Example 28.9: Testing Marginal Homogeneity with Cochran's Q

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_C
Controlling 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_C
Controlling 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_C
Controlling 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_C
Controlling for Drug_A

Cochran's Q, for Drug_A by Drug_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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