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 |
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 |
| 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 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).
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
