TESTING FOR ASSOCIATION IN CONTINGENCY-TABLES WITH MULTIPLE COLUMN RESPONSES

In many studies, multiple categorical responses or measurements are ma de on members of different populations or treatment groups. This arise s often in surveys where individuals may mark all answers that apply w hen responding to a multiple-choice question. Frequently, it is of int erest to determine whether the distributions of responses differ among groups. In this situation, the test statistic of the usual Pearson ch i-square test no longer measures a scaled distance between observed an d hypothesized cell counts in a contingency table, and its distributio n is no longer the familiar chi-square. This paper presents a modifica tion to the Pearson statistic that measures the appropriate distance f or multiple-response tables. The asymptotic distribution is shown to b e that of a linear combination of chi-square random variables with coe fficients depending on the true probabilities. A bootstrap resampling method is proposed instead to obtain a null-hypothesis sampling distri bution. Simulations show that this bootstrap method maintains its size under a variety of circumstances, while a naively applied Pearson chi -square test is severely affected by multiple responses.