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U of M PUBH 7440 - Bayesian Sample Size Computations

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Review of classical sample size calculationsBayesian sample size calculationsBayesian Assurance CurveConcluding remarksBayesian Sample Size ComputationsSudipto BanerjeeDivision of BiostatisticsSchool of Public HealthUniversity of MinnesotaApril 20, 20081Review of classical sample size calculationsSuppose we want to take a sample y1, . . . , yniid∼ N(θ, σ2).So ¯y ∼ N(θ,σ2n). Assume σ2is known. n is not known.Consider the classical hypothesis testing problem:H0: θ = θ0against the alternative H1: θ = θ1> θ0.Decision rule: Reject H0if ¯y > θ0+σ√nz1−α, whereΦ(zα) = α and Φ(·) is the standard normal cdf.2Review of classical sample size calculationsRequiring the procedure to have a power of at least 1 − β,we have:1 − β ≤ P (Rej H0|H1) = P¯y > θ0+σ√nz1−α|θ = θ1= P√nσ(¯y −θ1) >√nσ(θ0− θ1) + z1−α= PZ > −√n∆σ+ z1−α= 1 − Φ−√n∆σ+ z1−α= Φ√n∆σ− z1−α= Φ√n∆σ+ zα,where ∆ = θ1− θ0and, in the last steps, we have used thefacts that Φ(−x) = 1 − Φ(x) and z1−α= −zα.3Review of classical sample size calculationsThe preceding computations lead to√n∆σ+ zα≥ z1−β=⇒√n∆σ≥ (z1−β− zα) = −(zα+ zβ)=⇒ n ≥ (zα+ zβ)2σ∆2.Thus, we arrive at the ubiquitous sample size formula:The sample size formula:n = (zα+ zβ)2σ∆2.For two-sided alternatives, one simply replaces α by α/2 inthe above expression.4Bayesian sample size calculationsSuppose, we take the prior θ ∼ N(θ1, τ2). We let τ2=σ2n0,where n0(prior sample size!) reflects the precision of theprior relative to the data. This simplifies the calculations:Nθ |θ1,σ2n0× N¯y |θ,σ2n= Nθ |n0n + n0θ1+nn + n0¯y,σ2n + n0.Let Aα(θ0, θ1) = {¯y : P (θ < θ0| ¯y) < α}. Note:P (θ < θ0| ¯y) =P√n + n0σθ −n0θ1+ n¯yn + n0<√n + n0σθ0−n0θ1+ n¯yn + n0= Φ√n + n0σθ0−n0θ1+ n¯yn + n05Bayesian sample size calculationsThus,Aα(θ0, θ1) =¯y :√n + n0σθ0−n0θ1+ n¯yn + n0< zα=¯y : θ0−n0θ1+ n¯yn + n0<σ√n + n0zα=¯y : θ0−n0n + n0θ1−nn + n0¯y <σ√n + n0zα=¯y : ¯y > θ0−n0n(θ1− θ0) −r1 +n0nσ√nzαNote that as n0→ 0 (i.e. the prior becomes vague)Aα(θ0, θ1) becomes identical to the the critical region fromclassical hypothesis testing.6Bayesian sample size calculationsThe Bayesian power or Bayesian assurance δ is definedas:Bayesian assurance:δ = P¯y(Aα(θ0, θ1))= P¯y{¯y : P (θ < θ0| ¯y) < α}= P¯y{¯y : P (θ > θ0| ¯y) > 1 − α}= P¯y¯y > θ0−n0n(θ1− θ0) −r1 +n0nσ√nzα7Bayesian sample size calculationsThe marginal distribution of ¯y is given by:ZNθ |θ1,σ2n0× N¯y |θ,σ2ndθ = N¯y |θ1,σ2n + n0.Therefore, the Bayesian power or assurance is:P¯y¯y > θ0−n0n(θ1− θ0) −√n + n0nσzα= P¯y¯y −θ1> −1 +n0n(θ1− θ0) −√n + n0nσzα= PZ > −√n + n01 +n0nθ1− θ0σ−1 +n0nzα= Φ√n + n01 +n0nθ1− θ0σ+1 +n0nzα8Bayesian Assurance CurveRewriting the Bayesian power in terms of the relativeprecision n0/n and n, we obtain:δ = Φ√n1 +n0n3/2∆σ+1 +n0nzα,where ∆ = θ1− θ0. This is often called the criticaldifference that needs to be detected.This leads to:The Bayesian Power (Assurance) Curveδ(∆, n) = Φ√n1 +n0n3/2∆σ+1 +n0nzα,Note: For study design purposes, σ2and n0are assumedknown and the Bayesian power curve is investigated as afunction of ∆ and n.9Concluding remarksGiven n0, the Bayesian will compute the sample sizeneeded to detect a critical difference of ∆ with probability1 − β asn = arg min{n : δ(∆, n) ≥ 1 − β}As the prior becomes vague, n0→ 0 and:limn0→0δ(∆, n) = Φ√n∆σ+ zαwhich is exactly the classical power curve. Now theBayesian sample size formula coincides with the classicalsample size formula:Bayesian sample size with vague prior information:n = (zα+


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