Bayesian probability of success for clinical trials using historical data

On the distribution of the power function for the scale parameter of exponential families

The expected value of the standard power function of a test, computed with respect to a design prior distribution, is often used to evaluate the probability of success of an experiment. However, looking only at the expected value might be reductive. Instead, the whole probability distribution of the power function induced by the design prior can be exploited. In this article we consider one-sided testing for the scale parameter of exponential families and we derive general unifying expressions for cumulative distribution and density functions of the random power. Sample size determination criteria based on alternative summaries of these functions are discussed. The study sheds light on the relevance of the choice of the design prior in order to construct a successful experiment.

A Bayesian approach to study design and analysis with type I error rate control for response variables of mixed types

There has been increased interest in the design and analysis of studies consisting of multiple response variables of mixed types. For example, in clinical trials, it is desirable to establish efficacy for a treatment effect in primary and secondary outcomes. In this article, we develop Bayesian approaches for hypothesis testing and study planning for data consisting of multiple response variables of mixed types with covariates. We assume that the responses are correlated via a Gaussian copula, and that the model for each response is, marginally, a generalized linear model (GLM). Taking a fully Bayesian approach, the proposed method enables inference based on the joint posterior distribution of the parameters. Under some mild conditions, we show that the joint distribution of the posterior probabilities under any Bayesian analysis converges to a Gaussian copula distribution as the sample size tends to infinity. Using this result, we develop an approach to control the type I error rate under multiple testing. Simulation results indicate that the method is more powerful than conducting marginal regression models and correcting for multiplicity using the Bonferroni-Holm Method. We also develop a Bayesian approach to sample size determination in the presence of response variables of mixed types, extending the concept of probability of success (POS) to multiple response variables of mixed types.

Updating the probability of study success for combination therapies using related combination study data

Combination therapies are becoming increasingly used in a range of therapeutic areas such as oncology and infectious diseases, providing potential benefits such as minimising drug resistance and toxicity. Sets of combination studies may be related, for example, if they have at least one treatment in common and are used in the same indication. In this setting, value can be gained by sharing information between related combination studies. We present a framework that allows the study success probabilities of a set of related combination therapies to be updated based on the outcome of a single combination study. This allows us to incorporate both direct and indirect data on a combination therapy in the decision-making process for future studies. We also provide a robustification that accounts for the fact that the prior assumptions on the correlation structure of the set of combination therapies may be incorrect. We show how this framework can be used in practice and highlight the use of the study success probabilities in the planning of clinical studies.

A hierarchical prior for generalized linear models based on predictions for the mean response

There has been increased interest in using prior information in statistical analyses. For example, in rare diseases, it can be difficult to establish treatment efficacy based solely on data from a prospective study due to low sample sizes. To overcome this issue, an informative prior to the treatment effect may be elicited. We develop a novel extension of the conjugate prior of Chen and Ibrahim (2003) that enables practitioners to elicit a prior prediction for the mean response for generalized linear models, treating the prediction as random. We refer to the hierarchical prior as the hierarchical prediction prior (HPP). For independent and identically distributed settings and the normal linear model, we derive cases for which the hyperprior is a conjugate prior. We also develop an extension of the HPP in situations where summary statistics from a previous study are available. The HPP allows for discounting based on the quality of individual level predictions, and simulation results suggest that, compared to the conjugate prior and the power prior, the HPP efficiency gains (e.g., lower mean squared error) where predictions are incompatible with the data. An efficient Monte Carlo Markov chain algorithm is developed. Applications illustrate that inferences under the HPP are more robust to prior-data conflict compared to selected nonhierarchical priors.

Bayesian multivariate probability of success using historical data with type I error rate control

In clinical trials, it is common to have multiple clinical outcomes (e.g., coprimary endpoints or a primary and multiple secondary endpoints). It is often desirable to establish efficacy in at least one of multiple clinical outcomes, which leads to a multiplicity problem. In the frequentist paradigm, the most popular methods to correct for multiplicity are typically conservative. Moreover, despite guidance from regulators, it is difficult to determine the sample size of a future study with multiple clinical outcomes. In this article, we introduce a Bayesian methodology for multiple testing that asymptotically guarantees type I error control. Using a seemingly unrelated regression model, correlations between outcomes are specifically modeled, which enables inference on the joint posterior distribution of the treatment effects. Simulation results suggest that the proposed Bayesian approach is more powerful than the method of Holm (1979), which is commonly utilized in practice as a more powerful alternative to the ubiquitous Bonferroni correction. We further develop multivariate probability of success, a Bayesian method to robustly determine sample size in the presence of multiple outcomes.

Assurance in vaccine efficacy clinical trial design based on immunological responses

The assurance of a future clinical trial is a key quantitative tool for decision-making in drug development. It is derived from prior knowledge (Bayesian approach) about the clinical endpoint of interest, typically from previous clinical trials. In this paper, we examine assurance in the specific context of vaccine development, where early development (Phase 2) is often based on immunological endpoints (e.g., antibody levels), while the confirmatory trial (Phase 3) is based on the clinical endpoint (very large sample sizes and long follow-up). Our proposal is to use the Phase 2 vaccine efficacy predicted by the immunological endpoint (using a model estimated from epidemiological studies) as prior information for the calculation of the assurance.

Bayesian survival analysis with BUGS

Survival analysis is one of the most important fields of statistics in medicine and biological sciences. In addition, the computational advances in the last decades have favored the use of Bayesian methods in this context, providing a flexible and powerful alternative to the traditional frequentist approach. The objective of this article is to summarize some of the most popular Bayesian survival models, such as accelerated failure time, proportional hazards, mixture cure, competing risks, multi-state, frailty, and joint models of longitudinal and survival data. Moreover, an implementation of each presented model is provided using a BUGS syntax that can be run with JAGS from the R programming language. Reference to other Bayesian R-packages is also discussed.

The contemplated average success probability for normally distributed models with an application to optimal sample sizes selection

We analytically obtain the average success probability (ASP) and the contemplated average success probability (CASP) for normally distributed observed differences in the treatment group and the placebo group means of the early trial and the confirmatory trial, assuming a uniform noninformative prior for the population treatment effect and a common known variance of the observations from both groups. For the CASP optimization problem with a fixed subtotal sample size of the early trial and the confirmatory trial of one arm larger than a threshold, we obtain the optimal plan of the sample sizes in a theorem. Moreover, in the theorem, we obtain the analytical formula of the optimal CASP as an increasing function of the subtotal sample size. After that, we calculate and compare the numerical values of the ASP with those in Table 1 of Chuang-Stein (2006). Finally, we investigate the numerical features of the CASP and find the optimal plan of the sample sizes for a given subtotal sample size.

Optimal Decision Criteria for the Study Design and Sample Size of a Biomarker-Driven Phase III Trial

BACKGROUND

The design and sample size of a phase III study for new medical technologies were historically determined within the framework of frequentist hypothesis testing. Recently, drug development using predictive biomarkers, which can predict efficacy based on the status of biomarkers, has attracted attention, and various study designs using predictive biomarkers have been suggested. Additionally, when choosing a study design, considering economic factors, such as the risk of development, expected revenue, and cost, is important.

METHODS

Here, we propose a method to determine the optimal phase III design and sample size and judge whether the phase III study will be conducted using the expected net present value (eNPV). The eNPV is defined using the probability of success of the study calculated based on historical data, the revenue that will be obtained after the success of the phase III study, and the cost of the study. Decision procedures of the optimal phase III design and sample size considering historical data obtained up to the start of the phase III study were considered using numerical examples.

RESULTS

Based on the numerical examples, the optimal study design and sample size depend on the mean treatment effect in the biomarker-positive and biomarker-negative populations obtained from historical data, the between-trial variance of response, the prevalence of the biomarker-positive population, and the threshold value of probability of success required to go to phase III study.

CONCLUSIONS

Thus, the design and sample size of a biomarker-driven phase III study can be appropriately determined based on the eNPV.

Assessment of Bayesian expected power via Bayesian bootstrap

The Bayesian expected power (BEP) has become increasingly popular in assessing the probability of success for a future trial. While the traditional power assumes a single value for the unknown effect size Δ and is thus highly dependent on the assumed value, the BEP embraces the uncertainty around Δ given the prior information and is therefore a less subjective measure for the probability of success than the traditional power especially when the prior information is not rich. Current methods for assessing BEP are often based in a parametric framework by imposing a model on the pilot data to derive and sample from the posterior distributions of Δ. The model-based approach can be analytically challenging and computationally costly especially for multivariate data sets, and it also runs the risk of generating misleading BEP if the model is misspecified. We propose an approach based on the Bayesian bootstrap (BBS) technique to simulate future trials in the presence of individual-level pilot data, based on which the empirical BEP can be calculated. The BBS approach is model-free with no assumptions about the distribution of the prior data and also circumvents the analytical and computational complexity associated with obtaining the posterior distribution of the Δ. Information from multiple pilot studies is also straightforward to combine. We also propose the double bootstrap technique, a frequentist counterpart to the BBS, that shares similar properties and achieves the same goal as the BBS for BEP assessment. Simulation and case studies are presented to demonstrate the implementation of the BBS technique and the double bootstrap technique and to compare the BEP results with model-based approach.

Predictive probability methods for interim monitoring in clinical trials with longitudinal outcomes

In clinical research and development, interim monitoring is critical for better decision-making and minimizing the risk of exposing patients to possible ineffective therapies. For interim futility or efficacy monitoring, predictive probability methods are widely adopted in practice. Those methods have been well studied for univariate variables. However, for longitudinal studies, predictive probability methods using univariate information from only completers may not be most efficient, and data from on-going subjects can be utilized to improve efficiency. On the other hand, leveraging information from on-going subjects could allow an interim analysis to be potentially conducted once a sufficient number of subjects reach an earlier time point. For longitudinal outcomes, we derive closed-form formulas for predictive probabilities, including Bayesian predictive probability, predictive power, and conditional power and also give closed-form solutions for predictive probability of success in a future trial and the predictive probability of success of the best dose. When predictive probabilities are used for interim monitoring, we study their distributions and discuss their analytical cutoff values or stopping boundaries that have desired operating characteristics. We show that predictive probabilities utilizing all longitudinal information are more efficient for interim monitoring than that using information from completers only. To illustrate their practical application for longitudinal data, we analyze 2 real data examples from clinical trials.