Testing for Simultaneous Pairwise Marginal Independence

1 Introduction What types of cars do you own? What are your sources of veterinary information? For what criminal offenses have you been arrested? These are all example questions appearing on surveys where the respondent is prompted to pick any number of responses from a set of variables that summarize this type of survey data have been called multiple-response (or pick any/c) categorical variables. The survey data arising from questions of this type present a unique challenge for analysis because of the dependence among responses provided by individual subjects. For categorical variables in general, testing for independence between variables is often of interest. When at least one of the 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. Furthermore, a special kind of independence, called marginal independence, becomes of interest in the presence of multiple-response categorical variables. The purpose of this paper is to develop new approaches to the testing of marginal independence in an r×c contingency table with two multiple-response categorical variables. Agresti and Liu (1999) call this a test for simultaneous pairwise marginal independence (SPMI). an example where SPMI can be tested. The data comes from a survey conducted by the Department of Animal Sciences at Kansas State University. Two questions in the survey asked Kansas farmers about their sources of veterinary information and their swine waste storage methods. For each of these two questions, the farmers were permitted to select as many responses as applied from a list of items. The data are summarized as a 4×5 contingency table in Table 1. For example, 34 farmers picked professional consultant as a source of veterinary information and lagoon as a waste storage method. A test for SPMI determines if each source of veterinary information is simultaneously independent of each swine waste storage method. More specifically, 4 * 5=20 different 2×2 tables can be formed marginally summarizing all possible responses to item (category) pairs. Table 2 shows the 2×2 table for professional consultant and lagoon. Independence is tested in each of the 20 2×2 tables simultaneously for a test of SPMI. The test is marginal because responses are summed over the other item choices for each of the multiple-response categorical variables. This test may be of interest to agricultural companies who would like to use the information for marketing purposes. Tests for marginal independence have …