Abstract
The analysis of R×C contingency tables usually features a test for independence between row and column counts. Throughout the social sciences, the adequacy of the independence hypothesis is generally evaluated by the outcome of a classical p-value null-hypothesis significance test. Unfortunately, however, the classical p-value comes with a number of well-documented drawbacks. Here we outline an alternative, Bayes factor method to quantify the evidence for and against the hypothesis of independence in R×C contingency tables. First we describe different sampling models for contingency tables and provide the corresponding default Bayes factors as originally developed by Gunel and Dickey (Biometrika, 61(3):545–557 (1974)). We then illustrate the properties and advantages of a Bayes factor analysis of contingency tables through simulations and practical examples. Computer code is available online and has been incorporated in the “BayesFactor” R package and the JASP program (jasp-stats.org).
Highlights
Contingency tables are ubiquitous throughout psychology and the social sciences
The main goal of this article is to outline an alternative, Bayes factor hypothesis test for the R × C contingency table that can be used to complement or replace the classical hypothesis tests based on p-values
To compute the evidence assuming that the total number of observations is fixed, we look at the change from the Bayes factor using only the first part of the data to the Bayes factor conditioned on the whole data set
Summary
None of the cell counts are fixed. Each cell count is assumed to be Poisson distributed: yrc ∼ Poisson(λrc). The p-value does not quantify how much these data should shift our belief To address this question we calculate the joint multinomial GD74 Bayes factor and obtain BFM10 = 373.13, indicating extreme evidence for the hypothesis that there exists an association between the satisfaction level of supervisors and workers. To address this question we calculate the independent multinomial GD74 Bayes factor and obtain log BFI10 = 23.03, indicating strong evidence for the hypothesis that there exists an association between children’s race and the color of the doll they preferred to play with.
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