Bayesian hypothesis testing for proportions

Most clinical trials contain tests on proportions, usually they are answered by means of the Frequentist approach, nevertheless another valid option could be to solve them using a Bayesian approach. The Bayesian approach has the advantage that it is not restricted to only one alternative hypothesis. Moreover, the hypotheses to be tested do not necessarily overlap. In this paper we show a SAS® macro to perform Bayesian hypothesis testing for proportions, that can be also extended to other kinds of endpoints and distributions. For simplicity only the null and one alternative hypothesis are shown. This macro is constructed assuming an improper prior distribution, the uniform (0,1), and a Beta as the posterior conjugate distribution. Therefore after calculating the proportion of successes in the trial, the probability of being under the null hypothesis or under the alternative hypothesis and a text label indicating the highest probability are shown. INTRODUCTION This paper has not the aim to confront Frequentist approach vs. Bayesian approach. In fact, both approaches can coexist and should be used indistinctly in the statistical interest. Consequently, we have implemented easy SAS macros to calculate the probabilities of different hypotheses using a Bayesian approach. TESTS ON PROPORTIONS Almost all, if not all, clinical trials contain tests on proportions. The proportion distribution is a collection of ‘‘n’’ Bernoulli experiments; i.e., it is counted as the sum of the number of successes/failures out of ‘‘n’’ independent samples. Proportion tests usually are solved by means of a Frequentist approach, but this is not the only way. In a Frequentist analysis, if the comparison p-value is lower than the significance level selected, then the null hypothesis is rejected. In a Bayesian approach, the probability to be under any hypothesis is estimated and then these probabilities can be compared to decide what is the most plausible alternative. THE BAYES’’ THEOREM The Bayesian approach is based on the Bayes’’ theorem (1763), and expresses the conditional probability of a random event A given that an event B has occurred in terms of the conditional probability distribution of the event B given that A has occurred and the marginal probability of only A. In other words, beginning with the prior experience/knowledge (i.e., ““a priori distribution””) and then joining it with the trial investigation, a posterior conjugate distribution is obtained to be used to produce probabilities once the clinical trial has been completed. BAYESIAN TESTS The sum of the Bernouilli experiments is a Binomial distribution, which combined with the ““a priori”” information should lead to a posterior known distribution that allow an easy calculation of probabilities. Then, in practice, we will need to model the prior information to find the probability distribution that better fits the ““a priori”” knowledge and to lead to a posterior distribution easy to handle. The Bayesian approach has the advantage that it is not restricted to only one alternative hypothesis. In addition, the hypotheses to be tested do not necessarily overlap and, therefore, probabilities associated under any hypotheses can be calculated in function of the different cutoffs selected as long as we know the conjugate distribution to be used.