Implementation of group sequential logrank tests in a maximum duration trial

To control the Type I error probability in a group sequential procedure using the logrank test, it is important to know the information times (fractions) at the times of interim analyses conducted for purposes of data monitoring. For the logrank test, the information time at an interim analysis is the fraction of the total number of events to be accrued in the entire trial. In a maximum information trial design, the trial is concluded when a prespecified total number of events has been accrued. For such a design, therefore, the information time at each interim analysis is known. However, many trials are designed to accrue data over a fixed duration of follow-up on a specified number of patients. This is termed a maximum duration trial design. Under such a design, the total number of events to be accrued is unknown at the time of an interim analysis. For a maximum duration trial design, therefore, these information times need to be estimated. A common practice is to assume that a fixed fraction of information will be accrued between any two consecutive interim analyses, and then employ a Pocock or O'Brien-Fleming boundary. In this article, we describe an estimate of the information time based on the fraction of total patient exposure, which tends to be slightly negatively biased (i.e., conservative) if survival is exponentially distributed. We then present a numerical exploration of the robustness of this estimate when nonexponential survival applies. We also show that the Lan-DeMets (1983, Biometrika 70, 659-663) procedure for constructing group sequential boundaries with the desired level of Type I error control can be computed using the estimated information fraction, even though it may be biased. Finally, we discuss the implications of employing a biased estimate of study information for a group sequential procedure.     

Closed testing procedures for group sequential clinical trials with multiple endpoints

A simple approach is given for conducting closed testing in clinical trials with multiple endpoints in which group sequential monitoring is planned. The approach allows a flexible stopping time; the earliest and latest stopping times are described. The paradigm is applicable both to clinical trials with multiple endpoints and to the one-sided multiple comparison problem of several treatments versus a control. The approach leads to enhancements of previous methods and suggestions for new methods. An example of a respiratory disease trial with four endpoints is given.     

Bayesian approaches to joint cure-rate and longitudinal models with applications to cancer vaccine trials

Complex issues arise when investigating the association between longitudinal immunologic measures and time to an event, such as time to relapse, in cancer vaccine trials. Unlike many clinical trials, we may encounter patients who are cured and no longer susceptible to the time-to-event endpoint. If there are cured patients in the population, there is a plateau in the survival function, S(t), after sufficient follow-up. If we want to determine the association between the longitudinal measure and the time-to-event in the presence of cure, existing methods for jointly modeling longitudinal and survival data would be inappropriate, since they do not account for the plateau in the survival function. The nature of the longitudinal data in cancer vaccine trials is also unique, as many patients may not exhibit an immune response to vaccination at varying time points throughout the trial. We present a new joint model for longitudinal and survival data that accounts both for the possibility that a subject is cured and for the unique nature of the longitudinal data. An example is presented from a cancer vaccine clinical trial.     

One-sided sequential stopping boundaries for clinical trials: a decision-theoretic approach

We address one-sided stopping rules for clinical trials, or more generally, drug development programs, from a decision-theoretic point of view. If efficacy results are sufficiently negative then the trial will be stopped. But regardless of how positive the efficacy results are, the trial will continue in order to demonstrate safety. We show how sequential decisions should be made by a pharmaceutical company attempting to maximize its expected profits.     

Monitoring the rates of composite events with censored data in phase II clinical trials

In many phase II clinical trials, interim monitoring is based on the probability of a binary event, response, defined in terms of one or more time-to-event variables within a time period of fixed length. Such outcome-adaptive methods may require repeated interim suspension of accrual in order to follow each patient for the time period required to evaluate response. This may increase trial duration, and eligible patients arriving during such delays either must wait for accrual to reopen or be treated outside the trial. Alternatively, monitoring may be done continuously by ignoring censored data each time the stopping rule is applied, which wastes information. We propose an adaptive Bayesian method that eliminates these problems. At each patient's accrual time, an approximate posterior for the response probability based on all of the event-time data is used to compute an early stopping criterion. Application to a leukemia trial with a composite event shows that the method can reduce trial duration substantially while maintaining the reliability of interim decisions.     

Design and analysis of group sequential clinical trials with multiple primary endpoints

In many phase III clinical trials, it is desirable to separately assess the treatment effect on two or more primary endpoints. Consider the MERIT-HF study, where two endpoints of primary interest were time to death and the earliest of time to first hospitalization or death (The International Steering Committee on Behalf of the MERIT-HF Study Group, 1997, American Journal of Cardiology 80[9B], 54J-58J). It is possible that treatment has no effect on death but a beneficial effect on first hospitalization time, or it has a detrimental effect on death but no effect on hospitalization. A good clinical trial design should permit early stopping as soon as the treatment effect on both endpoints becomes clear. Previous work in this area has not resolved how to stop the study early when one or more endpoints have no treatment effect or how to assess and control the many possible error rates for concluding wrong hypotheses. In this article, we develop a general methodology for group sequential clinical trials with multiple primary endpoints. This method uses a global alpha-spending function to control the overall type I error and a multiple decision rule to control error rates for concluding wrong alternative hypotheses. The method is demonstrated with two simulated examples based on the MERIT-HF study.     

Data adaptive interim modification of sample sizes for candidate-gene association studies

OBJECTIVES: The use of conventional Transmission/Disequilibrium tests in the analysis of candidate-gene association studies requires the precise and complete pre-specification of the total number of trios to be sampled to obtain sufficient power at a certain significance level (type I error risk). In most of these studies, very little information about the genetic effect size will be available beforehand and thus it will be difficult to calculate a reasonable sample size. One would therefore wish to reassess the sample size during the course of a study. METHOD: We propose an adaptive group sequential procedure which allows for both early stopping of the study with rejection of the null hypothesis (H0) and for recalculation of the sample size based on interim effect size estimates when H0 cannot be rejected. The applicability of the method which was developed by Müller and Sch?fer [Biometrics 2001;57:886-891] in a clinical context is demonstrated by a numerical example. Monte Carlo simulations are performed comparing the adaptive procedure with a fixed sample and a conventional group sequential design. RESULTS: The main advantage of the adaptive procedure is its flexibility to allow for design changes in order to achieve a stabilized power characteristic while controlling the overall type I error and using the information already collected. CONCLUSIONS: Given these advantages, the procedure is a promising alternative to traditional designs.     

Sample Sizes in Sequential Survival Clinical Trials

In survival clinical trials the response, i.e. (time to) death is delayed. As a result there are at any time patients that have not responded, yet. Given a stopping rule based on the number of deaths, the distribution of the total number of patients that will enter the trial is examined. Besides, a simple approximation to the expected number of patients is presented.     

Nonparametric comparison of two survival-time distributions in the presence of dependent censoring

When testing the null hypothesis that treatment arm-specific survival-time distributions are equal, the log-rank test is asymptotically valid when the distribution of time to censoring is conditionally independent of randomized treatment group given survival time. We introduce a test of the null hypothesis for use when the distribution of time to censoring depends on treatment group and survival time. This test does not make any assumptions regarding independence of censoring time and survival time. Asymptotic validity of this test only requires a consistent estimate of the conditional probability that the survival event is observed given both treatment group and that the survival event occurred before the time of analysis. However, by not making unverifiable assumptions about the data-generating mechanism, there exists a set of possible values of corresponding sample-mean estimates of these probabilities that are consistent with the observed data. Over this subset of the unit square, the proposed test can be calculated and a rejection region identified. A decision on the null that considers uncertainty because of censoring that may depend on treatment group and survival time can then be directly made. We also present a generalized log-rank test that enables us to provide conditions under which the ordinary log-rank test is asymptotically valid. This generalized test can also be used for testing the null hypothesis when the distribution of censoring depends on treatment group and survival time. However, use of this test requires semiparametric modeling assumptions. A simulation study and an example using a recent AIDS clinical trial are provided.     

Stopping rules for clinical trials incorporating clinical opinion

A method is described of eliciting a 'range of equivalence', i.e. a range of differences between two treatments over which a group of clinical trial participants would have no clear preference for either treatment. This range of equivalence is then incorporated into a formal stopping rule for the trial using an extension of the group sequential design. Tables for the implementation of the design are presented. The method is discussed in the context of other sequential-trial designs.     

Estimation of a Secondary Parameter in a Group Sequential Clinical Trial

In this article, we investigate estimation of a secondary parameter in group sequential tests. We study the model in which the secondary parameter is the mean of the normal distribution in a subgroup of the subjects. The bias of the naive secondary parameter estimator is studied. It is shown that the sampling proportions of the subgroup have a crucial effect on the bias: As the sampling proportion of the subgroup at or just before the stopping time increases, the bias of the naive subgroup parameter estimator increases as well. An unbiased estimator for the subgroup parameter and an unbiased estimator for its variance are derived. Using simulations, we compare the mean squared error of the unbiased estimator to that of the naive estimator, and we show that the differences are negligible. As an example, the methods of estimation are applied to an actual group sequential clinical trial, The Beta-Blocker Heart Attack Trial.     

Sequential methods for comparing years of life saved in the two-sample censored data problem

This research develops nonparametric strategies for sequentially monitoring clinical trial data where detecting years of life saved is of interest. The recommended test statistic looks at integrated differences in survival estimates during the time frame of interest. In many practical situations, the test statistic presented has an independent increments covariance structure. Hence, with little additional work, we may apply these testing procedures using available methodology. In the case where an independent increments covariance structure is present, we suggest how clinical trial data might be monitored using these statistics in an information-based design. The resulting study design maintains the desired stochastic operating characteristics regardless of the shapes of the survival curves being compared. This offers an advantage over the popular log-rank-based design strategy since more restrictive assumptions relating to the behavior of the hazards are required to guarantee the planned power of the test. Recommendations for how to sequentially monitor clinical trial progress in the nonindependent increments case are also provided along with an example.     

Supplementary analysis at the conclusion of a sequential clinical trial

Suppose that a clinical trial has been carried out using a sequential design. Any analysis that neglects the monitoring of the trial is potentially biased, and this holds for analyses of responses not directly used to determine whether the study should be stopped. Although valid methods exist to deal with the responses actually used in the sequential design, little has been written about secondary analyses of further responses. In this paper, two approaches to secondary analyses are explored. One is a conditional approach that avoids the problems of bias, the other an unconditional approach that allows for the sequential nature of the trial. The two methods are illustrated by application to three examples, each concerning a different type of response variable.     

Improved Repeated Confidence Bounds in Trials with a Maximal Goal

Repeated confidence intervals can be computed at every interim analysis of a flexible group sequential design without the need to stop the trial with a pre‐planned stopping rule. Often, however, there is a maximal goal such that the trial is surely stopped if this goal is reached. This induces a maximal stopping rule, and repeated confidence intervals are strictly conservative, when adhering to it. A modification is proposed which uniformly improves the one sided repeated confidence interval in such a situation. It preserves the monitoring character, and leads to uniformly smaller intervals, when reaching the maximal goal at an interim analysis. The modification is worked out for two stage designs and is indicated for multi‐stage trials. The extent of the improvement is quantified for two simple scenarios.     

Incorporating data received after a sequential trial has stopped into the final analysis: implementation and comparison of methods

In a sequential clinical trial, accrual of data on patients often continues after the stopping criterion for the study has been met. This is termed "overrunning." Overrunning occurs mainly when the primary response from each patient is measured after some extended observation period. The objective of this article is to compare two methods of allowing for overrunning. In particular, simulation studies are reported that assess the two procedures in terms of how well they maintain the intended type I error rate. The effect on power resulting from the incorporation of "overrunning data" using the two procedures is evaluated.     

Flexible implementations of group sequential stopping rules using constrained boundaries

Group sequential stopping rules are often used during the conduct of clinical trials in order to attain more ethical treatment of patients and to better address efficiency concerns. Because the use of such stopping rules materially affects the frequentist operating characteristics of the hypothesis test, it is necessary to choose an appropriate stopping rule during the planning of the study. It is often the case, however, that the number and timing of interim analyses are not precisely known at the time of trial design, and thus the implementation of a particular stopping rule must allow for flexible determination of the schedule of interim analyses. In this article, we consider the use of constrained stopping boundaries in the implementation of stopping rules. We compare this approach when used on various scales for the test statistic. When implemented on the scale of boundary crossing probabilities, this approach is identical to the error spending function approach of Lan and DeMets (1983).     

Testing a Primary and a Secondary Endpoint in a Group Sequential Design

Summary We consider a clinical trial with a primary and a secondary endpoint where the secondary endpoint is tested only if the primary endpoint is significant. The trial uses a group sequential procedure with two stages. The familywise error rate (FWER) of falsely concluding significance on either endpoint is to be controlled at a nominal level α. The type I error rate for the primary endpoint is controlled by choosing any α‐level stopping boundary, e.g., the standard O'Brien–Fleming or the Pocock boundary. Given any particular α‐level boundary for the primary endpoint, we study the problem of determining the boundary for the secondary endpoint to control the FWER. We study this FWER analytically and numerically and find that it is maximized when the correlation coefficient ρ between the two endpoints equals 1. For the four combinations consisting of O'Brien–Fleming and Pocock boundaries for the primary and secondary endpoints, the critical constants required to control the FWER are computed for different values of ρ. An ad hoc boundary is proposed for the secondary endpoint to address a practical concern that may be at issue in some applications. Numerical studies indicate that the O'Brien–Fleming boundary for the primary endpoint and the Pocock boundary for the secondary endpoint generally gives the best primary as well as secondary power performance. The Pocock boundary may be replaced by the ad hoc boundary for the secondary endpoint with a very little loss of secondary power if the practical concern is at issue. A clinical trial example is given to illustrate the methods.     

Robustness of an odds-ratio test in a stratified group sequential trial with a binary outcome measure

Many clinical trials compare two or more treatment groups by using a binary outcome measure. For example, the goal could be to determine whether the frequency of pain episodes is significantly reduced in the treatment group (arm A) as compared to the control group (arm B). However, for ethical or regulatory reasons, group sequential designs are commonly employed. Then, based on a binomial distribution, the stopping boundaries for the interim analyses are constructed for assessing the difference in the response probabilities between the two groups. This is easily accomplished by using any of the standard procedures, e.g., those discussed by Jennison and Turnbull (2000), and using one of the most commonly used software packages, East (2000). Several factors are known to often affect the primary outcome of interest, but their true distributions are not known in advance. In addition, these factors may cause heterogeneous treatment responses among individuals in a group, and their exact effect size may be unknown. To limit the effect of such factors on the comparison of the two arms, stratified randomization is used in the actual conduct of the trial. Then, a stratified analysis based on the odds ratio proposed in Jennison and Turnbull (2000, pages 251-252) and consistent with the stratified design is undertaken. However, the stopping rules used for the interim analyses are those obtained for determining the differences in response rates in a design that was not stratified. The purpose of this paper is to assess the robustness of such an approach on the performance of the odds ratio test when the underlying distribution and effect size of the factors that influence the outcome may vary. The simulation studies indicate that, in general, the stratified approach offers consistently better results than does the unstratified approach, as long as the difference in the weighted average of the response probabilities across strata between the two groups remains closer to the hypothesized values, irrespective of the differences in the (allocation) distributions and heterogeneous response rate. However, if the response probabilities deviate significantly from the hypothesized values so that the difference in the weighted average is less than the hypothesized value, then the proposed study could be significantly underpowered.     

Analyzing time-to-event data in a clinical trial when an unknown proportion of subjects has experienced the event at entry

In some clinical trials, where the outcome is the time until development of a silent event, an unknown proportion of subjects who have already experienced the event will be unknowingly enrolled due to the imperfect nature of the diagnostic tests used to screen potential subjects. For example, commonly used diagnostic tests for evaluating HIV infection status in infants, such as DNA PCR and HIV Culture, have low sensitivity when given soon after infection. This can lead to the inclusion of an unknown proportion of HIV-infected infants into clinical trials aimed at the prevention of transmission from HIV-positive mothers to their infants through breastfeeding. The infection status of infants at the end of the trial, when they are more than a year of age, can be determined with certainty. For those infants found to be infected with HIV at the end of the trial, it cannot be determined whether this occurred during the study or whether they were already infected when they were enrolled. In these settings, estimates of the cumulative risk of the event by the end of the study will overestimate the true probability of event during the study period and hypothesis tests comparing two or more intervention strategies can also be biased. We present inference methods for the distribution of time until the event of interest in these settings, and investigate issues in the design of such trials when there is a choice of using both imperfect and perfect diagnostic tests.     

Sequential Analysis of Survival Times in Clinical Trials

This paper provides a synopsis of statistical methods which can be used for the sequential analysis of possibly censored survival times in clinical trials. Especially, results on the asymptotic behaviour of the Breslow-Haug statistic and on the sequential version of the logrank statistic are presented in a standardized terminology. In addition, formulae for the explicit calculation of linear and square-root boundaries for sequential plans are given and illustrated by an example. Practical problems of applying these methods when monitoring a fixed-sample clinical trial as well as group sequential methods and calculation of P-values are also discussed.