Calculation of sample size in survival trials: the impact of informative noncompliance

Sample size calculations for survival trials typically include an adjustment to account for the expected rate of noncompliance, or discontinuation from study medication. Existing sample size methods assume that when patients discontinue, they do so independently of their risk of an endpoint; that is, that noncompliance is noninformative. However, this assumption is not always true, as we illustrate using results from a published clinical trial database. In this article, we introduce a modified version of the method proposed by Lakatos (1988, Biometrics 44, 229-241) that can be used to calculate sample size under informative noncompliance. This method is based on the concept of two subpopulations: one with high rates of endpoint and discontinuation and another with low rates. Using this new method, we show that failure to consider the impact of informative noncompliance can lead to a considerably underpowered study.     

Variance and sample size calculations in quality-of-life--adjusted survival analysis (Q-TWiST)

The Quality-Adjusted Time Without Symptoms or Toxicity (Q-TWiST) statistic previously introduced by Glasziou, Simes and Gelber (1990, Statistics in Medicine 9, 1259-1276) combines toxicity, disease-free survival, and overall survival information in assessing the impact of treatments on the lives of patients. This methodology has received positive reviews from clinicians as intuitive and useful, but to date, the variance of this statistic has remained unspecified. We review aspects of the Q-TWiST method for analyzing clinical trial data, extend the method to accommodate multiple treatment arms, and provide closed-form asymptotic variance formulas. We also provide a framework for designing Q-TWiST clinical trials with sample sizes determined using the derived asymptotic variance formulas. Trials currently collecting quality of life data did not have the benefit of these sample size calculation techniques in designing their studies.     

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.     

A proportional hazards model for interval-censored failure time data

This paper develops a method for fitting the proportional hazards regression model when the data contain left-, right-, or interval-censored observations. Results given for testing the hypothesis of a zero regression coefficient lead to a generalization of the log-rank test for comparison of several survival curves. The method is used to analyze data from an animal tumorigenicity study and also a clinical trial.     

A sample size formula for the supremum log-rank statistic

An advantage of the supremum log-rank over the standard log-rank statistic is an increased sensitivity to a wider variety of stochastic ordering alternatives. In this article, we develop a formula for sample size computation for studies utilizing the supremum log-rank statistic. The idea is to base power on the proportional hazards alternative, so that the supremum log rank will have the same power as the standard log rank in the setting where the standard log rank is optimal. This results in a slight increase in sample size over that required for the standard log rank. For example, a 5.733% increase occurs for a two-sided test having type I error 0.05 and power 0.80. This slight increase in sample size is offset by the significant gains in power the supremum log-rank test achieves for a wide range of nonproportional hazards alternatives. A small simulation study is used for illustration. These results should facilitate the wider use of the supremum log-rank statistic in clinical trials.     

Exact Tests using Two Correlated Binomial Variables in Contemporary Cancer Clinical Trials

New therapy strategies for the treatment of cancer are rapidly emerging because of recent technology advances in genetics and molecular biology. Although newer targeted therapies can improve survival without measurable changes in tumor size, clinical trial conduct has remained nearly unchanged. When potentially efficacious therapies are tested, current clinical trial design and analysis methods may not be suitable for detecting therapeutic effects. We propose an exact method with respect to testing cytostatic cancer treatment using correlated bivariate binomial random variables to simultaneously assess two primary outcomes. The method is easy to implement. It does not increase the sample size over that of the univariate exact test and in most cases reduces the sample size required. Sample size calculations are provided for selected designs.     

A simple approximation for calculating sample sizes for detecting linear trend in proportions

A simple approximate formula for sample sizes for detecting a linear trend in proportions is derived. The formulas for both the uncorrected and corrected Cochran-Armitage test are given. For two binomial proportions these reduce to those given by Casagrande, Pike, and Smith (1978, Biometrics 34, 483-486). Some numerical results of a power study for small sample sizes show that the nominal power corresponding to the approximate sample size is a reasonably good approximation to the actual power.     

Comparing treatments in the presence of crossing survival curves: an application to bone marrow transplantation

Summary .   In some clinical studies comparing treatments in terms of their survival curves, researchers may anticipate that the survival curves will cross at some point, leading to interest in a long-term survival comparison. However, simple comparison of the survival curves at a fixed point may be inefficient, and use of a weighted log-rank test may be overly sensitive to early differences in survival. We formulate the problem as one of testing for differences in survival curves after a prespecified time point, and propose a variety of techniques for testing this hypothesis. We study these methods using simulation and illustrate them on a study comparing survival for autologous and allogeneic bone marrow transplants.     

On the use of historical control data to estimate dose response trends in quantal bioassay.

New tests for trend in proportions, in the presence of historical control data, are proposed. One such test is a simple score statistic based on a binomial likelihood for the "current" study and beta-binomial likelihoods for each historical control series. A closely related trend statistic based on estimating equations is also proposed. Trend statistics that allow overdispersed proportions in the current study are also developed, including a version of Tarone's (1982, Biometrics 38, 215-220) test that acknowledges sampling variation in the beta distribution parameters, and a trend statistic based on estimating equations. Each such trend test is evaluated with respect to size and power under both binomial and beta-binomial sampling conditions for the current study, and illustrations are provided.     

Quantile regression models with multivariate failure time data

As an alternative to the mean regression model, the quantile regression model has been studied extensively with independent failure time data. However, due to natural or artificial clustering, it is common to encounter multivariate failure time data in biomedical research where the intracluster correlation needs to be accounted for appropriately. For right-censored correlated survival data, we investigate the quantile regression model and adapt an estimating equation approach for parameter estimation under the working independence assumption, as well as a weighted version for enhancing the efficiency. We show that the parameter estimates are consistent and asymptotically follow normal distributions. The variance estimation using asymptotic approximation involves nonparametric functional density estimation. We employ the bootstrap and perturbation resampling methods for the estimation of the variance-covariance matrix. We examine the proposed method for finite sample sizes through simulation studies, and illustrate it with data from a clinical trial on otitis media.     

Supremum weighted log-rank test and sample size for comparing two-stage adaptive treatment strategies

In two-stage adaptive treatment strategies, patients receivean induction treatment followed by a maintenance therapy, giventhat the patient responded to the induction treatment they received.To test for a difference in the effects of different inductionand maintenance treatment combinations, a modified supremumweighted log-rank test is proposed. The test is applied to adataset from a two-stage randomized trial and the results arecompared to those obtained using a standard weighted log-ranktest. A sample-size formula is proposed based on the limitingdistribution of the supremum weighted log-rank statistic. Thesample-size formula reduces to Eng and Kosorok's sample-sizeformula for a two-sample supremum log-rank test when there isno second randomization. Monte Carlo studies show that the proposedtest provides sample sizes that are close to those obtainedby standard weighted log-rank test under a proportional hazardsalternative. However, the proposed test is more powerful thanthe standard weighted log-rank test under non-proportional hazardsalternatives.     

Exact log-rank tests for unequal follow-up

The asymptotic log-rank and generalized Wilcoxon tests are the standard procedures for comparing samples of possibly censored survival times. For comparison of samples of very different sizes, an exact test is available that is based on a complete permutation of log-rank or Wilcoxon scores. While the asymptotic tests do not keep their nominal sizes if sample sizes differ substantially, the exact complete permutation test requires equal follow-up of the samples. Therefore, we have developed and present two new exact tests also suitable for unequal follow-up. The first of these is an exact analogue of the asymptotic log-rank test and conditions on observed risk sets, whereas the second approach permutes survival times while conditioning on the realized follow-up in each group. In an empirical study, we compare the new procedures with the asymptotic log-rank test, the exact complete permutation test, and an earlier proposed approach that equalizes the follow-up distributions using artificial censoring. Results confirm highly satisfactory performance of the exact procedure conditioning on realized follow-up, particularly in case of unequal follow-up. The advantage of this test over other options of analysis is finally exemplified in the analysis of a breast cancer study.     

Accurate Computation of Survival Statistics in Genome-Wide Studies

A key challenge in genomics is to identify genetic variants that distinguish patients with different survival time following diagnosis or treatment. While the log-rank test is widely used for this purpose, nearly all implementations of the log-rank test rely on an asymptotic approximation that is not appropriate in many genomics applications. This is because: the two populations determined by a genetic variant may have very different sizes; and the evaluation of many possible variants demands highly accurate computation of very small p-values. We demonstrate this problem for cancer genomics data where the standard log-rank test leads to many false positive associations between somatic mutations and survival time. We develop and analyze a novel algorithm, Exact Log-rank Test (ExaLT), that accurately computes the p-value of the log-rank statistic under an exact distribution that is appropriate for any size populations. We demonstrate the advantages of ExaLT on data from published cancer genomics studies, finding significant differences from the reported p-values. We analyze somatic mutations in six cancer types from The Cancer Genome Atlas (TCGA), finding mutations with known association to survival as well as several novel associations. In contrast, standard implementations of the log-rank test report dozens-hundreds of likely false positive associations as more significant than these known associations.     

Asymptotic Efficiency of the Chi-Square Test for Heterogeneity of Proportions in Some Sparse-Data Cases

We consider the problem of testing for heterogeneity of K proportions when K is not small and the binomial sample sizes may not be large. We assume that the binomial proportions are normally distributed with variance σ². The asymptotic relative efficiency (ARE) of the usual chi-square test is found relative to the likelihood-based tests for σ²=0. The chi-square test is found to have ARE = 1 when the binomial sample sizes are all equal and high relative efficiency for other cases. The efficiency is low only in cases where there is insufficient data to use the chi-square test.     

Adaptive design: estimation and inference with censored data in a semiparametric model

In this article, we provide a method of estimation for the treatment effect in the adaptive design for censored survival data with or without adjusting for risk factors other than the treatment indicator. Within the semiparametric Cox proportional hazards model, we propose a bias-adjusted parameter estimator for the treatment coefficient and its asymptotic confidence interval at the end of the trial. The method for obtaining an asymptotic confidence interval and point estimator is based on a general distribution property of the final test statistic from the weighted linear rank statistics at the interims with or without considering the nuisance covariates. The computation of the estimates is straightforward. Extensive simulation studies show that the asymptotic confidence intervals have reasonable nominal probability of coverage, and the proposed point estimators are nearly unbiased with practical sample sizes.     

Calculation of narrower confidence intervals for tree mortality rates when we know nothing but the location of the death/survival events

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A Bootstrap Procedure for Adaptive Selection of the Test Statistic in Flexible Two‐Stage Designs

Adaptive two‐stage designs allow a data‐driven change of design characteristics during the ongoing trial. One of the available options is an adaptive choice of the test statistic for the second stage of the trial based on the results of the interim analysis. Since there is often only a vague knowledge of the distribution shape of the primary endpoint in the planning phase of a study, a change of the test statistic may then be considered if the data indicate that the assumptions underlying the initial choice of the test are not correct. Collings and Hamilton proposed a bootstrap method for the estimation of the power of the two‐sample Wilcoxon test for shift alternatives. We use this approach for the selection of the test statistic. By means of a simulation study, we show that the gain in terms of power may be considerable when the initial assumption about the underlying distribution was wrong, whereas the loss is relatively small when in the first instance the optimal test statistic was chosen. The results also hold true for comparison with a one‐stage design. Application of the method is illustrated by a clinical trial example.     

On Approximate and Exact Sample Sizes of Equivalence Tests for Binomial Proportions

There exist several formulae for sample sizes for testing the equivalence of binomial proportions which are based on approximations by the normal distribution. Quite often these formulae produce drastically different results. In this paper the validity of the approximate sample sizes is investigated with respect to the exact distributions.     

Evaluation of sample size and power for analyses of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification

When designing a clinical trial to test the equality of survival distributions for two treatment groups, the usual assumptions are exponential survival, uniform patient entry, full compliance, and censoring only administratively at the end of the trial. Various authors have presented methods for estimation of sample size or power under these assumptions, some of which allow for an R-year accrual period with T total years of study, T greater than R. The method of Lachin (1981, Controlled Clinical Trials 2, 93-113) is extended to allow for cases where patients enter the trial in a nonuniform manner over time, patients may exit from the trial due to loss to follow-up (other than administrative), other patients may continue follow-up although failing to comply with the treatment regimen, and a stratified analysis may be planned according to one or more prognostic covariates.     

Inference for the effect of treatment on survival probability in randomized trials with noncompliance and administrative censoring

In many clinical studies with a survival outcome, administrative censoring occurs when follow-up ends at a prespecified date and many subjects are still alive. An additional complication in some trials is that there is noncompliance with the assigned treatment. For this setting, we study the estimation of the causal effect of treatment on survival probability up to a given time point among those subjects who would comply with the assignment to both treatment and control. We first discuss the standard instrumental variable (IV) method for survival outcomes and parametric maximum likelihood methods, and then develop an efficient plug-in nonparametric empirical maximum likelihood estimation (PNEMLE) approach. The PNEMLE method does not make any assumptions on outcome distributions, and makes use of the mixture structure in the data to gain efficiency over the standard IV method. Theoretical results of the PNEMLE are derived and the method is illustrated by an analysis of data from a breast cancer screening trial. From our limited mortality analysis with administrative censoring times 10 years into the follow-up, we find a significant benefit of screening is present after 4 years (at the 5% level) and this persists at 10 years follow-up.