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 The CATMOD Procedure

## 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

 The CATMOD Procedure

 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

 The CATMOD Procedure

 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

 The CATMOD Procedure

 Sample FunctionNumber ResponseFunction 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

 The CATMOD Procedure

 Analysis of Variance Source DF Chi-Square 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

 The CATMOD Procedure

 Analysis of Weighted Least Squares Estimates Effect Parameter Estimate StandardError 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 left-eye vision and right-eye vision. In other words, the distribution of the quality of right-eye vision differs significantly from the quality of left-eye vision in the same subjects. The test of the Side effect is equivalent to Bhapkar's test (Agresti 1990).

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