Collaborative Research: Testing for Marginal Independence Between Two or More Multiple-Response Categorical Variables

调查中经常会出现这样的问题，即要求受访者从一组项目中选择所有适用的项目。总结这类调查数据的类别变量称为多重响应(或选择任意/c)类别变量。检验两个分类变量之间的独立性通常是有意义的。当类别变量可能有多个反应时，由于反应之间存在被试内部依赖关系，不应该使用传统的皮尔逊卡方检验来检验独立性。本研究将提供检验两个或多个多反应分类变量之间独立性的方法。皮尔逊统计量的修改版本将执行测试，而Bootstrap过程将提供近似的抽样分布。一阶和二阶修正将允许抽样分布的卡方分布近似。广义对数线性模型和多元二项对数正态模型将为独立性检验提供一种基于模型的方法。许多调查问题都是以多重回答的方式提出的。比如：“你有什么类型的车？”以及“你因什么刑事犯罪而被捕？”其他问题自然属于多重回答形式，但由于统计分析问题，一些研究人员避免使用这种形式。例如，大多数涉及种族的调查问题只允许受访者做出一个选择，这在当今高度多元文化的人口中是完全不合适的。其他研究人员可能会分析多个反应的问题，就好像它们来自单一反应的分类变量，这可能会导致非常保守的独立性测试。这项研究将允许研究人员将这些类型的问题纳入调查，并使用统计上正确的分析方法。这项研究得到了方法学、测量和统计计划以及调查和统计方法学资助机会研究下的一个联邦统计机构联盟的支持。

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 multiple response (or pick any/c) categorical variables. It is often of interest to test for independence between two categorical variables. When 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. This research will provide methods to test for independence between two or more multiple-response categorical variables. A modified version of the Pearson statistic will perform the test, and bootstrap procedures will provide approximate sampling distributions. First and second-order corrections will allow for chi-square distribution approximations to the sampling distribution. Generalized log linear models and multivariate binomial logit-normal models will provide a model-based approach for the test of independence. Many survey questions are asked in a multiple-response manner. Examples include: "What types of cars do you own?" and "For what criminal offenses have you been arrested?" Other questions naturally fall into a multiple-response format, but some researchers avoid asking them in this format due to statistical analysis problems. For example, most survey questions dealing with ethnicity allow respondents to make only one choice, which is entirely inappropriate in today's highly multicultural population. Other researchers may analyze multiple-response questions as if they came from single-response categorical variables, which can lead to very conservative tests of independence. This research will allow researchers to incorporate these types of questions into surveys and use statistically correct methods of analysis. The research is supported by the Methodology, Measurement, and Statistics Program and a consortium of federal statistical agencies under the Research on Survey and Statistical Methodology Funding Opportunity.