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 The CATMOD Procedure

## Example 28.6 Repeated Measures, 2 Response Levels, 3 Populations

In this multiple-population repeated measures example, from Guthrie (1981), subjects from three groups have their responses (0 or 1) recorded in each of four trials. The analysis of the marginal probabilities is directed at assessing the main effects of the repeated measurement factor (Trial) and the independent variable (Group), as well as their interaction. Although the contingency table is incomplete (only 13 of the 16 possible responses are observed), this poses no problem in the computation of the marginal probabilities. The following statements produce Output 28.6.1:

```data group;
input a b c d Group wt @@;
datalines;
1 1 1 1 2 2     0 0 0 0 2 2     0 0 1 0 1 2     0 0 1 0 2 2
0 0 0 1 1 4     0 0 0 1 2 1     0 0 0 1 3 3     1 0 0 1 2 1
0 0 1 1 1 1     0 0 1 1 2 2     0 0 1 1 3 5     0 1 0 0 1 4
0 1 0 0 2 1     0 1 0 1 2 1     0 1 0 1 3 2     0 1 1 0 3 1
1 0 0 0 1 3     1 0 0 0 2 1     0 1 1 1 2 1     0 1 1 1 3 2
1 0 1 0 1 1     1 0 1 1 2 1     1 0 1 1 3 2
;
```

```title 'Multiple-Population Repeated Measures';
proc catmod data=group;
weight wt;
response marginals;
model a*b*c*d=Group _response_ Group*_response_
/ freq;
repeated Trial 4;
title2 'Saturated Model';
run;
```

Output 28.6.1 Analysis of Multiple-Population Repeated Measures
 Multiple-Population Repeated Measures Saturated Model

The CATMOD Procedure

Data Summary
Response a*b*c*d Response Levels 13
Weight Variable wt Populations 3
Data Set GROUP Total Frequency 45
Frequency Missing 0 Observations 23

Population Profiles
Sample Group Sample Size
1 1 15
2 2 15
3 3 15

Response Profiles
Response a b c d
1 0 0 0 0
2 0 0 0 1
3 0 0 1 0
4 0 0 1 1
5 0 1 0 0
6 0 1 0 1
7 0 1 1 0
8 0 1 1 1
9 1 0 0 0
10 1 0 0 1
11 1 0 1 0
12 1 0 1 1
13 1 1 1 1

Response Frequencies
Sample Response Number
1 2 3 4 5 6 7 8 9 10 11 12 13
1 0 4 2 1 4 0 0 0 3 0 1 0 0
2 2 1 2 2 1 1 0 1 1 1 0 1 2
3 0 3 0 5 0 2 1 2 0 0 0 2 0

Analysis of Variance
Source DF Chi-Square Pr > ChiSq
Intercept 1 354.88 <.0001
Group 2 24.79 <.0001
Trial 3 21.45 <.0001
Group*Trial 6 18.71 0.0047
Residual 0 . .

Analysis of Weighted Least Squares Estimates
Effect Parameter Estimate Standard
Error
Chi-
Square
Pr > ChiSq
Intercept 1 0.5833 0.0310 354.88 <.0001
Group 2 0.1333 0.0335 15.88 <.0001
3 -0.0333 0.0551 0.37 0.5450
Trial 4 0.1722 0.0557 9.57 0.0020
5 0.1056 0.0647 2.66 0.1028
6 -0.0722 0.0577 1.57 0.2107
Group*Trial 7 -0.1556 0.0852 3.33 0.0679
8 -0.0556 0.0800 0.48 0.4877
9 -0.0889 0.0953 0.87 0.3511
10 0.0111 0.0866 0.02 0.8979
11 0.0889 0.0822 1.17 0.2793
12 -0.0111 0.0824 0.02 0.8927

The analysis of variance table in Output 28.6.1 shows that there is a significant interaction between the independent variable Group and the repeated measurement factor Trial. An intermediate model (not shown) is fit in which the effects Trial and Group* Trial are replaced by Trial(Group=1), Trial(Group=2), and Trial(Group=3). Of these three effects, only the last is significant, so it is retained in the final model. The following statements produce Output 28.6.2 and Output 28.6.3:

```   model a*b*c*d=Group _response_(Group=3)
/ noprofile noparm design;
title2 'Trial Nested within Group 3';
quit;
```

Output 28.6.2 displays the design matrix resulting from retaining the nested effect.

Output 28.6.2 Final Model: Design Matrix
 Multi-Population Repeated Measures Trial Nested within Group 3

The CATMOD Procedure

Data Summary
Response a*b*c*d Response Levels 13
Weight Variable wt Populations 3
Data Set GROUP Total Frequency 45
Frequency Missing 0 Observations 23

Response Functions and Design Matrix
Sample Function
Number
Response
Function
Design Matrix
1 2 3 4 5 6
1 1 0.73333 1 1 0 0 0 0
2 0.73333 1 1 0 0 0 0
3 0.73333 1 1 0 0 0 0
4 0.66667 1 1 0 0 0 0
2 1 0.66667 1 0 1 0 0 0
2 0.66667 1 0 1 0 0 0
3 0.46667 1 0 1 0 0 0
4 0.40000 1 0 1 0 0 0
3 1 0.86667 1 -1 -1 1 0 0
2 0.66667 1 -1 -1 0 1 0
3 0.33333 1 -1 -1 0 0 1
4 0.06667 1 -1 -1 -1 -1 -1

The residual goodness-of-fit statistic tests the joint effect of Trial(Group=1) and Trial(Group=2). The analysis of variance table in Output 28.6.3 shows that the final model fits, that there is a significant Group effect, and that there is a significant Trial effect in Group 3.

Output 28.6.3 ANOVA Table
Analysis of Variance
Source DF Chi-Square Pr > ChiSq
Intercept 1 386.94 <.0001
Group 2 25.42 <.0001
Trial(Group=3) 3 75.07 <.0001
Residual 6 5.09 0.5319

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