Example 36.10 Cochran’s Q Test
When a binary response is measured several times or under different conditions, Cochran’s 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 and other measures of agreement for the three-way table. These statements produce Output 36.10.1 through Output 36.10.5.
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 36.10.1 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 36.10.2 and Output 36.10.3 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 36.10.1
One-Way Frequency Tables
Output 36.10.2
Measures of Agreement for Drug A Favorable
-0.0328 |
0.1167 |
-0.2615 |
0.1960 |
Output 36.10.3
Measures of Agreement for Drug A Unfavorable
-0.1538 |
0.2230 |
-0.5909 |
0.2832 |
Output 36.10.4 displays the overall kappa coefficient. The small negative value of kappa indicates no agreement between drug B response and drug C response.
Cochran’s is statistically significant (=0.0144 in Output 36.10.5), which leads to rejection of the hypothesis that the probability of favorable response is the same for the three drugs.
Output 36.10.4
Overall Measures of Agreement
-0.0588 |
0.1034 |
-0.2615 |
0.1439 |
Output 36.10.5
Cochran’s Q Test