INCORPORATING PRIOR BELIEFS INTO CLINICAL DECISIONS

Purpose: Clinicians often have sound prior beliefs but limited data when deciding whether or not to give a patient a novel treatment.    Converting vague prior beliefs into one explicit Bayesian prior distribution is not robust.  Instead, specifying a few prior constraints such as the prior mean and mode, we find the range for the posterior mean, Mu, over all distributions that fit the constraints.   With data from a normal, binomial, or Poisson distribution, a package such as SAS computes the range in several seconds without recourse to massive simulation such as MCMC methods.

Method: Beliefs expressed as prior constraints specify not one, but an infinite class of distributions.    We compute the range of the posterior mean over C(m), the class of all prior distributions with a pre-specified mode, m, that fit any specific set of K elicited moment constraints (the expected value of a piecewise-continuous function).   We focus on two cases:  1) elicit the prior mean and variance and 2) elicit a probability on each of a set of adjacent subintervals that support the prior distribution: the ‘histogram’ prior for binomial data with p = P(success), e.g., P(0 <= p <= 0.3) =0.025, P(0.3 <= p <= 0.625) =0.520, P(0.625 <= p <= 0.8) =0.368, and P(0.8 <= p <= 1.0) =0.085.  Setting p’ = g(p)  the loss function, we obtain the range of the posterior expected loss.

Result: To illustrate, let  p =P(novel treatment succeeds).   A clinician believes that p has prior mean, variance, and mode of 0.6, 0.02, and 0.625.  These fit a beta prior distribution proportionate to p⁵(1-p)³ . With 10 patients, then with 3, 5, or 7 successes, the corresponding posterior range of Mu is 0.41-0.51, 0.53-0.63, or 0.60-0.67.   For the histogram prior, as specified above that fit this beta distribution, for 3, 5, or 7 successes, the range is 0.43-0.50, 0.55-62, or 0.60-0.65.   Histogram prior ranges are usually narrower.

Conclusion: Fast stable algorithms compute the range of the posterior mean.   Experts, naïve about statistics, can specify constraints by drawing their histogram prior.  They can make a decision or else modify the constraints, redo the analysis, and decide later.  This simplicity and documentation should lead to better more robust decisions.