Sample size re-estimation in a superiority clinical trial using a hybrid classical and Bayesian procedure.

When designing studies involving a continuous endpoint, the hypothesized difference in means ( θA ) and the assumed variability of the endpoint ( σ2 ) play an important role in sample size and power calculations. Traditional methods of sample size re-estimation often update one or both of these parameters using statistics observed from an internal pilot study. However, the uncertainty in these estimates is rarely addressed. We propose a hybrid classical and Bayesian method to formally integrate prior beliefs about the study parameters and the results observed from an internal pilot study into the sample size re-estimation of a two-stage study design. The proposed method is based on a measure of power called conditional expected power (CEP), which averages the traditional power curve using the prior distributions of θ and σ2 as the averaging weight, conditional on the presence of a positive treatment effect. The proposed sample size re-estimation procedure finds the second stage per-group sample size necessary to achieve the desired level of conditional expected interim power, an updated CEP calculation that conditions on the observed first-stage results. The CEP re-estimation method retains the assumption that the parameters are not known with certainty at an interim point in the trial. Notional scenarios are evaluated to compare the behavior of the proposed method of sample size re-estimation to three traditional methods.

Sample size determination for a binary response in a superiority clinical trial using a hybrid classical and Bayesian procedure.

BACKGROUND: When designing studies that have a binary outcome as the primary endpoint, the hypothesized proportion of patients in each population experiencing the endpoint of interest (i.e., π 1,π 2) plays an important role in sample size and power calculations. Point estimates for π 1 and π 2 are often calculated using historical data. However, the uncertainty in these estimates is rarely addressed. METHODS: This paper presents a hybrid classical and Bayesian procedure that formally integrates prior information on the distributions of π 1 and π 2 into the study's power calculation. Conditional expected power (CEP), which averages the traditional power curve using the prior distributions of π 1 and π 2 as the averaging weight conditional on the presence of a positive treatment effect (i.e., π 2>π 1), is used, and the sample size is found that equates the pre-specified frequentist power (1-ß) and the conditional expected power of the trial. RESULTS: Notional scenarios are evaluated to compare the probability of achieving a target value of power with a trial design based on traditional power and a design based on CEP. We show that if there is uncertainty in the study parameters and a distribution of plausible values for π 1 and π 2, the performance of the CEP design is more consistent and robust than traditional designs based on point estimates for the study parameters. Traditional sample size calculations based on point estimates for the hypothesized study parameters tend to underestimate the required sample size needed to account for the uncertainty in the parameters. The greatest marginal benefit of the proposed method is achieved when the uncertainty in the parameters is not large. CONCLUSIONS: Through this procedure, we are able to formally integrate prior information on the uncertainty and variability of the study parameters into the design of the study while maintaining a frequentist framework for the final analysis. Solving for the sample size that is necessary to achieve a high level of CEP given the available prior information helps protect against misspecification of hypothesized treatment effect and provides a substantiated estimate that forms the basis for discussion about the study's feasibility during the design phase.

Selection of the effect size for sample size determination for a continuous response in a superiority clinical trial using a hybrid classical and Bayesian procedure.

BACKGROUND: When designing studies that have a continuous outcome as the primary endpoint, the hypothesized effect size ([Formula: see text]), that is, the hypothesized difference in means ([Formula: see text]) relative to the assumed variability of the endpoint ([Formula: see text]), plays an important role in sample size and power calculations. Point estimates for [Formula: see text] and [Formula: see text] are often calculated using historical data. However, the uncertainty in these estimates is rarely addressed. METHODS: This article presents a hybrid classical and Bayesian procedure that formally integrates prior information on the distributions of [Formula: see text] and [Formula: see text] into the study's power calculation. Conditional expected power, which averages the traditional power curve using the prior distributions of [Formula: see text] and [Formula: see text] as the averaging weight, is used, and the value of [Formula: see text] is found that equates the prespecified frequentist power ([Formula: see text]) and the conditional expected power of the trial. This hypothesized effect size is then used in traditional sample size calculations when determining sample size for the study. RESULTS: The value of [Formula: see text] found using this method may be expressed as a function of the prior means of [Formula: see text] and [Formula: see text], [Formula: see text], and their prior standard deviations, [Formula: see text]. We show that the "naïve" estimate of the effect size, that is, the ratio of prior means, should be down-weighted to account for the variability in the parameters. An example is presented for designing a placebo-controlled clinical trial testing the antidepressant effect of alprazolam as monotherapy for major depression. CONCLUSION: Through this method, we are able to formally integrate prior information on the uncertainty and variability of both the treatment effect and the common standard deviation into the design of the study while maintaining a frequentist framework for the final analysis. Solving for the effect size which the study has a high probability of correctly detecting based on the available prior information on the difference [Formula: see text] and the standard deviation [Formula: see text] provides a valuable, substantiated estimate that can form the basis for discussion about the study's feasibility during the design phase.

Selection of the treatment effect for sample size determination in a superiority clinical trial using a hybrid classical and Bayesian procedure.

Specification of the treatment effect that a clinical trial is designed to detect (θA) plays a critical role in sample size and power calculations. However, no formal method exists for using prior information to guide the choice of θA. This paper presents a hybrid classical and Bayesian procedure for choosing an estimate of the treatment effect to be detected in a clinical trial that formally integrates prior information into this aspect of trial design. The value of θA is found that equates the pre-specified frequentist power and the conditional expected power of the trial. The conditional expected power averages the traditional frequentist power curve using the conditional prior distribution of the true unknown treatment effect θ as the averaging weight. The Bayesian prior distribution summarizes current knowledge of both the magnitude of the treatment effect and the strength of the prior information through the assumed spread of the distribution. By using a hybrid classical and Bayesian approach, we are able to formally integrate prior information on the uncertainty and variability of the treatment effect into the design of the study, mitigating the risk that the power calculation will be overly optimistic while maintaining a frequentist framework for the final analysis. The value of θA found using this method may be written as a function of the prior mean µ0 and standard deviation τ0, with a unique relationship for a given ratio of µ0/τ0. Results are presented for Normal, Uniform, and Gamma priors for θ.