Restricted mean survival time as a summary measure of time‐to‐event outcome

Many clinical research studies evaluate a time‐to‐event outcome, illustrate survival functions, and conventionally report estimated hazard ratios to express the magnitude of the treatment effect when comparing between groups. However, it may not be straightforward to interpret the hazard ratio clinically and statistically when the proportional hazards assumption is invalid. In some recent papers published in clinical journals, the use of restricted mean survival time (RMST) or τ ‐year mean survival time is discussed as one of the alternative summary measures for the time‐to‐event outcome. The RMST is defined as the expected value of time to event limited to a specific time point corresponding to the area under the survival curve up to the specific time point. This article summarizes the necessary information to conduct statistical analysis using the RMST, including the definition and statistical properties of the RMST, adjusted analysis methods, sample size calculation, information fraction for the RMST difference, and clinical and statistical meaning and interpretation. Additionally, we discuss how to set the specific time point to define the RMST from two main points of view. We also provide developed SAS codes to determine the sample size required to detect an expected RMST difference with appropriate power and reconstruct individual survival data to estimate an RMST reference value from a reported survival curve.     

A new method for the comparison of survival distributions

The assessment of overall homogeneity of time‐to‐event curves is a key element in survival analysis in biomedical research. The currently commonly used testing methods, e.g. log‐rank test, Wilcoxon test, and Kolmogorov–Smirnov test, may have a significant loss of statistical testing power under certain circumstances. In this paper we propose a new testing method that is robust for the comparison of the overall homogeneity of survival curves based on the absolute difference of the area under the survival curves using normal approximation by Greenwood's formula. Monte Carlo simulations are conducted to investigate the performance of the new testing method compared against the log‐rank, Wilcoxon, and Kolmogorov–Smirnov tests under a variety of circumstances. The proposed new method has robust performance with greater power to detect the overall differences than the log‐rank, Wilcoxon, and Kolmogorov–Smirnov tests in many scenarios in the simulations. Furthermore, the applicability of the new testing approach is illustrated in a real data example from a kidney dialysis trial. Copyright © 2009 John Wiley & Sons, Ltd.     

Study design of single‐arm phase II immunotherapy trials with long‐term survivors and random delayed treatment effect

In the traditional study design of a single‐arm phase II cancer clinical trial, the one‐sample log‐rank test has been frequently used. A common practice in sample size calculation is to assume that the event time in the new treatment follows exponential distribution. Such a study design may not be suitable for immunotherapy cancer trials, when both long‐term survivors (or even cured patients from the disease) and delayed treatment effect are present, because exponential distribution is not appropriate to describe such data and consequently could lead to severely underpowered trial. In this research, we proposed a piecewise proportional hazards cure rate model with random delayed treatment effect to design single‐arm phase II immunotherapy cancer trials. To improve test power, we proposed a new weighted one‐sample log‐rank test and provided a sample size calculation formula for designing trials. Our simulation study showed that the proposed log‐rank test performs well and is robust of misspecified weight and the sample size calculation formula also performs well.     

Hazard ratio inference in stratified clinical trials with time‐to‐event endpoints and limited sample size

The stratified Cox model is commonly used for stratified clinical trials with time‐to‐event endpoints. The estimated log hazard ratio is approximately a weighted average of corresponding stratum‐specific Cox model estimates using inverse‐variance weights; the latter are optimal only under the (often implausible) assumption of a constant hazard ratio across strata. Focusing on trials with limited sample sizes (50‐200 subjects per treatment), we propose an alternative approach in which stratum‐specific estimates are obtained using a refined generalized logrank (RGLR) approach and then combined using either sample size or minimum risk weights for overall inference. Our proposal extends the work of Mehrotra et al, to incorporate the RGLR statistic, which outperforms the Cox model in the setting of proportional hazards and small samples. This work also entails development of a remarkably accurate plug‐in formula for the variance of RGLR‐based estimated log hazard ratios. We demonstrate using simulations that our proposed two‐step RGLR analysis delivers notably better results through smaller estimation bias and mean squared error and larger power than the stratified Cox model analysis when there is a treatment‐by‐stratum interaction, with similar performance when there is no interaction. Additionally, our method controls the type I error rate while the stratified Cox model does not in small samples. We illustrate our method using data from a clinical trial comparing two treatments for colon cancer.     

Properties of the weighted log‐rank test in the design of confirmatory studies with delayed effects

Proportional hazards are a common assumption when designing confirmatory clinical trials in oncology. This assumption not only affects the analysis part but also the sample size calculation. The presence of delayed effects causes a change in the hazard ratio while the trial is ongoing since at the beginning we do not observe any difference between treatment arms, and after some unknown time point, the differences between treatment arms will start to appear. Hence, the proportional hazards assumption no longer holds, and both sample size calculation and analysis methods to be used should be reconsidered. The weighted log‐rank test allows a weighting for early, middle, and late differences through the Fleming and Harrington class of weights and is proven to be more efficient when the proportional hazards assumption does not hold. The Fleming and Harrington class of weights, along with the estimated delay, can be incorporated into the sample size calculation in order to maintain the desired power once the treatment arm differences start to appear. In this article, we explore the impact of delayed effects in group sequential and adaptive group sequential designs and make an empirical evaluation in terms of power and type‐I error rate of the of the weighted log‐rank test in a simulated scenario with fixed values of the Fleming and Harrington class of weights. We also give some practical recommendations regarding which methodology should be used in the presence of delayed effects depending on certain characteristics of the trial.     

Reference‐based sensitivity analysis for time‐to‐event data

The analysis of time‐to‐event data typically makes the censoring at random assumption, ie, that—conditional on covariates in the model—the distribution of event times is the same, whether they are observed or unobserved (ie, right censored). When patients who remain in follow‐up stay on their assigned treatment, then analysis under this assumption broadly addresses the de jure, or “while on treatment strategy” estimand. In such cases, we may well wish to explore the robustness of our inference to more pragmatic, de facto or “treatment policy strategy,” assumptions about the behaviour of patients post‐censoring. This is particularly the case when censoring occurs because patients change, or revert, to the usual (ie, reference) standard of care. Recent work has shown how such questions can be addressed for trials with continuous outcome data and longitudinal follow‐up, using reference‐based multiple imputation. For example, patients in the active arm may have their missing data imputed assuming they reverted to the control (ie, reference) intervention on withdrawal. Reference‐based imputation has two advantages: (a) it avoids the user specifying numerous parameters describing the distribution of patients' postwithdrawal data and (b) it is, to a good approximation, information anchored, so that the proportion of information lost due to missing data under the primary analysis is held constant across the sensitivity analyses. In this article, we build on recent work in the survival context, proposing a class of reference‐based assumptions appropriate for time‐to‐event data. We report a simulation study exploring the extent to which the multiple imputation estimator (using Rubin's variance formula) is information anchored in this setting and then illustrate the approach by reanalysing data from a randomized trial, which compared medical therapy with angioplasty for patients presenting with angina.     

Cancer immunotherapy trial design with delayed treatment effect

A challenge arising in cancer immunotherapy trial design is the presence of a delayed treatment effect wherein the proportional hazard assumption no longer holds true. As a result, a traditional survival trial design based on the standard log‐rank test, which ignores the delayed treatment effect, will lead to substantial loss of statistical power. Recently, a piecewise weighted log‐rank test is proposed to incorporate the delayed treatment effect into consideration of the trial design. However, because the sample size formula was derived under a sequence of local alternative hypotheses, it results in an underestimated sample size when the hazard ratio is relatively small for a balanced trial design and an inaccurate sample size estimation for an unbalanced design. In this article, we derived a new sample size formula under a fixed alternative hypothesis for the delayed treatment effect model. Simulation results show that the new formula provides accurate sample size estimation for both balanced and unbalanced designs.     

Designing cancer immunotherapy trials with delayed treatment effect using maximin efficiency robust statistics

The indirect mechanism of action of immunotherapy causes a delayed treatment effect, producing delayed separation of survival curves between the treatment groups, and violates the proportional hazards assumption. Therefore using the log‐rank test in immunotherapy trial design could result in a severe loss efficiency. Although few statistical methods are available for immunotherapy trial design that incorporates a delayed treatment effect, recently, Ye and Yu proposed the use of a maximin efficiency robust test (MERT) for the trial design. The MERT is a weighted log‐rank test that puts less weight on early events and full weight after the delayed period. However, the weight function of the MERT involves an unknown function that has to be estimated from historical data. Here, for simplicity, we propose the use of an approximated maximin test, the V₀ test, which is the sum of the log‐rank test for the full data set and the log‐rank test for the data beyond the lag time point. The V₀ test fully uses the trial data and is more efficient than the log‐rank test when lag exits with relatively little efficiency loss when no lag exists. The sample size formula for the V₀ test is derived. Simulations are conducted to compare the performance of the V₀ test to the existing tests. A real trial is used to illustrate cancer immunotherapy trial design with delayed treatment effect.     

Sample size calculation for the one‐sample log‐rank test

In this paper, an exact variance of the one‐sample log‐rank test statistic is derived under the alternative hypothesis, and a sample size formula is proposed based on the derived exact variance. Simulation results showed that the proposed sample size formula provides adequate power to design a study to compare the survival of a single sample with that of a standard population. Copyright © 2014 John Wiley & Sons, Ltd.     

Survival analysis using inverse probability of treatment weighted methods based on the generalized propensity score

In survival analysis, treatment effects are commonly evaluated based on survival curves and hazard ratios as causal treatment effects. In observational studies, these estimates may be biased due to confounding factors. The inverse probability of treatment weighted (IPTW) method based on the propensity score is one of the approaches utilized to adjust for confounding factors between binary treatment groups. As a generalization of this methodology, we developed an exact formula for an IPTW log‐rank test based on the generalized propensity score for survival data. This makes it possible to compare the group differences of IPTW Kaplan–Meier estimators of survival curves using an IPTW log‐rank test for multi‐valued treatments. As causal treatment effects, the hazard ratio can be estimated using the IPTW approach. If the treatments correspond to ordered levels of a treatment, the proposed method can be easily extended to the analysis of treatment effect patterns with contrast statistics. In this paper, the proposed method is illustrated with data from the Kyushu Lipid Intervention Study (KLIS), which investigated the primary preventive effects of pravastatin on coronary heart disease (CHD). The results of the proposed method suggested that pravastatin treatment reduces the risk of CHD and that compliance to pravastatin treatment is important for the prevention of CHD. Copyright © 2009 John Wiley & Sons, Ltd.     

Sensitivity to censored‐at‐random assumption in the analysis of time‐to‐event endpoints

Over the past years, significant progress has been made in developing statistically rigorous methods to implement clinically interpretable sensitivity analyses for assumptions about the missingness mechanism in clinical trials for continuous and (to a lesser extent) for binary or categorical endpoints. Studies with time‐to‐event outcomes have received much less attention. However, such studies can be similarly challenged with respect to the robustness and integrity of primary analysis conclusions when a substantial number of subjects withdraw from treatment prematurely prior to experiencing an event of interest. We discuss how the methods that are widely used for primary analyses of time‐to‐event outcomes could be extended in a clinically meaningful and interpretable way to stress‐test the assumption of ignorable censoring. We focus on a ‘tipping point’ approach, the objective of which is to postulate sensitivity parameters with a clear clinical interpretation and to identify a setting of these parameters unfavorable enough towards the experimental treatment to nullify a conclusion that was favorable to that treatment. Robustness of primary analysis results can then be assessed based on clinical plausibility of the scenario represented by the tipping point. We study several approaches for conducting such analyses based on multiple imputation using parametric, semi‐parametric, and non‐parametric imputation models and evaluate their operating characteristics via simulation. We argue that these methods are valuable tools for sensitivity analyses of time‐to‐event data and conclude that the method based on piecewise exponential imputation model of survival has some advantages over other methods studied here. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonparametric covariate adjustment in estimating hazard ratios

In randomized clinical trials with time‐to‐event outcomes, the hazard ratio is commonly used to quantify the treatment effect relative to a control. The Cox regression model is commonly used to adjust for relevant covariates to obtain more accurate estimates of the hazard ratio between treatment groups. However, it is well known that the treatment hazard ratio based on a covariate‐adjusted Cox regression model is conditional on the specific covariates and differs from the unconditional hazard ratio that is an average across the population. Therefore, covariate‐adjusted Cox models cannot be used when the unconditional inference is desired. In addition, the covariate‐adjusted Cox model requires the relatively strong assumption of proportional hazards for each covariate. To overcome these challenges, a nonparametric randomization‐based analysis of covariance method was proposed to estimate the covariate‐adjusted hazard ratios for multivariate time‐to‐event outcomes. However, empirical evaluations of the performance (power and type I error rate) of the method have not been studied. Although the method is derived for multivariate situations, for most registration trials, the primary endpoint is a univariate outcome. Therefore, this approach is applied to univariate outcomes, and performance is evaluated through a simulation study in this paper. Stratified analysis is also investigated. As an illustration of the method, we also apply the covariate‐adjusted and unadjusted analyses to an oncology trial. Copyright © 2015 John Wiley & Sons, Ltd.     

Design considerations in clinical trials with cure rate survival data: A case study in oncology

For clinical trials with time‐to‐event as the primary endpoint, the clinical cutoff is often event‐driven and the log‐rank test is the most commonly used statistical method for evaluating treatment effect. However, this method relies on the proportional hazards assumption in that it has the maximal power in this circumstance. In certain disease areas or populations, some patients can be curable and never experience the events despite a long follow‐up. The event accumulation may dry out after a certain period of follow‐up and the treatment effect could be reflected as the combination of improvement of cure rate and the delay of events for those uncurable patients. Study power depends on both cure rate improvement and hazard reduction. In this paper, we illustrate these practical issues using simulation studies and explore sample size recommendations, alternative ways for clinical cutoffs, and efficient testing methods with the highest study power possible.     

The win ratio: Impact of censoring and follow‐up time and use with nonproportional hazards

The win ratio has been studied methodologically and applied in data analysis and in designing clinical trials. Researchers have pointed out that the results depend on follow‐up time and censoring time, which are sometimes used interchangeably. In this article, we distinguish between follow‐up time and censoring time, show theoretically the impact of censoring on the win ratio, and illustrate the impact of follow‐up time. We then point out that, if the treatment has long‐term benefit from a more important but less frequent endpoint (eg, death), the win ratio can show that benefit by following patients longer, avoiding masking by more frequent but less important outcomes, which occurs in conventional time‐to‐first‐event analyses. For the situation of nonproportional hazards, we demonstrate that the win ratio can be a good alternative to methods such as landmark survival rate, restricted mean survival time, and weighted log‐rank tests.     

Two‐stage phase II survival trial design

Recently, molecularly targeted agents and immunotherapy have been advanced for the treatment of relapse or refractory cancer patients, where disease progression‐free survival or event‐free survival is often a primary endpoint for the trial design. However, methods to evaluate two‐stage single‐arm phase II trials with a time‐to‐event endpoint are currently processed under an exponential distribution, which limits application of real trial designs. In this paper, we developed an optimal two‐stage design, which is applied to the four commonly used parametric survival distributions. The proposed method has advantages compared with existing methods in that the choice of underlying survival model is more flexible and the power of the study is more adequately addressed. Therefore, the proposed two‐stage design can be routinely used for single‐arm phase II trial designs with a time‐to‐event endpoint as a complement to the commonly used Simon's two‐stage design for the binary outcome.     

Two‐Sample Test Against One‐Sided Alternatives

Abstract. This paper proposes, implements and investigates a new non‐parametric two‐sample test for detecting stochastic dominance. We pose the question of detecting the stochastic dominance in a non‐standard way. This is motivated by existing evidence showing that standard formulations and pertaining procedures may lead to serious errors in inference. The procedure that we introduce matches testing and model selection. More precisely, we reparametrize the testing problem in terms of Fourier coefficients of well‐known comparison densities. Next, the estimated Fourier coefficients are used to form a kind of signed smooth rank statistic. In such a setting, the number of Fourier coefficients incorporated into the statistic is a smoothing parameter. We determine this parameter via some flexible selection rule. We establish the asymptotic properties of the new test under null and alternative hypotheses. The finite sample performance of the new solution is demonstrated through Monte Carlo studies and an application to a set of survival times.     

Predicting analysis time in events‐driven clinical trials using accumulating time‐to‐event surrogate information

For clinical trials with time‐to‐event endpoints, predicting the accrual of the events of interest with precision is critical in determining the timing of interim and final analyses. For example, overall survival (OS) is often chosen as the primary efficacy endpoint in oncology studies, with planned interim and final analyses at a pre‐specified number of deaths. Often, correlated surrogate information, such as time‐to‐progression (TTP) and progression‐free survival, are also collected as secondary efficacy endpoints. It would be appealing to borrow strength from the surrogate information to improve the precision of the analysis time prediction. Currently available methods in the literature for predicting analysis timings do not consider utilizing the surrogate information. In this article, using OS and TTP as an example, a general parametric model for OS and TTP is proposed, with the assumption that disease progression could change the course of the overall survival. Progression‐free survival, related both to OS and TTP, will be handled separately, as it can be derived from OS and TTP. The authors seek to develop a prediction procedure using a Bayesian method and provide detailed implementation strategies under certain assumptions. Simulations are performed to evaluate the performance of the proposed method. An application to a real study is also provided. Copyright © 2015 John Wiley & Sons, Ltd.     

Two-sample inference procedures under nonproportional hazards

We introduce a new two-sample inference procedure to assess the relative performance of two groups over time. Our model-free method does not assume proportional hazards, making it suitable for scenarios where nonproportional hazards may exist. Our procedure includes a diagnostic tau plot to identify changes in hazard timing and a formal inference procedure. The tau-based measures we develop are clinically meaningful and provide interpretable estimands to summarize the treatment effect over time. Our proposed statistic is a U-statistic and exhibits a martingale structure, allowing us to construct confidence intervals and perform hypothesis testing. Our approach is robust with respect to the censoring distribution. We also demonstrate how our method can be applied for sensitivity analysis in scenarios with missing tail information due to insufficient follow-up. Without censoring, Kendall's tau estimator we propose reduces to the Wilcoxon-Mann–Whitney statistic. We evaluate our method using simulations to compare its performance with the restricted mean survival time and log-rank statistics. We also apply our approach to data from several published oncology clinical trials where nonproportional hazards may exist.     

Single‐arm phase II trial design under parametric cure models

The current practice of designing single‐arm phase II survival trials is limited under the exponential model. Trial design under the exponential model may not be appropriate when a portion of patients are cured. There is no literature available for designing single‐arm phase II trials under the parametric cure model. In this paper, a test statistic is proposed, and a sample size formula is derived for designing single‐arm phase II trials under a class of parametric cure models. Extensive simulations showed that the proposed test and sample size formula perform very well under different scenarios. Copyright © 2015 John Wiley & Sons, Ltd.     

Likelihood‐based Inference with Missing Data Under Missing‐at‐Random

Likelihood‐based inference with missing data is challenging because the observed log likelihood is often an (intractable) integration over the missing data distribution, which also depends on the unknown parameter. Approximating the integral by Monte Carlo sampling does not necessarily lead to a valid likelihood over the entire parameter space because the Monte Carlo samples are generated from a distribution with a fixed parameter value. We consider approximating the observed log likelihood based on importance sampling. In the proposed method, the dependency of the integral on the parameter is properly reflected through fractional weights. We discuss constructing a confidence interval using the profile likelihood ratio test. A Newton–Raphson algorithm is employed to find the interval end points. Two limited simulation studies show the advantage of the Wilks inference over the Wald inference in terms of power, parameter space conformity and computational efficiency. A real data example on salamander mating shows that our method also works well with high‐dimensional missing data.