Testing for marginal independence between two categorical variables with multiple responses.

Questions that ask respondents to "choose all that apply" from a set of items occur frequently in surveys. Categorical variables that summarize this type of survey data are called both pick any/c variables and multiple-response categorical variables. It is often of interest to test for independence between two categorical variables. When both categorical variables can have multiple responses, traditional Pearson chi-square tests for independence should not be used because of the within-subject dependence among responses. An intuitively constructed version of the Pearson statistic is proposed to perform the test using bootstrap procedures to approximate its sampling distribution. First- and second-order adjustments to the proposed statistic are given in order to use a chi-square distribution approximation. A Bonferroni adjustment is proposed to perform the test when the joint set of responses for individual subjects is unavailable. Simulations show that the bootstrap procedures hold the correct size more consistently than the other procedures.