A discussion on significance indices for contingency tables under small sample sizes

Area of Science:

• Statistics
• Biostatistics
• Genetics

Background:

• Hypothesis testing in contingency tables typically relies on asymptotic results, limiting its application to large sample sizes.
• Small sample sizes pose challenges for traditional statistical tests, necessitating alternative approaches.

Purpose of the Study:

• To evaluate the performance of the frequentist Likelihood Ratio Test (LRT) and Bayesian Full Bayesian Significance Test (FBST) under small-sample conditions.
• To define an accurate index for testing hypotheses of homogeneity, independence, and Hardy-Weinberg equilibrium in contingency tables.
• To compare the proposed exact LRT p-value with other indices and exact tests.

Main Methods:

• Utilized the Likelihood Ratio Test (LRT) and defined an exact p-value as a benchmark.
• Analyzed performance across various sample sizes and contingency table dimensions.
• Included comparisons with Fisher's exact test and Barnard's exact test for 2x2 tables.

Main Results:

• The asymptotic p-value from the LRT demonstrated high power and accuracy, even in small samples.
• Most tested indices exhibited similar behavior, with Fisher's and Barnard's tests showing discrete patterns.
• The asymptotic LRT p-value proved to be a robust alternative for small sample hypothesis testing.

Conclusions:

• The asymptotic p-value derived from the Likelihood Ratio Test is a reliable and powerful tool for hypothesis testing in contingency tables, even with small sample sizes.
• Exact tests like Fisher's and Barnard's have distinct, discrete behaviors compared to other methods.
• The findings support the use of the asymptotic LRT p-value as a viable alternative when dealing with limited data.