Hypothesis Testing in Contingency Tables with Small Samples.

Abstract

The aim of this research is to present solutions to the problem of small samples for hypothesis testing in a contingency table framework. Regarding the use of a chi-squared test statistic, we discuss the impact of a variance estimator choice on the conservativeness or liberalism of the procedure with respect to the Type I error. We then propose new continuity corrections for testing homogeneity in a (2 x 2) comparative trial with arbitrary sample sizes. These continuity corrections are based on the average interval between successive differences of binomial proportions. In a second step, we derive unconditional or approximate exact tests for different sampling designs. The sizes of these unconditional exact tests are then obtained by associating an estimated sampling distribution with an ordering criterion, i.e., a chi-squared test statistic, in order to construct an empirical cumulative distribution function on that statistic. We demonstrate the conservativeness of the conditional exact tests when applied on the same sampling designs. Monte Carlo simulations are performed to study the closeness of the emipirical significance level to the nominal significance level with the X('2)'s corrected for the proposed continuity corrections. Results show that the new continuity corrections are to be preferred over the uncorrected X('2) or the Yates' or Pirie and Hamdan's corrected X('2)'s when sample sizes are unequal. For equal sample sizes, the uncorrected X('2) should be used. These results hold for sample sizes ranging from 2 to 20. Formulas based on the proposed continuity corrections are derived for sample size determination. Finally, suggestions are presented for future research.