Likelihood Ratio Test and the Evidential Approach for 2 × 2 Tables

Categorical data analysis of 2 × 2 contingency tables is extremely common, not least because they provide risk difference, risk ratio, odds ratio, and log odds statistics in medical research. A χ2 test analysis is most often used, although some researchers use likelihood ratio test (LRT) analysis. Does it matter which test is used? A review of the literature, examination of the theoretical foundations, and analyses of simulations and empirical data are used by this paper to argue that only the LRT should be used when we are interested in testing whether the binomial proportions are equal. This so-called test of independence is by far the most popular, meaning the χ2 test is widely misused. By contrast, the χ2 test should be reserved for where the data appear to match too closely a particular hypothesis (e.g., the null hypothesis), where the variance is of interest, and is less than expected. Low variance can be of interest in various scenarios, particularly in investigations of data integrity. Finally, it is argued that the evidential approach provides a consistent and coherent method that avoids the difficulties posed by significance testing. The approach facilitates the calculation of appropriate log likelihood ratios to suit our research aims, whether this is to test the proportions or to test the variance. The conclusions from this paper apply to larger contingency tables, including multi-way tables.