Cochran Q Test

Motivation

Cochran's Q test has enjoyed widespread use in behavioral, business, educational, medical, and social science research when it may be desirable to evaluate possible significance of differences in the proportion of successes over several treatment groups.

As an example of a study in which each participant offers repeated measurements, one each for the c levels of the treatment or condition under evaluation, the efficacy of c different drugs prescribed for providing relief from some chronic condition may be investigated. Other examples of this type could involve taste testing wines or other consumer preference studies in which each of c items is classified as “acceptable” or “not acceptable.” As an example of a randomized complete block experiment in which the c homogeneous members of a block are randomly assigned, one each, to the c levels of the treatment, an educator may wish to form sets of homogeneous blocks of students and randomly assign the members of the block to each of c learning methods with the goal of assessing whether there are significant differences among the learning methods based on the proportions of successes observed.

Development

The layout for the dichotomous responses from a sample of either r participants or r blocks of matched participants over c levels of a treatment condition is shown in Table 1.

Cochran's Q test statistic is given by

where

c is the number of treatment groups (i.e., columns),

r is the number of blocks (i.e., rows of subjects),

xij is the binary response (success = 1, failure = 0) for the jth treatment in the ith block,

x.j is the total number of successes for treatment j,

xi. is the total number of successes for block i, and

= the total number of successes.

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