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 |
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) .