March 28, 30

• Return exam• Analyses of covariance• 2-way ANOVA• Analyses of binary outcomes

Exam 1 ScoresN = 23

75% Q3 9550% Median 8325% Q1 74

Mean = 83.7SD = 11.1

90-100 A89-89 B70-79 C

30% of Grade30% Exam 230% Assignments10% Project

Analysis of Variance

ANOVA simultaneously tests for difference in k means

• Y - continuous• k samples from k normal distributions

• each size ni, not necessarily equal

• each with possibly different mean• each with constant variance 2

Analyses of Covariance

• Comparing k means adjusting for 1 or more other variables (covariates)

• Uses– Randomized students to adjust for an imbalance among

treatments in a baseline factor.– In observational studies controlling for a confounding

factor– To throw light on the nature of treatment effects in a

randomized study– Improve precision of estimated differences among

treatments

Analyses of Covariance

• Comparing k means adjusting for 1 or more other variables (covariates)

• Compute adjusted (or least square) means. • Sometimes called ANCOVA

Original Use

• Fisher (1941) the Y were yields of tea bushes in an experiment.

• But the luck of the draw, some treatments will have been allotted to a more productive set of bushes than other treatments

• Use as an adjustment variable the yields of the bushes in the previous year.

Adjusted Means Computation

YBARi = Mean of Y for group I

XBARi = Mean of X for group I

XBAR = Mean of X for all groups combined

= Regression slope of X with Y

YBAR(A)i = Adjusted mean for group I

YBAR(A)i = YBARi – XBARi – XBAR)

Adjustment

Adjusted Means ComputationObservations

YBAR(A)i = YBARi – XBARi – XBAR)

1) If then adjusted mean equals unadjusted mean

2) If mean of X is same for all group then adjusted mean equals unadjusted mean

Adjusted Mean Interpretation

The mean of Y for the group if the mean of X for the group was at the overall mean.

Uses a model to make the mean of X the same for all groups

What would the means have been if all groups had the same mean of X?

Example from TOMHS

Compare 12-month visit mean serum cholesterol between diuretic group and placebo group.

12-mo Avg. Baseline Avg.Diuretic 231.7 230.7Placebo 219.7 224.9

Diff: 12.0 5.8

Note: Diuretic group started out with higher cholesterols so may want to adjust for this difference.

Computing the Adjusted Means

12-mo Avg. Baseline Avg.Diuretic 231.7 230.7Placebo 219.7 224.9Total 227.0

=0.894 Regression slope of 12-month cholesterol onbaseline cholesterol

YBAR(A) (Diur) = 231.7 – 0.894 (230.7 – 227.0)= 231.7 – 0.894 (3.7) = 228.4

YBAR(A) (Plac) = 219.7 – 0.894 (224.9 – 227.0)= 219.7 – 0.894 (-3.7) = 221.6

6.8

SAS Code

PROC GLM; CLASS group; MODEL chol12 = group cholbl/SS3 SOLUTION; MEANS group; LSMEANS group; ESTIMATE ‘Adjusted Mean Dif' group 1 -1;RUN;

SAS GLM Output

Source DF Type III SS Mean Square F Value Pr > F

GROUP 1 3666.8314 3666.8314 6.29 0.0126cholbl 1 393014.2008 393014.2008 674.26 <.0001

StandardParameter Estimate Error t Value Pr > |t|

Intercept 18.77666226 B 7.90780076 2.37 0.0181GROUP 3 6.79524641 B 2.70925949 2.51 0.0126GROUP 6 0.00000000 B . . .cholbl 0.89351891 0.03441046 25.97 <.0001

Regression slope

SOLUTION

SAS GLM Output

Level of ------------CHOL12----------- ------------cholbl-----------GROUP N Mean Std Dev Mean Std Dev

3 125 231.696000 46.2561633 230.688000 38.96947036 221 219.737557 38.5904356 224.909502 37.1702588

Least Squares Means

CHOL12GROUP LSMEAN

3 228.3981206 221.602873

StandardParameter Estimate Error t Value Pr > |t|

dif 6.79524641 2.70925949 2.51 0.0126

LSMEANS group;

ESTIMATE 'dif' group 1 -1;

dif

Two-Way ANOVA

• Two categorical factors related to a continuous outcome (Factor A and factor B). If factors are allocated randomly to all combinations of A and B then design called factorial design

• Questions asked– Overall is A related to Y– Overall is B related to Y– Does the effect of A on Y depend on level of B

• Example– A = Race; B = BP drug; Y = BP response– A = Vitamin E (y/n); aspirin use (y/n)

Factorial Design Example

Aspirin +

Vitamin E

Aspirin +

Placebo forVitamin E

Placebo for Aspirin

+Vitamin E

Placebo for Aspirin

+Placebo forVitamin E

A = Aspirin use (yes or no)

B = Vitamin E use (yes or no)

Placebo for aspirin and placebo for Vitamin E

TOMHS Example

Question:Do certain BP medications differ in lowering blood pressure in blacks compared to whites?

Change in SBP (mm Hg)Diuretic Blocker

Blacks -23.6 -8.7Whites -21.4 -17.8Difference -2.2 +9.1

Is the difference –2.2 significantly different from +9.1

SAS Code

LIBNAME tomhs 'C:\my documents\ph5415\';

DATA temp; SET tomhs.bpstudy;* Choose diuretic and alpha blocker groups;* Variable black = 1 or 2; if group in(3,4); sbpdif = sbp12 - sbpbl;RUN;

PROC GLM DATA=temp; CLASS black group; MODEL sbpdif = black group black*group; MEANS black*group;RUN;

Tests for interaction

SAS OUTPUTThe GLM Procedure

Level of Level of ------------sbpdif-----------

GROUP BLACK N Mean Std Dev

3 1 27 -23.6296296 14.6442379

3 2 97 -21.3505155 14.4939220

4 1 24 -8.7291667 17.5802379

4 2 104 -17.8125000 12.5978091

SAS OUTPUTThe GLM Procedure

Dependent Variable: sbpdif

Sum ofSource DF Squares Mean Square F Value Pr > F

Model 3 3791.89107 1263.96369 6.37 0.0004

Error 248 49197.96210 198.37888

Corrected Total 251 52989.85317

Source DF Type III SS Mean Square F Value Pr > F

GROUP 1 3447.055834 3447.055834 17.38 <.0001

BLACK 1 469.412605 469.412605 2.37 0.1253

GROUP*BLACK 1 1309.006907 1309.006907 6.60 0.0108

Your Turn

• Using TOMHS data test