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.

In many applications, we assume that two random observations x and y are generated according to independent Poisson distributions $𝒫\left(\lambda S\right)$ and $𝒫\left(\mu T\right)$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.

DOI : https://doi.org/10.1051/ps/2010018
Classification : 62F15,  62F03,  62F25,  62P10
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
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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},
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url = {http://archive.numdam.org/articles/10.1051/ps/2010018/}
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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://archive.numdam.org/articles/10.1051/ps/2010018/

[1] N. Balakrishnan, N.L. Johnson and S. Kotz, Continuous Univariate Distributions, 2nd edition. John Wiley, New York 1 (1995). | Zbl 0821.62001

[2] M.J. Bayarri and J. Berger, The interplay of Bayesian and frequentist analysis. Stat. Sci. 19 (2004) 58-80. | MR 2082147 | Zbl 1062.62001

[3] J.O. Berger and J.M. Bernardo, Ordered Group Reference Priors With Applications to Multinomial and Variance Component Problems. Technical Report Dept. of Statistics, Purdue University (1989).

[4] J.O. Berger and J.M. Bernardo, Estimating a product of means : Bayesian analysis with reference priors. J. Amer. Statist. Assoc. 84 (1989) 200-207. | MR 999679 | Zbl 0682.62018

[5] J.O. Berger and J.M. Bernardo, Ordered group reference priors, with applications to multinomial problems. Biometrika 79 (1992) 25-37. | MR 1158515 | Zbl 0763.62014

[6] J.O. Berger and J.M. Bernardo, 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 1380269 | Zbl 1194.62019

[7] J.O. Berger and D. Sun, Reference priors with partial information. Biometrika 85 (1998) 55-71. | MR 1627242 | Zbl 1067.62521

[8] J.O. Berger and R. Yang, A catalog of noninformative priors. ISDS Discussion Paper, Duke Univ. (1997) 97-42.

[9] J.O. Berger, J.M. Bernardo and D. Sun, The formal definition of reference priors. Ann. Stat. 37 (2009). | MR 2502655 | Zbl 1162.62013

[10] J.M. Bernardo, Reference posterior distributions for Bayesian inference (with discussion). J. R. Stat. Soc. B 41 (1979) 113-148. | MR 547240 | Zbl 0428.62004

[11] J.M. Bernardo, Noninformative priors do not exist : a discussion. (with discussion) J. Stat. Plann. Inference 65 (1997) 159-189. | MR 1619672

[12] J.M. Bernardo, Reference Analysis, edited by D.K. Dey and C.R. Rao. Handbook of Stat. 25 (2005) 17-90. | MR 2490522

[13] J.M. Bernardo, Intrinsic credible regions : an objective Bayesian approach to interval estimation (with discussion). Test 14 (2005) 317-384. | MR 2211385 | Zbl 1087.62036

[14] J.M. Bernardo and J.M. Ramon, An introduction to Bayesian reference analysis : inference on the ratio of multinomial parameters. J. R. Stat. Soc. D 47 (1998) 101-135.

[15] J.M. Bernardo and A.F.M. Smith, Bayesian Theory. Wiley, Chichester (1994). | MR 1274699 | Zbl 0943.62009

[16] D.A. Berry, M.C. Wolff and D. Sack, Decision making during a phase III randomized controlled trial. Control. Clin. Trials 15 (1994) 360-378.

[17] L.D. Brown, T.T. Cai and A. Dasgupta, Interval estimation for a binomial proportion (with discussion). Stat. Sci. 16 (2001) 101-133. | MR 1861069 | Zbl 1059.62533

[18] L.D. Brown, T.T. Cai and A. Dasgupta, Confidence intervals for a binomial proportion and edgeworth expansions. Ann. Stat. 30 (2002) 160-201. | MR 1892660 | Zbl 1012.62026

[19] H. Chu and M.E. Halloran, Bayesian estimation of vaccine efficacy. Clin. Trials 1 (2004) 306-314.

[20] R.D. Cousins, Improved central confidence intervals for the ratio of Poisson means. Nucl. Instrum. Methods Phys. Res. A 417 (1998) 391-399.

[21] G.S. Datta and R. Mukerjee, Probability Matching Priors : Higher Order Asymptotics. Springer, New-York (2004). | MR 2053794 | Zbl 1044.62031

[22] M. Ewell, Comparing methods for calculating confidence intervals for vaccine efficacy. Stat. Med. 15 (1996) 2379-2392.

[23] M.E. Halloran, I.M. Jr. Longini and C.J. Struchiner, Design and interpretation of vaccine field studies. Epidemiol. Rev. 21 (1999) 73-88.

[24] N.L. Johnson, A.W. Kemp and S. Kotz, Univariate Discrete Distributions, 3rd edition. John Wiley, New York (2005). | MR 2163227 | Zbl 0773.62007

[25] R.E. Kass and L. Wasserman, The selection of prior distributions by formal rules. J. Am. Statist. Assoc. 91 (1996) 1343-1370. | Zbl 0884.62007

[26] C. Kleiber and S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences, Wiley (2003). | MR 1994050 | Zbl 1044.62014

[27] K. Krishnamoorthy and M. Lee, 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 2581121 | Zbl 1181.62028

[28] K. Krishnamoorthy and J. Thomson, A more powerful test for comparing two Poisson means. J. Statist. Plann. Inference 119 (2004) 23-35. | MR 2018448 | Zbl 1031.62013

[29] B. Lecoutre, 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 1014.00013

[30] E.L. Lehmann and J.P. Romano, Testing Statistical Hypotheses, 3rd edition. Springer, New York (2005). | MR 2135927 | Zbl 1076.62018

[31] B. Liseo, Elimination of Nuisance Parameters with Reference Noninformative Priors. Technical Report #90-58C, Purdue University, Department of Statistics (1990).

[32] R.M. Price and D.G. Bonett, Estimating the ratio of two Poisson rates. Comput. Stat. Data Anal. 34 (2000) 345-356. | Zbl 1061.62523

[33] C. Robert, The Bayesian Choice : From Decision-Theoretic Foundations to Computational Implementation, 2nd edition. Springer Texts in Statistics (2001). | MR 1835885 | Zbl 1129.62003

[34] J. Robins and L. Wasserman, Conditioning, likelihood and coherence : A review of some foundational concepts. J. Amer. Statist. Assoc. 95 (2000) 1340-1346. | MR 1825290 | Zbl 1072.62507

[35] H. Sahai and A. Khurshid, Confidence intervals for the ratio of two Poisson means. Math. Sci. 18 (1993) 43-50. | MR 1227240 | Zbl 0770.62022

[36] J.D. Stamey, D.M. Young, T.L. Bratcher, Bayesian sample-size determination for one and two Poisson rate parameters with applications to quality control. J. Appl. Stat. 33 (2006) 583-594. | MR 2240928 | Zbl 1118.62315

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