Example 22.7: Repeated Measures, 4 Response Levels, 1 Population
This example illustrates a repeated measurement analysis in
which there are more than two levels of response. In this
study, from Grizzle, Starmer, and Koch (1969, p. 493), 7477
women aged 30 39 are tested for vision in both right
and left eyes. Since there are four response levels for
each dependent variable, the
RESPONSE statement computes three marginal probabilities for
each dependent variable, resulting in six response functions
for analysis. Since the model contains a repeated
measurement factor (Side) with two levels (
Right, Left), PROC CATMOD groups the functions into
sets of three (=6/2). Therefore, the Side effect has
three degrees of freedom (one for each marginal
probability), and it is the appropriate test of marginal
homogeneity. The following statements produce Output 22.7.1
through Output 22.7.5:
title 'Vision Symmetry';
data vision;
input Right Left count @@;
datalines;
1 1 1520 1 2 266 1 3 124 1 4 66
2 1 234 2 2 1512 2 3 432 2 4 78
3 1 117 3 2 362 3 3 1772 3 4 205
4 1 36 4 2 82 4 3 179 4 4 492
;
proc catmod data=vision;
weight count;
response marginals;
model Right*Left=_response_ / freq;
repeated Side 2;
title2 'Test of Marginal Homogeneity';
quit;
Output 22.7.1: Vision Study: Analysis of Marginal Homogeneity
Vision Symmetry 
Test of Marginal Homogeneity 
Response 
Right*Left 
Response Levels 
16 
Weight Variable 
count 
Populations 
1 
Data Set 
VISION 
Total Frequency 
7477 
Frequency Missing 
0 
Observations 
16 
Sample 
Sample Size 
1 
7477 

Output 22.7.2: Response Profiles
Vision Symmetry 
Test of Marginal Homogeneity 
Response Profiles 
Response 
Right 
Left 
1 
1 
1 
2 
1 
2 
3 
1 
3 
4 
1 
4 
5 
2 
1 
6 
2 
2 
7 
2 
3 
8 
2 
4 
9 
3 
1 
10 
3 
2 
11 
3 
3 
12 
3 
4 
13 
4 
1 
14 
4 
2 
15 
4 
3 
16 
4 
4 
Response Frequencies 
Sample 
Response Number 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
1 
1520 
266 
124 
66 
234 
1512 
432 
78 
117 
362 
1772 
205 
36 
82 
179 
492 

Output 22.7.3: Design Matrix
Vision Symmetry 
Test of Marginal Homogeneity 
Sample 
Function Number 
Response Function 
Design Matrix 
1 
2 
3 
4 
5 
6 
1 
1 
0.26428 
1 
0 
0 
1 
0 
0 

2 
0.30173 
0 
1 
0 
0 
1 
0 

3 
0.32847 
0 
0 
1 
0 
0 
1 

4 
0.25505 
1 
0 
0 
1 
0 
0 

5 
0.29718 
0 
1 
0 
0 
1 
0 

6 
0.33529 
0 
0 
1 
0 
0 
1 

Output 22.7.4: ANOVA Table
Vision Symmetry 
Test of Marginal Homogeneity 
Analysis of Variance 
Source 
DF 
ChiSquare 
Pr > ChiSq 
Intercept 
3 
78744.17 
<.0001 
Side 
3 
11.98 
0.0075 
Residual 
0 
. 
. 

Output 22.7.5: Parameter Estimates
Vision Symmetry 
Test of Marginal Homogeneity 
Analysis of Weighted Least Squares Estimates 
Effect 
Parameter 
Estimate 
Standard Error 
Chi Square 
Pr > ChiSq 
Intercept 
1 
0.2597 
0.00468 
3073.03 
<.0001 

2 
0.2995 
0.00464 
4160.17 
<.0001 

3 
0.3319 
0.00483 
4725.25 
<.0001 
Side 
4 
0.00461 
0.00194 
5.65 
0.0174 

5 
0.00227 
0.00255 
0.80 
0.3726 

6 
0.00341 
0.00252 
1.83 
0.1757 

The analysis of variance table in Output 22.7.4 shows that
the Side effect is significant, so there is not marginal
homogeneity between lefteye vision and righteye vision.
In other words, the distribution of the quality of righteye
vision differs significantly from the quality of lefteye
vision in the same subjects. The test of the Side
effect is equivalent to Bhapkar's test (Agresti 1990).
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.