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A two stage adaptive allocation design for survival outcome with informative censoring
Abstract
An adaptive two stage allocation design is developed for survival responses subject to independent informative censoring. Asymptotic p value of a score test related to a hypothesis of treatment effectiveness is used to set the assignment probability of the second stage. Several characteristics of the design and the follow-up inference are studied, both numerically and theoretically, and are compared with those of an existing competitor. Applicability of the proposed design is also illustrated through a real data.
In the context of clinical trials, a multi-treatment two-stage adaptive randomization procedure is developed for survival responses with copula-based nonrandom censoring. Because no such allocation design is available for survival responses with nonrandom censoring, this is probably the first work in this direction. After allocating the first-stage patients by complete randomization (CR), asymptotic p- values of relevant statistical tests are used to derive the allocation probabilities for the second-stage incoming patients. Several design and precision-based operating characteristics of the proposed design are studied both theoretically and empirically and compared with those of the CR procedure and the two-stage design of Bhattacharya and Shome (2019). Applicability of the proposed design is further envisaged through redesigning a real clinical trial with cancer patients.
Compromising ethics and precision in the context of a multiarmed clinical trial, an optimal order adjusted response adaptive design is proposed for survival outcomes subject to independent random censoring. The operating characteristics of the proposed design and the follow-up inference are studied both theoretically as well as empirically and are compared with those of the competitors. Applicability of the developed design is further illustrated through redesigning a real clinical trial with survival responses.
Randomized trials with dropouts or censored data and discrete time-to-event type outcomes are frequently analyzed using the
Kaplan–Meier or product limit (PL) estimation method. However, the PL method assumes that the censoring mechanism is noninformative
and when this assumption is violated, the inferences may not be valid. We propose an expanded PL method using a Bayesian framework
to incorporate informative censoring mechanism and perform sensitivity analysis on estimates of the cumulative incidence curves.
The expanded method uses a model, which can be viewed as a pattern mixture model, where odds for having an event during the
follow-up interval ( t k − 1 , t k ] (tk−1,tk], conditional on being at risk at t k − 1 tk−1, differ across the patterns of missing data. The sensitivity parameters relate the odds of an event, between subjects from
a missing-data pattern with the observed subjects for each interval. The large number of the sensitivity parameters is reduced
by considering them as random and assumed to follow a log-normal distribution with prespecified mean and variance. Then we
vary the mean and variance to explore sensitivity of inferences. The missing at random (MAR) mechanism is a special case of
the expanded model, thus allowing exploration of the sensitivity to inferences as departures from the inferences under the
MAR assumption. The proposed approach is applied to data from the TRial Of Preventing HYpertension.
Prehypertension is considered a precursor of stage 1 hypertension and a predictor of excessive cardiovascular risk. We investigated whether pharmacologic treatment of prehypertension prevents or postpones stage 1 hypertension.
Participants with repeated measurements of systolic pressure of 130 to 139 mm Hg and diastolic pressure of 89 mm Hg or lower, or systolic pressure of 139 mm Hg or lower and diastolic pressure of 85 to 89 mm Hg, were randomly assigned to receive two years of candesartan (Atacand, AstraZeneca) or placebo, followed by two years of placebo for all. When a participant reached the study end point of stage 1 hypertension, treatment with antihypertensive agents was initiated. Both the candesartan group and the placebo group were instructed to make changes in lifestyle to reduce blood pressure throughout the trial.
A total of 409 participants were randomly assigned to candesartan, and 400 to placebo. Data on 772 participants (391 in the candesartan group and 381 in the placebo group; mean age, 48.5 years; 59.6 percent men) were available for analysis. During the first two years, hypertension developed in 154 participants in the placebo group and 53 of those in the candesartan group (relative risk reduction, 66.3 percent; P<0.001). After four years, hypertension had developed in 240 participants in the placebo group and 208 of those in the candesartan group (relative risk reduction, 15.6 percent; P<0.007). Serious adverse events occurred in 3.5 percent of the participants assigned to candesartan and 5.9 percent of those receiving placebo.
Over a period of four years, stage 1 hypertension developed in nearly two thirds of patients with untreated prehypertension (the placebo group). Treatment of prehypertension with candesartan appeared to be well tolerated and reduced the risk of incident hypertension during the study period. Thus, treatment of prehypertension appears to be feasible. (ClinicalTrials.gov number, NCT00227318.).
Randomised Response-Adaptive Designs in Clinical Trials presents methods for the randomised allocation of treatments to patients in sequential clinical trials. Emphasizing the practical application of clinical trial designs, the book is designed for medical and applied statisticians, clinicians, and statisticians in training. After introducing clinical trials in drug development, the authors assess a simple adaptive design for binary responses without covariates. They discuss randomisation and covariate balance in normally distributed responses and cover many important response-adaptive designs for binary responses. The book then develops response-adaptive designs for continuous and longitudinal responses, optimum designs with covariates, and response-adaptive designs with covariates. It also covers response-adaptive designs that are derived by optimising an objective function subject to constraints on the variance of estimated parametric functions. The concluding chapter explores future directions in the development of adaptive designs.
This greatly expanded second edition of Survival Analysis- A Self-learning Text provides a highly readable description of state-of-the-art methods of analysis of survival/event-history data. This text is suitable for researchers and statisticians working in the medical and other life sciences as well as statisticians in academia who teach introductory and second-level courses on survival analysis. The second edition continues to use the unique "lecture-book" format of the first (1996) edition with the addition of three new chapters on advanced topics:
Chapter 7: Parametric Models
Chapter 8: Recurrent events
Chapter 9: Competing Risks.
Also, the Computer Appendix has been revised to provide step-by-step instructions for using the computer packages STATA (Version 7.0), SAS (Version 8.2), and SPSS (version 11.5) to carry out the procedures presented in the main text.
The original six chapters have been modified slightly
to expand and clarify aspects of survival analysis in response to suggestions by students, colleagues and reviewers, and
to add theoretical background, particularly regarding the formulation of the (partial) likelihood functions for proportional hazards, stratified, and extended Cox regression models
David Kleinbaum is Professor of Epidemiology at the Rollins School of Public Health at Emory University, Atlanta, Georgia. Dr. Kleinbaum is internationally known for innovative textbooks and teaching on epidemiological methods, multiple linear regression, logistic regression, and survival analysis. He has provided extensive worldwide short-course training in over 150 short courses on statistical and epidemiological methods. He is also the author of ActivEpi (2002), an interactive computer-based instructional text on fundamentals of epidemiology, which has been used in a variety of educational environments including distance learning.
Mitchel Klein is Research Assistant Professor with a joint appointment in the Department of Environmental and Occupational Health (EOH) and the Department of Epidemiology, also at the Rollins School of Public Health at Emory University. Dr. Klein is also co-author with Dr. Kleinbaum of the second edition of Logistic Regression- A Self-Learning Text (2002). He has regularly taught epidemiologic methods courses at Emory to graduate students in public health and in clinical medicine. He is responsible for the epidemiologic methods training of physicians enrolled in Emory’s Master of Science in Clinical Research Program, and has collaborated with Dr. Kleinbaum both nationally and internationally in teaching several short courses on various topics in epidemiologic methods.
Convex HullsClosure and Interior of a SetWeierstrass's TheoremSeparation and Support of SetsConvex Cones and PolarityPolyhedral Sets, Extreme Points, and Extreme DirectionsLinear Programming and the Simplex Method
ExercisesNotes and References
A randomized two treatment allocation design, conducted in two stages, is proposed for a class of continuous response trials. Patients are assigned to each treatment in equal numbers in the first stage and p value of a test of equality of treatment effects based on these data is used to determine the assignment probability of second stage patients. Relevant properties of the proposed allocation design are investigated and compared with suitable competitors.
Probability. Measure. Integration. Random Variables and Expected Values. Convergence of Distributions. Derivatives and Conditional Probability. Stochastic Processes. Appendix. Notes on the Problems. Bibliography. List of Symbols. Index.
In epidemiological studies, survival analyses are often carried out in order to better understand the onset of an event. The data have the particularity of being incomplete due to the different censoring phenomena. Traditional methods make the hypothesis of censoring being independent from the event, which may be a source of bias in certain pathologies. The Inverse Probability of Censoring Weighted (IPCW) method adapts the Kaplan-Meier estimators and the Cox partial likelihood method to cases, with non-independent censoring. This method uses the information resulting from censoring to modify the contribution of individuals in the estimators. This method is applied to asthma, a case in which therapists believe that patients lost to follow-up are patients who are in otherwise good health, and do not feel the necessity to consult a doctor (informative censoring).
A simple cost function approach is proposed for designing an optimal clinical trial when a total of N patients with a disease are to be treated with one of two medical treatments. The cost function is constructed with but one cost, the consequences of treating a patient with the superior or inferior of the two treatments. Fixed sample size and sequential trials are considered. Minimax, maximin, and Bayesian approaches are used for determining the optimal size of a fixed sample trial and the optimal position of the boundaries of a sequential trial. Comparisons of the different approaches are made as well as comparisons of the results for the fixed and sequential plans.
A clinical trial setting is considered in which two treatments are available for a particular ailment. The responses to the treatments are normally distributed with unknown means and a common known variance. A two-stage trial and a three-stage trial are studied. In the two-stage trial, patients are randomised equally in the first stage, and the better treatment at the end of this stage is used exclusively in the second stage. The three-stage trial permits a second randomised stage before a single treatment is selected. For these designs, the exact bias and variance of the estimated treatment difference at the end of the trial are derived. These quantities are also derived when there are time trends in the data. Numerical results indicate that the presence of time trends can seriously bias the estimated treatment difference and can also lead to an increase in its variance.
An up-to-date approach to understanding statistical inference Statistical inference is finding useful applications in numerous fields, from sociology and econometrics to biostatistics. This volume enables professionals in these and related fields to master the concepts of statistical inference under inequality constraints and to apply the theory to problems in a variety of areas. Constrained Statistical Inference: Order, Inequality, and Shape Constraints provides a unified and up-to-date treatment of the methodology. It clearly illustrates concepts with practical examples from a variety of fields, focusing on sociology, econometrics, and biostatistics. The authors also discuss a broad range of other inequality-constrained inference problems that do not fit well in the contemplated unified framework, providing a meaningful way for readers to comprehend methodological resolutions. Chapter coverage includes: • Population means and isotonic regression • Inequality-constrained tests on normal means • Tests in general parametric models • Likelihood and alternatives • Analysis of categorical data • Inference on monotone density function, unimodal density function, shape constraints, and DMRL functions • Bayesian perspectives, including Stein's Paradox, shrinkage estimation, and decision theory.
In the present work, we formulate a two-treatment single period two-stage adaptive allocation design for achieving larger allocation proportion to the better treatment arm in the course of the trial with increased precision of the parameter estimator. We examine some properties of the proposed rule and compare it with some of the existing allocation rules and report substantial gain in efficiency with a considerably larger number of allocations to the better treatment even for moderate sample sizes.
A clinical trial setting is considered in which two treatments are available for a particular ailment. A two—stage trial is studied, in which patients are randomised equally in the first stage, and the better treatment at the end of this stage is used exclusively in the second stage. For exponential and Bernoulli responses, the exact bias and variance of the estimated treatment difference at the end of the trial are derived. Corresponding results for normal responses with unequal variances are also obtained, and the numerical accuracy of a normal approximation is investigated. The results indicate that the bias in estimation can be up to 25% when the size of the first stage is small, reducing to less than 7% for moderate first—stage sizes. For both exponential and Bernoulli responses, a normal approximation works well for moderate first—stage sizes, with the approximation for Bernoulli responses being slightly better.
A randomized two-stage adaptive design is proposed and studied for allocation of patients to treatments and comparison in a phase III clinical trial with survival time as treatment responses. We consider the possibility of several covariates in the design and analysis. Several exact and limiting properties of the design and the follow-up inference are studied, both numerically and theoretically. The applicability of the proposed methodology is illustrated by using some real data.
A randomized two-stage adaptive Bayesian design is proposed and studied for allocation and comparison in a phase III clinical trial with survival time as treatment response. Several exact and limiting properties of the design and the follow-up inference are studied, both numerically and theoretically, and are compared with a single-stage randomized procedure. The applicability of the proposed methodology is illustrated by using some real data.
SUMMARY The asymptotic distributions of camér-von Mises type statistics based on the productlimit estimate of the distribution function
of a certain class of randomly censored observations are derived; the asymptotic significance points of the statistics for
various degrees of censoring are given. The statistics are also partitioned into orthogonal components in the manner of Durbin
& Knott (1972). The asymptotic powers of the statistics and their components against normal mean and variance shifts, exponential
scale shifts, and Weibull alternatives to exponentiality are compared. Data arising in a competing risk situation are examined,
using the Cramér-von Mises statistic.
This randomized, controlled, multicenter, open-label, phase III study compared docetaxel versus paclitaxel in patients with advanced breast cancer that had progressed after an anthracycline-containing chemotherapy regimen.
Patients (n = 449) were randomly assigned to receive either docetaxel 100 mg/m2 (n = 225) or paclitaxel 175 mg/m2 (n = 224) on day 1, every 21 days until tumor progression, unacceptable toxicity, or withdrawal of consent.
In the intent-to-treat population, both the median overall survival (OS, 15.4 v 12.7 months; hazard ratio [HR], 1.41; 95% CI, 1.15 to 1.73; P = .03) and the median time to progression (TTP, 5.7 months v 3.6 months; HR, 1.64; 95% CI, 1.33 to 2.02; P < .0001) for docetaxel were significantly longer than for paclitaxel, and the overall response rate (ORR, 32% v 25%; P = .10) was higher for docetaxel. These results were confirmed by multivariate analyses. The incidence of treatment-related hematologic and nonhematologic toxicities was greater for docetaxel than for paclitaxel; however, quality-of-life scores were not statistically different between treatment groups over time.
Docetaxel was superior to paclitaxel in terms of OS and TTP. ORR was higher for docetaxel. Hematologic and nonhematologic toxicities occurred more frequently in the docetaxel group. The global quality-of-life scores were similar for both agents over time.