Point estimation in adaptive enrichment designs

Adaptive enrichment designs are an attractive option for clinical trials that aim at demonstrating efficacy of therapies, which may show different benefit for the full patient population and a prespecified subgroup. In these designs, based on interim data, either the subgroup or the full population is selected for further exploration. When selection is based on efficacy data, this introduces bias to the commonly used maximum likelihood estimator. For the situation of two‐stage designs with a single prespecified subgroup, we present six alternative estimators and investigate their performance in a simulation study. The most consistent reduction of bias over the range of scenarios considered was achieved by a method combining the uniformly minimum variance conditionally unbiased estimator with a conditional moment estimator. Application of the methods is illustrated by a clinical trial example.     

Estimation after subpopulation selection in adaptive seamless trials

During the development of new therapies, it is not uncommon to test whether a new treatment works better than the existing treatment for all patients who suffer from a condition (full population) or for a subset of the full population (subpopulation). One approach that may be used for this objective is to have two separate trials, where in the first trial, data are collected to determine if the new treatment benefits the full population or the subpopulation. The second trial is a confirmatory trial to test the new treatment in the population selected in the first trial. In this paper, we consider the more efficient two‐stage adaptive seamless designs (ASDs), where in stage 1, data are collected to select the population to test in stage 2. In stage 2, additional data are collected to perform confirmatory analysis for the selected population. Unlike the approach that uses two separate trials, for ASDs, stage 1 data are also used in the confirmatory analysis. Although ASDs are efficient, using stage 1 data both for selection and confirmatory analysis introduces selection bias and consequently statistical challenges in making inference. We will focus on point estimation for such trials. In this paper, we describe the extent of bias for estimators that ignore multiple hypotheses and selecting the population that is most likely to give positive trial results based on observed stage 1 data. We then derive conditionally unbiased estimators and examine their mean squared errors for different scenarios.©2015 The Authors. Statistics in Medicine Published by JohnWiley & Sons Ltd.     

Identifying combined design and analysis procedures in two‐stage trials with a binary end point

Two‐stage trial designs provide the flexibility to stop early for efficacy or futility and are popular because they have a smaller sample size on average than a traditional trial has with the same type I and II error rates. This makes them financially attractive but also has the ethical benefit of reducing, in the long run, the number of patients who are given ineffective treatments. Designs that minimise the expected sample size are often referred to as ‘optimal’. However, two‐stage designs can impart a substantial bias into the parameter estimate at the end of the trial. In this paper, we argue that the expected performance of one's chosen estimation method should also be considered when deciding on a two‐stage trial design. We review the properties of standard and bias‐adjusted maximum likelihood estimators as well as mean and median unbiased estimators. We then identify optimal two‐stage design and analysis procedures that balance projected sample size considerations with those of estimator performance. We make available software to implement this new methodology. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparison of conditional bias‐adjusted estimators for interim analysis in clinical trials with survival data

Group sequential designs are widely used in clinical trials to determine whether a trial should be terminated early. In such trials, maximum likelihood estimates are often used to describe the difference in efficacy between the experimental and reference treatments; however, these are well known for displaying conditional and unconditional biases. Established bias‐adjusted estimators include the conditional mean‐adjusted estimator (CMAE), conditional median unbiased estimator, conditional uniformly minimum variance unbiased estimator (CUMVUE), and weighted estimator. However, their performances have been inadequately investigated. In this study, we review the characteristics of these bias‐adjusted estimators and compare their conditional bias, overall bias, and conditional mean‐squared errors in clinical trials with survival endpoints through simulation studies. The coverage probabilities of the confidence intervals for the four estimators are also evaluated. We find that the CMAE reduced conditional bias and showed relatively small conditional mean‐squared errors when the trials terminated at the interim analysis. The conditional coverage probability of the conditional median unbiased estimator was well below the nominal value. In trials that did not terminate early, the CUMVUE performed with less bias and an acceptable conditional coverage probability than was observed for the other estimators. In conclusion, when planning an interim analysis, we recommend using the CUMVUE for trials that do not terminate early and the CMAE for those that terminate early. Copyright © 2017 John Wiley & Sons, Ltd.     

Adjusting for treatment selection in phase II/III clinical trials with time to event data

Phase II/III clinical trials are efficient two-stage designs that test multiple experimental treatments. In stage 1, patients are allocated to the control and all experimental treatments, with the data collected from them used to select experimental treatments to continue to stage 2. Patients recruited in stage 2 are allocated to the selected treatments and the control. Combined data of stage 1 and stage 2 are used for a confirmatory phase III analysis. Appropriate analysis needs to adjust for selection bias of the stage 1 data. Point estimators exist for normally distributed outcome data. Extending these estimators to time to event data is not straightforward because treatment selection is based on correlated treatment effects and stage 1 patients who do not get events in stage 1 are followed-up in stage 2. We have derived an approximately uniformly minimum variance conditional unbiased estimator (UMVCUE) and compared its biases and mean squared errors to existing bias adjusted estimators. In simulations, one existing bias adjusted estimator has similar properties as the practically unbiased UMVCUE while the others can have noticeable biases but they are less variable than the UMVCUE. For confirmatory phase II/III clinical trials where unbiased estimators are desired, we recommend the UMVCUE or the existing estimator with which it has similar properties.     

Combining biomarkers for classification with covariate adjustment

Combining multiple markers can improve classification accuracy compared with using a single marker. In practice, covariates associated with markers or disease outcome can affect the performance of a biomarker or biomarker combination in the population. The covariate‐adjusted receiver operating characteristic (ROC) curve has been proposed as a tool to tease out the covariate effect in the evaluation of a single marker; this curve characterizes the classification accuracy solely because of the marker of interest. However, research on the effect of covariates on the performance of marker combinations and on how to adjust for the covariate effect when combining markers is still lacking. In this article, we examine the effect of covariates on classification performance of linear marker combinations and propose to adjust for covariates in combining markers by maximizing the nonparametric estimate of the area under the covariate‐adjusted ROC curve. The proposed method provides a way to estimate the best linear biomarker combination that is robust to risk model assumptions underlying alternative regression‐model‐based methods. The proposed estimator is shown to be consistent and asymptotically normally distributed. We conduct simulations to evaluate the performance of our estimator in cohort and case/control designs and compare several different weighting strategies during estimation with respect to efficiency. Our estimator is also compared with alternative regression‐model‐based estimators or estimators that maximize the empirical area under the ROC curve, with respect to bias and efficiency. We apply the proposed method to a biomarker study from an human immunodeficiency virus vaccine trial. Copyright © 2017 John Wiley & Sons, Ltd.     

Supplementary analysis of probabilities at the termination of a group sequential phase II trial

We consider estimation of various probabilities after termination of a group sequential phase II trial. A motivating example is that the stopping rule of a phase II oncologic trial is determined solely based on response to a drug treatment, and at the end of the trial estimating the rate of toxicity and response is desirable. The conventional maximum likelihood estimator (sample proportion) of a probability is shown to be biased, and two alternative estimators are proposed to correct for bias, a bias-reduced estimator obtained by using Whitehead's bias-adjusted approach, and an unbiased estimator from the Rao-Blackwell method of conditioning. All three estimation procedures are shown to have certain invariance property in bias. Moreover, estimators of a probability and their bias and precision can be evaluated through the observed response rate and the stage at which the trial stops, thus avoiding extensive computation.     

Estimation and sample design in prevalence surveys of dementia.

Population prevalence rates of dementia using stratified sampling have previously been estimated using two methods: standard weighted estimates and a logistic model-based approach. An earlier study described this application of the model-based approach and reported a small computer simulation comparing the performance of this estimator to the standard weighted estimator. In this article we use large-scale computer simulations based on data from the recently completed Kame survey of prevalent dementia in the Japanese-American residents of King County, Washington, to describe the performance of these estimators. We found that the standard weighted estimator was unbiased. This estimator performed well for a sample design with proportional allocation, but performed poorly for a sample design that included large strata that were lightly sampled. The logistic model-based estimator performed consistently well for all sample designs considered in terms of the extent of variability in estimation, although some modest bias was observed.     

Point estimation for adaptive trial designs I: A methodological review

Recent FDA guidance on adaptive clinical trial designs defines bias as “a systematic tendency for the estimate of treatment effect to deviate from its true value,” and states that it is desirable to obtain and report estimates of treatment effects that reduce or remove this bias. The conventional end-of-trial point estimates of the treatment effects are prone to bias in many adaptive designs, because they do not take into account the potential and realized trial adaptations. While much of the methodological developments on adaptive designs have tended to focus on control of type I error rates and power considerations, in contrast the question of biased estimation has received relatively less attention. This article is the first in a two-part series that studies the issue of potential bias in point estimation for adaptive trials. Part I provides a comprehensive review of the methods to remove or reduce the potential bias in point estimation of treatment effects for adaptive designs, while part II illustrates how to implement these in practice and proposes a set of guidelines for trial statisticians. The methods reviewed in this article can be broadly classified into unbiased and bias-reduced estimation, and we also provide a classification of estimators by the type of adaptive design. We compare the proposed methods, highlight available software and code, and discuss potential methodological gaps in the literature.