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: where h1(y),h2(y),h3(y) and h4(y) are the densities of Beta(a, b), and random variables, respectively, evaluated at y. The link parameters introduce correlations between pCR and pC and pER and pE so that under the proposed model, the related data help us learn about remission rates in both trials. However, because there is uncertainty about the precise relationships between parameters in each trial, the related data are discounted for learning about pC and pE. Using Bayes theorem to combine prior beliefs with the related data, we obtain the joint distribution for pE and pC as (2) capturing the state of knowledge about pC and pE before the MYPAN trial is conducted. The marginal prior density for θ can be found by applying a transformation of variables to Equation (2). Prior distributions for parameters incorporating the MYCYC data are not of standard forms. Figure 3 shows densities for pC, pE and θ that result from updating prior distributions pC ∼ Beta(3.6, 2.1) and θ ∼ N( − 0.26,0.25) with the MYCYC data for the consensus responses to questions about λC and λE, which were as follows: (a) 0.55, (b) 0.25, (c) 0.5 and (d) 0.25. According to these responses, λC ∼ N(0.12, 0.86) and λE ∼ N(0,0.60), consistent with the opinion that remission rates on CYC might be slightly higher in adults with ANCA-associated vasculitis than in children with PAN but that remission rates on MMF would be similar in these two populations. Incorporating the MYCYC data shifts the location of the prior for pC only slightly, as the MYCYC remission rate is similar to the mode of the prior elicited without reference to these data. The updated prior for pC has mode 0.74, mean 0.70 and standard deviation 0.11. The location of the prior for pE increases upon inclusion of the MYCYC data as is larger than the mode of the day 1 prior, and expert opinion indicates that remission rates in the MYCYC and MYPAN patient groups are likely to be similar. The updated prior for pE has mode 0.71, mean 0.67 and standard deviation 0.12. Incorporating the MYCYC data has clearly reduced uncertainty about the absolute remission rates on the two treatments: 90% credibility intervals for pE and pC are (0.45, 0.85) and (0.51, 0.86), respectively. The related data have less impact on the prior for θ because beliefs about θ were already rather precise before their inclusion.
Affiliation: Medical and Pharmaceutical Statistics Research Unit, Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, U.K.