# An intriguing problem in Bayesian analysis of binomial clinical trails

Motivating example: a recently completed clinical trial on migraine pains in the Johnson and Johnson pharmaceutical company

• Binomial Clinical trial on migrant headache
• Randomized trial of two treatment groups:
• n1 subjects in A group; n0 subjects in B group (control)
• Parameter of interest:  treatment improvement d = p1 – p0
• Expert opinions on the improvement (d = p1 – p0)
• Being solicited from 11 experts, prior to the clinical trial.
• Following the design of Parmar et al. (1994), Spiegelhalter et al. (1994)
• Goal: Incorporate expert opinions with information in the clinical data
• Generating hypothesis testing questions for future studies

A Bi-variate Bayesian Approach

• Bi-Beta prior (Olkin and Liu, 2003)
• Marginals are beta-distributed; Correlation ranges from 0 to 1.
• Additional information (or assumption) on p0 is required
• Likelihood function
• Two independent binomial distributions (or normal approximations)
• Posterior
• No explicit formula and need to use an MCMC algorithm

Discussions

Although the contour plot of the posterior distribution sits between those of the prior distribution and the likelihood function, its projected peak is more extreme than the other two. Further examination suggests that this phenomenon is genuine in binomial clinical trials and it would not go away even if we adopt other (skewed) priors (for example, the independent beta priors used in Joseph et al. (1997)). In fact, as long as the center of a posterior distribution is not on the line joining the two centers of the joint prior and likelihood function (as it is often the case with skewed distributions), there exists a direction along which the marginal posterior fails to fall between the prior and likelihood function of the same parameter. It would be interesting to know what ramifications this counter-intuitive (or paradoxical) phenomenon may have in inferences. In any case, it is certainly not easy to explain this phenomenon to clinicians or general practitioners of statistics.