On the first-order Rao-Scott correction of the Umesh-Loughin-Scherer statistic

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.

Permutation analysis of data with multiple binary category choices

In many studies, respondents may mark all answers that apply when responding to a multiple-choice question, i.e., a cafeteria or multiple-response question. One exact and two approximate permutation methods are described to analyze multiple-response questions. The methods provide the probability, under the null hypothesis, that the multiple binary responses do not differ among specified groups.

Testing for conditional multiple marginal independence

Survey respondents are often prompted to pick any number of responses from a set of possible responses. Categorical variables that summarize this kind of data are called pick any/c variables. Counts from surveys that contain a pick any/c variable along with a group variable (r levels) and stratification variable (q levels) can be marginally summarized into an r x c x q contingency table. A question that may naturally arise from this setup is to determine if the group and pick any/c variable are marginally independent given the stratification variable. A test for conditional multiple marginal independence (CMMI) can be used to answer this question. Since subjects may pick any number out of c possible responses, the Cochran (1954, Biometrics 10, 417-451) and Mantel and Haenszel (1959, Journal of the National Cancer Institute 22, 719-748) tests cannot be used directly because they assume that units in the contingency table are independent of each other. Therefore, new testing methods are developed. Cochran's test statistic is extended to r x 2 x q tables, and a modified version of this statistic is proposed to test CMMI. Its sampling distribution can be approximated through bootstrapping. Other CMMI testing methods discussed are bootstrap p-value combination methods and Bonferroni adjustments. Simulation findings suggest that the proposed bootstrap procedures and the Bonferroni adjustments consistently hold the correct size and provide power against various alternatives.