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 ) 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.
Mots clés : 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://www.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://www.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://www.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, Tome 16 (2012), pp. 375-398. doi : 10.1051/ps/2010018. http://www.numdam.org/articles/10.1051/ps/2010018/
[1] Continuous Univariate Distributions, 2nd edition. John Wiley, New York 1 (1995). | Zbl
, and ,[2] The interplay of Bayesian and frequentist analysis. Stat. Sci. 19 (2004) 58-80. | MR | Zbl
and ,[3] Ordered Group Reference Priors With Applications to Multinomial and Variance Component Problems. Technical Report Dept. of Statistics, Purdue University (1989).
and ,[4] Estimating a product of means : Bayesian analysis with reference priors. J. Amer. Statist. Assoc. 84 (1989) 200-207. | MR | Zbl
and ,[5] Ordered group reference priors, with applications to multinomial problems. Biometrika 79 (1992) 25-37. | MR | Zbl
and ,[6] On the development of reference priors, edited by J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith, Bayesian Statistics. University Press, Oxford (with discussion) 4 (1992) 35-60. | MR | Zbl
and ,[7] Reference priors with partial information. Biometrika 85 (1998) 55-71. | MR | Zbl
and ,[8] A catalog of noninformative priors. ISDS Discussion Paper, Duke Univ. (1997) 97-42.
and ,[9] The formal definition of reference priors. Ann. Stat. 37 (2009). | MR | Zbl
, and ,[10] Reference posterior distributions for Bayesian inference (with discussion). J. R. Stat. Soc. B 41 (1979) 113-148. | MR | Zbl
,[11] Noninformative priors do not exist : a discussion. (with discussion) J. Stat. Plann. Inference 65 (1997) 159-189. | MR
,[12] Reference Analysis, edited by D.K. Dey and C.R. Rao. Handbook of Stat. 25 (2005) 17-90. | MR
,[13] Intrinsic credible regions : an objective Bayesian approach to interval estimation (with discussion). Test 14 (2005) 317-384. | MR | Zbl
,[14] An introduction to Bayesian reference analysis : inference on the ratio of multinomial parameters. J. R. Stat. Soc. D 47 (1998) 101-135.
and ,[15] Bayesian Theory. Wiley, Chichester (1994). | MR | Zbl
and ,[16] Decision making during a phase III randomized controlled trial. Control. Clin. Trials 15 (1994) 360-378.
, and ,[17] Interval estimation for a binomial proportion (with discussion). Stat. Sci. 16 (2001) 101-133. | MR | Zbl
, and ,[18] Confidence intervals for a binomial proportion and edgeworth expansions. Ann. Stat. 30 (2002) 160-201. | MR | Zbl
, and ,[19] Bayesian estimation of vaccine efficacy. Clin. Trials 1 (2004) 306-314.
and ,[20] Improved central confidence intervals for the ratio of Poisson means. Nucl. Instrum. Methods Phys. Res. A 417 (1998) 391-399.
,[21] Probability Matching Priors : Higher Order Asymptotics. Springer, New-York (2004). | MR | Zbl
and ,[22] Comparing methods for calculating confidence intervals for vaccine efficacy. Stat. Med. 15 (1996) 2379-2392.
,[23] Design and interpretation of vaccine field studies. Epidemiol. Rev. 21 (1999) 73-88.
, and ,[24] Univariate Discrete Distributions, 3rd edition. John Wiley, New York (2005). | MR | Zbl
, and ,[25] The selection of prior distributions by formal rules. J. Am. Statist. Assoc. 91 (1996) 1343-1370. | Zbl
and ,[26] Statistical Size Distributions in Economics and Actuarial Sciences, Wiley (2003). | MR | Zbl
and ,[27] Inference for functions of parameters in discrete distributions based on fiducial approach : Binomial and Poisson cases. J. Statist. Plann. Inference 140 (2009) 1182-1192. | MR | Zbl
and ,[28] A more powerful test for comparing two Poisson means. J. Statist. Plann. Inference 119 (2004) 23-35. | MR | Zbl
and ,[29] And if you were a Bayesian without knowing it? Bayesian inference and maximum entropy methods in science and engineering. AIP Conf. Proc. 872 (2006) 15-22. | Zbl
,[30] Testing Statistical Hypotheses, 3rd edition. Springer, New York (2005). | MR | Zbl
and ,[31] Elimination of Nuisance Parameters with Reference Noninformative Priors. Technical Report #90-58C, Purdue University, Department of Statistics (1990).
,[32] Estimating the ratio of two Poisson rates. Comput. Stat. Data Anal. 34 (2000) 345-356. | Zbl
and ,[33] The Bayesian Choice : From Decision-Theoretic Foundations to Computational Implementation, 2nd edition. Springer Texts in Statistics (2001). | MR | Zbl
,[34] Conditioning, likelihood and coherence : A review of some foundational concepts. J. Amer. Statist. Assoc. 95 (2000) 1340-1346. | MR | Zbl
and ,[35] Confidence intervals for the ratio of two Poisson means. Math. Sci. 18 (1993) 43-50. | MR | Zbl
and ,[36] Bayesian sample-size determination for one and two Poisson rate parameters with applications to quality control. J. Appl. Stat. 33 (2006) 583-594. | MR | Zbl
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