On implementing Jeffreys' substitution likelihood for Bayesian inference concerning the medians of unknown distributions

When statisticians are uncertain as to which parametric statistical model to use to analyse experimental data, they will often resort to a non-parametric approach. The purpose of this paper is to provide insight into a simple approach to take when it is unclear as to the appropriate parametric model and plan to conduct a Bayesian analysis. I introduce an approximate, or substitution likelihood, first proposed by Harold Jeffreys in 1939 and show how to implement the approach combined with both a non-informative and an informative prior to provide a random sample from the posterior distribution of the median of the unknown distribution. The first example I use to demonstrate the approach is a within-patient bioequivalence design and then show how to extend the approach to a parallel group design.

Automated summary

When statisticians are uncertain about which parametric statistical model to use for analyzing experimental data, they often turn to non-parametric approaches. This paper presents a simple method for conducting Bayesian analysis when the appropriate parametric model is unclear. The author introduces an approximate likelihood function proposed by Harold Jeffreys in 1939 and demonstrates how to implement the approach using both non-informative and informative priors to obtain a random sample from the posterior distribution of the median of the unknown distribution. The method is illustrated using examples of within-patient bioequivalence design and parallel group design.