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

Abstract

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

Some Poisson mixtures distributions with a hyperscale parameter

“…Using the factorial moments, we get that the mean of τ PGIB(a, c, d) is τ ad(c + d) −1 and its variance is Laurent and Legrand (2011) defined a natural semi-conjugate family of priors for the "two Poisson samples" model, which contains the Berger & Bernardo reference prior for the ratio of the two Poisson rates. The results we give in this section provide the prior and posterior predictive distributions for this family of priors, and also the prior predictive distributions for the larger family of priors defined in Lemma 3.1.…”

“…Throughout this section, we consider the statistical model given by two independent observations x ∼ P(λS) and y ∼ P(μT ) with unknown incidence rates λ, μ, and fixed "observation-opportunity sizes," or "sample sizes," S and T , and we denote by φ := λ/μ the so-called relative risk. When μ and φ have independent prior distributions with μ ∼ G(a, b), then, as shown by Laurent and Legrand (2011), the conditional joint prior predictive distribution of (x, y) given φ is the bivariate Poisson-Gamma distribution having the marginal-conditional factorization…”

“…The semi-conjugate family defined by Laurent and Legrand (2011) is the case when ρ = (T + b)/S, that is, when φ has the B 2 (c + x, d + a + y, ρ) posterior distribution. Hereafter, we will simply call it the semi-conjugate family.…”

“…The natural conjugate family of priors for the "two Poisson samples" model is formed by the independent products of Gamma distributions on μ and λ. This family contains the Jeffreys prior which is the case when μ ∼ G( 1 2 , 0) and λ ∼ G( 1 2 , 0), and, as noticed by Laurent and Legrand (2011), the Jeffreys prior and the φ-reference prior yield the same posterior on φ, but do not yield the same posterior predictive distributions. We note however that these posterior predictive distributions are close, because of (τ, t) Bailey(a, c, d, 1) ≈ τ PG(c, 1) ⊗ t PG(d, 1) when a ≈ c + d,…”

“…In Section 3, the univariate Poisson mixtures help us to derive the predictive distributions corresponding to the semi-conjugate family of priors defined by Laurent and Legrand (2011) for the "two Poisson samples" model, and also the prior (but not the posterior) predictive distributions for a larger family we define (by adding a scale parameter for the prior on the ratio of the two Poisson rates). In particular, we get the posterior predictive distributions corresponding to the Berger & Bernardo reference prior for the ratio of the Poisson rates.…”