Application of a sequential multiple assignment randomized trial (SMART) design in older patients with chronic lymphocytic leukemia.

BACKGROUND: Ibrutinib therapy is safe and effective in patients with chronic lymphocytic leukemia (CLL). Currently, ibrutinib is administered continuously until disease progression. Combination regimens with ibrutinib are being developed to deepen response which could allow for ibrutinib maintenance (IM) discontinuation. Among untreated older patients with CLL, clinical investigators had the following questions: (i) does ibrutinib + venetoclax + obinutuzumab (IVO) with IM have superior progression-free survival (PFS) compared with ibrutinib + obinutuzumab (IO) with IM, and (ii) does the treatment strategy of IVO + IM for patients without minimal residual disease complete response (MRD- CR) or IVO + IM discontinuation for patients with MRD- CR have superior PFS compared with IO + IM. DESIGN: Conventional designs randomize patients to IO with IM or IVO with IM to address the first objective, or randomize patients to each treatment strategy to address the second objective. A sequential multiple assignment randomized trial (SMART) design and analysis is proposed to address both objectives. RESULTS: A SMART design strategy is appropriate when comparing adaptive interventions, which are defined by an individual's sequence of treatment decisions and guided by intermediate outcomes, such as response to therapy. A review of common applications of SMART design strategies is provided. Specific to the SMART design previously considered for Alliance study A041702, the general structure of the SMART is presented, an approach to sample size and power calculations when comparing adaptive interventions embedded in the SMART with a time-to-event end point is fully described, and analyses plans are outlined. CONCLUSION: SMART design strategies can be used in cancer clinical trials with adaptive interventions to identify optimal treatment strategies. Further, standard software exists to provide sample size, power calculations, and data analysis for a SMART design.

Testing equality of survival functions of quality-adjusted lifetime.

We present a method for comparing the survival functions of quality-adjusted lifetime from two treatments. This test statistic becomes the ordinary log-rank test when quality-adjusted lifetime is the same as the survival time. Simulation experiments are conducted to examine the behavior of our proposed test statistic under both null and alternative hypotheses. In addition, we apply our method to a breast cancer trial for comparing the distribution of quality-adjusted lifetime between two treatment regimes.

Using auxiliary time-dependent covariates to recover information in nonparametric testing with censored data.

Murray and Tsiatis (1996) described a weighted survival estimate that incorporates prognostic time-dependent covariate information to increase the efficiency of estimation. We propose a test statistic based on the statistic of Pepe and Fleming (1989, 1991) that incorporates these weighted survival estimates. As in Pepe and Fleming, the test is an integrated weighted difference of two estimated survival curves. This test has been shown to be effective at detecting survival differences in crossing hazards settings where the logrank test performs poorly. This method uses stratified longitudinal covariate information to get more precise estimates of the underlying survival curves when there is censored information and this leads to more powerful tests. Another important feature of the test is that it remains valid when informative censoring is captured by the incorporated covariate. In this case, the Pepe-Fleming statistic is known to be biased and should not be used. These methods could be useful in clinical trials with heavy censoring that include collection over time of covariates, such as laboratory measurements, that are prognostic of subsequent survival or capture information related to censoring.

Causal inference on the difference of the restricted mean lifetime between two groups.

When comparing survival times between two treatment groups, it may be more appropriate to compare the restricted mean lifetime, i.e., the expectation of lifetime restricted to a time L, rather than mean lifetime in order to accommodate censoring. When the treatments are not assigned to patients randomly, as in observational studies, we also need to account for treatment imbalances in confounding factors. In this article, we propose estimators for the difference of the restricted mean lifetime between two groups that account for treatment imbalances in prognostic factors assuming a proportional hazards relationship. Large-sample properties of our estimators based on martingale theory for counting processes are also derived. Simulation studies were conducted to compare these estimators and to assess the adequacy of the large-sample approximations. Our methods are also applied to an observational database of acute coronary syndrome patients from Duke University Medical Center to estimate the treatment effect on the restricted mean lifetime over 5 years.

Multiple imputation methods for estimating regression coefficients in the competing risks model with missing cause of failure.

We propose a method to estimate the regression coefficients in a competing risks model where the cause-specific hazard for the cause of interest is related to covariates through a proportional hazards relationship and when cause of failure is missing for some individuals. We use multiple imputation procedures to impute missing cause of failure, where the probability that a missing cause is the cause of interest may depend on auxiliary covariates, and combine the maximum partial likelihood estimators computed from several imputed data sets into an estimator that is consistent and asymptotically normal. A consistent estimator for the asymptotic variance is also derived. Simulation results suggest the relevance of the theory in finite samples. Results are also illustrated with data from a breast cancer study.

Utilizing propensity scores to estimate causal treatment effects with censored time-lagged data.

Observational studies frequently are conducted to compare long-term effects of treatments. Without randomization, patients receiving one treatment are not guaranteed to be prognostically comparable to those receiving another treatment. Furthermore, the response of interest may be right-censored because of incomplete follow-up. Statistical methods that do not account for censoring and confounding may lead to biased estimates. This article presents a method for estimating treatment effects in nonrandomized studies with right-censored responses. We review the assumptions required to estimate average causal effects and derive an estimator for comparing two treatments by applying inverse weights to the complete cases. The weights are determined according to the estimated probability of receiving treatment conditional on covariates and the estimated treatment-specific censoring distribution. By utilizing martingale representations, the estimator is shown to be asymptotically normal and an estimator for the asymptotic variance is derived. Simulation results are presented to evaluate the properties of the estimator. These methods are applied to an observational data set of acute coronary syndrome patients from Duke University Medical Center to estimate the effect of a treatment strategy on the mean 5-year medical cost.

Using conditional logistic regression to fit proportional odds models to interval censored data.

An easily implemented approach to fitting the proportional odds regression model to interval-censored data is presented. The approach is based on using conditional logistic regression routines in standard statistical packages. Using conditional logistic regression allows the practitioner to sidestep complications that attend estimation of the baseline odds ratio function. The approach is applicable both for interval-censored data in settings in which examinations continue regardless of whether the event of interest has occurred and for current status data. The methodology is illustrated through an application to data from an AIDS study of the effect of treatment with ZDV+ddC versus ZDV alone on 50% drop in CD4 cell count from baseline level. Simulations are presented to assess the accuracy of the procedure.

Testing for differences in survival with delayed ascertainment.

During the interim stages of most large-scale clinical trials, knowledge that a patient is alive or dead is usually not up-to-date. This is due to the pattern of patient visits to hospitals as well as the administrative set-up used by the study to obtain information on vital status. On a two-armed study, if the process of ascertaining vital status is not the same in both treatment groups, then the standard method of testing based on the logrank statistic may not be applicable. Instead, an ad hoc modification to the logrank test, which artificially truncates follow-up prior to the time of analysis, is often used. These approaches have not been formally addressed in the literature. In the early stages of a clinical trial, severe bias or loss of power may result. For this situation, we propose a class of test statistics that extends the usual class of U statistics. Asymptotic normality is derived by reformulating the statistics in terms of counting processes and employing the theory of U statistics along with martingale techniques. For early interim analyses, a numerical study indicates that the new tests can be more powerful than the current practice when differential ascertainment is present. To illustrate the potential loss of information when lagging follow-up to control for ascertainment delays, we reanalyze an AIDS clinical trial with the truncated logrank and the new statistics.

Cost-effectiveness of platelet glycoprotein IIb/IIIa inhibition with eptifibatide in patients with non-ST-elevation acute coronary syndromes.

BACKGROUND: In the PURSUIT trial, eptifibatide significantly reduced the 30-day incidence of death and myocardial infarction relative to placebo in 9461 patients with an acute coronary syndrome (unstable angina or non-Q-wave myocardial infarction). METHODS AND RESULTS: We conducted a 2-part prospective economic substudy of the 3522 US patients enrolled in PURSUIT: (1) an empirical intention-to-treat comparison of medical costs (hospital plus physician) up to 6 months after hospitalization and (2) a lifetime cost-effectiveness analysis. The base-case cost-effectiveness ratio was expressed as the 1996 US dollars required to add 1 life-year with eptifibatide therapy. The 2 treatment arms had equivalent resource consumption and medical costs (exclusive of the cost of the eptifibatide regimen) during the index (enrollment) hospitalization (P=0.78) and up to 6 months afterward (P=0.60). The average wholesale price of the eptifibatide regimen was $1217, but a typical hospital discounted price was $1014. The estimated life expectancy from randomization in the US patients was 15.96 years for eptifibatide and 15.85 years for placebo, an incremental difference of 0.111. The incremental cost-effectiveness ratio for eptifibatide therapy in US PURSUIT patients was $16 491 per year of life saved. This result was robust through a wide range of sensitivity analyses. The cost-utility ratio for eptifibatide (using time trade-off defined utilities) was $19 693 per added quality-adjusted life-year. CONCLUSIONS: Based on the results observed in the US PURSUIT patients, the routine addition of eptifibatide to standard care for non-ST-elevation acute coronary syndrome patients is economically attractive by conventional standards.

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.

Efficient estimation of the distribution of quality-adjusted survival time.

Quality of life is an important aspect in evaluation of clinical trials of chronic diseases, such as cancer and AIDS. Quality-adjusted survival analysis is a method that combines both the quantity and quality of a patient's life into one single measure. In this paper, we discuss the efficiency of weighted estimators for the distribution of quality-adjusted survival time. Using the general representation theorem for missing data processes, we are able to derive an estimator that is more efficient than the one proposed in Zhao and Tsiatis (1997, Biometrika 84, 339-348). Simulation experiments are conducted to assess the small sample properties of this estimator and to compare it with the semiparametric efficiency bound. The value of this estimator is demonstrated from an application of the method to a data set obtained from a breast cancer clinical trial.

The use of simulation and bootstrap in information-based group sequential studies.

In this paper, we present an information-based design and monitoring procedure which applies to any type of model for any type of group sequential study provided there is a unique parameter of interest one can estimate efficiently. Simulation techniques are described to handle the design phase of this procedure. Since designs depend on potentially unreliable guesses of nuisance parameters, we propose a bootstrap method that uses the information available at the interim analysis times to generate projections and prediction intervals for the time at which the study will be fully powered. A monitoring board can use this information to decide whether a redesign of the trial is warranted. We also show how to use simulation to redesign studies in progress. We illustrate all of these techniques with data from AIDS Clinical Trial Group Protocol 021.

Semiparametric efficient estimation in the generalized odds-rate class of regression models for right-censored time-to-event data.

The generalized odds-rate class of regression models for time to event data is indexed by a non-negative constant rho and assumes that [formula: see text] where g: rho(s) = log(rho-1(s-rho - 1)) for rho > 0, g0(s) = log(-logs), S(t[symbol: see text]Z) is the survival function of the time to event for an individual with q x 1 covariate vector Z, beta is a q x 1 vector of unknown regression parameters, and alpha(t) is some arbitrary increasing function of t. When rho = 0, this model is equivalent to the proportional hazards model and when rho = 1, this model reduces to the proportional odds model. In the presence of right censoring, we construct estimators for beta and exp(alpha(t)) and show that they are consistent and asymptotically normal. In addition, we show that the estimator for beta is semiparametric efficient in the sense that it attains the semiparametric variance bound.

Estimating the parameters in the Cox model when covariate variables are measured with error.

The Cox proportional hazards model is commonly used to model survival data as a function of covariates. Because of the measuring mechanism or the nature of the environment, covariates are often measured with error and are not directly observable. A naive approach is to use the observed values of the covariates in the Cox model, which usually produces biased estimates of the true association of interest. An alternative strategy is to take into account the error in measurement, which may be carried out for the Cox model in a number of ways. We examine several such approaches and compare and contrast them through several simulation studies. We introduce a likelihood-based approach, which we refer to as the semiparametric method, and show that this method is an appealing alternative. The methods are applied to analyze the relationship between survival and CD4 count in patients with AIDS.

Evaluating surrogate markers of clinical outcome when measured with error.

In most clinical trials, markers are measured periodically with error. In the presence of measurement error, the naive method of using the observed marker values in the Cox model to evaluate the relationship between the marker and clinical outcome can produce biased estimates and lead to incorrect conclusions when evaluating a potential surrogate. We propose a two-stage approach to account for the measurement error and reduce the bias of the estimate. In the first stage, an empirical Bayes estimate of the time-dependent covariate is computed at each event time. In the second stage, these estimates are imputed in the Cox proportional hazards model to estimate the regression parameter of interest. We demonstrate through extensive simulations that this methodology reduces the bias of the regression estimate and correctly identifies good surrogate markers more often than the naive approach. An application evaluating CD4 count as a surrogate of disease progression in an AIDS clinical trial is presented.

A joint model for survival and longitudinal data measured with error.

The relationship between a longitudinal covariate and a failure time process can be assessed using the Cox proportional hazards regression model. We consider the problem of estimating the parameters in the Cox model when the longitudinal covariate is measured infrequently and with measurement error. We assume a repeated measures random effects model for the covariate process. Estimates of the parameters are obtained by maximizing the joint likelihood for the covariate process and the failure time process. This approach uses the available information optimally because we use both the covariate and survival data simultaneously. Parameters are estimated using the expectation-maximization algorithm. We argue that such a method is superior to naive methods where one maximizes the partial likelihood of the Cox model using the observed covariate values. It also improves on two-stage methods where, in the first stage, empirical Bayes estimates of the covariate process are computed and then used as time-dependent covariates in a second stage to find the parameters in the Cox model that maximize the partial likelihood.

Nonparametric survival estimation using prognostic longitudinal covariates.

One of the primary problems facing statisticians who work with survival data is the loss of information that occurs with right-censored data. This research considers trying to recover some of this endpoint information through the use of a prognostic covariate which is measured on each individual. We begin by defining a survival estimate which uses time-dependent covariates to more precisely get at the underlying survival curves in the presence of censoring. This estimate has a smaller asymptotic variance than the usual Kaplan-Meier in the presence of censoring and reduces to the Kaplan-Meier (1958, Journal of the American Statistical Association 53, 457-481) in situations where the covariate is not prognostic or no censoring occurs. In addition, this estimate remains consistent when the incorporated covariate contains information about the censoring process as well as survival information. Because the Kaplan-Meier estimate is known to be biased in this situation due to informative censoring, we recommend use of our estimate.

Design and analysis of group sequential logrank tests in maximum duration versus information trials.

When monitoring a clinical trial with failure time data using the logrank test and the type I error spending function approach, the information time has to be estimated as a fraction of the maximum number of failures. In maximum duration trials, the denominator of this fraction is a random quantity and has to be estimated; besides, there are two candidates for the denominator, one under the null hypothesis of no treatment difference and the other under the specified alternative hypothesis. Either way, some adjustments are necessary in determining group sequential boundaries in order to maintain type I error at a desired significance level. As a consequence, the type I error spending function will be altered from the one chosen for the design, thus affecting the operating characteristics of the subsequent group sequential logrank tests. In maximum information trials, however, the maximum amount of information is fixed, and thus the estimate of the information time is always unbiased. The net effect is that computation of group sequential boundaries becomes straightforward, with a potential saving in study durations as compared to maximum duration trials. We will illustrate how adjustments are made in maximum duration trials to maintain type I error when the information times are estimated with the information horizons under the null and alternative hypotheses and present numerical explorations to compare robustness of two different estimates of the information times. We then propose a design procedure for maximum information trials and investigate the properties of maximum information trials for different group sequential boundaries. We also compare maximum information trials and maximum duration trials based on an example.

CD4 count and the risk for death in patients infected with HIV receiving antiretroviral therapy.

OBJECTIVE: To investigate the relation between CD4 count and the immediate hazard of dying in patients receiving zidovudine (azidothymidine [AZT])-based antiretroviral therapy. SETTING: A research hospital that recruits patients from the entire United States. DESIGN: Retrospective analysis of a cohort of patients with the acquired immunodeficiency syndrome (AIDS) or AIDS-related complex participating in long-term zidovudine-based antiretroviral protocols. PATIENTS: Fifty-five patients with human immunodeficiency virus (HIV) infection and either AIDS or severe AIDS-related complex who were followed for as many as 4 years while they received antiretroviral therapy. MEASUREMENTS: CD4 counts were measured. MAIN RESULTS: Ten patients are known to be alive and 1 was lost to follow-up. Of the 44 patients who are known to have died, the CD4 range was known within 6 months of death in 41. All but 1 of these 41 assessable deaths occurred in patients whose CD4 counts were known to have fallen below 50 CD4 cells/mm3 (P less than 10(-10)). The hazard of dying in the cohort ranged from 0 deaths/patient-month (95% CI, 0 to 0.008 deaths/patient-month) in patients with 200 or more CD4 cells/mm3 to 0.07 deaths/patient-month (CI, 0.050 to 0.094 deaths/patient-month) in patients with fewer than 50 CD4 cells/mm3. For the patients who died and whose cases were assessable, the mean of the last three CD4 counts obtained before death was 7.7 CD4 cells/mm3 (CI, 0.9 to 63.3 cells/mm3). The median survival of patients once their CD4 counts fell below 50 CD4 cells/mm3 was 12.1 months (CI, 7.2 to 19.4 months). CONCLUSIONS: In a carefully followed cohort treated with zidovudine-based antiretroviral therapy, nearly all deaths occurred in patients with fewer than 50 CD4 cells/mm3. These findings may have implications in the monitoring of patients with AIDS and in the use of CD4 count as a clinical trials end point for the antiretroviral therapy of HIV infection.