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.     

Quantifying treatment differences in confirmatory trials under non-proportional hazards

Proportional hazards are a common assumption when designing confirmatory clinical trials in oncology. With the emergence of immunotherapy and novel targeted therapies, departure from the proportional hazard assumption is not rare in nowadays clinical research. Under non-proportional hazards, the hazard ratio does not have a straightforward clinical interpretation, and the log-rank test is no longer the most powerful statistical test even though it is still valid. Nevertheless, the log-rank test and the hazard ratio are still the primary analysis tools, and traditional approaches such as sample size increase are still proposed to account for the impact of non-proportional hazards. The weighed log-rank test and the test based on the restricted mean survival time (RMST) are receiving a lot of attention as a potential alternative to the log-rank test. We conduct a simulation study comparing the performance and operating characteristics of the log-rank test, the weighted log-rank test and the test based on the RMST, including a treatment effect estimation, under different non-proportional hazards patterns. Results show that, under non-proportional hazards, the hazard ratio and weighted hazard ratio have no straightforward clinical interpretation whereas the RMST ratio can be interpreted regardless of the proportional hazards assumption. In terms of power, the RMST achieves a similar performance when compared to the log-rank test.     

Bias and precision of measures of survival gain from right‐censored data

In cost‐effectiveness analyses of drugs or health technologies, estimates of life years saved or quality‐adjusted life years saved are required. Randomised controlled trials can provide an estimate of the average treatment effect; for survival data, the treatment effect is the difference in mean survival. However, typically not all patients will have reached the endpoint of interest at the close‐out of a trial, making it difficult to estimate the difference in mean survival. In this situation, it is common to report the more readily estimable difference in median survival. Alternative approaches to estimating the mean have also been proposed. We conducted a simulation study to investigate the bias and precision of the three most commonly used sample measures of absolute survival gain – difference in median, restricted mean and extended mean survival – when used as estimates of the true mean difference, under different censoring proportions, while assuming a range of survival patterns, represented by Weibull survival distributions with constant, increasing and decreasing hazards. Our study showed that the three commonly used methods tended to underestimate the true treatment effect; consequently, the incremental cost‐effectiveness ratio (ICER) would be overestimated. Of the three methods, the least biased is the extended mean survival, which perhaps should be used as the point estimate of the treatment effect to be inputted into the ICER, while the other two approaches could be used in sensitivity analyses. More work on the trade‐offs between simple extrapolation using the exponential distribution and more complicated extrapolation using other methods would be valuable. Copyright © 2015 John Wiley & Sons, Ltd.     

Semiparametric inference on the absolute risk reduction and the restricted mean survival difference

For time-to-event data, when the hazards are non-proportional, in addition to the hazard ratio, the absolute risk reduction and the restricted mean survival difference can be used to describe the time-dependent treatment effect. The absolute risk reduction measures the direct impact of the treatment on event rate or survival, and the restricted mean survival difference provides a way to evaluate the cumulative treatment effect. However, in the literature, available methods are limited for flexibly estimating these measures and making inference on them. In this article, point estimates, pointwise confidence intervals and simultaneous confidence bands of the absolute risk reduction and the restricted mean survival difference are established under a semiparametric model that can be used in a sufficiently wide range of applications. These methods are motivated by and illustrated for data from the Women’s Health Initiative estrogen plus progestin clinical trial.     

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.     

Comparison of the restricted mean survival time with the hazard ratio in superiority trials with a time‐to‐event end point

With the emergence of novel therapies exhibiting distinct mechanisms of action compared to traditional treatments, departure from the proportional hazard (PH) assumption in clinical trials with a time‐to‐event end point is increasingly common. In these situations, the hazard ratio may not be a valid statistical measurement of treatment effect, and the log‐rank test may no longer be the most powerful statistical test. The restricted mean survival time (RMST) is an alternative robust and clinically interpretable summary measure that does not rely on the PH assumption. We conduct extensive simulations to evaluate the performance and operating characteristics of the RMST‐based inference and against the hazard ratio–based inference, under various scenarios and design parameter setups. The log‐rank test is generally a powerful test when there is evident separation favoring 1 treatment arm at most of the time points across the Kaplan‐Meier survival curves, but the performance of the RMST test is similar. Under non‐PH scenarios where late separation of survival curves is observed, the RMST‐based test has better performance than the log‐rank test when the truncation time is reasonably close to the tail of the observed curves. Furthermore, when flat survival tail (or low event rate) in the experimental arm is expected, selecting the minimum of the maximum observed event time as the truncation timepoint for the RMST is not recommended. In addition, we recommend the inclusion of analysis based on the RMST curve over the truncation time in clinical settings where there is suspicion of substantial departure from the PH assumption.     

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.     

Regression Analysis of Restricted Mean Survival Time Based on Pseudo-Observations

Regression models for survival data are often specified from the hazard function while classical regression analysis of quantitative outcomes focuses on the mean value (possibly after suitable transformations). Methods for regression analysis of mean survival time and the related quantity, the restricted mean survival time, are reviewed and compared to a method based on pseudo-observations. Both Monte Carlo simulations and two real data sets are studied. It is concluded that while existing methods may be superior for analysis of the mean, pseudo-observations seem well suited when the restricted mean is studied.     

Survival estimation through the cumulative hazard function with monotone natural cubic splines

In this paper we explore the estimation of survival probabilities via a smoothed version of the survival function, in the presence of censoring. We investigate the fit of a natural cubic spline on the cumulative hazard function under appropriate constraints. Under the proposed technique the problem reduces to a restricted least squares one, leading to convex optimization. The approach taken in this paper is evaluated and compared via simulations to other known methods such as the Kaplan Meier and the logspline estimator. Our approach is easily extended to address estimation of survival probabilities in the presence of covariates when the proportional hazards model assumption holds. In this case the method is compared to a restricted cubic spline approach that involves maximum likelihood. The proposed approach can be also adjusted to accommodate left censoring.     

Risk patterns in drug safety study using relative times by accelerated failure time models when proportional hazards assumption is questionable: an illustrative case study of cancer risk of patients on glucose‐lowering therapies

Observational drug safety studies may be susceptible to confounding or protopathic bias. This bias may cause a spurious relationship between drug exposure and adverse side effect when none exists and may lead to unwarranted safety alerts. The spurious relationship may manifest itself through substantially different risk levels between exposure groups at the start of follow‐up when exposure is deemed too short to have any plausible biological effect of the drug. The restrictive proportional hazards assumption with its arbitrary choice of baseline hazard function renders the commonly used Cox proportional hazards model of limited use for revealing such potential bias. We demonstrate a fully parametric approach using accelerated failure time models with an illustrative safety study of glucose‐lowering therapies and show that its results are comparable against other methods that allow time‐varying exposure effects. Our approach includes a wide variety of models that are based on the flexible generalized gamma distribution and allows direct comparisons of estimated hazard functions following different exposure‐specific distributions of survival times. This approach lends itself to two alternative metrics, namely relative times and difference in times to event, allowing physicians more ways to communicate patient's prognosis without invoking the concept of risks, which some may find hard to grasp. In our illustrative case study, substantial differences in cancer risks at drug initiation followed by a gradual reduction towards null were found. This evidence is compatible with the presence of protopathic bias, in which undiagnosed symptoms of cancer lead to switches in diabetes medication. Copyright © 2015 John Wiley & Sons, Ltd.     

Estimation of odds of concordance based on the Aalen additive model

The Cox regression model is often used when analyzing survival data as it provides a convenient way of summarizing covariate effects in terms of relative risks. The proportional hazards assumption may not hold, however. A typical violation of the assumption is time-changing covariate effects. Under such scenarios one may use more flexible models but the results from such models may be complicated to communicate and it is desirable to have simple measures of a treatment effect, say. In this paper we focus on the odds-of-concordance measure that was recently studied by Schemper et al. (Stat Med 28:2473?C2489, 2009). They suggested to estimate this measure using weighted Cox regression (WCR). Although WCR may work in many scenarios no formal proof can be established. We suggest an alternative estimator of the odds-of-concordance measure based on the Aalen additive hazards model. In contrast to the WCR, one may derive the large sample properties for this estimator making formal inference possible. The estimator also allows for additional covariate effects.     

Delayed treatment effects,treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology

In the analysis of survival times, the logrank test and the Cox model have been established as key tools, which do not require specific distributional assumptions. Under the assumption of proportional hazards, they are efficient and their results can be interpreted unambiguously. However, delayed treatment effects, disease progression, treatment switchers or the presence of subgroups with differential treatment effects may challenge the assumption of proportional hazards. In practice, weighted logrank tests emphasizing either early, intermediate or late event times via an appropriate weighting function may be used to accommodate for an expected pattern of non-proportionality. We model these sources of non-proportional hazards via a mixture of survival functions with piecewise constant hazard. The model is then applied to study the power of unweighted and weighted log-rank tests, as well as maximum tests allowing different time dependent weights. Simulation results suggest a robust performance of maximum tests across different scenarios, with little loss in power compared to the most powerful among the considered weighting schemes and huge power gain compared to unfavorable weights. The actual sources of non-proportional hazards are not obvious from resulting populationwise survival functions, highlighting the importance of detailed simulations in the planning phase of a trial when assuming non-proportional hazards.We provide the required tools in a software package, allowing to model data generating processes under complex non-proportional hazard scenarios, to simulate data from these models and to perform the weighted logrank tests.     

The use of restricted cubic splines to approximate complex hazard functions in the analysis of time-to-event data: a simulation study

If interest lies in reporting absolute measures of risk from time-to-event data then obtaining an appropriate approximation to the shape of the underlying hazard function is vital. It has previously been shown that restricted cubic splines can be used to approximate complex hazard functions in the context of time-to-event data. The degree of complexity for the spline functions is dictated by the number of knots that are defined. We highlight through the use of a motivating example that complex hazard function shapes are often required when analysing time-to-event data. Through the use of simulation, we show that provided a sufficient number of knots are used, the approximated hazard functions given by restricted cubic splines fit closely to the true function for a range of complex hazard shapes. The simulation results also highlight the insensitivity of the estimated relative effects (hazard ratios) to the correct specification of the baseline hazard.     

Choice of Parametric Accelerated Life and Proportional Hazards Models for Survival Data: Asymptotic Results

We discuss the impact of misspecifying fully parametric proportional hazards and accelerated life models. For the uncensored case, misspecified accelerated life models give asymptotically unbiased estimates of covariate effect, but the shape and scale parameters depend on the misspecification. The covariate, shape and scale parameters differ in the censored case. Parametric proportional hazards models do not have a sound justification for general use: estimates from misspecified models can be very biased, and misleading results for the shape of the hazard function can arise. Misspecified survival functions are more biased at the extremes than the centre. Asymptotic and first order results are compared. If a model is misspecified, the size of Wald tests will be underestimated. Use of the sandwich estimator of standard error gives tests of the correct size, but misspecification leads to a loss of power. Accelerated life models are more robust to misspecification because of their log-linear form. In preliminary data analysis, practitioners should investigate proportional hazards and accelerated life models; software is readily available for several such models.     

Prognostic score matching methods for estimating the average effect of a non-reversible binary time-dependent treatment on the survival function

In evaluating the benefit of a treatment on survival, it is often of interest to compare post-treatment survival with the survival function that would have been observed in the absence of treatment. In many practical settings, treatment is time-dependent in the sense that subjects typically begin follow-up untreated, with some going on to receive treatment at some later time point. In observational studies, treatment is not assigned at random and, therefore, may depend on various patient characteristics. We have developed semi-parametric matching methods to estimate the average treatment effect on the treated (ATT) with respect to survival probability and restricted mean survival time. Matching is based on a prognostic score which reflects each patient’s death hazard in the absence of treatment. Specifically, each treated patient is matched with multiple as-yet-untreated patients with similar prognostic scores. The matched sets do not need to be of equal size, since each matched control is weighted in order to preserve risk score balancing across treated and untreated groups. After matching, we estimate the ATT non-parametrically by contrasting pre- and post-treatment weighted Nelson–Aalen survival curves. A closed-form variance is proposed and shown to work well in simulation studies. The proposed methods are applied to national organ transplant registry data.

     

Continuous threshold models with two-way interactions in survival analysis

Proportional hazards model with the biomarker–treatment interaction plays an important role in the survival analysis of the subset treatment effect. A threshold parameter for a continuous biomarker variable defines the subset of patients who can benefit or lose from a certain new treatment. In this article, we focus on a continuous threshold effect using the rectified linear unit and propose a gradient descent method to obtain the maximum likelihood estimation of the regression coefficients and the threshold parameter simultaneously. Under certain regularity conditions, we prove the consistency, asymptotic normality and provide a robust estimate of the covariance matrix when the model is misspecified. To illustrate the finite sample properties of the proposed methods, we simulate data to evaluate the empirical biases, the standard errors and the coverage probabilities for both the correctly specified models and misspecified models. The proposed continuous threshold model is applied to a prostate cancer data with serum prostatic acid phosphatase as a biomarker.     

Confidence Bands for the Difference of Two Survival Curves Under Proportional Hazards Model

A common approach to testing for differences between the survival rates of two therapies is to use a proportional hazards regression model which allows for an adjustment of the two survival functions for any imbalance in prognostic factors in the comparison. When the relative risk of one treatment to the other is not constant over time the question of which therapy has a survival advantage is difficult to determine from the Cox model. An alternative approach to this problem is to plot the difference between the two predicted survival functions with a confidence band that provides information about when these two treatments differ. Such a band will depend on the covariate values of a given patient. In this paper we show how to construct a confidence band for the difference of two survival functions based on the proportional hazards model. A simulation approach is used to generate the bands. This approach is used to compare the survival probabilities of chemotherapy and allogeneic bone marrow transplants for chronic leukemia.     

Weighted Mean Survival Test Statistics: a Class of Distance Tests for Censored Survival Data

A class of test statistics is introduced which is sensitive against the alternative of stochastic ordering in the two-sample censored data problem. The test statistics for evaluating a cumulative weighted difference in survival distributions are developed while taking into account the imbalances in base-line covariates between two groups. This procedure can be used to test the null hypothesis of no treatment effect, especially when base-line hazards cross and prognostic covariates need to be adjusted. The statistics are semiparametric, not rank based, and can be written as integrated weighted differences in estimated survival functions, where these survival estimates are adjusted for covariate imbalances. The asymptotic distribution theory of the tests is developed, yielding test procedures that are shown to be consistent under a fixed alternative. The choice of weight function is discussed and relies on stability and interpretability considerations. An example taken from a clinical trial for acquired immune deficiency syndrome is presented.     

Choice of parametric models in survival analysis: applications to monotherapy for epilepsy and cerebral palsy

Summary. In the analysis of medical survival data, semiparametric proportional hazards models are widely used. When the proportional hazards assumption is not tenable, these models will not be suitable. Other models for covariate effects can be useful. In particular, we consider accelerated life models, in which the effect of covariates is to scale the quantiles of the base-line distribution. Solomon and Hutton have suggested that there is some robustness to misspecification of survival regression models. They showed that the relative importance of covariates is preserved under misspecification with assumptions of small coefficients and orthogonal transformation of covariates. We elucidate these results by applications to data from five trials which compare two common anti-epileptic drugs (carbamazepine versus sodium valporate monotherapy for epilepsy) and to survival of a cohort of people with cerebral palsy. Results on the robustness against model misspecification depend on the assumptions of small coefficients and on the underlying distribution of the data. These results hold in cerebral palsy but do not hold in epilepsy data which have early high hazard rates. The orthogonality of coefficients is not important. However, the choice of model is important for an estimation of the magnitude of effects, particularly if the base-line shape parameter indicates high initial hazard rates.     

Improved precision in the analysis of randomized trials with survival outcomes,without assuming proportional hazards

We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the Kaplan–Meier and inverse probability weighted estimators, under identifiability conditions; (iii) it remains consistent under violations of independent censoring (unlike the Kaplan–Meier estimator) when either the censoring or survival distributions, conditional on covariates, are estimated consistently; and (iv) it achieves the nonparametric efficiency bound when both of these distributions are consistently estimated. We illustrate the performance of our method using simulations based on resampling data from a completed, phase 3 randomized clinical trial of a new surgical treatment for stroke; the proposed estimator achieves a 12% gain in relative efficiency compared to the Kaplan–Meier estimator. The proposed estimator has potential advantages over existing approaches for randomized trials with time-to-event outcomes, since existing methods either rely on model assumptions that are untenable in many applications, or lack some of the efficiency and consistency properties (i)–(iv). We focus on estimation of the restricted mean survival time, but our methods may be adapted to estimate any treatment effect measure defined as a smooth contrast between the survival curves for each study arm. We provide R code to implement the estimator.