Alternatives to the P value: connotations of significance

The statistical significance of a clinical trial analysis result is determined by a mathematical calculation and probability based on null hypothesis significance testing. However, statistical significance does not always align with meaningful clinical effects; thus, assigning clinical relevance to statistical significance is unreasonable. A statistical result incorporating a clinically meaningful difference is a better approach to present statistical significance. Thus, the minimal clinically important difference (MCID), which requires integrating minimum clinically relevant changes from the early stages of research design, has been introduced. As a follow-up to the previous statistical round article on P values, confidence intervals, and effect sizes, in this article, we present hands-on examples of MCID and various effect sizes and discuss the terms statistical significance and clinical relevance, including cautions regarding their use.

Empirical Bayes factors for common hypothesis tests

Bayes factors for composite hypotheses have difficulty in encoding vague prior knowledge, as improper priors cannot be used and objective priors may be subjectively unreasonable. To address these issues I revisit the posterior Bayes factor, in which the posterior distribution from the data at hand is re-used in the Bayes factor for the same data. I argue that this is biased when calibrated against proper Bayes factors, but propose adjustments to allow interpretation on the same scale. In the important case of a regular normal model, the bias in log scale is half the number of parameters. The resulting empirical Bayes factor is closely related to the widely applicable information criterion. I develop test-based empirical Bayes factors for several standard tests and propose an extension to multiple testing closely related to the optimal discovery procedure. When only a P-value is available, an approximate empirical Bayes factor is 10p. I propose interpreting the strength of Bayes factors on a logarithmic scale with base 3.73, reflecting the sharpest distinction between weaker and stronger belief. This provides an objective framework for interpreting statistical evidence, and realises a Bayesian/frequentist compromise.