.2316 0.037353

b4 0.05506 0.09610 572 0.57 0.5669 0.05 -0.1337 0.2438 -0.03355

b5 0.9481 6.7158 572 0.14 0.8878 0.05 -12.2425 14.1386 0.034565

b6 5.3107 7.7034 572 0.69 0.4909 0.05 -9.8198 20.4412 0.128487

b7 -0.05733 0.1284 572 -0.45 0.6553 0.05 -0.3095 0.1948 -0.0321

b8 0.02616 0.1014 572 0.26 0.7965 0.05 -0.1730 0.2253 0.00277

b9 -0.05388 0.1142 572 -0.47 0.6373 0.05 -0.2782 0.1704 -0.03374

b10 -0.01817 0.1378 572 -0.13 0.8952 0.05 -0.2889 0.2526 -0.00886

b11 -0.1165 0.1568 572 -0.74 0.4579 0.05 -0.4245 0.1915 -0.024

V1 2.1285 0.2091 572 10.18 <.0001 0.05 1.7177 2.5392 0.001664

6.3.3 Discussion

Part 1

2. Full model with unstructured mean and covariance structure: Consider year and status as categoricalvariables (put them in the class statement), all other covariates can be considered continuous.


3. Final model after reducing mean structure (for both datasets): Only year, indicating the time at whichthe interview was taken, seems to have a significant effect on the marital satisfaction, both for fathers andmothers, meaning that the marital satisfaction changes over time. Taking a look at the parameter estimates(Tables 6.1-6.2), we can conclude that marital satisfaction increased from 1990 to 1995, and decreased againbetween 1995 and 2000 (but less than the increase 1990-1995), both for father and mother.

Table 6.1: Parameter Estimates for the Final Model for Father

Par. MLE (s.e.)

Year 1990 5.2266 (0.06773)

Year 1995 6.0236 (0.08209)

Year 2000 5.7469 (0.08731)

Table 6.2: Parameter Estimates for the Final Model for Mother

Par. MLE (s.e.)

Year 1990 5.2588 (0.07476)

Year 1995 5.9184 (0.08878)

Year 2000 5.7037 (0.08520)

4. Reducing the covariance structure (for both datasets): The different covariance structures can be putin a nesting scheme (Figure 6.3). When reducing the covariance structure, it is only allowed to reduce toa structure which is connected with (nested in) the original covariance structure. A likelihood ratio testis used to choose the best covariance structure (degrees of freedom equal to the difference in number ofparameters). We performed this reduction both for father and mother. Results are presented in Tables 6.3-6.4. For father, the covariance structure can be reduced to the heterogeneous compound symmetry (CSH),assuming 4 parameters, namely σ2

1 = 0.5793, σ22 = 0.8066, σ2

3 = 0.9165 and ρ = 0.6067. For mother, weborderline choose the compound symmetry (CS) covariance structure with 2 parameters, σ2

1 = 0.4468 andσ2 = 0.3962.

Figure 6.3: Nesting Scheme of the Different Covariance Structures

UN

UN(2) ARH(1) CSH TOEPH

UN(1) AR(1) CS TOEP

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Table 6.3: Summary of Covariance Structure Reduction for Father

Model Covariance −2` Par Ref G2 df p-value

1 UN 799.2 6

2 UN(2) 845.3 5 1 46.1 1 < 0.0001

3 ARH(1) 812.1 4 1 12.9 2 0.0016

4 TOEPH 802.7 5 1 3.5 1 0.0614

5 CSH 804.4 4 1 5.2 2 0.0743

4 1.7 1 0.4270

6 CS 814.8 2 5 10.4 2 0.0055

Table 6.4: Summary of Covariance Structure Reduction for Mother


1 UN 872.0 6

2 UN(2) 902.2 5 1 30.2 1 < 0.0001

3 ARH(1) 880.1 4 1 8.1 2 0.0174

4 TOEPH 873.9 5 1 1.9 1 0.1680

5 CSH 876.3 4 1 4.3 2 0.1220

4 2.4 1 0.3012

6 CS 880.1 2 5 3.8 1 0.0512

7 SIMPLE 976.2 1 6 96.1 1 < 0.0001

5. Modelling longitudinal data of both parents simultaneously: As mean structure, only year, parentand their interaction are included into the model, since year is the only covariate that seemed to have asignificant effect on marital satisfaction, when analyzing father and mother separately. To allow this effectto differ between parents, parent and its interaction with year is also included into the model.When both parents are analyzed together, an extra clustering is present, namely not only three measurementsper person (father or mother), but also both parents in a family. Therefore, the 6 × 6 covariance structurewill now consist of 6 (= 3 × 2) variances and 15 covariances. They can be estimated, using for example afully unstructured covariance matrix. However, it is more natural to estimate the covariance separately forthe two clustering variables (a 2×2 matrix for parent, and a 3×3 matrix for year), and afterwards takingthe Kronecker product of both matrices, also resulting in a 6×6 covariance matrix. In this way, SAS allowsthree possibilities, UN@UN, UN@AR(1) and UN@CS. From Table 6.5, we can conclude that reducing thecovariance structure using these special structures with the Kronecker product, does not fit the data well.Therefore, we stick to the fully 6 × 6 unstructured (UN) covariance matrix. However, this can be reducedalong the edges of Figure 6.3 in a similar way as the data for father and mother separately.

Table 6.5: Summary of Covariance Structure Reduction


1 UN 1603.4 21

2 UN@UN 1633.8 8 1 30.4 13 0.0041

3 UN@AR(1) 1663.7 4 2 29.9 4 < 0.0001

4 UN@CS 1649.4 4 2 15.6 4 0.0036


Part 2

1. Covariance structure: Different choices can be considered for the covariance structure, e.g., unstructured,exchangeable or independence. Recall that the parameter estimators are consistent even if the workingcorrelation matrix is incorrect. However, an appropriate choice can improve the efficiency of the estimators.In the LMM analysis, different complicated covariance structures in the form of Kronecker products wereconsidered for the joint analysis of the two clustering variables. Since the covariance structure is treatedas a nuisance parameter in the GEE analysis, as e.g. an unstructured 6 × 6 matrix for the within classcorrelation structure in the REPEATED statement.

2. Univariate vs bivariate: The univariate and bivariate analyses lead essentially to the same final modelfor males and females: for males a very simple model with only an intercept, for females slightly morecomplicated. Note that taking into account the correlation between members of the same family does havean impact on the size of the parameter estimates.

3. Method In this section only the completers were used. However GEE is not a likelihood method and thereforethis model is only valid under MCAR. Other models will have to be considered when this assumption isquestionable.

Part 3

1. Multilevel model formulation: The dataset considered here is a typical example of an hierarchical datastructure with nested classification factors. In particular, a random sample of families has been collected andmarital satisfaction has been recorded in 1990, 1995 and 2000 for both parents resulting in 6 measurementsper family. In this setting, the marital satisfaction is expected to differ from one parent to the other withineach family and from family to family. Thus, the “family” and the “parent” are considered to be nestedclassification factors. A random effects term is then associated with the “family” factor and one with the“parent” factor nested within the “family”. This implies that the probability of marital satisfaction variesfrom family to family and in the same family the probability of marital satisfaction varies from husband towife.

2. Nested random effects results: The solution obtained from fitting the nested random effects modelunder PQL and MQL seems questionable since both methods lead to a non-positive definite random effectscovariance matrix. This could be attributed to the dichotomous nature of the data and the small numberof repeated measurements. On the contrary the fit using the adaptive Gauss Hermite method is successful.The likelihood ratio test suggests that the two-level random effects structure is not statistically significant(p-value = 0.0578). However, caution in needed in using such a test since the null hypothesis lies on theboundary of the parameter space.

3. PQL, MQL under both REML and ML: PQL and MQL behave similarly in parameter estimates andstandard errors under either REML and ML (see Table 6.6). However, it is known that they behave poorly forbinary outcomes with relatively small number of repeated measurements, as is the case in this example with6 measurements per family. Observe also that either in PQL or MQL, ML gives slightly smaller estimatesfor the covariance parameters than REML. This is due to the fact that REML accounts for the variabilityin estimating the fixed-effects coefficients. Finally, the information criteria should be treated with cautionsince they do not refer to the real data but to pseudo data.

4. Gaussian quadrature vs Adaptive quadrature: The number of nodes chosen in the Gaussian and adaptiveGaussian quadrature rule can have a direct impact on the estimated parameters, as it can be seen in Tables6.7 and 6.8. The impact is greater in the simple Gaussian rule than in the adaptive, in which slight changesare observed in the parameter estimates for more than 5 nodes.

5. Approximation methods in GLMMs: Table 6.9 shows severe differences in parameter estimates betweenthe PQL/MQL and the quadrature method. The Laplace approximation method, which is equivalent tothe adaptive Gauss-Hermite with 1 node, gives different estimates from the adaptive Gauss-Hermite with20 nodes revealing the importance of the number of nodes in quadrature methods.


Table 6.6: PQL and MQL under REML and ML

Parameter PQL (REML) PQL (ML) MQL (REML) MQL (ML)

Int 3.053 (0.8803) 3.050 (0.8773) 2.853 (0.8596) 2.853 (0.8575)

year90 -0.620 (0.4397) -0.620 (0.4387) -0.546 (0.4179) -0.547 (0.4174)

year95 0.347 (0.4058) 0.346 (0.4052) 0.300 (0.3751) 0.300 (0.3750)

parent0 0.137 (0.4310) 0.137 (0.4306) 0.121 (0.3880) 0.121 (0.3881)

parent0×year90 -0.031 (0.4723) -0.031 (0.4718) -0.029 (0.4288) -0.029 (0.4289)

parent0×year95 -0.294 (0.5409) -0.293 (0.5403) -0.266 (0.4914) -0.266 (0.4915)

child90 0.084 (0.0958) 0.083 (0.0955) 0.066 (0.0941) 0.066 (0.0938)

yearsmarr90 -0.029 (0.0296) -0.029 (0.0295) -0.031 (0.0294) -0.031 (0.0293)

σ2

b1.664 (0.2396) 1.639 (0.2370) 2.396 (0.3247) 2.366 (0.3220)

Table 6.7: Gauss Hermite

Parameter Q = 3 Q = 5 Q = 10 Q = 20 Q = 50

Int 4.290 (1.2570) 4.358 (1.2366) 4.320 (1.2595) 4.354 (1.2535) 4.339 (1.2545)

year90 -0.724 (0.5634) -0.758 (0.5597) -0.729 (0.5677) -0.759 (0.5643) -0.755 (0.5658)

year95 0.419 (0.4717) 0.424 (0.4795) 0.471 (0.4825) 0.457 (0.4796) 0.458 (0.4805)

parent0 0.147 (0.4845) 0.173 (0.4801) 0.155 (0.4825) 0.151 (0.4824) 0.151 (0.4824)

parent0×year90 -0.018 (0.5337) -0.045 (0.5287) -0.025 (0.5319) -0.020 (0.5316) -0.020 (0.5315)

parent0×year95 -0.363 (0.5999) -0.387 (0.6021) -0.378 (0.6063) -0.372 (0.6064) -0.372 (0.6063)

child90 0.064 (0.1029) 0.190 (0.1386) 0.164 (0.1425) 0.153 (0.1364) 0.153 (0.1381)

yearsmarr90 -0.048 (0.0411) -0.053 (0.0408) -0.046 (0.0422) -0.046 (0.0424) -0.045 (0.0425)

σb 1.839 (0.1242) 2.131 (0.2011) 2.189 (0.2266) 2.142 (0.2178) 2.137 (0.2191)

−2` 1201.9 1205.3 1202.7 1202.9 1203.0

Table 6.8: Adaptive Gauss Hermite

Parameter Q = 3 Q = 5 Q = 10 Q = 20 Q = 50

Int 4.033 (1.1280) 4.246 (1.2074) 4.324 (1.2469) 4.338 (1.2543) 4.339 (1.2545)

year90 -0.741 (0.5269) -0.751 (0.5514) -0.755 (0.5636) -0.755 (0.5657) -0.755 (0.5658)

year95 0.431 (0.4609) 0.442 (0.4725) 0.456 (0.4792) 0.458 (0.4805) 0.458 (0.4805)

parent0 0.152 (0.4733) 0.151 (0.4783) 0.151 (0.4817) 0.151 (0.4824) 0.151 (0.4824)

parent0×year90 -0.022 (0.5214) -0.021 (0.5268) -0.020 (0.5307) -0.020 (0.5315) -0.020 (0.5315)

parent0×year95 -0.354 (0.5938) -0.364 (0.6005) -0.371 (0.6054) -0.372 (0.6063) -0.372 (0.6063)

child90 0.131 (0.1240) 0.145 (0.1322) 0.152 (0.1371) 0.153 (0.1381) 0.153 (0.1381)

yearsmarr90 -0.042 (0.0381) -0.044 (0.0409) -0.045 (0.0422) -0.045 (0.0425) -0.045 (0.0425)

σb 1.786 (0.1652) 2.007 (0.1799) 2.116 (0.2082) 2.137 (0.2189) 2.137 (0.2191)

−2` 1219.0 1205.2 1203.2 1203.0 1203.0

Table 6.9: PQL, MQL, Adaptive Gauss Hermite, Laplace

Parameter PQL MQL AQ (Q = 20) Laplace

Int 3.050 (0.8773) 2.853 (0.8575) 4.338 (1.2543) 6.058 (1.7343)

year90 -0.620 (0.4387) -0.547 (0.4174) -0.755 (0.5657) -0.757 (0.6993)

year95 0.346 (0.4052) 0.300 (0.3750) 0.458 (0.4805) 0.626 (0.5567)

parent0 0.137 (0.4306) 0.121 (0.3881) 0.151 (0.4824) 0.158 (0.5220)

parent0×year90 -0.031 (0.4718) -0.029 (0.4289) -0.020 (0.5315) -0.011 (0.5792)

parent0×year95 -0.293 (0.5403) -0.266 (0.4915) -0.372 (0.6063) -0.458 (0.6614)

child90 0.083 (0.0955) 0.066 (0.0938) 0.153 (0.1381) 0.219 (0.1843)

yearsmarr90 -0.029 (0.0295) -0.031 (0.0293) -0.045 (0.0425) -0.047 (0.0565)

σ2

b1.639 (0.2370) 2.366 (0.3220) 4.565 (0.9354) 16.536 (6.9512)

6. Test for the parent effect: The likelihood ratio test suggests that the parent effect on the maritalsatisfaction is not statistically significant (p-value = 0.8495).


Part 4

1. Exploring incomplete data.In Table 6.10 an overview of the missingness patterns together with the frequencies with which they occuris shown. The group of dropouts is of considerable magnitude, while the intermittent missingness group ismuch smaller. They are all included into the analyses. We performed analyses on the continuous and binary

Table 6.10: Marital Satisfaction Data. Overview of missingness patterns and the frequencies with whichthey occur. ’O’ indicates observed and ’M’ indicates missing.


Group 1990 1995 2000 Freq Percent

1 O O O 268 23.39

Dropout

2 O O M 392 34.21

3 O M M 428 37.35

4 M M M 14 1.22

Intermittent Missingness

5 M O O 1 0.09

6 O M O 41 3.58

7 M O M 2 0.17

response. For the continuous response, we compared the complete case analysis with the direct likelihoodapproach and multiple imputation using mixed models. The results of a single imputation are shown forreasons of comparison. For the binary response, GEE on the completers, on the available cases, WGEE andmultiple imputation GEE were performed. In these analyses, the focus was not on model building but ratheron the comparison of the resulting estimates.

2. Continuous response using MIXEDA mixed model using ‘year’, ‘parent’ and their interaction for the mean structure and unstructured (‘parent’)and compound symmetry (‘year’) for the covariance structure is fitted using the complete cases (CC), directlikelihood approach, single and multiple imputation. Table 6.11 shows only moderate differences amongparameter estimates based on the various missing data approaches and the complete cases.

Table 6.11: Marital Satisfaction Data. Parameter estimates and standard errors based on a mixed modelusing the complete cases (CC), direct likelihood approach, single and multiple imputation.

Par. CC Direct Single Multiple

Likelihood Imputation Imputation

Year 1990 5.26(0.08) 5.19(0.04) 5.19(0.04) 5.19(0.04)

Year 1995 5.92(0.08) 5.93(0.05) 5.80(0.04) 5.82(0.05)

Year 2000 5.70(0.08) 5.60(0.07) 5.93(0.04) 5.90(0.06)

Parent 0 0.04(0.09) 0.06(0.07) -0.03(0.05) 0.04(0.09)

Year 1990 Parent 0 -0.08(0.09) -0.06(0.07) 0.03(0.06) -0.03(0.09)

Year 1995 Parent 0 0.06(0.09) -0.02(0.07) 0.14(0.07) 0.01(0.11)

3. Binary response using GEEA GEE analysis was performed on the completers and available cases and the resulting parameter estimatescompared with those based on WGEE, single and multiple imputation GEE. The mean structure used,


consisted of ‘parent year birth parent*year parent*birth birth*year parent*birth*year’ where ‘year’ was takencategorical. Table 6.12 shows considerable differences among the different modelling approaches.By means of logistic regression a dropout model was fitted (Table 6.13). Among the variables considered:previous measurement, parent, year and birth; parent and year contributed significantly. For the now theother variables were retained in the model.

Table 6.12: Marital Satisfaction Data: GEE analyses based on completers, available cases, WGEE, singleand multiple imputation.

Par. CC Available WGEE Single (1) Multiple

Cases Imputation Imputation

Int -3.94(3.49) -1.79(2.41) -3.08(3.63) 0.74(2.37) -0.34(2.64)

Parent 0 8.29(3.64) 6.52(2.87) 14.03(4.90) 2.89(3.43) 2.88(3.62)

Year 1990 10.54(4.20) 1.44(2.64) 2.28(3.67) -1.68(2.80) -0.64(3.01)

Year 1995 11.11(4.99) 6.00(2.93) 10.78(4.85) 2.67(3.29) 2.79(3.36)

Birth 0.13(0.07) 0.08(0.05) 0.13(0.08) 0.05(0.05) 0.06(0.06)

Year 1990 Parent 0 -12.49(4.74) -4.20(3.08) -11.65(5.13) -0.44(3.95) -0.29(4.13)

Year 1995 Parent 0 -17.39(5.29) -3.98(3.90) -14.40(8.10) -0.04(4.55) -1.69(4.19)

Birth Parent 0 -0.18(0.07) -0.13(0.06) -0.29(0.10) -0.05(0.07) -0.06(0.08)

Birth Year 1990 -0.22(0.08) -0.04(0.05) -0.07(0.08) 0.01(0.06) -0.00(0.06)

Birth Year 1995 -0.22(0.10) -0.11(0.06) -0.21(0.10) -0.06(0.07) -0.06(0.07)

Birth Year 1990 Parent 0 0.26(0.10) 0.09(0.06) 0.25(0.11) 0.00(0.08) 0.01(0.09)

Birth Year 1995 Parent 0 0.36(0.11) 0.08(0.08) 0.29(0.16) -0.01(0.09) 0.03(0.09)

Table 6.13: Marital Satisfaction Data: Parameter estimates for the dropout model used to determine theweights for the WGEE.

Par. Estimate

Int -1.32(0.76)

Prev 0.23(0.25)

Parent 1 -4.33(0.71)

Year 1995 -0.33(0.16)

Birth 0.01(0.02)

4. Binary response using GLMMAs an alternative to GEE, a generalized linear mixed model was performed. Again considerable differencesare noted between the complete case estimates and the direct likelihood approach. Note that the estimatesare not comparable to those based on GEE.

5. General Conclusion:Further analyses are needed to investigate the nature of the missingness process and the appropriateness ofthe models used.


Table 6.14: Marital Satisfaction Data: Parameter estimates based on a generalized linear mixed model usingcompleters and direct likelihood.

Par. CC Direct

Likelihood

Int 1.79(6.89) 0.81(4.77)

Parent 0 3.11(7.98) 2.94(6.26)

Year 1990 1.94(8.13) -1.68(5.03)

Year 1995 -0.07(9.31) 3.30(5.68)

Birth 0.05(0.14) 0.06(0.10)

Year 1990 Parent 0 2.57(10.44) 0.95(6.72)

Year 1995 Parent 0 2.76(11.99) 5.31(7.70)

Birth Parent 0 -0.07(0.16) -0.06(0.13)

Birth Year 1990 -0.05(0.16) 0.03(0.10)

Birth Year 1995 0.01(0.19) -0.05(0.11)

Birth Year 1990 Parent 0 -0.04(0.21) -0.02(0.14)

Birth Year 1995 Parent 0 -0.05(0.25) -0.12(0.16)

Variance 2.16(0.38) 2.13(0.21)

esalq course on models for longitudinal and incomplete

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