weekend workshop i proc mixed. random or fixed ?randomfixedlevels: selected at random from infinite...
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Weekend Workshop IWeekend Workshop I
PROC MIXED
Random or Fixed ?
RANDOMRANDOM FIXEDFIXED
Levels:Levels: Selected at random Selected at random from infinite populationfrom infinite population
Finite number of Finite number of possibilitiespossibilities
Another ExperimentAnother Experiment Different selections Different selections from same populationfrom same population
Same LevelsSame Levels
GoalGoal Estimate variance Estimate variance componentscomponents
Compare meansCompare means
InferenceInference All levels in populationAll levels in population Only levels used in the Only levels used in the experiment.experiment.
Twins: One gets SAS training method 1, the other gets method 2
Response Y = programming times
PROC MIXED ModelPROC MIXED Model
38.25
33.00
28.75
30.00
46.50
55.25
1
2
1 0 1
1 1 0
1 0 1
1 1 0
1 1 0
1 0 1
M
M
1
2
3
1 0 0
1 0 0
0 1 0
0 1 0
0 0 1
0 0 1
F
F
F
1
2
3
4
5
6
e
e
e
e
e
e
Y = X Z + e
Variance of is G = ,Variance of e is R = 2
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
2
1 0 0
0 1 0
0 0 1F
Model ; Random ; Repeated ;
PROC MIXED DATA=TWINS; CLASS FAMILY METHOD; MODEL TIME = METHOD; * fixed; RANDOM FAMILY; *<- family ~ N(0, 2
F) ; Covariance Parameter Estimates
Cov Parm Estimate family 21.2184 Residual 40.8338
Type 3 Tests of Fixed Effects
Num Den Effect DF DF F Value Pr > F method 1 19 9.60 0.0059
Intraclass correlation (related to heritability) 2
F /(2F + 2)
Estimated as 21.2/62 or about 1/3. Q: Why not usual (Pearson) correlation?
DemoDemoGet_Twins.sasGet_Twins.sas
Twins_MIXED.sasTwins_MIXED.sas
BLUPBLUP
Yij = + Fi + eij
Di = Family mean – Fi + ei.best estimate of Fi = ?
Variance of (Fi – b Di) is (1-b)22F + b2 2/2
Use b = 2F /(2
F + 2/2) Estimate: b = 21.2/(21.2 + 40.8/2) = 0.510
Overall mean + 0.510(Family i mean – Overall mean) PROC MIXED DATA=TWINS; CLASS FAMILY METHOD; MODEL TIME = METHOD; RANDOM FAMILY; ESTIMATE "1 " intercept 1 | family 1;ESTIMATE "2 " intercept 1 | family 0 1;
PROC GLM DATA=TWINS; CLASS FAMILY METHOD; MODEL TIME = FAMILY METHOD; LSMEANS FAMILY;
MEANS andMEANS and BLUPsBLUPs
(MIXED)(GLM)
DemoDemo Twins_BLUP.sasTwins_BLUP.sasTwins_TEST.sasTwins_TEST.sas
REML EstimationREML Estimation(1)(1)Regress out fixed effectsRegress out fixed effects(2)(2)Maximze likelihood of residuals (mean known: 0)Maximze likelihood of residuals (mean known: 0)(3)(3)Variance estimates less biased (unbiased in some Variance estimates less biased (unbiased in some simple cases) simple cases)
ML EstimationML Estimation Search over all (fixed Search over all (fixed andand random) parameters random) parameters
Estimates of variances biased low! Estimates of variances biased low!
Unbalanced DataUnbalanced Data
SUBJ Ear plug
A B C D E F G
I 25 (L) 19 (L) 29 (R) 16 (R) 25 (L)
II 8 (R) 7 (L) 23 (L) 16 (R) 24 (R)
III 22 (R) 7 (R) 14 (L) 12 (L)
I vs. III free of subject effects for red data. Misses info in other data.
proc glm; class plug worker; model loss = worker plug; Random Worker;Estimate "I vs III - GLM" Plug -1 0 1; run;proc mixed; class plug worker; model Loss=Plug; Random Worker; Estimate "I vs III - Mixed" Plug -1 0 1; run;
GLMSource DF Type III SS F Value Pr > F worker 6 451.9062500 12.21 0.0074 plug 2 62.6562500 5.08 0.0625 StandardParameter Estimate Error t Value Pr > |t|I vs III - GLM -4.8125 1.9635 -2.45 0.0579
Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F plug 2 5 5.79 0.0499 Estimates StandardLabel Estimate Error DF t Value Pr > |t|I vs III - Mixed -5.2448 1.9347 5 -2.71 0.0422
Covariance Parameter Estimates
Cov Parm Estimate
worker 37.578 Residual 6.1674
DemoDemo Earplugs.sasEarplugs.sas
Soil Variety 1 1 1 2 2 1 2 2 3 1 3 2
4 Aquariums, 2 aerated2 notsix dishes / aquarium
one plant / dish soil x variety combinations
ANOVA
SourceAir Air Error AError A V S VA VS AS AVSError B
SPLIT PLOT
PROC MIXED; CLASS VAR AQUARIUM SOIL AIR; MODEL YIELD = AIR SOIL VAR SOIL*VAR AIR*SOIL AIR*VAR AIR*SOIL*VAR / DDFM=SATTERTHWAITE;RANDOM AQUARIUM(AIR);
ESTIMATE "SOIL 1: AIR EFFECT" AIR -1 1 AIR*SOIL -1 1 0 0 0 0; RUN;
Compare Air to No Air within soil 1Variance of this contrast is hard to figure out:
(1/3)[MS(A)+2 MS(B)]
Need Satterthwaite df AUTOMATIC IN MIXED!!!
Covariance Parameter Estimates
Cov Parm Estimate
AQUARIUM(AIR) 2.1833
Residual 7.7333
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
AIR 1 2 16.20 0.0565 SOIL 2 10 7.87 0.0088
VAR 1 10 24.91 0.0005
VAR*SOIL 2 10 0.04 0.9631
SOIL*AIR 2 10 1.08 0.3752
VAR*AIR 1 10 4.22 0.0669
VAR*SOIL*AIR 2 10 0.23 0.7973
Standard Label Estimate Error DF t Value Pr > |t|
SOIL 1: AIR EFFECT 5.2500 2.4597 5.47 2.13 0.0812
DemoDemo Aquarium.sasAquarium.sas
Random Coefficient ModelsRandom Coefficient Models the basic ideathe basic idea
mistakesmistakes
Program writing time
Average programmer
Dave
Line for individual j: (a0 + aj) + ( b0 + bj )t
2
2
,0
0~
BAB
ABA
j
j Nb
a
a0 + b0 t
Hierarchial ModelsHierarchial Models(1)(1)Same as split plot - Same as split plot - almost almost (2)(2)Whole and split level Whole and split level continuouscontinuous predictor predictor variables (typically)variables (typically)
(1)(1)Aquarium level (level i): pHAquarium level (level i): pHii
(2)(2)Dish level: Soil nitrogen test (NDish level: Soil nitrogen test (Nijij))
YYijij = a = aii + b + bii N Nijij+e+eijij
(3) Idea: a(3) Idea: aii = = 00 + + 11 pH pHii + a + aii**
bbii = = 00 + + 11 pH pHii + b + bii* *
YYijij = a = aii + b + bii N Nijij+e+eijijYYijij = = 00 + + 11 pH pHii + a + aii* * + b+ bii N Nijij+e+eijijYYijij = = 00 + + 11 pH pHii + a + aii
* * + + ((00 + + 11 pH pHii + b + bii* * ) ) NNijij+e+eijij
YYijij = = [[00 + + 11 pH pHii + + 00 N Nij ij + + 11 pH pHii N Nijij] ] + + [a[aii** +b +bii
* * NNijij+e+eijij]]
fixed fixed randomrandom
PROC MIXED DATA = UNDERWATER; MODEL GROWTH = N P N*P; RANDOM INTERCEPT N / SUBJECT = TANK TYPE=UN;
p
Num DenEffect DF DF F Value Pr > FN 1 2 3.50 0.2018pH 1 2.05 6.76 0.1186N*pH 1 2 1.31 0.3702
aquarium N pH growth
1 2.21 5.5 27.05 1 1.25 5.5 25.92 1 4.36 5.5 30.09 1 7.14 5.5 33.66 1 8.61 5.5 36.13 1 6.53 5.5 33.00 2 6.58 4.7 35.72 2 3.12 4.7 31.17 2 5.28 4.7 34.35 2 1.09 4.7 28.34 2 4.83 4.7 33.56 2 9.61 4.7 40.25 3 7.99 4.2 47.04 3 7.79 4.2 46.56 3 8.32 4.2 48.27 3 2.53 4.2 34.20 3 6.85 4.2 44.59 3 4.73 4.2 39.29 4 0.95 5.1 24.94 4 2.00 5.1 27.33 4 9.99 5.1 43.84 4 0.23 5.1 23.54 4 0.13 5.1 23.56 4 1.17 5.1 25.68
Num DenEffect DF DF F Value Pr > FN 1 3 50.19 0.0058pH 1 2.03 14.68 0.0603
Cov Parm EstimateUN(1,1) 1.8976UN(2,1) -0.5563UN(2,2) 0.2596Residual 0.0286
pHN
DemoDemo Hierarchial.sasHierarchial.sas
Next: Repeated Measures
Notes in pdf from NCSU experimental design class(ST 711)
DemoDemo SURGERY.sasSURGERY.sas