Contingencies for analysis of contingency tables: More on the chi‐squared test

Abstract

Analysis of the chi‐squared statistic for contingency tables is basic to all branches of science and social science. Rejection of a null hypothesis of no association, especially for large samples, provides little information about the table being studied. Methods, developed by Diaconis & Efron (1985), for interpreting a chi‐squared statistic with regard to its power against a family of alternative distributions arc discussed and illustrated. Higher power is not necessarily implied by higher values for the observed chi‐squared. Power is a function of the computed chi‐squared, the sample size, the row and column marginal totals and the particular alternative hypothesis under consideration. Intermediate models, the relation of one‐way analysis of variance to the random effects model, and the concept of effective sample size are elaborated in the context of actual tables from the literature. Diagnostics for protrusion effects and the topic of granularity are discussed and studied graphically.