__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.

- This counter-intuitive result is reported in the paper Incorporating External Information in
Analysis of Clinical Trials with Binary Outcomes (AOAS, 2012) [Journal Link].

- A related talk is A General Framework for Combining Information
& A Frequentist Approach to Incorporate Expert Opinions.