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

Stéphane Laurent; Catherine Legrand

ESAIM: Probability and Statistics (2012)

  • Volume: 16, page 375-398
  • ISSN: 1292-8100

Abstract

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In many applications, we assume that two random observations x and y are generated according to independent Poisson distributions ( λ S ) 𝒫(λS) and ( μ 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.

How to cite

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Laurent, Stéphane, and Legrand, Catherine. "A Bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials∗." ESAIM: Probability and Statistics 16 (2012): 375-398. <http://eudml.org/doc/222458>.

@article{Laurent2012,
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.},
author = {Laurent, Stéphane, Legrand, Catherine},
journal = {ESAIM: Probability and Statistics},
keywords = {Poisson rates; relative risk; vaccine efficacy; partial immunity model; semi-conjugate family; reference prior; Jeffreys’ prior; frequentist coverage; beta prime distribution; beta-negative binomial distribution; Jeffreys prior},
language = {eng},
month = {9},
pages = {375-398},
publisher = {EDP Sciences},
title = {A Bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials∗},
url = {http://eudml.org/doc/222458},
volume = {16},
year = {2012},
}

TY - JOUR
AU - Laurent, Stéphane
AU - Legrand, Catherine
TI - A Bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials∗
JO - ESAIM: Probability and Statistics
DA - 2012/9//
PB - EDP Sciences
VL - 16
SP - 375
EP - 398
AB - 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.
LA - eng
KW - Poisson rates; relative risk; vaccine efficacy; partial immunity model; semi-conjugate family; reference prior; Jeffreys’ prior; frequentist coverage; beta prime distribution; beta-negative binomial distribution; Jeffreys prior
UR - http://eudml.org/doc/222458
ER -

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