Wald’s SPRTGeorge WoodworthHypotheses (big is good)Inferior (H0): θ ≤ θ0−Δ Superior (H2): θ ≥ θ0Inferior (H0)θNoninferior (H1)θ0−Δθ0Superior (H2)Loss and Penalty FunctionsCost/penalty function: C(t) = C(R(t))represents cost of data + penalty for delayInferior Neither SuperiorH0: H2:Decisionθ ≤ θ0 − Δ θ0 − Δ < θ < θ0 θ0 < θAccept (acc)001Reject (rej)k 00True status of the new interventionLoss functionWald SPRTSequential Likelihood Ratio TestObservations: # of successes in n trialsS1, S2, …, Sn, …Hypotheses: H0p = p0, H1p = p1Likelihood Ratio:()()11100011exp ln , where ln11nnpppLR S npppλλ⎛⎞⎛⎞⎛⎞−−⎟⎟⎜⎟⎜⎜⎟⎟⎟⎜⎜= ⋅ + ⋅ =⎜⎟⎟⎟⎜⎜⎜⎟⎟⎟⎜⎟⎟⎜⎜−−⎝⎠⎝⎠⎝⎠Note: λ is the log of the odds ratioDecision RuleIf LRn< B, then stop and accept H0If LRn> A, then stop and reject H0Else take another observation.Wald suggested–A = (1−β)/α–B = β/(1-α)Stopping Boundaries for Sn()()10ln ln1, where C = ln1nBnC AnCpSpλλ⎛⎞−−−⎟⎜⎟<<⎜⎟⎜⎟⎜−⎝⎠AssignmentThe excel file SPRT.xls is set up to compute lower and upper stopping boundaries for an SPRT with p0= 0.92 and p1= 0.96 . For example at n=30, the LSB “accept”boundary is Sn= 26 and the USB “reject”boundary is Sn= 33. Assuming a beta(1,1) prior, compute P( p < .92 | Sn) and P(p > .96 | Sn) for n = 20,40,60,80,100 and Sn=LSB and
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