The FREQ Procedure

Example 40.10 Cochran’s Q Test

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 2002). 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';
run;
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 40.10.1 through Output 40.10.5.

proc freq data=Drugs;
   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.;
   weight Count;
   title 'Study of Three Drug Treatments for a Chronic Disease';
run;

The one-way frequency tables in Output 40.10.1 provide the marginal response for each drug. For drugs A and B, 61% of the subjects reported a favorable response; for drug C, 35% of the subjects reported a favorable response. Output 40.10.2 and Output 40.10.3 display measures of agreement for the 'Favorable' and 'Unfavorable' levels of drug A, respectively. McNemar’s test shows a strong discordance between drugs B and C when the response to drug A is favorable.

Output 40.10.1: 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



Output 40.10.2: Measures of Agreement 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



Output 40.10.3: Measures of Agreement 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



Output 40.10.4 displays the overall kappa coefficient. The small negative value of kappa indicates no agreement between drug B response and drug C response.

Output 40.10.4: Overall Measures of Agreement

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



Cochran’s Q is statistically significant (p=0.0145 in Output 40.10.5), which leads to rejection of the hypothesis that the probability of favorable response is the same for the three drugs.

Output 40.10.5: Cochran’s Q Test

Cochran's Q, for Drug_A by
Drug_B by Drug_C
Statistic (Q) 8.4706
DF 2
Pr > Q 0.0145