On Invariance of Chi-squared Tests Under Different Probability Models

A chi-squared test is a popular test for assessing relationship between two factors or categorical variables summarized in the form of a contingency table. In this study, we establish the invariance of a chi-squared test under three different study designs, namely, cohort, case-control and cross-sectional studies involving distinct probabilistic models. By the invariance of the chi-squared test, we refer to the fact that the form of a chi-squared test remains unchanged under different probabilistic models. The theoretical derivation of expected cell frequencies carried out in this study, under different study designs and probability models, will be exemplary and invaluable to researchers to understand as to why they can use an identical form of the chi-squared test for a contingency table resulting from case-control, cohort or cross-sectional study design for testing independence. This study is also useful in academia to demonstrate why contingency table resulting under different study designs is subject to identical form of a chi-squared test, which has not been well documented in existing literature. The examples and applications utilized in this study provide directions as to how differently formulated studies are implemented via a chi-squared test.