The Calibrated Bayesian Hypothesis Test for Directional Hypotheses of the Odds Ratio in $$2\times 2$$ 2 × 2 Contingency Tables

The $$\chi ^{2}$$ χ 2 test is among the most widely used statistical hypothesis tests in medical research. Often, the statistical analysis deals with the test of row-column independence in a $$2\times 2$$ 2 × 2 contingency table, and the statistical parameter of interest is the odds ratio. A novel Bayesian analogue to the frequentist $$\chi ^{2}$$ χ 2 test is introduced. The test is based on a Dirichlet-multinomial model under a joint sampling scheme and works with balanced and unbalanced randomization. The test focusses on the quantity of interest in a variety of medical research, the odds ratio in a $$2\times 2$$ 2 × 2 contingency table. A computational implementation of the test is developed and R code is provided to apply the test. To meet the demands of regulatory agencies, a calibration of the Bayesian test is introduced which allows to calibrate the false-positive rate and power. The latter provides a Bayes-frequentist compromise which ensures control over the long-term error rates of the test. Illustrative examples using clinical trial data and simulations show how to use the test in practice. In contrast to existing Bayesian tests for $$2\times 2$$ 2 × 2 tables, calibration of the acceptance threshold for the hypothesis of interest allows to achieve a bound on the false-positive rate and minimum power for a prespecified odds ratio of interest. The novel Bayesian test provides an attractive choice for Bayesian biostatisticians who face the demands of regulatory agencies which usually require formal control over false-positive errors and power under the alternative. As such, it constitutes an easy-to-apply addition to the arsenal of already existing Bayesian tests.