Bayesian methods for the design and interpretation of clinical trials in very rare diseases.
Bottom Line: For such studies, the sample size needed to meet a conventional frequentist power requirement is clearly infeasible.A systematic elicitation from clinicians of their beliefs concerning treatment efficacy is used to establish Bayesian priors for unknown model parameters.As sample sizes are small, it is possible to compute all possible posterior distributions of the two success rates.
Affiliation: Medical and Pharmaceutical Statistics Research Unit, Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, U.K.Show MeSH
Related in: MedlinePlus
Mentions: Here, the preference of most clinicians for probability differences rather than odds ratios is acknowledged, although it would be more direct to ask the questions in terms of the latter. The answer to question (iii) can be equated to the prior probability that pE > pC, which is Φ(μ/σ), where Φ is the distribution function of a standard normal variate. Question (iv) asks for the prior probability that pE − pC < − 0.1; that is, MMF is inferior to CYC by at least the pre-specified non-inferiority margin. To reflect the prior uncertainty about pC and pE, we write this probability as an integral of the prior joint density g0(pC,pE). As this joint density can be expressed in terms of σ and parameters that have already been fixed by answers to questions (i)–(iii), numerical integration of g0(pC,pE) can be used in a univariate search to determine a suitable value of σ and hence μ. We have proposed a simple approach to determining prior distributions for the Bayesian model parameters. Although the chosen beta and normal models will not accommodate all opinion, it is unlikely that prior knowledge is so detailed that they will not provide an acceptable approximation. Graphical interpretations of hypothetical answers to questions (iii) and (iv) (as shown in Figure 1) helped to clarify experts’ understanding of the quantities sought. Figure 1 illustrates the ordering that answers should follow; that is, the answer to (iii) should be less than 1 minus the answer to (iv). Experts were also advised that the opinion that MMF and CYC would have similar efficacy could be represented by an answer to question (iii) of 0.5 and an answer to question (iv) of close to 0. It is useful to ask more questions than there are model hyperparameters to allow the model's goodness of fit to be assessed and inconsistencies in the experts’ opinions to be detected (17, Section 6.3 and 26). Therefore, each expert was asked two further questions about their opinion for p E, which were expressed as follows:
Affiliation: Medical and Pharmaceutical Statistics Research Unit, Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, U.K.