HYPOTHESIS ESTIMATES AND ACCEPTABILITY PROFILES FOR 2X2-CONTINGENCY-TABLES

By estimating, rather than testing hypotheses regarding the degree of dependence between the factors in 2 x 2 tables, the technical difficul ties associated with small sample sizes are avoided. The estimators pr oposed here attempt to estimate I when the alternative hypothesis is t rue and 0 when the null hypothesis is true, subject to a bound on the squared error loss under the hypothesis. Such estimators provide guard ed weights of evidence for the alternative hypothesis. Guarded weights of evidence based on the likelihood ratio are compared with those bas ed on the p value or mid-p value, and they are shown to have lower ris k functions except when the alternative is far from the hypothesis. Fo r the case of two independent binomial distributions, it is shown that the conditional likelihood ratio estimator for the hypothesis of homo geneity against the two-sided alternative has a smaller unconditional risk than the unconditional likelihood ratio estimator, except when th e binomial probabilities are far apart. Inversion of a family of guard ed weights of evidence leads to acceptability profiles. These profiles provide more information than traditional confidence intervals regard ing the unknown parameter. Two-sided profiles are found for the degree of dependence as measured by the odds ratio and log-odds ratio, and o ne-sided profiles are found for Yule's Q.