U of M PUBH 7440 - Bayesian Assessment of Sample Size for Clinical Trials of Cost Effectiveness

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O’Hagan, Stevens MEDICAL DECISION MAKINGVOL 21/NO 3, MAY–JUNE 2001 Bayesian Assessment of Sample SizeMethodologyBayesian Assessment of Sample Size forClinical Trials of Cost-EffectivenessANTHONY O’HAGAN, JOHN W. STEVENSThe authors present an analysis of the choice of sample sizes for demonstratingcost-effectiveness of a new treatment or procedure, when data on both cost and efficacy willbe collected in a clinical trial. The Bayesian approach to statistics is employed, as well as anovel Bayesian criterion that provides insight into the sample size problem and offers a veryflexible formulation.Key words:Bayesian statistics; cost-effectiveness; experimental design;net benefits; pharmacoeconomics; sample size. (Med Decis Making 2001;21:219–230)1. IntroductionTraditionally, health economic evaluation ofresource usage has been considered of secondaryinterest to the clinical objectives of confirmatoryclinical trials regarding the efficacy of a treatment,mainly because health economic data are generallynot required for the registration of a new treatment.However, Australia1and Canada2do requireinformation on the cost-effectiveness of a newtreatment before granting reimbursement, therebyraising the importance of health economicevaluation in the marketing of treatments. Othercountries are also beginning to require informationon the cost-effectiveness of treatments beforegranting reimbursement, including Norway,3whichmakes this a requirement as of January 2002, andNICE4in England and Wales.Confirmatory clinical trials are generallypowered to detect clinically relevant treatmenteffects based upon 1 primary endpoint. In addition,confirmatory clinical trials usually involve manycenters across several countries to be able to recruitthe required number of patients. This aspect ofclinical study design does not usually cause anyproblems when estimating treatment effectsproviding that the treatment effect is a constantadditive term across centers. However, differentclinical practices across centers may lead to verydifferent patterns of resource usage so that thepooling of resource usage and cost data is notstraightforward. In practice, this problem isfrequently ignored or glossed over. Where it isrecognized, resource usage is often summarizedseparately by center in health economic evaluation.Together, the increasing requirement todemonstrate the cost-effectiveness of newtreatments and the need to consider clinical trialdesign issues means that health economists mustaddress the ability of a study to provide sufficientinformation to support a health economic claim.Much of the literature on comparing the cost-effectiveness of 2 treatments has been based on theincremental cost-effectiveness ratio, both from atheoretical perspective5–7and in practicalapplications.8–10However, Stinnett and Mullahy11argued for inference and decisions to be basedinstead on a net benefits approach; Van Hout andothers12had earlier proposed the C/E AcceptabilityCurve (CEAC), which plots the probability of apositive net benefit against the threshold cost of aunit increase in efficacy. O’Hagan and others13endorsed the use of the CEAC and pointed out thatits formulation is intrinsically Bayesian. Practicaluses of this approach are now appearing.14,15Analyses of power considerations and the choiceof sample size have been published both in thecontext of subsequent analysis based onincremental cost-effectiveness ratios16,17and in thecontext of a net benefits formulation.18–20All ofthese methods are based on the classical approachof frequentist statistics. There is a considerable219Received 6 June 2000 from the University of Sheffield, United Kingdom(AO’H) and AstraZeneca R&D Charnwood, United Kingdom (JWS).Revision accepted for publication 10 January 2001.Address correspondence and reprint requests to Dr. O’Hagan: StatisticalServices Unit, University of Sheffield, Hicks Building, Sheffield S2 7RHUnited Kingdom; telephone: 0114 222 3900; fax: 0114 222 3909; e-mail:[email protected] on Bayesian design of experimentsgenerally21,22and some work in the context ofclinical trials.23–25The purpose of the present articleis to give a novel alternative solution usingBayesian statistics and to demonstrate that thisleads to a rather more flexible framework forconsidering both “power” and sample size. Ourwork is presented in the context of identifyingsample size to demonstrate cost-effectiveness,although it is equally applicable to choosing thesample size to demonstrate efficacy.2. Problem Formulation2.1 MODELWe suppose that the study to be designed willcomprise a randomized clinical trial in which n1patients are given treatment 1 and n2are giventreatment 2. The sample size problem is todetermine n1and n2under some suitablespecification of model and study objectives.We assume that the data will yield a measure ofefficacy and a cost for each patient, and we denotethe observed efficacy for patient j given treatment iby eij, whereas the cost for that patient is denoted bycij. The range of subscripts is i = 1,2 and j =1,2 . . . ,ni. The expectation of eijis E(eij)=µi, thepopulation mean efficacy under treatment i.Similarly, E(eij)=γiis the population mean costunder treatment i.An important input to a sample size assessmentis the population variances,Var eij i()=σ2andVar cij i()=τ2, since these determine the amount ofinformation provided by a single observation.However, it is necessary in practice to allow forstatistical dependence between the efficacy and costmeasures, and so we define Corr(eij, cij)=ρi.Although dependence between efficacy and costmay be more complex than is measured by thecorrelation coefficient alone, our calculationsdepend only on the correlation coefficient (becauseof the linear nature of the net benefit measureintroduced below).We suppose that the objective of the study is todemonstrate that treatment 2 is more cost-effectivethan treatment 1 if this is the true situation. Let ∆e=µ2– µ1be the true (population) mean increment ofefficacy for treatment 2 compared with treatment 1.Let ∆c= γ2– γ1denote the corresponding meanincrement in cost. The cost-effectivenessassessment will be based on the net monetarybenefitβ = K∆e– ∆c= K(µ2– µ1)–(γ2– γ1)of changing from treatment 1 to treatment 2, whereK is the threshold unit cost. Thus, K is themaximum price (in appropriate monetary units) thatthe health care provider is prepared to pay to obtaina unit increase


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U of M PUBH 7440 - Bayesian Assessment of Sample Size for Clinical Trials of Cost Effectiveness

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