Hierarchical Power Prior Models forAdaptive Incorporation ofHistorical Information in Clinical TrialsBrian P. Hobbs, Bradley P. Carlin, Sumithra Mandrekar, andDaniel SargentDivision of BiostatisticsSchool of Public HealthUniversity of MinnesotaMayo Clinic, Rochester, MN, U.S.AMarch 21, 2010Broad BackgroundIFDA CDRH has long been encouraging the use of Bayesianmethods with informative priors derived from historical data indevice trialsIAdvantagesIreduced sample size (at least in control group), hence lowercost and ethical hazardIHigher power (both frequentist and Bayesian)IDisadvantages if the informative prior turns out to be wrongIHigher Type I errorIpossibility of a lengthier trial (to resolve the conflict)IWe need models for incorporating historical data that areadaptively robust, to prior knowledge that reveals itself to beinconsistent with the accumulating experimental dataIProposed solution: Power Priors (Ibrahim and Chen, 2000,Statistical Science)Introduction to Power PriorsIIC (2000) define historical data as “data arising from previoussimilar studies”ILet D0= (n0, x0) denote historical data, suppose θ is theparameter of interest, and let L(D0|θ) denote the generallikelihoodISuppose π0(θ) is the prior distribution on θ before D0isobserved, the initial priorIThe conditional power prior on θ for the current study is thehistorical likelihood, L(D0|θ), raised to power α0, whereα0∈ [0, 1] multiplied by the initial prior:π(θ|D0, α0) ∝ L(D0|θ)α0π0(θ) ,Iα0is the power parameter that controls the “degree ofborrowing” from D0Power Priors (cont’d)IThe power parameter, α0, “can be interpreted as a relativeprecision parameter for the historical data” (IC, 2000, p.48)ICertainly apparent for normal data: x0iiid∼ N(θ, σ20), sinceunder flat initial prior,π0(θ|D0, α0) = N¯x0, σ20/(α0n0)IThink of α0n0as the “effective” historical sample sizeIGiven current data, D = (n, x), the conditional posteriorq(θ|D, D0, α0) ∝ L(D0|θ)α0L(D|θ)π0(θ)Iα0→ 1, q(θ|D, D0, α0) → approaches full borrowing from D0Iα0→ 0, q(θ|D, D0, α0) → approaches no borrowing from D0Power Priors (cont’d)IFix α0∈ [0, 1] and assume consistency among D0and Dknown a prioriIHowever, erroneous presumptions will result in models withpoor frequentist operating characteristicsIChoosing a hyperprior, π(α0), for α0enables the data to helpdetermine probable values for α0IIbrahim-Chen (2000) propose joint power priors of formπ(θ, α0|D0) ∝ L(D0|θ)α0π0(θ)π(α0)IDuan et al. (2006) and Pericchi (2009) propose modified jointpower priors (MPP) which follow the Likelihood Principle,π(θ, α0|D0) ∝L(D0|θ)α0π0(θ)RL(D0|θ)α0π0(θ)dθπ(α0)Potential Benefits of Joint Power Priors in Clinical TrialsISpecifying π(α0) = Beta(a, b) for fixed (a, b) is appealingI(a, b) control likely “degree of borrowing”IAs data from the current trial accumulates the joint posterioradapts to protect against type I error given the data isinconsistent or improve power given consistencyIInvaluable when exchangeability is in question and n is smallISimulations using simple Gaussian models suggest power priorscan dominate “full borrowing” for type I error and coveragePower Prior Model for a Single Arm TrialSuppose a pilot study suggests that a true treatment effect for aparticular drug exists indicated by µ 6= 0Historical DataISuppose x0= (x01, ..., x0n0) ∼ Normal(µ, σ20) i.i.d.where D0= (x0, n0, σ20) and σ20is knownIπ0(µ) ∝ 1Power Prior ModelISuppose x = (x1, ..., xn) ∼ Normal(µ, σ2) i.i.d.where D = (x, n, σ2) and σ2is knownIπ(α0) = Beta(a, b)Iq(µ|D0, D, α0) = Normalσ20n¯x+σ2α0n0¯x0nσ20+σ2α0n0,σ2σ20nσ20+σ2α0n0.Modified Power PriorsJoint prior for µ, α0Iπ(µ, α0|D0) ∝ Normal¯x0| µ,σ20α0n0× Beta (α0|a, b)IUnder normalized power prior α0is a relative precisionparameter for the historical dataMarginal posterior for α0Iq(α0|D0, D) ∝ Normal¯x − ¯x0| 0 ,σ2n+σ20α0n0×Beta (α0|a, b)ILess skewed than ICILess sensitive to the hyperparameters (a, b) than ICISlightly less “overattenuation” than IC¯x = 0 ¯x = 1 ¯x = 20.0 0.2 0.4 0.6 0.8 1.00.00 0.01 0.02 0.03 0.04density0.0 0.2 0.4 0.6 0.8 1.00.000 0.001 0.002 0.003 0.004 0.005density0.0 0.2 0.4 0.6 0.8 1.00.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012density0.0 0.2 0.4 0.6 0.8 1.00.00 0.01 0.02 0.03 0.04 0.05density0.0 0.2 0.4 0.6 0.8 1.00.000 0.002 0.004 0.006 0.008 0.010 0.012density0.0 0.2 0.4 0.6 0.8 1.00.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012densityMarginal posteriors for α0, MPP (red); IC (black), under Uniform Beta Hy-perprior (a = 1, b = 1), where ¯x0= 2, n0= 30, σ20= σ2= 1. Each columncorresponds to ¯x = (0, 1, 2). Across the top row, n = 5 and bottom row, n = 30.Problems with Current Power Prior ApproachIDo not directly parametrize the “commensurability” of thehistorical and current dataIWide, bimodal, or otherwise irregular posterior densities for α0IThe posterior of α0degenerates rapidly to 0 as prioruncertainty about the parameter increasesIOverattenuates impact of the historical data, forcing use ofinformative hyperpriors to deliver sufficient borrowingIIncapable of “borrowing fully” from consistent historical dataIMinimal gain in power over frequentist models that ignore thehistorical data completelyCommensurate Power Priors (CPP)IWe propose adaptive modification based on MPPIDifferent parameters in historical and current group, θ0and θIExtend hierarchical model to include parameter, τ , thatdirectly measures similarity of θ and θ0IConstruct prior for θ dependent upon θ0and τIτ parametrizes commensurabilityIUse information in τ to guide prior on α0πC(θ, α0, τ|D0) ∝Z[L(D0|θ0) × N(θ| θ0, τ)]α0RR(L(D0|θ0) × N(θ| θ0, τ))α0dθ0dθdθ0×Beta(α0|aτ, 1)p(τ)Commensurate Power Prior for Single Arm TrialILet x0iiid∼ Normal(µ0, σ2) and xiiid∼ Normal(µ, σ2), with σ2again assumed knownIFormalize commensurate as µ0near µ by adopting Normalprior on µ with mean µ0and variance τ2IBeta(aτ2, 1) prior on α0for some a > 0Iτ2close to 0 corresponds to very high commensurability, whilevery large τ2implies the two datasets do not arise fromsimilar populationsIτ2→ 0,aτ2→ ∞, leading to a point-mass prior on α0at 1Iτ2becomes large,aτ2→ 0, strongly discouraging incorporatingany historical informationCommensurate Power Prior for Single Arm Trial (cont’d)IAssume a