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

Published online by Cambridge University Press:  03 September 2012

Stéphane Laurent  and
Catherine Legrand
Stéphane Laurent
Affiliation:
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
Catherine Legrand
Affiliation:
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
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Abstract

In many applications, we assume that two random observations x and y are generated according to independent Poisson distributions \hbox{$\PPP(\lambda S)$}𝒫(λS) and \hbox{$\PPP(\mu T)$}𝒫(μT) 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.

Keywords

Poisson ratesrelative riskvaccine efficacypartial immunity modelsemi-conjugate familyreference priorJeffreys’ priorfrequentist coveragebeta prime distributionbeta-negative binomial distribution
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 375 - 398
Copyright
© EDP Sciences, SMAI, 2012

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