Simulation-based sequential Bayesian design

Nonmyopic and pseudo-nonmyopic approaches to optimal sequential design in the presence of covariates

In sequential experiments, subjects become available for the study over a period of time, and covariates are often measured at the time of arrival. We consider the setting where the sample size is fixed but covariate values are unknown until subjects enrol. Given a model for the outcome, a sequential optimal design approach can be used to allocate treatments to minimize the variance of the estimator of the treatment effect. We extend existing optimal design methodology so it can be used within a nonmyopic framework, where treatment allocation for the current subject depends not only on the treatments and covariates of the subjects already enrolled in the study, but also the impact of possible future treatment assignments within a specified horizon. The nonmyopic approach requires recursive formulae and suffers from the curse of dimensionality. We propose a pseudo-nonmyopic approach which has a similar aim to the nonmyopic approach, but does not involve recursion and instead relies on simulating trajectories of future possible decisions. Our simulation studies show that, for the simple case of a logistic regression with a single binary covariate and a binary treatment, and a more realistic case with four binary covariates, binary treatment and treatment-covariate interactions, the nonmyopic and pseudo-nonmyopic approaches provide no competitive advantage over the myopic approach, both in terms of the size of the estimated treatment effect and also the efficiency of the designs. Results are robust to the size of the horizon used in the nonmyopic approach, and the number of simulated trajectories used in the pseudo-nonmyopic approach.

Reflecting on Andy Grieve's influence and innovation: A personal perspective

Throughout his career, Andy Grieve has developed and implemented many novel methods and has been involved in trial design and analysis for many trials that have broken new ground in the statistics field. His record of innovation is clear, but it is the way that he also applies these innovations in practice, reaches pragmatic solutions to problems and then shares and disseminates those findings that mark him out as a true leader in the statistical field. In this short article, I will discuss my own views of Andy's innovation, pragmatism and influence and how it has left its mark in my own career.

From innovative thinking to pharmaceutical industry implementation: Some success stories

In industry, successful innovation involves not only developing new statistical methodology, but also ensuring that this methodology is implemented successfully. This includes enabling applied statisticians to understand the method, its benefits and limitations and empowering them to implement the new method. This will include advocacy, influencing in-house and external stakeholders, such that these stakeholders are receptive to the new methodology. In this paper, we describe some industry successes and focus on our colleague, Andy Grieve's role in these.

Simulation-based sequential design

We review some simulation-based methods to implement optimal decisions in sequential design problems as they naturally arise in clinical trial design. As a motivating example we use a stylized version of a dose-ranging design in the ASTIN trial. The approach can be characterized as constrained backward induction. The nature of the constraint is a restriction of the decisions to a set of actions that are functions of the current history only implicitly through a low-dimensional summary statistic. In addition, the action set is restricted to time-invariant policies. Time-dependence is only introduced indirectly through the change of the chosen summary statistic over time. This restriction allows computationally efficient solutions to the sequential decision problem. A further simplification is achieved by restricting optimal actions to be described by decision boundaries on the space of such summary statistics.

Optimal Sampling Regimes for Estimating Population Dynamics

Ecologists are interested in modeling the population growth of species in various ecosystems. Specifically, logistic growth arises as a common model for population growth. Studying such growth can assist environmental managers in making better decisions when collecting data. Traditionally, ecological data is recorded on a regular time frequency and is very well-documented. However, sampling can be an expensive process due to available resources, money and time. Limiting sampling makes it challenging to properly track the growth of a population. Thus, this design study proposes an approach to sampling based on the dynamics associated with logistic growth. The proposed method is demonstrated via a simulation study across various theoretical scenarios to evaluate its performance in identifying optimal designs that best estimate the curves. Markov Chain Monte Carlo sampling techniques are implemented to predict the probability of the model parameters using Bayesian inference. The intention of this study is to demonstrate a method that can minimize the amount of time ecologists spend in the field, while maximizing the information provided by the data.

SSNdesign-An R package for pseudo-Bayesian optimal and adaptive sampling designs on stream networks

Streams and rivers are biodiverse and provide valuable ecosystem services. Maintaining these ecosystems is an important task, so organisations often monitor the status and trends in stream condition and biodiversity using field sampling and, more recently, autonomous in-situ sensors. However, data collection is often costly, so effective and efficient survey designs are crucial to maximise information while minimising costs. Geostatistics and optimal and adaptive design theory can be used to optimise the placement of sampling sites in freshwater studies and aquatic monitoring programs. Geostatistical modelling and experimental design on stream networks pose statistical challenges due to the branching structure of the network, flow connectivity and directionality, and differences in flow volume. Geostatistical models for stream network data and their unique features already exist. Some basic theory for experimental design in stream environments has also previously been described. However, open source software that makes these design methods available for aquatic scientists does not yet exist. To address this need, we present SSNdesign, an R package for solving optimal and adaptive design problems on stream networks that integrates with existing open-source software. We demonstrate the mathematical foundations of our approach, and illustrate the functionality of SSNdesign using two case studies involving real data from Queensland, Australia. In both case studies we demonstrate that the optimal or adaptive designs outperform random and spatially balanced survey designs implemented in existing open-source software packages. The SSNdesign package has the potential to boost the efficiency of freshwater monitoring efforts and provide much-needed information for freshwater conservation and management.

Do we need to adjust for interim analyses in a Bayesian adaptive trial design?

Background

Bayesian adaptive methods are increasingly being used to design clinical trials and offer several advantages over traditional approaches. Decisions at analysis points are usually based on the posterior distribution of the treatment effect. However, there is some confusion as to whether control of type I error is required for Bayesian designs as this is a frequentist concept.

Methods

We discuss the arguments for and against adjusting for multiplicities in Bayesian trials with interim analyses. With two case studies we illustrate the effect of including interim analyses on type I/II error rates in Bayesian clinical trials where no adjustments for multiplicities are made. We propose several approaches to control type I error, and also alternative methods for decision-making in Bayesian clinical trials.

Results

In both case studies we demonstrated that the type I error was inflated in the Bayesian adaptive designs through incorporation of interim analyses that allowed early stopping for efficacy and without adjustments to account for multiplicity. Incorporation of early stopping for efficacy also increased the power in some instances. An increase in the number of interim analyses that only allowed early stopping for futility decreased the type I error, but also decreased power. An increase in the number of interim analyses that allowed for either early stopping for efficacy or futility generally increased type I error and decreased power.

Conclusions

Currently, regulators require demonstration of control of type I error for both frequentist and Bayesian adaptive designs, particularly for late-phase trials. To demonstrate control of type I error in Bayesian adaptive designs, adjustments to the stopping boundaries are usually required for designs that allow for early stopping for efficacy as the number of analyses increase. If the designs only allow for early stopping for futility then adjustments to the stopping boundaries are not needed to control type I error. If one instead uses a strict Bayesian approach, which is currently more accepted in the design and analysis of exploratory trials, then type I errors could be ignored and the designs could instead focus on the posterior probabilities of treatment effects of clinically-relevant values.

Bayesian Methods in Regulatory Science

Regulatory science comprises the tools, standards, and approaches that regulators use to assess safety, efficacy, quality, and performance of drugs and medical devices. A major focus of regulatory science is the design and analysis of clinical trials. Clinical trials are an essential part of clinical research programs that aim to improve therapies and reduce the burden of disease. These clinical experiments help us learn about what works clinically and what does not work. The results of clinical trials support therapeutic and policy decisions. When designing clinical trials, investigators make many decisions regarding various aspects of how they will carry out the study, such as the primary objective of the study, primary and secondary endpoints, methods of analysis, sample size, etc. This paper provides a brief review of the clinical development of new treatments and argues for the use of Bayesian methods and decision theory in clinical research.

A Review of Clinical Trials With an Adaptive Design and Health Economic Analysis

OBJECTIVE

An adaptive design uses data collected as a clinical trial progresses to inform modifications to the trial. Hence, adaptive designs and health economics aim to facilitate efficient and accurate decision making. Nevertheless, it is unclear whether the methods are considered together in the design, analysis, and reporting of trials. This review aims to establish how health economic outcomes are used in the design, analysis, and reporting of adaptive designs.

METHODS

Registered and published trials up to August 2016 with an adaptive design and health economic analysis were identified. The use of health economics in the design, analysis, and reporting was assessed. Summary statistics are presented and recommendations formed based on the research team's experiences and a practical interpretation of the results.

RESULTS

Thirty-seven trials with an adaptive design and health economic analysis were identified. It was not clear whether the health economic analysis accounted for the adaptive design in 17/37 trials where this was thought necessary, nor whether health economic outcomes were used at the interim analysis for 18/19 of trials with results. The reporting of health economic results was suboptimal for the (17/19) trials with published results.

CONCLUSIONS

Appropriate consideration is rarely given to the health economic analysis of adaptive designs. Opportunities to use health economic outcomes in the design and analysis of adaptive trials are being missed. Further work is needed to establish whether adaptive designs and health economic analyses can be used together to increase the efficiency of health technology assessments without compromising accuracy.

The Performance of Fixed-Horizon, Look-Ahead Procedures Compared to Backward Induction in Bayesian Adaptive-Randomization Decision-Theoretic Clinical Trial Design

Designing optimal, Bayesian decision-theoretic trials has traditionally required the use of computationally-intensive backward induction. While methods for addressing this barrier have been put forward, few are both computationally tractable and non-myopic, with applications of the Gittins index being one notable example. Here we explore the look-ahead approach with adaptive-randomization, with designs ranging from the fully myopic to the fully informed. We compare the operating characteristics of the look-ahead designed trials, in which decision rules are based on a fixed number of future blocks, with those of trials designed using traditional backward induction. The less-myopic designs performed well. As the designs become more myopic or the trials longer, there were disparities in regions of the decision space that are transition zones between continuation and stopping decisions. The more myopic trials generally suffered from early stopping as compared to the less myopic and backward induction trials. Myopic trials with adaptive randomization also saw as many as 28 % of their continuation decisions change to a different randomization ratio as compared to the backward induction designs. Finally, early stages of myopic-designed trials may have disproportionate effect on trial characteristics.

Clinical Trial Design as a Decision Problem

The intent of this discussion is to highlight opportunities and limitations of utility-based and decision theoretic arguments in clinical trial design. The discussion is based on a specific case study, but the arguments and principles remain valid in general. The example concerns the design of a randomized clinical trial to compare a gel sealant versus standard care for resolving air leaks after pulmonary resection. The design follows a principled approach to optimal decision making, including a probability model for the unknown distributions of time to resolution of air leaks under the two treatment arms, and an explicit utility function that quantifies clinical preferences for alternative outcomes. As is typical for any real application, the final implementation includes some compromises from the initial principled setup. In particular, we use the formal decision problem only for the final decision, but use reasonable ad-hoc decision boundaries for making interim group sequential decisions that stop the trial early. Beyond the discussion of the particular study, we review more general considerations of using a decision theoretic approach for clinical trial design and summarize some of the reasons why such approaches are not commonly used.

Bayesian randomized clinical trials: From fixed to adaptive design

Randomized controlled studies are the gold standard for phase III clinical trials. Using α-spending functions to control the overall type I error rate, group sequential methods are well established and have been dominating phase III studies. Bayesian randomized design, on the other hand, can be viewed as a complement instead of competitive approach to the frequentist methods. For the fixed Bayesian design, the hypothesis testing can be cast in the posterior probability or Bayes factor framework, which has a direct link to the frequentist type I error rate. Bayesian group sequential design relies upon Bayesian decision-theoretic approaches based on backward induction, which is often computationally intensive. Compared with the frequentist approaches, Bayesian methods have several advantages. The posterior predictive probability serves as a useful and convenient tool for trial monitoring, and can be updated at any time as the data accrue during the trial. The Bayesian decision-theoretic framework possesses a direct link to the decision making in the practical setting, and can be modeled more realistically to reflect the actual cost-benefit analysis during the drug development process. Other merits include the possibility of hierarchical modeling and the use of informative priors, which would lead to a more comprehensive utilization of information from both historical and longitudinal data. From fixed to adaptive design, we focus on Bayesian randomized controlled clinical trials and make extensive comparisons with frequentist counterparts through numerical studies.

Planning Beyond the Next Trial in Adaptive Experiments: A Dynamic Programming Approach

Experimentation is at the heart of scientific inquiry. In the behavioral and neural sciences, where only a limited number of observations can often be made, it is ideal to design an experiment that leads to the rapid accumulation of information about the phenomenon under study. Adaptive experimentation has the potential to accelerate scientific progress by maximizing inferential gain in such research settings. To date, most adaptive experiments have relied on myopic, one-step-ahead strategies in which the stimulus on each trial is selected to maximize inference on the next trial only. A lingering question in the field has been how much additional benefit would be gained by optimizing beyond the next trial. A range of technical challenges has prevented this important question from being addressed adequately. This study applies dynamic programming (DP), a technique applicable for such full-horizon, "global" optimization, to model-based perceptual threshold estimation, a domain that has been a major beneficiary of adaptive methods. The results provide insight into conditions that will benefit from optimizing beyond the next trial. Implications for the use of adaptive methods in cognitive science are discussed.

Bayesian designs and the control of frequentist characteristics: a practical solution

Frequentist concepts, such as the control of the type I error or the false discovery rate, are well established in the medical literature and often required by regulators. Most Bayesian designs are defined without explicit considerations of frequentist characteristics. Once the Bayesian design is structured, statisticians use simulations and adjust tuning parameters to comply with a set of targeted operating characteristics. These adjustments affect the use of prior information and utility functions. Here we consider a Bayesian decision theoretic approach for experimental designs with explicit frequentist requisites. We define optimal designs under a set of constraints required by a regulator. Our approach combines the use of interpretable utility functions with frequentist criteria, and selects an optimal design that satisfies a set of required operating characteristics. We illustrate the approach using a group-sequential multi-arm Phase II trial and a bridging trial.

A Bayesian decision-theoretic sequential response-adaptive randomization design

We propose a class of phase II clinical trial designs with sequential stopping and adaptive treatment allocation to evaluate treatment efficacy. Our work is based on two-arm (control and experimental treatment) designs with binary endpoints. Our overall goal is to construct more efficient and ethical randomized phase II trials by reducing the average sample sizes and increasing the percentage of patients assigned to the better treatment arms of the trials. The designs combine the Bayesian decision-theoretic sequential approach with adaptive randomization procedures in order to achieve simultaneous goals of improved efficiency and ethics. The design parameters represent the costs of different decisions, for example, the decisions for stopping or continuing the trials. The parameters enable us to incorporate the actual costs of the decisions in practice. The proposed designs allow the clinical trials to stop early for either efficacy or futility. Furthermore, the designs assign more patients to better treatment arms by applying adaptive randomization procedures. We develop an algorithm based on the constrained backward induction and forward simulation to implement the designs. The algorithm overcomes the computational difficulty of the backward induction method, thereby making our approach practicable. The designs result in trials with desirable operating characteristics under the simulated settings. Moreover, the designs are robust with respect to the response rate of the control group.

A Predictive Approach to Nonparametric Inference for Adaptive Sequential Sampling of Psychophysical Experiments

We present a predictive account on adaptive sequential sampling of stimulus-response relations in psychophysical experiments. Our discussion applies to experimental situations with ordinal stimuli when there is only weak structural knowledge available such that parametric modeling is no option. By introducing a certain form of partial exchangeability, we successively develop a hierarchical Bayesian model based on a mixture of Pólya urn processes. Suitable utility measures permit us to optimize the overall experimental sampling process. We provide several measures that are either based on simple count statistics or more elaborate information theoretic quantities. The actual computation of information theoretic utilities often turns out to be infeasible. This is not the case with our sampling method, which relies on an efficient algorithm to compute exact solutions of our posterior predictions and utility measures. Finally, we demonstrate the advantages of our framework on a hypothetical sampling problem.

Bayesian decision theoretic two-stage design in phase II clinical trials with survival endpoint

In this paper, we consider two-stage designs with failure-time endpoints in single-arm phase II trials. We propose designs in which stopping rules are constructed by comparing the Bayes risk of stopping at stage I with the expected Bayes risk of continuing to stage II using both the observed data in stage I and the predicted survival data in stage II. Terminal decision rules are constructed by comparing the posterior expected loss of a rejection decision versus an acceptance decision. Simple threshold loss functions are applied to time-to-event data modeled either parametrically or nonparametrically, and the cost parameters in the loss structure are calibrated to obtain desired type I error and power. We ran simulation studies to evaluate design properties including types I and II errors, probability of early stopping, expected sample size, and expected trial duration and compared them with the Simon two-stage designs and a design, which is an extension of the Simon's designs with time-to-event endpoints. An example based on a recently conducted phase II sarcoma trial illustrates the method.

Model discrimination through adaptive experimentation

An ideal experiment is one in which data collection is efficient and the results are maximally informative. This standard can be difficult to achieve because of uncertainties about the consequences of design decisions. We demonstrate the success of a Bayesian adaptive method (adaptive design optimization, ADO) in optimizing design decisions when comparing models of the time course of forgetting. Across a series of testing stages, ADO intelligently adapts the retention interval in order to maximally discriminate power and exponential models. Compared with two different control (non-adaptive) methods, ADO distinguishes the models decisively, with the results unambiguously favoring the power model. Analyses suggest that ADO's success is due in part to its flexibility in adjusting to individual differences. This implementation of ADO serves as an important first step in assessing its applicability and usefulness to psychology.

Adaptive design optimization: a mutual information-based approach to model discrimination in cognitive science

Discriminating among competing statistical models is a pressing issue for many experimentalists in the field of cognitive science. Resolving this issue begins with designing maximally informative experiments. To this end, the problem to be solved in adaptive design optimization is identifying experimental designs under which one can infer the underlying model in the fewest possible steps. When the models under consideration are nonlinear, as is often the case in cognitive science, this problem can be impossible to solve analytically without simplifying assumptions. However, as we show in this letter, a full solution can be found numerically with the help of a Bayesian computational trick derived from the statistics literature, which recasts the problem as a probability density simulation in which the optimal design is the mode of the density. We use a utility function based on mutual information and give three intuitive interpretations of the utility function in terms of Bayesian posterior estimates. As a proof of concept, we offer a simple example application to an experiment on memory retention.

The future of drug development: advancing clinical trial design

Declining pharmaceutical industry productivity is well recognized by drug developers, regulatory authorities and patient groups. A key part of the problem is that clinical studies are increasingly expensive, driven by the rising costs of conducting Phase II and III trials. It is therefore crucial to ensure that these phases of drug development are conducted more efficiently and cost-effectively, and that attrition rates are reduced. In this article, we argue that moving from the traditional clinical development approach based on sequential, distinct phases towards a more integrated view that uses adaptive design tools to increase flexibility and maximize the use of accumulated knowledge could have an important role in achieving these goals. Applications and examples of the use of these tools--such as Bayesian methodologies--in early- and late-stage drug development are discussed, as well as the advantages, challenges and barriers to their more widespread implementation.

Bayesian optimal design for phase II screening trials

Most phase II screening designs available in the literature consider one treatment at a time. Each study is considered in isolation. We propose a more systematic decision-making approach to the phase II screening process. The sequential design allows for more efficiency and greater learning about treatments. The approach incorporates a Bayesian hierarchical model that allows combining information across several related studies in a formal way and improves estimation in small data sets by borrowing strength from other treatments. The design incorporates a utility function that includes sampling costs and possible future payoff. Computer simulations show that this method has high probability of discarding treatments with low success rates and moving treatments with high success rates to phase III trial.