A Bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials^∗

Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA), Université catholique de Louvain, Voie du Roman Pays, 20, 1348 Louvain la Neuve, Belgium
stephane.laurent@uclouvain.be; catherine.legrand@uclouvain.be

Received: 4 May 2009
Revised: 14 June 2010

Abstract

In many applications, we assume that two random observations x and y are generated according to independent Poisson distributions and and we are interested in performing statistical inference on the ratio φ = λ / μ of the two incidence rates. In vaccine efficacy trials, x and y are typically the numbers of cases in the vaccine and the control groups respectively, φ is called the relative risk and the statistical model is called ‘partial immunity model’. In this paper we start by defining a natural semi-conjugate family of prior distributions for this model, allowing straightforward computation of the posterior inference. Following theory on reference priors, we define the reference prior for the partial immunity model when φ is the parameter of interest. We also define a family of reference priors with partial information on μ while remaining uninformative about φ. We notice that these priors belong to the semi-conjugate family. We then demonstrate using numerical examples that Bayesian credible intervals for φ enjoy attractive frequentist properties when using reference priors, a typical property of reference priors.