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 3.10.1 through Output 3.10.5.
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 3.10.1 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 3.10.2 and Output 3.10.3 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 3.10.1: One-Way Frequency Tables
Study of Three Drug Treatments for a Chronic Disease |
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 3.10.2: 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 3.10.3: 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 3.10.4 displays the overall kappa coefficient. The small negative value of kappa indicates no agreement between drug B response and drug C response.
Output 3.10.4: 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 3.10.5), which leads to rejection of the hypothesis that the probability of favorable response is the same for the three drugs.
Output 3.10.5: 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 |