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Hypothesis testing in contingency tables is usually based on asymptoticresults, thereby restricting its proper use to large samples. To study thesetests in small samples, we consider the likelihood ratio test and define anaccurate index, the P-value, for the celebrated hypotheses of homogeneity,independence, and Hardy-Weinberg equilibrium. The aim is to understand the useof the asymptotic results of the frequentist Likelihood Ratio Test and theBayesian FBST -- Full Bayesian Significance Test -- under small-samplescenarios. The proposed exact P-value is used as a benchmark to understand theother indices. We perform analysis in different scenarios, consideringdifferent sample sizes and different table dimensions. The exact Fisher testfor $2 \times 2$ tables that drastically reduces the sample space is alsodiscussed. The main message of this paper is that all indices have very similarbehavior, so the tests based on asymptotic results are very good to be used inany circumstance, even with small sample sizes.

The Likelihood Ratio Test and Full Bayesian Significance Test under small sample sizes for contingency tables