Bayesian Phase II Decision with Prior Sensitivity
A phase II oncology trial plans to enroll 40 patients. The historical response rate for standard therapy is 0.20. The investigator elicits a Beta(4,16) prior, which has mean 0.20 and prior sample size 20. After 40 patients, 12 respond. A skeptical reviewer proposes a Beta(1,1) prior and says the trial should be declared promising only if the posterior probability that response rate exceeds 0.20 is at least 0.90. A second reviewer wants a 95% credible interval and a prior sensitivity check. The trial team also asks for the posterior predictive probability that a future patient responds. The prior was elicited before unblinding, but the historical series came from a different disease subtype and a different line of therapy. No interim analysis was planned.
1. Compute the posterior under the Beta(4,16) prior, including the posterior mean, approximate 95% credible interval, and posterior probability that p exceeds 0.20.
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Prior Beta(4,16) plus 12 responses in 40 gives posterior Beta(4+12,16+28)=Beta(16,44). Posterior mean =16/60=0.2667. Posterior variance = (16×44)/(60^2×61)=704/219,600=0.003205, SD=0.0566. An approximate 95% credible interval is 0.2667±1.96(0.0566)=(0.156,0.378). The posterior probability that p>0.20 is approximately P(Z>(0.20-0.2667)/0.0566)=P(Z>-1.18)=0.88, below the 0.90 threshold.
2. Repeat the decision calculation under Beta(1,1) and assess prior sensitivity.
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With Beta(1,1), posterior is Beta(13,29), mean 13/42=0.3095, variance=(13×29)/(42^2×43)=377/75,852=0.00497, SD=0.0705. Then P(p>0.20)≈P(Z>(0.20-0.3095)/0.0705)=P(Z>-1.55)=0.94. The decision crosses 0.90 depending on the prior. The historical prior is skeptical (effective sample size 20, one-third of the posterior's 60), so this is material sensitivity. The historical series was from a different subtype, which weakens the exchangeability assumption. Report both analyses and do not claim robustness.
3. What is the posterior predictive probability under each prior, and how should the trial be reported?
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Under Beta(16,44), the posterior predictive probability of response for a future patient is the posterior mean, 0.2667. Under Beta(13,29) it is 0.3095. The trial is not clearly promising by the pre-specified 0.90 rule under the elicited prior; it is under the uniform prior. A defensible conclusion states prior sensitivity, gives the credible interval, and avoids a binary claim. If the 0.90 rule was pre-specified, the primary analysis should use the elicited prior, with the uniform prior as sensitivity.