Hypothesis Estimates and Acceptability Profiles for 2 × 2 Contingency Tables

Abstract

By estimating, rather than testing, hypotheses regarding the degree of dependence between the factors in 2 × 2 tables, the technical difficulties associated with small sample sizes are avoided. The estimators proposed here attempt to estimate 1 when the alternative hypothesis is true and 0 when the null hypothesis is true, subject to a bound on the squared error loss under the hypothesis. Such estimators provide guarded weights of evidence for the alternative hypothesis. Guarded weights of evidence based on the likelihood ratio are compared with those based on the p value or mid-p value, and they are shown to have lower risk functions except when the alternative is far from the hypothesis. For the case of two independent binomial distributions, it is shown that the conditional likelihood ratio estimator for the hypothesis of homogeneity against the two-sided alternative has a smaller unconditional risk than the unconditional likelihood ratio estimator, except when the binomial probabilities are far apart. Inversion of a family of guarded weights of evidence leads to acceptability profiles. These profiles provide more information than traditional confidence intervals regarding the unknown parameter. Two-sided profiles are found for the degree of dependence as measured by the odds ratio and log-odds ratio, and one-sided profiles are found for Yule's Q.