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The CATMOD Procedure

Example 28.9 Repeated Measures, Two Repeated Measurement Factors

This example, from MacMillan et al. (1981), illustrates a repeated measures analysis in which there are two repeated measurement factors. Two diagnostic procedures (standard and test) are performed on each subject, and the results of both are evaluated at each of two times as being positive or negative. In the following DATA step, std1 and std2 are the two measurements of the standard procedure, and test1 and test2 are the two measurements of the test procedure:

   data a;
      input std1 $ test1 $ std2 $ test2 $ wt @@;
      datalines;
   neg neg neg neg 509  neg neg neg pos  4  neg neg pos neg  17
   neg neg pos pos   3  neg pos neg neg 13  neg pos neg pos   8
   neg pos pos pos   8  pos neg neg neg 14  pos neg neg pos   1
   pos neg pos neg  17  pos neg pos pos  9  pos pos neg neg   7
   pos pos neg pos   4  pos pos pos neg  9  pos pos pos pos 170
   ;

For the initial model, the response functions are marginal probabilities, and the repeated measurement factors are Time and Treatment. The model is a saturated one, containing effects for Time, Treatment, and Time*Treatment. The following statements produce Output 28.9.1:

   proc catmod data=a;
      title2 'Marginal Symmetry, Saturated Model';
      weight wt;
      response marginals;
      model std1*test1*std2*test2=_response_ / freq design noparm;
      repeated Time 2, Treatment 2 / _response_=Time Treatment 
               Time*Treatment;
   run;

The analysis of variance table in Output 28.9.1 shows that there is no significant effect of Time, either by itself or in its interaction with Treatment. The second model includes only the Treatment effect. Again, the response functions are marginal probabilities, and the repeated measurement factors are Time and Treatment.

Output 28.9.1 Diagnosis Data: Two Repeated Measurement Factors

Diagnostic Procedure Comparison

Marginal Symmetry, Saturated Model

The CATMOD Procedure

Data Summary
Response	std1*test1*std2*test2	Response Levels	15
Weight Variable	wt	Populations	1
Data Set	A	Total Frequency	793
Frequency Missing	0	Observations	15

Population Profiles
Sample	Sample Size
1	793

Response Profiles
Response	std1	test1	std2	test2
1	neg	neg	neg	neg
2	neg	neg	neg	pos
3	neg	neg	pos	neg
4	neg	neg	pos	pos
5	neg	pos	neg	neg
6	neg	pos	neg	pos
7	neg	pos	pos	pos
8	pos	neg	neg	neg
9	pos	neg	neg	pos
10	pos	neg	pos	neg
11	pos	neg	pos	pos
12	pos	pos	neg	neg
13	pos	pos	neg	pos
14	pos	pos	pos	neg
15	pos	pos	pos	pos

Response Frequencies
Sample	Response Number
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	509	4	17	3	13	8	8	14	1	17	9	7	4	9	170

Response Functions and Design Matrix
Sample	Function Number	Response Function	Design Matrix
			1	2	3	4
1	1	0.70870	1	1	1	1
	2	0.72383	1	1	-1	-1
	3	0.70618	1	-1	1	-1
	4	0.73897	1	-1	-1	1

Analysis of Variance
Source	DF	Chi-Square	Pr > ChiSq
Intercept	1	2385.34	<.0001
Time	1	0.85	0.3570
Treatment	1	8.20	0.0042
Time*Treatment	1	2.40	0.1215
Residual	0	.	.

A main effect model with respect to Treatment is fit. The following statements produces Output 28.9.2:

      title2 'Marginal Symmetry, Reduced Model';
      model std1*test1*std2*test2=_response_ / corrb design noprofile;
      repeated Time 2, Treatment 2 / _response_=Treatment;
   run;

The analysis of variance table for the reduced model (Output 28.9.2) shows that the model fits (since the residual chi-square is nonsignificant) and that the treatment effect is significant. The negative parameter estimate for Treatment shows that the first level of treatment (std) has a smaller probability of the first response level (neg) than the second level of treatment (test). In other words, the standard diagnostic procedure gives a significantly higher probability of a positive response than the test diagnostic procedure.

The next example illustrates a RESPONSE statement that, at each time, computes the sensitivity and specificity of the test diagnostic procedure with respect to the standard procedure. Since these are measures of the relative accuracy of the two diagnostic procedures, the repeated measurement factors in this case are labeled Time and Accuracy. Only 15 of the 16 possible responses are observed, so additional care must be taken in formulating the RESPONSE statement for computation of sensitivity and specificity.

The following statements produce Output 28.9.3 and Output 28.9.4:

      title2 'Sensitivity and Specificity Analysis, ' 
             'Main-Effects Model';
      model std1*test1*std2*test2=_response_ / covb design noprofile;
      repeated Time 2, Accuracy 2 / _response_=Time Accuracy;
      response exp 1 -1  0  0  0  0  0  0,
                   0  0  1 -1  0  0  0  0,
                   0  0  0  0  1 -1  0  0,
                   0  0  0  0  0  0  1 -1
   
               log 0 0 0 0   0 0  0   0 0 0 0   1 1 1 1,
                   0 0 0 0   0 0  0   1 1 1 1   1 1 1 1,
                   1 1 1 1   0 0  0   0 0 0 0   0 0 0 0,
                   1 1 1 1   1 1  1   0 0 0 0   0 0 0 0,
                   0 0 0 1   0 0  1   0 0 0 1   0 0 0 1,
                   0 0 1 1   0 0  1   0 0 1 1   0 0 1 1,
                   1 0 0 0   1 0  0   1 0 0 0   1 0 0 0,
                   1 1 0 0   1 1  0   1 1 0 0   1 1 0 0;
   quit;

For the sensitivity and specificity analysis, the four response functions displayed next to the design matrix (Output 28.9.3) represent the following:

1. sensitivity, time 1

2. specificity, time 1

3. sensitivity, time 2

4. specificity, time 2

The sensitivities and specificities are for the test diagnostic procedure relative to the standard procedure.

The ANOVA table in Output 28.9.3 shows that an additive model fits, that there is a significant effect of time, and that the sensitivity is significantly different from the specificity.

Output 28.9.4 shows that the predicted sensitivities and specificities are lower for time 1 (since parameter 2 is negative). It also shows that the sensitivity is significantly less than the specificity.

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