Bayesian two-interval test

TL;DR

This paper introduces a Bayesian two-interval test (2IT) that explicitly tests hypotheses as intervals, offering a scientifically sound alternative to traditional null hypothesis testing with practical advantages in various statistical scenarios.

Contribution

The paper proposes the 2IT, a Bayesian testing method using interval hypotheses, addressing limitations of classical tests and enabling sample size calculation and sequential analysis.

Findings

The 2IT effectively distinguishes between hypotheses with increasing sample size.

Bias in mean estimation with 2IT is low and diminishes rapidly.

The method applies to superiority, non-inferiority, and equivalence testing.

Abstract

The null hypothesis test (NHT) is widely used for validating scientific hypotheses but is actually highly criticized. Although Bayesian tests overcome several criticisms, some limits remain. We propose a Bayesian two-interval test (2IT) in which two hypotheses on an effect being present or absent are expressed as prespecified joint or disjoint intervals and their posterior probabilities are computed. The same formalism can be applied for superiority, non-inferiority, or equivalence tests. The 2IT was studied for three real examples and three sets of simulations (comparison of a proportion and a mean to a reference and comparison of two proportions). Several scenarios were created (with different sample sizes), and simulations were conducted to compute the probabilities of the parameter of interest being in the interval corresponding to either hypothesis given the data generated under one of the hypotheses. Posterior estimates were obtained using conjugacy with a low-informative prior. Bias was also estimated. The probability of accepting a hypothesis when that hypothesis is true progressively increases the sample size, tending towards 1, while the probability of accepting the other hypothesis is always very low (less than 5%) and tends towards 0. The speed of convergence varies with the gap between the hypotheses and with their width. In the case of a mean, the bias is low and rapidly becomes negligible. We propose a Bayesian test that follows a scientifically sound process, in which two interval hypotheses are explicitly used and tested. The proposed test has almost none of the limitations of the NHT and suggests new features, such as a rationale for serendipity or a justification for a "trend in data". The conceptual framework of the 2-IT also allows the calculation of a sample size and the use of sequential methods in numerous contexts.