### Example 30.7 Repeated Measures, 4 Response Levels, 1 Population

This example illustrates a repeated measures analysis in which there are more than two levels of response. In this study, from Grizzle, Starmer, and Koch (1969, p. 493), 7,477 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 30.7.1:

```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 design;
repeated Side 2;
title2 'Test of Marginal Homogeneity';
quit;
```

Output 30.7.1: Vision Study: Analysis of Marginal Homogeneity

 Vision Symmetry Test of Marginal Homogeneity

The CATMOD Procedure

Data Summary
Response Right*Left Response Levels 16
Weight Variable count Populations 1
Data Set VISION Total Frequency 7477
Frequency Missing 0 Observations 16

Population Profiles
Sample Sample Size
1 7477

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

Response Functions and Design Matrix
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

Analysis of Variance
Source DF Chi-Square Pr > ChiSq
Intercept 3 78744.17 <.0001
Side 3 11.98 0.0075
Residual 0 . .

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 30.7.1 shows that the `Side` effect is significant, so there is not marginal homogeneity between left-eye vision and right-eye vision. In other words, the distribution of the quality of right-eye vision differs significantly from the distribution of the quality of left-eye vision in the same subjects. The test of the `Side` effect is equivalent to Bhapkar’s test (Agresti, 1990) .