Multiple marginal independence testing for pick any/C variables

Many survey questions allow respondents to pick any number out of c possible categorical responses or "items". These kinds of survey questions often use the terminology "choose all that apply" or "pick any". Often of interest is determining if the marginal response distributions of each item differ among r different groups of respondents. Agresti and Liu (1998, 1999) call this a test for multiple marginal independence (MMI). If respondents are allowed to pick only 1 out of c responses, the hypothesis test may be performed using the Pearson chi-square test of independence. However, since respondents may pick more or less than 1 response, the test's assumptions that responses are made independently of each other is violated. Recently, a few MMI testing methods have been proposed. Loughin and Scherer (1998) propose using a bootstrap method based on a modified version of the Pearson chi-square test statistic. Agresti and Liu (1998, 1999) propose using marginal logit models, quasisymmetric loglinear models, and a few methods based on Pearson chi-square test statistics. Decady and Thomas (1999) propose using a Rao-Scott adjusted chi-squared test statistic. There has not been a full investigation of these MMI testing methods. The purpose here is to evaluate the proposed methods and propose a few new methods. Recommendations are given to guide the practitioner in choosing which MMI testing methods to use.