In many applications, we assume that two random observations x and y are generated according to independent Poisson distributions $\mathcal{P}\left(\lambda S\right)$ and $\mathcal{P}\left(\mu T\right)$and we are interested in performing statistical inference on the ratio ) 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 rates, relative risk, vaccine efficacy, partial immunity model, semi-conjugate family, reference prior, Jeffreys' prior, frequentist coverage, beta prime distribution, beta-negative binomial distribution

@article{PS_2012__16__375_0, author = {Laurent, St\'ephane and Legrand, Catherine}, title = {A bayesian framework for the ratio of two {Poisson} rates in the context of vaccine efficacy trials}, journal = {ESAIM: Probability and Statistics}, pages = {375--398}, publisher = {EDP-Sciences}, volume = {16}, year = {2012}, doi = {10.1051/ps/2010018}, mrnumber = {2972499}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps/2010018/} }

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 PY - 2012 SP - 375 EP - 398 VL - 16 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2010018/ DO - 10.1051/ps/2010018 LA - en ID - PS_2012__16__375_0 ER -

%0 Journal Article %A Laurent, Stéphane %A Legrand, Catherine %T A bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials %J ESAIM: Probability and Statistics %D 2012 %P 375-398 %V 16 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2010018/ %R 10.1051/ps/2010018 %G en %F PS_2012__16__375_0

Laurent, Stéphane; Legrand, Catherine. A bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials. ESAIM: Probability and Statistics, Volume 16 (2012), pp. 375-398. doi : 10.1051/ps/2010018. http://archive.numdam.org/articles/10.1051/ps/2010018/

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