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 The FREQ Procedure

Example 28.9: Testing Marginal Homogeneity with Cochran's Q

When a binary response is measured several times or under different conditions, Cochran's Q tests that the marginal probability of a positive response is unchanged across the times or conditions. When there are more than two response categories, you can use the CATMOD procedure to fit a repeated-measures model.

The data set Drugs contains data for a study of three drugs to treat a chronic disease (Agresti 1990). Forty-six subjects receive drugs A, B, and C. The response to each drug is either favorable ('F') or unfavorable ('U').

```   proc format;
value \$ResponseFmt 'F'='Favorable'
'U'='Unfavorable';

data drugs;
input Drug_A \$ Drug_B \$ Drug_C \$ Count @@;
datalines;
F F F  6   U F F  2
F F U 16   U F U  4
F U F  2   U U F  6
F U U  4   U U U  6
;
```

The following statements create one-way frequency tables of the responses to each drug. The AGREE option produces Cochran's Q and other measures of agreement for the three-way table. These statements produce Output 28.9.1 through Output 28.9.3.

```   proc freq data=Drugs;
weight Count;
tables Drug_A Drug_B Drug_C / nocum;
tables Drug_A*Drug_B*Drug_C / agree noprint;
format Drug_A Drug_B Drug_C \$ResponseFmt.;
title 'Study of Three Drug Treatments for a Chronic Disease';
run;
```

Output 28.9.1: One-Way Frequency Tables

 Study of Three Drug Treatments for a Chronic Disease
 The FREQ Procedure
 Drug_A Frequency Percent Favorable 28 60.87 Unfavorable 18 39.13

 Drug_B Frequency Percent Favorable 28 60.87 Unfavorable 18 39.13

 Drug_C Frequency Percent Favorable 16 34.78 Unfavorable 30 65.22

The one-way frequency tables in Output 28.9.1 provide the marginal response for each drug. For drugs A and B, 61% of the subjects reported a favorable response while 35% of the subjects reported a favorable response to drug C.

Output 28.9.2: Measures of Agreement

 Study of Three Drug Treatments for a Chronic Disease
 The FREQ Procedure
 Statistics for Table 1 of Drug_B by Drug_CControlling for Drug_A=Favorable

 McNemar's Test Statistic (S) 10.8889 DF 1 Pr > S 0.0010

 Simple Kappa Coefficient Kappa -0.0328 ASE 0.1167 95% Lower Conf Limit -0.2615 95% Upper Conf Limit 0.1960
 Sample Size = 28

 Statistics for Table 2 of Drug_B by Drug_CControlling for Drug_A=Unfavorable

 McNemar's Test Statistic (S) 0.4000 DF 1 Pr > S 0.5271

 Simple Kappa Coefficient Kappa -0.1538 ASE 0.2230 95% Lower Conf Limit -0.5909 95% Upper Conf Limit 0.2832
 Sample Size = 18

 Study of Three Drug Treatments for a Chronic Disease
 The FREQ Procedure
 Summary Statistics for Drug_B by Drug_CControlling for Drug_A

 Overall Kappa Coefficient Kappa -0.0588 ASE 0.1034 95% Lower Conf Limit -0.2615 95% Upper Conf Limit 0.1439

 Test for Equal Kappa Coefficients Chi-Square 0.2314 DF 1 Pr > ChiSq 0.6305
 Total Sample Size = 46

McNemar's test (Output 28.9.2) shows strong discordance between drugs B and C when the response to drug A is favorable. The small negative value of the simple kappa indicates no agreement between drug B response and drug C response.

Output 28.9.3: Cochran's Q

 Study of Three Drug Treatments for a Chronic Disease
 The FREQ Procedure
 Summary Statistics for Drug_B by Drug_CControlling for Drug_A

 Cochran's Q, for Drug_A byDrug_B by Drug_C Statistic (Q) 8.4706 DF 2 Pr > Q 0.0145
 Total Sample Size = 46

Cochran's Q is statistically significant (p=0.0144 in Output 28.9.3), which leads to rejection of the hypothesis that the probability of favorable response is the same for the three drugs.

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