ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 2 Issue 1 - 2014
Bayesian Sample Size Determination in Two-Sample Poisson Models
Ryan Sides, David Kahle* and James Stamey
Statistical Science, Baylor University, USA
Received: February 10, 2015 | Published: March 09, 2015
*Corresponding author: David Kahle, Department of Statistical Science, Baylor University, Waco, TX, USA, Tel: 254-710-6102; E-mail:
Citation: yan S, David K, James S (2015) Bayesian Sample Size Determination in Two-Sample Poisson Models. Biom Biostat Int J 2(1): 00023. DOI: 10.15406/bbij.2014.2.00023

Abstract

Sample size determination is a vital part of clinical studies where cost and safety concerns lead to greater importance of not using more subjects and resources than are required. The Bayesian approach to sample size determination has the advantages of being able to use prior data and expert opinion to possibly reduce the total sample size while also acknowledging all uncertainty at the design stage. We apply a Bayesian decision theoretic approach to the problem of comparing two Poisson rates and find the required sample size to obtain a desired power while controlling the Type I error rate.

Keywords: Bayes factor; Poisson rate; power; Type I error

Introduction

Sample size determination continues to be an important research area of statistics. Perhaps nowhere is this truer than pharmaceutical statistics, where cost and time constraints have made finding the appropriate sample size before conducting a study of the utmost importance. The problem is quite simple: too small a sample can lead to under-powered studies, while too large a sample size wastes precious resources. In this article we consider the sample size determination problem as it pertains to the two-sample testing of Poisson rates from a Bayesian perspective subject to operating characteristics constraints.

There are several advantages to the Bayesian perspective when trying to determine a study's requisite sample size, a topic that is expounded in Adcock [1]. Their construction does not depend on asymptotic approximations or bounds. Classical solutions to the sample size determination problem typically hinge on asymptotic arguments that require the researcher to specify one parameter value (perhaps vector valued) as representative for the entire parameter space, a process that is typically done using bounding arguments. This, for example, is what is done when determining the requisite sample size for a confidence interval of a fixed level and given length. The resulting sample sizes are consequently conservative. On the other hand, the Bayesian approach provides the statistician with the ability to model his indecision about the parameter through expert knowledge or previous studies. As noted by Bayarri and Berger [2], this can allow the Bayesian approach to have better operating characteristics, such as a smaller required sample sizes or better Type I and II error rates.

Various Bayesian sample size determination methods have been studied for binomial and Poisson data. Stamey et al. [3] considered one and two sample Poisson rates from the perspective of interval based criteria such as coverage and width. Hand et al. [4] extend those ideas by considering both interval-based and test-based criteria, albeit without considering power. Katsis and Toman [5] used more decision theoretic criteria for the two sample binomial case, but only to the extent of controlling the posterior risk with a pre specified bound. Zhao et al. [6] extend those ideas by using computational methods to consider expected Bayesian power of the test. In this article, we extend these results to the Poisson data model. We also consider the problem the subject to Type I and Type II constraints. This is thus an important extension of [6], because it

  • Extends the ideas to the Poisson case
  • Enables the incorporation of operating characteristics.

A subtle difference between the classical and Bayesian methods of sample size determination merits discussion before proceeding. One of the novel contributions of this article is an algorithmic solution to the sample size determination problem subject to operating characteristics constraints for Poisson data. However, since the entire problem is treated in a Bayesian context, the concept of Type I and Type II error rates is understood in an average, or expected, sense; see [7]. For example, the ``power of a test'' retains the interpretation of the probability the decision rule rejects when the null hypothesis when it is false; but rather than being a function defined over the alternative space, here it is averaged over that space and weighted by the prior distribution specified on the alternative hypothesis. To make the distinction more clear, we refer to this as the expected Bayesian power (EBP), as is done in [8]; alternatively, it may be referred to as the probability of a successful test. These ideas, though apparently not considered in the literature previously, can also apply to the concept of the significance level of a test. While frequentist methods typically report one value for significance level, what they are really doing (in non point null hypotheses) is taking the largest possible significance level; thus, taking an expectation of a significance level curve could be done as well. Consequently, in this article we also consider the expected Bayesian significance level (EBSL), defined as the expected value of the test under the prior distribution given on the null space. In both cases, any particular instance of the actual Type I and Type II error rates can be greater than or less than nominal.

This article proceeds as follows. In Section 2 we introduce the theoretical formulation of the sample size determination problem for two Poisson variates including consideration of operating characteristics. In Section 3, we present an algorithmic solution to the sample size determination problem posed in Section 4. Section 5 contains an application of the method in the area of pharmaceutical statistics. We then conclude with a discussion.

Problem specification and the bayes rule
We now follow the general framework of [6] in the development of this problem, adapting the binomial case to fit the Poisson data model. Suppose Y 1 ~Pois(t λ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaaIXaaabeaakiaac6hacaWGqbGaam4BaiaadMgacaWGZbGa aiikaiaadshaiiaacqWF7oaBdaWgaaWcbaGaaGymaaqabaGccaGGPa aaaa@40F3@  and Y 2 ~Pois(t λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaaIYaaabeaakiaac6hacaWGqbGaam4BaiaadMgacaWGZbGa aiikaiaadshaiiaacqWF7oaBdaWgaaWcbaGaaGOmaaqabaGccaGGPa aaaa@40F5@ , independently, where λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae83UdW 2aaSbaaSqaaiaaigdaaeqaaaaa@3817@  and λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae83UdW 2aaSbaaSqaaiaaikdaaeqaaaaa@3818@  represent the rate parameters of interest and t represents a common sample (or “opportunity”) size. Together, we write these Y=( Y 1 , Y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmGaa8xwai abg2da9iaacIcacaWGzbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaa dMfadaWgaaWcbaGaaGOmaaqabaGcceGGPaGbauaaaaa@3D17@  with observations y=( y 1 , y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmGaa8xEai abg2da9iaacIcacaWG5bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaa dMhadaWgaaWcbaGaaGOmaaqabaGcceGGPaGbauaaaaa@3D77@ . The sample size determination problem is to calculate the necessary sample size required to test the hypotheses
H 0 : λ 1 = λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIWaaabeaakiaacQdacqaH7oaBdaWgaaWcbaGaaGymaaqa baGccqGH9aqpcqaH7oaBdaWgaaWcbaGaaGOmaaqabaaaaa@3E38@ vs    H 1 : λ 1 λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIXaaabeaakiaacQdacqaH7oaBdaWgaaWcbaGaaGymaaqa baGccqGHGjsUcqaH7oaBdaWgaaWcbaGaaGOmaaqabaaaaa@3EFA@                                                            
using a given decision rule; here we use the Bayes rule. Denoting the parameter pair λ=( λ 1 , λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacceGae83UdW Maeyypa0JaaiikaGGaaiab+T7aSnaaBaaaleaacaaIXaaabeaakiaa cYcacqGF7oaBdaWgaaWcbaGaaGOmaaqabaGcceGGPaGbauaaaaa@3F95@ , the associated null and alternative spaces are therefore Λ 0 ={λ R + 2 : λ 1 = λ 2 }, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaaicdaaeqaaOGaeyypa0Jaai4EaGGabiab=T7aSjabgIGi olaadkfadaqhaaWcbaGaey4kaScabaGaaGOmaaaakiaacQdaiiaacq GF7oaBdaWgaaWcbaGaaGymaaqabaGccqGH9aqpcqGF7oaBdaWgaaWc baGaaGOmaaqabaGccaGG9bGaaiilaaaa@4889@  which we identify with R + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacqGHRaWkaeqaaaaa@375C@  with elements generically denoted λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@372B@ , and Λ 1 ={λ R + 2 : λ 1 λ 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaaigdaaeqaaOGaeyypa0Jaai4EaGGabiab=T7aSjabgIGi olaadkfadaqhaaWcbaGaey4kaScabaGaaGOmaaaakiaacQdaiiaacq GF7oaBdaWgaaWcbaGaaGymaaqabaGccqGHGjsUcqGF7oaBdaWgaaWc baGaaGOmaaqabaGccaGG9baaaa@489B@ .
As this problem is being considered from the Bayesian perspective, we place prior probabilities of π 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaaicdaaeqaaaaa@381A@  and π 1 =1 π 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGymaiabgkHiTiabec8aWnaa BaaaleaacaaIWaaabeaaaaa@3D76@  on H 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIWaaabeaaaaa@372A@  and H 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIXaaabeaaaaa@372B@ , respectively. Conditional on the null being true, we represent the expert opinion regarding λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@3729@  as p 0 (λ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaaIWaaabeaakiaacIcaiiaacqWF7oaBcaGGPaaaaa@3A6C@ , defined over the set Λ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaaicdaaeqaaaaa@37D2@ . Alternatively, conditionally on H 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIXaaabeaaaaa@372B@  being true, we represent the belief concerning λ 1 λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae83UdW 2aaSbaaSqaaiaaigdaaeqaaOGaeyiyIKRae83UdW2aaSbaaSqaaiaa ikdaaeqaaaaa@3C7F@  with a joint prior p( λ 1 , λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI caiiaacqWF7oaBdaWgaaWcbaGaaGymaaqabaGccaGGSaGae83UdW2a aSbaaSqaaiaaikdaaeqaaOGaaiykaaaa@3DBE@  with support Λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaaigdaaeqaaaaa@37D3@ . Marginalizing over the hypotheses, we have the unconditional prior
p( λ 1 , λ 2 )= p 0 (λ) π 0 I H 0 + p 1 ( λ 1 , λ 2 ) π 1 I H 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI caiiaacqWF7oaBdaWgaaWcbaGaaGymaaqabaGccaGGSaGae83UdW2a aSbaaSqaaiaaikdaaeqaaOGaaiykaiabg2da9iaadchadaWgaaWcba GaaGimaaqabaGccaGGOaGae83UdWMaaiykaiab=b8aWnaaBaaaleaa caaIWaaabeaakiaadMeadaWgaaWcbaGaamisamaaBaaameaacaaIWa aabeaaaSqabaGccqGHRaWkcaWGWbWaaSbaaSqaaiaaigdaaeqaaOGa aiikaiab=T7aSnaaBaaaleaacaaIXaaabeaakiaacYcacqWF7oaBda WgaaWcbaGaaGOmaaqabaGccaGGPaGae8hWda3aaSbaaSqaaiaaigda aeqaaOGaamysamaaBaaaleaacaWGibWaaSbaaWqaaiaaigdaaeqaaa WcbeaakiaacYcaaaa@594B@ Where I H 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGibWaaSbaaWqaaiaaicdaaeqaaaWcbeaaaaa@382E@  and I H 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGibWaaSbaaWqaaiaaigdaaeqaaaWcbeaaaaa@382F@  are the indicator function of H 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIWaaabeaaaaa@372A@  and H 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIXaaabeaaaaa@372B@ , respectively.
In practice, the prior distributions on λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW gaaa@372E@ , λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW 2aaSbaaSqaaiaaigdaaeqaaaaa@3815@ , and λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW 2aaSbaaSqaaiaaikdaaeqaaaaa@3816@  summarize expert opinion concerning the parameters in each of the two scenarios H 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIWaaabeaaaaa@372A@  and H 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIXaaabeaaaaa@372B@ . This information can be obtained in a number of ways including past data, prior elicitation of expert opinion (see especially [9]), or based on uninformative criteria. For simplicity, we consider conjugate priors for all three λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW gaaa@372E@ ’s, so that under H 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIWaaabeaaaaa@372A@ , λGamma(α,β) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW Mae8hpI4Naae4raiaabggacaqGTbGaaeyBaiaabggacaGGOaGae8xS deMaaiilaiab=j7aIjaacMcaaaa@4240@ , and under H 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIXaaabeaaaaa@372B@ , λ i Gamma( α i , β i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW 2aaSbaaSqaaiaadMgaaeqaaOGae8hpI4Naae4raiaabggacaqGTbGa aeyBaiaabggacaGGOaGae8xSde2aaSbaaSqaaiaadMgaaeqaaOGaai ilaiab=j7aInaaBaaaleaacaWGPbaabeaakiaacMcaaaa@45AC@  independently. This assumption is not very restrictive and allows us to specify parameters of prior distribution sin stead of distributions themselves.
We now derive the optimal Bayes (decision) rule in deciding between the hypotheses presented in (1). In solving the related problem between two binomial proportions, [6] use the classical decision theoretic setup using the 0-1 loss function [10]. Here we use the more general unequal loss function
L(H,a)={ c 1 if  H 0  is true and a = 1, c 2 if  H 1  is true and a = 0, 0,otherwise                     MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamitaiaacI cacaWGibGaaiilaiaadggacaGGPaGaeyypa0ZaaiqaaeaafaqabeWa baaabaGaam4yamaaBaaaleaacaaIXaaabeaakiaaykW7caaMc8UaaG PaVlaaykW7caqGPbGaaeOzaiaabccacaWGibWaaSbaaSqaaiaaicda aeqaaOGaaeiiaiaabMgacaqGZbGaaeiiaiaabshacaqGYbGaaeyDai aabwgacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaadggacaqGGaGa aeypaiaabccacaqGXaGaaeilaaqaaiaadogadaWgaaWcbaGaaGOmaa qabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaeyAaiaabAgacaqGGaGa amisamaaBaaaleaacaaIXaaabeaakiaabccacaqGPbGaae4Caiaabc cacaqG0bGaaeOCaiaabwhacaqGLbGaaeiiaiaabggacaqGUbGaaeiz aiaabccacaWGHbGaaeiiaiaab2dacaqGGaGaaeimaiaabYcaaeaaca aIWaGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caqGVbGaaeiDaiaa bIgacaqGLbGaaeOCaiaabEhacaqGPbGaae4CaiaabwgacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaaaaaiaawUhaaaaa@8FF6@
Where a=δ(y) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9iabes7aKjaacIcaieWacaWF5bGaaiykaaaa@3B65@  is the decision rule with a = 0 representing selection of H 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIWaaabeaaaaa@372A@  and a = 1 selection of H 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIXaaabeaaaaa@372B@ . Thus, c1 represents the loss associated with a Type I error, and c2 that of a Type II error.
The Bayes action is simply the one that minimizes posterior expected loss [11]. Since the posterior expected loss associated with an action a{0,1} MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyaiabgI GiolaacUhacaaIWaGaaiilaiaaigdacaGG9baaaa@3C04@  is
ρ(a)= E λ|y [L(φ,a)] ={ c 1 P( H 0 |Y=y),   if a=1, c 2 P( H 1 |Y=y),   if a=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGceaqabeaaiiaacq WFbpGCcaGGOaGaamyyaiaacMcacqGH9aqpcaWGfbWaaSbaaSqaaGGa biab+T7aSjaacYhaieWacaqF5baabeaakiaacUfacaWGmbGaaiikai abeA8aQjaacYcacaWGHbGaaiykaiaac2faaeaacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlabg2da9maaceaabaqbaeqabiqaaaqaaiaadogadaWgaaWcbaGa aGymaaqabaGccaWGqbGaaiikaiaadIeadaWgaaWcbaGaaGimaaqaba GccaGG8bGaa0xwaiabg2da9iaa9LhacaGGPaGaaiilaiaabccacaqG GaGaaeiiaiaabMgacaqGMbGaaeiiaiaadggacqGH9aqpcaaIXaGaai ilaaqaaiaadogadaWgaaWcbaGaaGOmaaqabaGccaWGqbGaaiikaiaa dIeadaWgaaWcbaGaaGymaaqabaGccaGG8bGaa0xwaiabg2da9iaa9L hacaGGPaGaaiilaiaabccacaqGGaGaaeiiaiaabMgacaqGMbGaaeii aiaadggacqGH9aqpcaaIWaGaaiilaaaaaiaawUhaaaaaaa@7DAA@ Setting c= c 1 / c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2 da9iaadogadaWgaaWcbaGaaGymaaqabaGccaGGVaGaam4yamaaBaaa leaacaaIYaaabeaaaaa@3BBF@  we can express the optimal decision rule, a * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaiOkaaaaaaa@3736@ , as
a * (y)={ 0,   if P( H 1 |Y=y)<cP( H 0 |Y=y), 1,   if P( H 1 |Y=y)cP( H 0 |Y=y). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaiOkaaaakiaacIcaieWacaWF5bGaaiykaiabg2da9maa ceaabaqbaeqabiqaaaqaaiaaicdacaGGSaGaaeiiaiaabccacaqGGa GaaeyAaiaabAgacaqGGaGaamiuaiaacIcacaWGibWaaSbaaSqaaiaa igdaaeqaaOGaaiiFaiaa=LfacqGH9aqpcaWF5bGaaiykaiabgYda8i aadogacaWGqbGaaiikaiaadIeadaWgaaWcbaGaaGimaaqabaGccaGG 8bGaa8xwaiabg2da9iaa=LhacaGGPaGaaiilaaqaaiaaigdacaGGSa GaaeiiaiaabccacaqGGaGaaeyAaiaabAgacaqGGaGaamiuaiaacIca caWGibWaaSbaaSqaaiaaigdaaeqaaOGaaiiFaiaa=LfacqGH9aqpca WF5bGaaiykaiabgwMiZkaadogacaWGqbGaaiikaiaadIeadaWgaaWc baGaaGimaaqabaGccaGG8bGaa8xwaiabg2da9iaa=LhacaGGPaGaai OlaaaaaiaawUhaaaaa@6C74@

The rejection region, W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4vaaaa@3651@ , is therefore
W={y:P( H 1 |Y=y)cP( H 0 |Y=y)}. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9iaacUhaieWacaWF5bGaaiOoaiaadcfacaGGOaGaamisamaaBaaa leaacaaIXaaabeaakiaacYhacaWFzbGaeyypa0Jaa8xEaiaacMcacq GHLjYScaWGJbGaamiuaiaacIcacaWGibWaaSbaaSqaaiaaicdaaeqa aOGaaiiFaiaa=LfacqGH9aqpcaWF5bGaaiykaiaac2hacaGGUaaaaa@4E05@ The optimal rule in (2) can be nicely represented in terms of the Bayes factor. The Bayes factor is defined as the ratio of the posterior odds of H 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIXaaabeaaaaa@3729@  to the prior odds of H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamisaaaa@3642@ , so that a large Bayes factor is evidence for rejecting the null hypothesis [12]. Specifically, the Bayes factor is defined

B= P( H 1 |Y=y)/P( H 0 |Y=y) π 1 / π 0 = P( H 1 |Y=y) π 0 P( H 0 |Y=y) π 1 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOqaiabg2 da9maalaaabaGaamiuaiaacIcacaWGibWaaSbaaSqaaiaaigdaaeqa aOGaaiiFaGqadiaa=LfacqGH9aqpcaWF5bGaaiykaiaac+cacaWGqb GaaiikaiaadIeadaWgaaWcbaGaaGimaaqabaGccaGG8bGaa8xwaiab g2da9iaa=LhacaGGPaaabaaccaGae4hWda3aaSbaaSqaaiaaigdaae qaaOGaai4laiab+b8aWnaaBaaaleaacaaIWaaabeaaaaGccqGH9aqp daWcaaqaaiaadcfacaGGOaGaamisamaaBaaaleaacaaIXaaabeaaki aacYhacaWFzbGaeyypa0Jaa8xEaiaacMcacqGFapaCdaWgaaWcbaGa aGimaaqabaaakeaacaWGqbGaaiikaiaadIeadaWgaaWcbaGaaGimaa qabaGccaGG8bGaa8xwaiabg2da9iaa=LhacaGGPaGae4hWda3aaSba aSqaaiaaigdaaeqaaaaakiaac6caaaa@6449@

This ratio is useful in Bayesian inference because it is often interpreted as partially eliminating the influence of the prior on the posterior, instead emphasizing the role of the data. Moreover, the decision rule is a function of a Bayes factor:
W={ y:P( H 1 |Y=y)cP( H 0 |Y=y) } ={ y:P( H 1 |Y=y) π 0 π 1 cP( H 0 |Y=y) π 0 π 1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGceaqabeaacaWGxb Gaeyypa0ZaaiWaaeaaieWacaWF5bGaaiOoaiaadcfacaGGOaGaamis amaaBaaaleaacaaIXaaabeaakiaacYhacaWFzbGaeyypa0Jaa8xEai aacMcacqGHLjYScaWGJbGaamiuaiaacIcacaWGibWaaSbaaSqaaiaa icdaaeqaaOGaaiiFaiaa=LfacqGH9aqpcaWF5bGaaiykaaGaay5Eai aaw2haaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab g2da9maacmaabaGaa8xEaiaacQdacaWGqbGaaiikaiaadIeadaWgaa WcbaGaaGymaaqabaGccaGG8bGaa8xwaiabg2da9iaa=LhacaGGPaWa aSaaaeaaiiaacqGFapaCdaWgaaWcbaGaaGimaaqabaaakeaacqGFap aCdaWgaaWcbaGaaGymaaqabaaaaOGaeyyzImRaam4yaiaadcfacaGG OaGaamisamaaBaaaleaacaaIWaaabeaakiaacYhacaWFzbGaeyypa0 Jaa8xEaiaacMcadaWcaaqaaiab+b8aWnaaBaaaleaacaaIWaaabeaa aOqaaiab+b8aWnaaBaaaleaacaaIXaaabeaaaaaakiaawUhacaGL9b aaaaaa@78BD@ = ={ y:Bc π 0 π 1 }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGPaVlabg2 da9maacmaabaacbmGaa8xEaiaacQdacaWGcbGaeyyzImRaam4yamaa laaabaaccaGae4hWda3aaSbaaSqaaiaaicdaaeqaaaGcbaGae4hWda 3aaSbaaSqaaiaaigdaaeqaaaaaaOGaay5Eaiaaw2haaiaac6caaaa@458A@

This is particularly useful because it allows for the interpretation of the Bayes factor B as the test statistic for the decision rule in (2); this is the condition in (4).

We now derive closed-form expression for the posterior probabilities of the null and alternative hypotheses. Using Bayes’ theorem, we have

P( H 0 |Y=y)= P(Y=y| H 0 ) π 0 P(Y=y| H 0 ) π 0 +P(Y=y| H 1 ) π 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI cacaWGibWaaSbaaSqaaiaaicdaaeqaaOGaaiiFaGqadiaa=LfacqGH 9aqpcaWF5bGaaiykaiabg2da9maalaaabaGaamiuaiaacIcacaWFzb Gaeyypa0Jaa8xEaiaacYhacaWGibWaaSbaaSqaaiaaicdaaeqaaOGa aiykaGGaaiab+b8aWnaaBaaaleaacaaIWaaabeaaaOqaaiaadcfaca GGOaGaa8xwaiabg2da9iaa=LhacaGG8bGaamisamaaBaaaleaacaaI WaaabeaakiaacMcacqGFapaCdaWgaaWcbaGaaGimaaqabaGccqGHRa WkcaWGqbGaaiikaiaa=LfacqGH9aqpcaWF5bGaaiiFaiaadIeadaWg aaWcbaGaaGymaaqabaGccaGGPaGae4hWda3aaSbaaSqaaiaaigdaae qaaaaaaaa@5E83@ and
P( H 1 |Y=y)= P(Y=y| H 1 ) π 1 P(Y=y| H 0 ) π 0 +P(Y=y| H 1 ) π 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI cacaWGibWaaSbaaSqaaiaaigdaaeqaaOGaaiiFaGqadiaa=LfacqGH 9aqpcaWF5bGaaiykaiabg2da9maalaaabaGaamiuaiaacIcacaWFzb Gaeyypa0Jaa8xEaiaacYhacaWGibWaaSbaaSqaaiaaigdaaeqaaOGa aiykaGGaaiab+b8aWnaaBaaaleaacaaIXaaabeaaaOqaaiaadcfaca GGOaGaa8xwaiabg2da9iaa=LhacaGG8bGaamisamaaBaaaleaacaaI WaaabeaakiaacMcacqGFapaCdaWgaaWcbaGaaGimaaqabaGccqGHRa WkcaWGqbGaaiikaiaa=LfacqGH9aqpcaWF5bGaaiiFaiaadIeadaWg aaWcbaGaaGymaaqabaGccaGGPaGae4hWda3aaSbaaSqaaiaaigdaae qaaaaaaaa@5E86@ .                                   
Consequently, the posterior probabilities have closed-form expressions if the likelihoods do. Computing these, we have
P(Y=y| H 0 )= 0 f(y|λ, H 0 ) p 0 (λ| H 0 )dλ = 0 (λt) y 1 e λt y 1 ! (λt) y 2 e λt y 2 ! β α Γ(α) λ α1 e βλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGceaqabeaacaWGqb GaaiikaGqadiaa=LfacqGH9aqpcaWF5bGaaiiFaiaadIeadaWgaaWc baGaaGimaaqabaGccaGGPaGaeyypa0Zaa8qCaeaacaWGMbGaaiikai aa=LhacaGG8baccaGae43UdWMaaiilaiaadIeadaWgaaWcbaGaaGim aaqabaGccaGGPaGaamiCamaaBaaaleaacaaIWaaabeaakiaacIcacq GF7oaBcaGG8bGaamisamaaBaaaleaacaaIWaaabeaakiaacMcacaWG KbGae43UdWgaleaacaaIWaaabaGaeyOhIukaniabgUIiYdaakeaaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeyypa0Zaa8 qCaeaadaWcaaqaaiaacIcacqGF7oaBcaWG0bGaaiykamaaCaaaleqa baGaamyEamaaBaaameaacaaIXaaabeaaaaGccaWGLbWaaWbaaSqabe aacqGHsislcqGF7oaBcaWG0baaaaGcbaGaamyEamaaBaaaleaacaaI XaaabeaakiaacgcaaaWaaSaaaeaacaGGOaGae43UdWMaamiDaiaacM cadaahaaWcbeqaaiaadMhadaWgaaadbaGaaGOmaaqabaaaaOGaamyz amaaCaaaleqabaGaeyOeI0Iae43UdWMaamiDaaaaaOqaaiaadMhada WgaaWcbaGaaGOmaaqabaGccaGGHaaaamaalaaabaGae4NSdi2aaWba aSqabeaacqGFXoqyaaaakeaacqqHtoWrcaGGOaGaeqySdeMaaiykaa aacqGF7oaBdaahaaWcbeqaaiab+f7aHjabgkHiTiaaigdaaaGccaWG LbWaaWbaaSqabeaacqGHsislcqGFYoGycqGF7oaBaaGccaWGKbGae4 3UdWgaleaacaaIWaaabaGaeyOhIukaniabgUIiYdaaaaa@BADF@ = t y 1 + y 2 β α Γ( y 1 + y 2 +α) y 1 ! y 2 !Γ(α) (2t+β) y 1 + y 2 +α , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGPaVlaayk W7caaMc8UaaGPaVlabg2da9maalaaabaGaamiDamaaCaaaleqabaGa amyEamaaBaaameaacaaIXaaabeaaliabgUcaRiaadMhadaWgaaadba GaaGOmaaqabaaaaGGaaOGae8NSdi2aaWbaaSqabeaacqWFXoqyaaGc cqqHtoWrcaGGOaGaamyEamaaBaaaleaacaaIXaaabeaakiabgUcaRi aadMhadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcqWFXoqycaGGPaaa baGaamyEamaaBaaaleaacaaIXaaabeaakiaacgcacaWG5bWaaSbaaS qaaiaaikdaaeqaaOGaaiyiaiabfo5ahjaacIcacqWFXoqycaGGPaGa aiikaiaaikdacaWG0bGaey4kaSIae8NSdiMaaiykamaaCaaaleqaba GaamyEamaaBaaameaacaaIXaaabeaaliabgUcaRiaadMhadaWgaaad baGaaGOmaaqabaWccqGHRaWkcqWFXoqyaaaaaOGaaiilaaaa@674D@ and
P(Y=y| H 1 )= 0 f(y|λ, H 1 ) p 1 (λ| H 1 )dλ = i=1 2 0 ( λ i t) y i e λ i t y i ! β i α i Γ( α i ) λ i α i 1 e β i λ i d λ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGceaqabeaacaWGqb GaaiikaGqadiaa=LfacqGH9aqpcaWF5bGaaiiFaiaadIeadaWgaaWc baGaaGymaaqabaGccaGGPaGaeyypa0Zaa8qCaeaacaWGMbGaaiikai aa=LhacaGG8baccaGae43UdWMaaiilaiaadIeadaWgaaWcbaGaaGym aaqabaGccaGGPaGaamiCamaaBaaaleaacaaIXaaabeaakiaacIcaii qacqqF7oaBcaGG8bGaamisamaaBaaaleaacaaIXaaabeaakiaacMca caWGKbGae03UdWgaleaacaaIWaaabaGaeyOhIukaniabgUIiYdaake aacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeyypa0 ZaaebCaeaadaWdXbqaamaalaaabaGaaiikaiab+T7aSnaaBaaaleaa caWGPbaabeaakiaadshacaGGPaWaaWbaaSqabeaacaWG5bWaaSbaaW qaaiaadMgaaeqaaaaakiaadwgadaahaaWcbeqaaiabgkHiTiab+T7a SnaaBaaameaacaWGPbaabeaaliaadshaaaaakeaacaWG5bWaaSbaaS qaaiaadMgaaeqaaOGaaiyiaaaadaWcaaqaaiab+j7aInaaBaaaleaa caWGPbaabeaakmaaCaaaleqabaGae4xSde2aaSbaaWqaaiaadMgaae qaaaaaaOqaaiabfo5ahjaacIcacqaHXoqydaWgaaWcbaGaamyAaaqa baGccaGGPaaaaiab+T7aSnaaBaaaleaacaWGPbaabeaakmaaCaaale qabaGae4xSde2aaSbaaWqaaiaadMgaaeqaaSGaeyOeI0IaaGymaaaa kiaadwgadaahaaWcbeqaaiabgkHiTiab+j7aInaaBaaameaacaWGPb aabeaaliab+T7aSnaaBaaameaacaWGPbaabeaaaaGccaWGKbGae43U dW2aaSbaaSqaaiaadMgaaeqaaaqaaiaaicdaaeaacqGHEisPa0Gaey 4kIipaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaaGOmaaqdcqGHpis1 aaaaaa@BEBD@
= i=1 2 t y i β i α i Γ( y i + α i ) y i !Γ( α i ) (t+ β i ) y i + α i . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH9aqpdaqe WbqaamaalaaabaGaamiDamaaCaaaleqabaGaamyEamaaBaaameaaca WGPbaabeaaaaaccaGccqWFYoGydaWgaaWcbaGaamyAaaqabaGcdaah aaWcbeqaaiab=f7aHnaaBaaameaacaWGPbaabeaaaaGccqqHtoWrca GGOaGaamyEamaaBaaaleaacaWGPbaabeaakiabgUcaRiab=f7aHnaa BaaaleaacaWGPbaabeaakiaacMcaaeaacaWG5bWaaSbaaSqaaiaadM gaaeqaaOGaaiyiaiabfo5ahjaacIcacqWFXoqydaWgaaWcbaGaamyA aaqabaGccaGGPaGaaiikaiaadshacqGHRaWkcqWFYoGydaWgaaWcba GaamyAaaqabaGccaGGPaWaaWbaaSqabeaacaWG5bWaaSbaaWqaaiaa dMgaaeqaaSGaey4kaSIae8xSde2aaSbaaWqaaiaadMgaaeqaaaaaaa aaleaacaWGPbGaeyypa0JaaGymaaqaaiaaikdaa0Gaey4dIunakiaa c6caaaa@6EDE@

Note that the probability of the data under the null hypothesis is the product of two independent negative binomial likelihoods.
Combining (3) with (5) and (6) allows us to represent W in terms of the null and alternative likelihoods as follows:

W={ y:P( H 1 |Y=y)cP( H 0 |Y=y) } ={ y:P(Y=y| H 1 ) π 1 cP(Y=y| H 0 ) π 0 }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGceaqabeaacaWGxb Gaeyypa0ZaaiWaaeaaieWacaWF5bGaaiOoaiaadcfacaGGOaGaamis amaaBaaaleaacaaIXaaabeaakiaacYhacaWFzbGaeyypa0Jaa8xEai aacMcacqGHLjYScaWGJbGaamiuaiaacIcacaWGibWaaSbaaSqaaiaa icdaaeqaaOGaaiiFaiaa=LfacqGH9aqpcaWF5bGaaiykaaGaay5Eai aaw2haaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab g2da9maacmaabaGaa8xEaiaacQdacaWGqbGaaiikaiaa=LfacqGH9a qpcaWF5bGaaiiFaiaadIeadaWgaaWcbaGaaGymaaqabaGccaGGPaac caGae4hWda3aaSbaaSqaaiaaigdaaeqaaOGaeyyzImRaam4yaiaadc facaGGOaGaa8xwaiabg2da9iaa=LhacaGG8bGaamisamaaBaaaleaa caaIWaaabeaakiaacMcacqGFapaCdaWgaaWcbaGaaGimaaqabaaaki aawUhacaGL9baacaGGUaaaaaa@7400@

Consequently, (7) and (8) give explicit conditions for the rejection of the optimal decision rule:
W={ y:B= Γ(α) (2t+β) y 1 + y 2 +α β α Γ( y 1 + y 2 +α) i=1 2 β i α i Γ( y i + α i ) Γ( α i ) (t+ β i ) y i + α i c π 0 π 1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maacmaabaacbmGaa8xEaiaacQdacaWGcbGaeyypa0ZaaSaaaeaa cqqHtoWrcaGGOaaccaGae4xSdeMae4xkaKIaaiikaiaaikdacaWG0b Gaey4kaSIae4NSdiMaaiykamaaCaaaleqabaGaamyEamaaBaaameaa caaIXaaabeaaliabgUcaRiaadMhadaWgaaadbaGaaGOmaaqabaWccq GHRaWkcqGFXoqyaaaakeaacqGFYoGydaahaaWcbeqaaiab+f7aHbaa kiabfo5ahjaacIcacaWG5bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaS IaamyEamaaBaaaleaacaaIYaaabeaakiabgUcaRiab+f7aHjaacMca aaWaaebCaeaadaWcaaqaaiab+j7aInaaBaaaleaacaWGPbaabeaakm aaCaaaleqabaGae4xSde2aaSbaaWqaaiaadMgaaeqaaaaakiabfo5a hjaacIcacaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIae4xSde 2aaSbaaSqaaiaadMgaaeqaaOGaaiykaaqaaiabfo5ahjaacIcacqGF XoqydaWgaaWcbaGaamyAaaqabaGccaGGPaGaaiikaiaadshacqGHRa WkcqGFYoGydaWgaaWcbaGaamyAaaqabaGccaGGPaWaaWbaaSqabeaa caWG5bWaaSbaaWqaaiaadMgaaeqaaSGaey4kaSIae4xSde2aaSbaaW qaaiaadMgaaeqaaaaaaaaaleaacaWGPbGaeyypa0JaaGymaaqaaiaa ikdaa0Gaey4dIunakiabgwMiZkaadogadaWcaaqaaiab+b8aWnaaBa aaleaacaaIWaaabeaaaOqaaiab+b8aWnaaBaaaleaacaaIXaaabeaa aaaakiaawUhacaGL9baaaaa@8966@
Note that the left side of (9) is our test statistic and Bayes factor, B, so that (9) is an explicit formulation of the condition presented in (4).

Sample Size Determination for the Bayes Rule
The explicit description of the decision rule in (9) allows us to compute all sorts of quantities of interest. For given prior parameters π 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaaicdaaeqaaaaa@3818@ , π 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaaigdaaeqaaaaa@3819@ , α, β, α 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde 2aaSbaaSqaaiaaigdaaeqaaaaa@3800@ , β 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8NSdi 2aaSbaaSqaaiaaikdaaeqaaaaa@3803@ , α 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde 2aaSbaaSqaaiaaikdaaeqaaaaa@3801@ , and β 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8NSdi 2aaSbaaSqaaiaaikdaaeqaaaaa@3803@  and loss penalties c 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaaIXaaabeaaaaa@3744@ and c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaaIYaaabeaaaaa@3745@ (or simply c), the Expected Bayesian Power (EBP) ω t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadshaaeqaaaaa@3867@ is defined
ω t =P(YW| H 1 )= yW P(Y=y| H 1 )= yW i=1 2 t y i β i α i Γ( y i + α i ) y i !Γ( α i ) (t+ β i ) y i + α i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadshaaeqaaOGaeyypa0JaamiuaiaacIcaieWacaWFzbGa eyicI4Saam4vaiaacYhacaWGibWaaSbaaSqaaiaaigdaaeqaaOGaai ykaiabg2da9maaqafabaGaamiuaiaacIcacaWFzbGaeyypa0Jaa8xE aiaacYhacaWGibWaaSbaaSqaaiaaigdaaeqaaaqaaiaa=LhacqGHii IZcaWGxbaabeqdcqGHris5aOGaaiykaiabg2da9maaqafabaWaaebC aeaadaWcaaqaaiaadshadaahaaWcbeqaaiaadMhadaWgaaadbaGaam yAaaqabaaaaGGaaOGae4NSdi2aaSbaaSqaaiaadMgaaeqaaOWaaWba aSqabeaacqGFXoqydaWgaaadbaGaamyAaaqabaaaaOGaeu4KdCKaai ikaiaadMhadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcqGFXoqydaWg aaWcbaGaamyAaaqabaGccaGGPaaabaGaamyEamaaBaaaleaacaWGPb aabeaakiaacgcacqqHtoWrcaGGOaGae4xSde2aaSbaaSqaaiaadMga aeqaaOGaaiykaiaacIcacaWG0bGaey4kaSIae4NSdi2aaSbaaSqaai aadMgaaeqaaOGaaiykamaaCaaaleqabaGaamyEamaaBaaameaacaWG PbaabeaaliabgUcaRiab+f7aHnaaBaaameaacaWGPbaabeaaaaaaaa WcbaGaamyAaiabg2da9iaaigdaaeaacaaIYaaaniabg+Givdaaleaa caWF5bGaeyicI4Saam4vaaqab0GaeyyeIuoakiaacYcaaaa@826D@
and the Expected Bayesian Significance Level (EBSL)   α t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde 2aaSbaaSqaaiaadshaaeqaaaaa@383E@ is
α t =P(YW| H 0 )= yW P(Y=y| H 0 )= yW t y 1 + y 2 β α Γ( y 1 + y 2 +α) y 1 ! y 2 !Γ(α) (2t+β) y 1 + y 2 +α . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde 2aaSbaaSqaaiaadshaaeqaaOGaeyypa0JaamiuaiaacIcaieWacaGF zbGaeyicI4Saam4vaiaacYhacaWGibWaaSbaaSqaaiaaicdaaeqaaO Gaaiykaiabg2da9maaqafabaGaamiuaiaacIcacaGFzbGaeyypa0Ja a4xEaiaacYhacaWGibWaaSbaaSqaaiaaicdaaeqaaaqaaiaa+Lhacq GHiiIZcaWGxbaabeqdcqGHris5aOGaaiykaiabg2da9maaqafabaWa aSaaaeaacaWG0bWaaWbaaSqabeaacaWG5bWaaSbaaWqaaiaaigdaae qaaSGaey4kaSIaamyEamaaBaaameaacaaIYaaabeaaaaGccqWFYoGy daahaaWcbeqaaiab=f7aHbaakiabfo5ahjaacIcacaWG5bWaaSbaaS qaaiaaigdaaeqaaOGaey4kaSIaamyEamaaBaaaleaacaaIYaaabeaa kiabgUcaRiab=f7aHjaacMcaaeaacaWG5bWaaSbaaSqaaiaaigdaae qaaOGaaiyiaiaadMhadaWgaaWcbaGaaGOmaaqabaGccaGGHaGaeu4K dCKaaiikaiab=f7aHjaacMcacaGGOaGaaGOmaiaadshacqGHRaWkcq WFYoGycaGGPaWaaWbaaSqabeaacaWG5bWaaSbaaWqaaiaaigdaaeqa aSGaey4kaSIaamyEamaaBaaameaacaaIYaaabeaaliabgUcaRiab=f 7aHbaaaaaabaGaa4xEaiabgIGiolaadEfaaeqaniabggHiLdGccaGG Uaaaaa@80CD@

Note three things. First the inclusion of the t subscripts highlights the fact that these quantities depend on t. Second, both ω t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadshaaeqaaaaa@3867@  and α t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde 2aaSbaaSqaaiaadshaaeqaaaaa@383E@ marginalize over the corresponding alternative and null spaces Λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaaigdaaeqaaaaa@37D1@  and Λ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaaicdaaeqaaaaa@37D0@ , respectively, this is the sense in which the power and significance level are expectations. Third, the constant c (or c1 and c2) is represented in the expressions through Wt, which is itself dependent on t.

In their articles, [5, 12] demonstrate that as the sample size tends to infinity, the Bayes factor  converges to either 0 or 1. As a consequence, in the current context as the sample size t tends to infinity the Bayes factor B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOqaaaa@363C@  converges to either 0 or 1, so that [12] implies that ω t a.s. 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadshaaeqaaOWaa4ajaSqaaiaadggacaGGUaGaam4Caiaa c6caaeqakiaawkziaiaaigdaaaa@3E06@  and α t a.s. 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde 2aaSbaaSqaaiaadshaaeqaaOWaa4ajaSqaaiaadggacaGGUaGaam4C aiaac6caaeqakiaawkziaiaaicdaaaa@3DDC@ . Thus, for any pre-specified power ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdChaaa@3742@  and significance level α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde gaaa@3719@ , there exists a t such that for all t t * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiDayaafa GaeyyzImRaamiDamaaCaaaleqabaGaaiOkaaaaaaa@3A14@ , ω t ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiqadshagaqbaaqabaGccqGHLjYScqaHjpWDaaa@3C10@  and α t α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde 2aaSbaaSqaaiqadshagaqbaaqabaGccqGHLjYScqWFXoqyaaa@3BB4@ . We define t α,ω * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaDa aaleaaiiaacqWFXoqycaGGSaGaeqyYdChabaGaaiOkaaaaaaa@3B6A@  to be the in fimum of this collection of lower bounds, i.e.

t α,ω * = inf t R + { t: α t αand  ω t ω }. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaDa aaleaaiiaacqWFXoqycaGGSaGaeqyYdChabaGaaiOkaaaakiabg2da 9maaxababaGaciyAaiaac6gacaGGMbaaleaacaWG0bGaeyicI4Saam OuamaaBaaameaacqGHRaWkaeqaaaWcbeaakmaacmaabaGaamiDaiaa cQdacqWFXoqydaWgaaWcbaGaamiDaaqabaGccqGHKjYOcqWFXoqyca aMc8Uaaeyyaiaab6gacaqGKbGaaeiiaiabeM8a3naaBaaaleaacaWG 0baabeaakiabgwMiZkabeM8a3bGaay5Eaiaaw2haaiaac6caaaa@5A21@

t α,ω * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaDa aaleaaiiaacqWFXoqycaGGSaGaeqyYdChabaGaaiOkaaaaaaa@3B6A@ is said to be the optimal sample size for ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdChaaa@3742@ and α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde gaaa@3719@ , and computing t * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaCa aaleqabaGaaiOkaaaaaaa@3749@  is called the sample size determination problem. If only a power is specified, or if only a significance level is specified, the other quantity is left off of the subscript and out of the definition. We often write simply t * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaCa aaleqabaGaaiOkaaaaaaa@3749@  for t α,ω * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaDa aaleaaiiaacqWFXoqycaGGSaGaeqyYdChabaGaaiOkaaaaaaa@3B6A@ .

Were (10) and (11) monotonic and continuous in t, the sample size determination problem would be quite straightforward. Simply run a numerical root-finder (e.g. Newton's method) on ω t ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadshaaeqaaOGaeyOeI0IaeqyYdChaaa@3B2B@ and α t α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde 2aaSbaaSqaaiaadshaaeqaaOGaeyOeI0IaeqySdegaaa@3AD4@  take the larger t. Unfortunately, however, as a function of t both the power and significance level are discontinuous functions that are not monotonic. As a consequence, it is possible, for example, for there to be two such sample sizes t1and t2 with t 1 < t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaaIXaaabeaakiabgYda8iaadshadaWgaaWcbaGaaGOmaaqa baaaaa@3A44@  such that ω t 1 ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadshadaWgaaadbaGaaGymaaqabaaaleqaaOGaeyyzImRa eqyYdChaaa@3CF7@  and yet ω t 2 <ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadshadaWgaaadbaGaaGOmaaqabaaaleqaaOGaeyipaWJa eqyYdChaaa@3C36@ . This is a consequence of the dependence of Wt on t: as t grows, the rejection region changes, and these changes result in discrete jumps in the rejection region.

Practically speaking, in our experience the appearance of these jumps is monotonically decreasing in magnitude and dissipate quite quickly in t so that, while there are jumps, they become relatively minor even for quite small t. Thus, while out-of-the-box numerical routines are insufficient for the task, a straight-forward heuristic algorithm suffices to certify t * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaCa aaleqabaGaaiOkaaaaaaa@3749@  to a reasonable level of accuracy.

Initialize t= t 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2 da9iaadshadaWgaaWcbaGaaGimaaqabaaaaa@3953@ with a value small enough to satisfy ω t <ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadshaaeqaaOGaeyipaWJaeqyYdChaaa@3B42@  and α t >α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadshaaeqaaOGaeyOpa4dccaGae8xSdegaaa@3AEE@ , typically a value like .1 will suffice. Then, double t until both conditions are met. Once both conditions are met, decrement t by 1 until the conditions are no longer satisfied; then decrement by .1, and so on to achieve the desired precision. Alternatively, one may move in a binary search manner; this method is faster but loses the certificate of the solution up to the highest evaluated point. By contrast, the first heuristic certifies that the sample size achieved is optimal up to the largest t observed, which is of the form 2 k t 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaam4AaaaakiaadshadaWgaaWcbaGaaGimaaqabaaaaa@3937@ .

One computational detail is relevant for implementing this procedure. By definition, since Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaaIXaaabeaaaaa@373A@  and Y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaaIYaaabeaaaaa@373B@  are Poisson variates, their sample spaces are infinitely large, and thus computing ω t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadshaaeqaaaaa@3867@  and α t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae8xSde 2aaSbaaSqaaiaadshaaeqaaaaa@383E@  is not numerically possible - one cannot check every y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaacbmGaa8xEaa aa@367B@  to determine whether or not it is included in Wt. There is, however, a very reasonable work-around for this problem using prior predictive distributions. Under H1, the prior predictive distributions on Y 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaaIXaaabeaaaaa@373A@  and Y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaaIYaaabeaaaaa@373B@  are both negative binomial. Specifically,
Y 1 ~NegBin( α 1 , β 1 t+ β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaaIXaaabeaakiaac6hacaqGobGaaeyzaiaabEgacaqGcbGa aeyAaiaab6gadaqadaqaaiabeg7aHnaaBaaaleaacaaIXaaabeaaki aacYcadaWcaaqaaGGaaiab=j7aInaaBaaaleaacaaIXaaabeaaaOqa aiaadshacqGHRaWkcqWFYoGydaWgaaWcbaGaaGymaaqabaaaaaGcca GLOaGaayzkaaaaaa@4963@ and
Y 2 ~NegBin( α 2 , β 2 t+ β 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaaIYaaabeaakiaac6hacaqGobGaaeyzaiaabEgacaqGcbGa aeyAaiaab6gadaqadaqaaiabeg7aHnaaBaaaleaacaaIYaaabeaaki aacYcadaWcaaqaaGGaaiab=j7aInaaBaaaleaacaaIYaaabeaaaOqa aiaadshacqGHRaWkcqWFYoGydaWgaaWcbaGaaGOmaaqabaaaaaGcca GLOaGaayzkaaGaaiOlaaaa@4A19@

While one cannot enumerate the entire sample space of , one can be confident they have a satisfactory approximation by computing small (e.g. .00001) and large (e.g. .99999) quintiles of these distributions and then simply taking every combination of the ranges from low to high. This is the approach we take to computing the sums listed in (10) and (11) above.

An Example from Cancer Therapeutics

  1. An example of the proposed methodology is readily available from the field of cancer therapeutics. Suppose (1) Drug A is an industry standard therapy for a certain type of cancer that is known to have the common side effect of mild seizures every hour of infusion
  2. (2) Drug B is a novel compound believed to have the same side effect but at a lessened rate
  3. (3) The goal is to design a design a clinical trial that compares the two using the minimum resources required to meet 5% significance level (EBSL) and 80% power (EBP). Moreover, suppose that the losses associated with Type I and Type II errors have a ratio of 1:1.

From past studies, it is known that the uncertainty in the rate of seizures with Drug A (per hour) is well-represented by a Gamma (4, 4) distribution so that λ A ~Gamma(4,4) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW 2aaSbaaSqaaiaadgeaaeqaaOGaaiOFaiaabEeacaqGHbGaaeyBaiaa b2gacaqGHbGaaiikaiaaisdacaGGSaGaaGinaiaacMcaaaa@4123@ . Drug B, by contrast, is believed to be a bit worse, with perhaps a rate that is double that of Drug A with experts 90% sure the value is less than about 3. This translates into roughly λ B ~Gamma(4,8) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW 2aaSbaaSqaaiaadkeaaeqaaOGaaiOFaiaabEeacaqGHbGaaeyBaiaa b2gacaqGHbGaaiikaiaaisdacaGGSaGaaGioaiaacMcaaaa@4128@ . Figure 1 shows both of these priors graphically.

Figure 1: Prior structures used in Poisson sample size determination example.

Assuming that the rates are the same, the past indicators of Drug A supersede the lack of evidence for Drug B, so that under the null hypothesis λ= λ 1 = λ 2 ~Gamma(4,4) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW Maeyypa0Jae83UdW2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0Jae83U dW2aaSbaaSqaaiaaikdaaeqaaOGaaiOFaiaabEeacaqGHbGaaeyBai aab2gacaqGHbGaaiikaiaaisdacaGGSaGaaGinaiaacMcaaaa@4774@  is most appropriate. Assuming that the null and alternative hypotheses are given the same belief (50%), the rejection rule from (9) is therefore

W={ y: Γ(4) (2t+4) y 1 + y 2 +4 4 4 Γ( y 1 + y 2 +4) 4 4 Γ( y 1 +4) Γ(4) (t+4) y 1 +4 4 8 Γ( y 2 +8) Γ(8) (t+4) y 1 +8 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maacmaabaacbmGaa8xEaiaacQdadaWcaaqaaiabfo5ahjaacIca caaI0aaccaGae4xkaKIaaiikaiaaikdacaWG0bGaey4kaSIaaGinai aacMcadaahaaWcbeqaaiaadMhadaWgaaadbaGaaGymaaqabaWccqGH RaWkcaWG5bWaaSbaaWqaaiaaikdaaeqaaSGaey4kaSIaaGinaaaaaO qaaiab+rda0maaCaaaleqabaGaaGinaaaakiabfo5ahjaacIcacaWG 5bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyEamaaBaaaleaaca aIYaaabeaakiabgUcaRiab+rda0iaacMcaaaWaaSaaaeaacqGF0aan daahaaWcbeqaaiaaisdaaaGccqqHtoWrcaGGOaGaamyEamaaBaaale aacaaIXaaabeaakiabgUcaRiaaisdacaGGPaaabaGaeu4KdCKaaiik aiaaisdacaGGPaGaaiikaiaadshacqGHRaWkcaaI0aGaaiykamaaCa aaleqabaGaamyEamaaBaaameaacaaIXaaabeaaliabgUcaRiaaisda aaaaaOWaaSaaaeaacqGF0aandaahaaWcbeqaaiaaiIdaaaGccqqHto WrcaGGOaGaamyEamaaBaaaleaacaaIYaaabeaakiabgUcaRiaaiIda caGGPaaabaGaeu4KdCKaaiikaiaaiIdacaGGPaGaaiikaiaadshacq GHRaWkcaaI0aGaaiykamaaCaaaleqabaGaamyEamaaBaaameaacaaI XaaabeaaliabgUcaRiaaiIdaaaaaaOGaeyyzImRaaG4maaGaay5Eai aaw2haaaaa@7F10@ (12)
={ y: (2t+4) y 1 + y 2 +4 (t+4) y 1 +4 15120 65536 ( y 1 + y 2 +3)! ( y 1 +3)!( y 2 +7)! } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaai WaaeaaieWacaWF5bGaaiOoamaalaaabaGaaiikaiaaikdacaWG0bGa ey4kaSIaaGinaiaacMcadaahaaWcbeqaaiaadMhadaWgaaadbaGaaG ymaaqabaWccqGHRaWkcaWG5bWaaSbaaWqaaiaaikdaaeqaaSGaey4k aSIaaGinaaaaaOqaaiaacIcacaWG0bGaey4kaSIaaGinaiaacMcada ahaaWcbeqaaiaadMhadaWgaaadbaGaaGymaaqabaWccqGHRaWkcaaI 0aaaaaaakiabgwMiZoaalaaabaGaaGymaiaaiwdacaaIXaGaaGOmai aaicdaaeaacaaI2aGaaGynaiaaiwdacaaIZaGaaGOnaaaadaWcaaqa aiaacIcacaWG5bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyEam aaBaaaleaacaaIYaaabeaakiabgUcaRiaaiodacaGGPaGaaiyiaaqa aiaacIcacaWG5bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaaG4mai aacMcacaGGHaGaaiikaiaadMhadaWgaaWcbaGaaGOmaaqabaGccqGH RaWkcaaI3aGaaiykaiaacgcaaaaacaGL7bGaayzFaaaaaa@6A4A@ (13)

To design a test with 80% power and at the 5% significance level, we need but to run the algorithm described in Section 3. To illustrate the scenario, we plot the significance level and power for every t from 1 to 80; these are included in Figures 2 and 3, respectively. Note how quickly (in t) the functions become smooth, nullifying any concern about jumps. The 80% level is achieved at t = 37, with a power of 80.1%, and the 5% significance level is achieved at t = 57, where the significance level is 4.9%. Thus, to achieve both, we select the higher sample size, t = 57.

Figure 2: EBP curve

Figure 3: EBSL curve

We can also verify these results via simulation by generating one million random values of λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW 2aaSbaaSqaaiaaigdaaeqaaaaa@3815@ and λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaccaGae83UdW 2aaSbaaSqaaiaaikdaaeqaaaaa@3816@  from the priors given above. To validate the significance level result, we simply (1) sample from the prior, (2) generate two Poisson observations with mean equal to the sample in (1) times 57, and (3) determine whether the test rejects (1) or not (0). Averaging the million test results, the value is confirmed to Monte Carlo error (code available upon request). To validate the power result, we (1) sample one number from a Gamma(4, 4)and one number from a Gamma(8, 4), (2) sample two Poisson observations from distributions with mean 37 times the two variates generated in (1), and (3) determine whether the test rejects or not. Averaging the million results validates the theoretical result.

Discussion

In this paper we have used conjugate prior structures in order to assess our beliefs about a rate parameter in a two sample Poisson trial a priori in order to find the minimal sample size needed to reach certain operating characteristics. By the use of a loss function constant, we are able to control for either the desired expected Bayesian significance level or the desired expected Bayesian power or both. This type of analysis had not been considered previously, specifically, simultaneously controlling for both Type I and Type II error.

Future work includes the consideration of analysis priors in this research. Ideally, we would be able to adapt this research to account for the fact that researchers often times use one set of priors when conducting sample size analyses, but a more vague or non-informative set of priors when actually analyzing the experiment. Though we used only mildly informative priors in our example, it may be the case where a substantially informative prior is required for the design stage but a less informative prior would be used in the analysis stage. Further, this process could be expanded to consider non-conjugate priors as well. However, the analytical tractability of conjugate priors made the Poisson/gamma model ideal, and modeling prior beliefs of a Poisson rate with a gamma distribution is not an unreasonable thing to do.

It also should be noted that time considerations, while already improved throughout the process, can always continue to improve. One improvement to current methods involves replacing the bi-sectional approaches with one that approximates the EBP and EBSL curves with some logarithmic function; this improvement should get us in the ballpark of a candidate solution much quicker. Future work also includes a more in depth look at how expected Bayesian error rates compare to typical frequentist ones, and potentially an in-depth look at the different sample size determination and testing methods in order to determine the relative advantages and disadvantages of each. Further, we could generalize the algorithm such that we are not looking at a common sample size t= t 1 = t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2 da9iaadshadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaWG0bWaaSba aSqaaiaaikdaaeqaaaaa@3C45@ , but rather two different sample sizes t 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaaIXaaabeaaaaa@3755@  and t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaaIYaaabeaaaaa@3756@  such that they do not need to be equal. Lastly, code is available upon request.

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