Calibration of prior variance in the Bayesian continual reassessment method

The continual reassessment method (CRM) is an adaptive model-based design used to estimate the maximum tolerated dose in phase I clinical trials. Asymptotically, the method has been shown to select the correct dose given that certain conditions are satisfied. When sample size is small, specifying a reasonable model is important. While an algorithm has been proposed for the calibration of the initial guesses of the probabilities of toxicity, the calibration of the prior distribution of the parameter for the Bayesian CRM has not been addressed. In this paper, we introduce the concept of least informative prior variance for a normal prior distribution. We also propose two systematic approaches to jointly calibrate the prior variance and the initial guesses of the probability of toxicity at each dose. The proposed calibration approaches are compared with existing approaches in the context of two examples via simulations. The new approaches and the previously proposed methods yield very similar results since the latter used appropriate vague priors. However, the new approaches yield a smaller interval of toxicity probabilities in which a neighboring dose may be selected.     

Adaptive prior variance calibration in the Bayesian continual reassessment method

The use of the continual reassessment method (CRM) and other model‐based approaches to design Phase I clinical trials has increased owing to the ability of the CRM to identify the maximum tolerated dose better than the 3 + 3 method. However, the CRM can be sensitive to the variance selected for the prior distribution of the model parameter, especially when a small number of patients are enrolled. Although methods have emerged to adaptively select skeletons and to calibrate the prior variance only at the beginning of a trial, there has not been any approach developed to adaptively calibrate the prior variance throughout a trial. We propose three systematic approaches to adaptively calibrate the prior variance during a trial and compare them via simulation with methods proposed to calibrate the variance at the beginning of a trial. Copyright © 2012 John Wiley & Sons, Ltd.     

The continual reassessment method: comparison of Bayesian stopping rules for dose-ranging studies

The continual reassessment method (CRM) provides a Bayesian estimation of the maximum tolerated dose (MTD) in phase I clinical trials and is also used to estimate the minimal efficacy dose (MED) in phase II clinical trials. In this paper we propose Bayesian stopping rules for the CRM, based on either posterior or predictive probability distributions that can be applied sequentially during the trial. These rules aim at early detection of either the mis-choice of dose range or a prefixed gain in the point estimate or accuracy of estimated probability of response associated with the MTD (or MED). They were compared through a simulation study under six situations that could represent the underlying unknown dose-response (either toxicity or failure) relationship, in terms of sample size, probability of correct selection and bias of the response probability associated to the MTD (or MED). Our results show that the stopping rules act correctly, with early stopping by using the two first rules based on the posterior distribution when the actual underlying dose-response relationship is far from that initially supposed, while the rules based on predictive gain functions provide a discontinuation of inclusions whatever the actual dose-response curve after 20 patients on average, that is, depending mostly on the accumulated data. The stopping rules were then applied to a data set from a dose-ranging phase II clinical trial aiming at estimating the MED dose of midazolam in the sedation of infants during cardiac catheterization. All these findings suggest the early use of the two first rules to detect a mis-choice of dose range, while they confirm the requirement of including at least 20 patients at the same dose to reach an accurate estimate of MTD (MED). A two-stage design is under study.     

Sequential continual reassessment method for two-dimensional dose finding

It is common to encounter two-dimensional dose finding in phase I trials, for example, in trials combining multiple drugs, or in single-agent trials that simultaneously search for the maximum tolerated dose (MTD) and the optimal treatment schedule. In these cases, the traditional single-agent dose-finding methods are not directly applicable. We propose a simple and adaptive two-dimensional dose-finding design that can accommodate any type of single-agent dose-finding method. In particular, we convert the two-dimensional dose-finding trial to a series of one-dimensional dose-finding subtrials along shortened line search segments by fixing the dose level of one drug. We then conduct the subtrials sequentially. Based on the MTD obtained from the completed one-dimensional trial, we eliminate the doses that lie outside of the search range based on the partial order, and thereby efficiently shrink the two-dimensional dose-finding space. The proposed design dramatically reduces the sample size and still maintains good performance. We illustrate the design through extensive simulation studies motivated by clinical trials evaluating multiple drugs or dose and schedule combinations.     

A comparison of model choices for the continual reassessment method in phase I cancer trials

Determination of the maximum tolerated dose (MTD) is the main objective of phase I trials. Trials are typically carried out with restricted sample sizes. Model‐based approaches proposed to identify the MTD (including the Continual Reassessment Method or CRM) suppose a simple model for the dose‐toxicity relation. At this early stage of clinical development, the true family of models is not known and several proposals have been done. Asymptotic convergence of the recommendation to the true MTD can be obtained with a one‐parameter model even in case of model misspecification. Nevertheless, operating characteristics with finite sample sizes can be largely affected by the choice of the model. In this paper, we evaluate and compare several models in a simulation framework. This framework includes a large class of dose‐toxicity relations against which to test the competing models, an ‘optimal’ method that provides efficient non‐parametric estimates of the probability of dose limiting toxicity to serve as a benchmark and as a graphic representation. In particular we explore the use of a one‐parameter versus a two‐parameter model, we compare the power and the logistic models and finally we investigate the impact of dose recoding on the operating characteristics. Comparisons are carried out with both a likelihood approach and a Bayesian approach for model estimations. We show that average performances of a one‐parameter model are superior and that the power model has good operating characteristics. Some models can speed up dose escalation and lead to more aggressive designs. We derive some behavior related to the choice of model and insist on the use of simulations under several scenarios before the initiation of each new trial in order to determine the best model to be used. Copyright © 2009 John Wiley & Sons, Ltd.     

The continual reassessment method and its applications: a Bayesian methodology for phase I cancer clinical trials

We discuss the continual reassessment method (CRM) and its extension with practical applications in phase I and I/II cancer clinical trials. The CRM has been proposed as an alternative design of a traditional cohort design and its essential features are the sequential (continual) selection of a dose level for the next patients based on the dose-toxicity relationship and the updating of the relationship based on patients' response data using Bayesian calculation. The original CRM has been criticized because it often tends to allocate too toxic doses to many patients and our proposal for overcoming this practical problem is to monitor a posterior density function of the occurrence of the dose limiting toxicity (DLT) at each dose level. A simulation study shows that strategies based on our proposal allocate a smaller number of patients to doses higher than the maximum tolerated dose (MTD) compared with the original method while the mean squared error of the probability of the DLT occurrence at the MTD is not inflated. We present a couple of extensions of the CRM with real prospective applications: (i) monitoring efficacy and toxicity simultaneously in a combination phase I/II trial; (ii) combining the idea of pharmacokinetically guided dose escalation (PKGDE) and utilization of animal toxicity data in determining the prior distribution. A stopping rule based on the idea of separation among the DLT density functions is discussed in the first example and a strategy for determining the model parameter of the dose-toxicity relationship is suggested in the second example.     

Some practical improvements in the continual reassessment method for phase I studies

The Continual Reassessment Method (CRM) is a Bayesian phase I design whose purpose is to estimate the maximum tolerated dose of a drug that will be used in subsequent phase II and III studies. Its acceptance has been hindered by the greater duration of CRM designs compared to standard methods, as well as by concerns with excessive experimentation at high dosage levels, and with more frequent and severe toxicity. This paper presents the results of a simulation study in which one assigns more than one subject at a time to each dose level, and each dose increase is limited to one level. We show that these modifications address all of the most serious criticisms of the CRM, reducing the duration of the trial by 50–67 per cent, reducing toxicity incidence by 20–35 per cent, and lowering toxicity severity. These are achieved with minimal effects on accuracy. Most important, based on our experience at our institution, such modifications make the CRM acceptable to clinical investigators.     

Adaptive design improvements in the continual reassessment method for phase I studies.

The continual reassessment method (CRM) enables full and efficient use of all data and prior information available in a phase I study. However, despite a number of recent enhancements to the method, its acceptance in actual clinical practice has been hampered by several practical difficulties. In this paper, we consider several further refinements in the context of phase I oncology trials. In particular, we allow the trial to stop when the width of the posterior 95 per cent probability interval for the maximum tolerated dose (MTD) becomes sufficiently narrow (that is, when the information accumulating from the trial data reaches a prespecified level). We employ a simulation study to evaluate five such stopping rules under three alternative states of prior knowledge regarding the MTD (accurate, too low and too high). Our results suggest our adaptive designs preserve the CRM's estimation ability while offering the possibility of earlier stopping of the trial.     

An extension of the continual reassessment method using decision theory.

The primary goal of a phase I trial is to find the maximally tolerated dose (MTD) of a treatment. The MTD is usually defined in terms of a tolerable probability, q(*), of toxicity. Our objective is to find the highest dose with toxicity risk that does not exceed q(*), a criterion that is often desired in designing phase I trials. This criterion differs from that of finding the dose with toxicity risk closest to q(*), that is used in methods such as the continual reassessment method. We use the theory of decision processes to find optimal sequential designs that maximize the expected number of patients within the trial allocated to the highest dose with toxicity not exceeding q(*), among the doses under consideration. The proposed method is very general in the sense that criteria other than the one considered here can be optimized and that optimal dose assignment can be defined in terms of patients within or outside the trial. It includes as an important special case the continual reassessment method. Numerical study indicates the strategy compares favourably with other phase I designs.     

Bridging continual reassessment method for phase I clinical trials in different ethnic populations

Accumulating evidence shows that the conventional one‐size‐fits‐all dose‐finding paradigm is problematic when applied to different ethnic populations. Because of inter‐ethnic heterogeneity, the dosage established in a landmark trial for a certain population may not be generalizable to a different ethnic population, and a follow‐up bridge trial is often needed to find the maximum tolerated dose for the new population. We propose the bridging continual reassessment method (B‐CRM) to facilitate dose finding for such follow‐up bridge trials. The B‐CRM borrows information from the landmark trial through a novel estimate of the dose‐toxicity curve and accommodates the inter‐ethnic heterogeneity using the Bayesian model averaging approach. Extensive simulation studies show that the B‐CRM has desirable operating characteristics with a high probability to select the target dose. This article focuses on ethnic heterogeneity, but the proposed method can be directly used to handle other types of patient heterogeneity, for example, patient subgroups defined by prognostic factors or biomarkers. The software to implement the B‐CRM design is available for free download at http://odin.mdacc.tmc.edu/~yyuan/ . Copyright © 2015 John Wiley & Sons, Ltd.     

A simulation-free approach to assessing the performance of the continual reassessment method

The continual reassessment method (CRM) is an adaptive design for Phase I trials whose operating characteristics, including appropriate sample size, probability of correctly identifying the maximum tolerated dose, and the expected proportion of participants assigned to each dose, can only be determined via simulation. The actual time to determine a final “best” design can take several hours or days, depending on the number of scenarios that are examined. The computational cost increases as the kernel of the one-parameter CRM design is expanded to other settings, including additional parameters, monitoring of both toxicity and efficacy, and studies of combinations of two agents. For a given vector of true DLT probabilities, we have developed an approach that replaces a simulation study of thousands of hypothetical trials with a single simulation. Our approach, which is founded on the consistency of the CRM, very accurately reflects the results produced by the simulation study, but does so in a fraction of time required by the simulation study. Relative to traditional simulations, we extensively examine how our method is able to assess the operating characteristics of a CRM design for a hypothetical trial whose characteristics are based upon a previously published Phase I trial. We also provide a metric of nonconsistency and demonstrate that although nonconsistency can impact the operating characteristics of our method, the degree of over- or under-estimation is unpredictable. As a solution, we provide an algorithm for maintaining the consistency of a chosen CRM design so that our method is applicable for any trial.     

A Bayesian model averaging approach for cost-effectiveness analyses

We consider the problem of assessing new and existing technologies for their cost-effectiveness in the case where data on both costs and effects are available from a clinical trial, and we address it by means of the cost-effectiveness acceptability curve. The main difficulty in these analyses is that cost data usually exhibit highly skew and heavy-tailed distributions so that it can be extremely difficult to produce realistic probabilistic models for the underlying population distribution, and in particular to model accurately the tail of the distribution, which is highly influential in estimating the population mean. Here, in order to integrate the uncertainty about the model into the analysis of cost data and into cost-effectiveness analyses, we consider an approach based on Bayesian model averaging: instead of choosing a single parametric model, we specify a set of plausible models for costs and estimate the mean cost with a weighted mean of its posterior expectations under each model, with weights given by the posterior model probabilities. The results are compared with those obtained with a semi-parametric approach that does not require any assumption about the distribution of costs.     

Improving safety of the continual reassessment method via a modified allocation rule

This article proposes a novel criterion for the allocation of patients in phase I dose-escalation clinical trials, aiming to find the maximum tolerated dose (MTD). Conventionally, using a model-based approach, the next patient is allocated to the dose with the toxicity estimate closest (in terms of the absolute or squared distance) to the maximum acceptable toxicity. This approach, however, ignores the uncertainty in point estimates and ethical concerns of assigning a lot of patients to overly toxic doses. In fact, balancing the trade-off between how accurately the MTD can be estimated and how many patients would experience adverse events is one of the primary challenges in phase I studies. Motivated by recent discussions in the theory of estimation in restricted parameter spaces, we propose a criterion that allows to balance these explicitly. The criterion requires a specification of one additional parameter only that has a simple and intuitive interpretation. We incorporate the proposed criterion into the one-parameter Bayesian continual reassessment method and show, using simulations, that it can result in similar accuracy on average as the original design, but with fewer toxic responses on average. A comparison with other model-based dose-escalation designs, such as escalation with overdose control and its modifications, demonstrates that the proposed design can result in either the same mean accuracy as alternatives but fewer toxic responses or in a higher mean accuracy but the same number of toxic responses. Therefore, the proposed design can provide a better trade-off between the accuracy and the number of patients experiencing adverse events, making the design a more ethical alternative over some of the existing methods for phase I trials.     

Comparison of Bayesian model averaging and stepwise methods for model selection in logistic regression

Logistic regression is the standard method for assessing predictors of diseases. In logistic regression analyses, a stepwise strategy is often adopted to choose a subset of variables. Inference about the predictors is then made based on the chosen model constructed of only those variables retained in that model. This method subsequently ignores both the variables not selected by the procedure, and the uncertainty due to the variable selection procedure. This limitation may be addressed by adopting a Bayesian model averaging approach, which selects a number of all possible such models, and uses the posterior probabilities of these models to perform all inferences and predictions. This study compares the Bayesian model averaging approach with the stepwise procedures for selection of predictor variables in logistic regression using simulated data sets and the Framingham Heart Study data. The results show that in most cases Bayesian model averaging selects the correct model and out-performs stepwise approaches at predicting an event of interest.     

Heterogeneity in phase I clinical trials: prior elicitation and computation using the continual reassessment method

Heterogeneity in a phase I clinical trial patient population may lead to distinctly different dose-response relationships along covariate values. For a given target probability of toxicity, this implies different maximum tolerated doses (MTDs) for each distinct subpopulation. Within the framework of O'Quigley, Pepe and Fisher's (1990) continual reassessment method, we propose the notion of average and patient-specific MTDs by augmenting the dose--response model with other covariates to account for such differences. A method to elicit prior distributions on the dose and other covariate parameters are proposed, based on the predictive approach of Ibrahim and Laud (1994), Laud and Ibrahim (1995), and Ibrahim, Ryan and Chen (1998). This approach relies on prior predictions for the response vector y(0) and a quantity a(0) specifying uncertainty in y(0). Then, y(0) and a(0) are used to specify a prior for the regression coefficients in a semi-automatic fashion. The elicitation scheme for y(0) uses results from previous phase I cancer clinical trials. The average and patient-specific MTDs and an elicitation method are demonstrated in logistic regression examples.     

Variable selection and Bayesian model averaging in case-control studies.

Covariate and confounder selection in case-control studies is often carried out using a statistical variable selection method, such as a two-step method or a stepwise method in logistic regression. Inference is then carried out conditionally on the selected model, but this ignores the model uncertainty implicit in the variable selection process, and so may underestimate uncertainty about relative risks. We report on a simulation study designed to be similar to actual case-control studies. This shows that p-values computed after variable selection can greatly overstate the strength of conclusions. For example, for our simulated case-control studies with 1000 subjects, of variables declared to be 'significant' with p-values between 0.01 and 0.05, only 49 per cent actually were risk factors when stepwise variable selection was used. We propose Bayesian model averaging as a formal way of taking account of model uncertainty in case-control studies. This yields an easily interpreted summary, the posterior probability that a variable is a risk factor, and our simulation study indicates this to be reasonably well calibrated in the situations simulated. The methods are applied and compared in the context of a case-control study of cervical cancer.     

The continual reassessment method in cancer phase i clinical trials: A simulation study

In cancer studies, the aim of phase I clinical trials is to identify an appropriate dose for experimentation in phase II and III studies. The continual reassessment method (CRM) has been developed recently and presented as the method of choice in the design and analysis of such phase I studies. However, to implement the method, some methodological and practical considerations must be addressed. This paper examines, through a simulation study, the sensitivity of CRM both to the initial modelling of the dose-toxicity relationship and the prior. It appears that the performance of CRM can be improved by using vague priors and initial tuning of the model to allow flexibility.     

Incorporating model uncertainty in cost-effectiveness analysis: a Bayesian model averaging approach

Recently, several authors have proposed the use of linear regression models in cost-effectiveness analysis. In this paper, by modelling costs and outcomes using patient and Health Centre covariates, we seek to identify the part of the cost or outcome difference that is not attributable to the treatment itself, but to the patients' condition or to characteristics of the Centres. Selection of the covariates to be included as predictors of effectiveness and cost is usually assumed by the researcher. This behaviour ignores the uncertainty associated with model selection and leads to underestimation of the uncertainty about quantities of interest. We propose the use of Bayesian model averaging as a mechanism to account for such uncertainty about the model. Data from a clinical trial are used to analyze the effect of incorporating model uncertainty, by comparing two highly active antiretroviral treatments applied to asymptomatic HIV patients. The joint posterior density of incremental effectiveness and cost and cost-effectiveness acceptability curves are proposed as decision-making measures.     

Using the time‐to‐event continual reassessment method in the presence of partial orders

The time‐to‐event continual reassessment method (TITE‐CRM) was proposed to handle the problem of long trial duration in Phase 1 trials as a result of late‐onset toxicities. Here, we implement the TITE‐CRM in dose‐finding trials of combinations of agents. When studying multiple agents, monotonicity of the dose–toxicity curve is not clearly defined. Therefore, the toxicity probabilities follow a partial order, meaning that there are pairs of treatments for which the ordering of the toxicity probabilities is not known at the start of the trial. A CRM design for partially ordered trials (PO‐CRM) was recently proposed. Simulation studies show that extending the TITE‐CRM to the partial order setting produces results similar to those of the PO‐CRM in terms of maximum tolerated dose recommendation yet reduces the duration of the trial. Copyright © 2012 John Wiley & Sons, Ltd.     

Practical modifications to the time-to-event continual reassessment method for phase I cancer trials with fast patient accrual and late-onset toxicities

The goal of phase I cancer trials is to determine the highest dose of a treatment regimen with an acceptable toxicity rate. Traditional designs for phase I trials, such as the Continual Reassessment Method (CRM) and the 3 + 3 design, require each patient or a cohort of patients to be fully evaluated for the dose-limiting toxicity (DLT) before new patients can be enrolled. As such, the trial duration may be prohibitively long. The Time-to-Event Continual Reassessment Method (TITE-CRM, Cheung and Chappell, 2000) circumvents this limitation by allowing staggered patient accrual without the need for complete DLT follow-up of previously treated patients. However, in the setting of fast patient accrual and late-onset toxicities, the TITE-CRM results in overly aggressive dose escalation and exposes a considerable number of patients to toxic doses. We examine a modification to the TITE-CRM proposed by the original TITE-CRM creator and propose an alternative approach useful in this setting by incorporating an accrual suspension rule. A simulation study designed based on a neuro-oncology trial indicates that the modified methods provide a much improved degree of safety than the TITE-CRM while maintaining desirable design accuracy. The practical aspects of the proposed designs are discussed. The modifications presented are useful when planning phase I trials involving chemoradiation therapy.