Clustered restricted mean survival time regression

For multicenter randomized trials or multilevel observational studies, the Cox regression model has long been the primary approach to study the effects of covariates on time-to-event outcomes. A critical assumption of the Cox model is the proportionality of the hazard functions for modeled covariates, violations of which can result in ambiguous interpretations of the hazard ratio estimates. To address this issue, the restricted mean survival time (RMST), defined as the mean survival time up to a fixed time in a target population, has been recommended as a model-free target parameter. In this article, we generalize the RMST regression model to clustered data by directly modeling the RMST as a continuous function of restriction times with covariates while properly accounting for within-cluster correlations to achieve valid inference. The proposed method estimates regression coefficients via weighted generalized estimating equations, coupled with a cluster-robust sandwich variance estimator to achieve asymptotically valid inference with a sufficient number of clusters. In small-sample scenarios where a limited number of clusters are available, however, the proposed sandwich variance estimator can exhibit negative bias in capturing the variability of regression coefficient estimates. To overcome this limitation, we further propose and examine bias-corrected sandwich variance estimators to reduce the negative bias of the cluster-robust sandwich variance estimator. We study the finite-sample operating characteristics of proposed methods through simulations and reanalyze two multicenter randomized trials.     

Bayesian nonparametric analysis of restricted mean survival time

The restricted mean survival time (RMST) evaluates the expectation of survival time truncated by a prespecified time point, because the mean survival time in the presence of censoring is typically not estimable. The frequentist inference procedure for RMST has been widely advocated for comparison of two survival curves, while research from the Bayesian perspective is rather limited. For the RMST of both right- and interval-censored data, we propose Bayesian nonparametric estimation and inference procedures. By assigning a mixture of Dirichlet processes (MDP) prior to the distribution function, we can estimate the posterior distribution of RMST. We also explore another Bayesian nonparametric approach using the Dirichlet process mixture model and make comparisons with the frequentist nonparametric method. Simulation studies demonstrate that the Bayesian nonparametric RMST under diffuse MDP priors leads to robust estimation and under informative priors it can incorporate prior knowledge into the nonparametric estimator. Analysis of real trial examples demonstrates the flexibility and interpretability of the Bayesian nonparametric RMST for both right- and interval-censored data.     

Regression models for the mean of the quality-of-life-adjusted restricted survival time using pseudo-observations

In this research we develop generalized linear regression models for the mean of a quality-of-life-adjusted restricted survival time. Parameter and standard error estimates could be obtained from generalized estimating equations applied to pseudo-observations. Simulation studies with moderate sample sizes are conducted and an example from the International Breast Cancer Study Group Ludwig Trial V is used to illustrate the newly developed methodology.     

On restricted mean time in favor of treatment

The restricted mean time in favor (RMT-IF) of treatment is a nonparametric effect size for complex life history data. It is defined as the net average time the treated spend in a more favorable state than the untreated over a prespecified time window. It generalizes the familiar restricted mean survival time (RMST) from the two-state life–death model to account for intermediate stages in disease progression. The overall estimand can be additively decomposed into stage-wise effects, with the standard RMST as a component. Alternate expressions of the overall and stage-wise estimands as integrals of the marginal survival functions for a sequence of landmark transitioning events allow them to be easily estimated by plug-in Kaplan–Meier estimators. The dynamic profile of the estimated treatment effects as a function of follow-up time can be visualized using a multilayer, cone-shaped “bouquet plot.” Simulation studies under realistic settings show that the RMT-IF meaningfully and accurately quantifies the treatment effect and outperforms traditional tests on time to the first event in statistical efficiency thanks to its fuller utilization of patient data. The new methods are illustrated on a colon cancer trial with relapse and death as outcomes and a cardiovascular trial with recurrent hospitalizations and death as outcomes. The R-package rmt implements the proposed methodology and is publicly available from the Comprehensive R Archive Network (CRAN).     

Time-dependent ROC curves for censored survival data and a diagnostic marker

ROC curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker, X, for a binary disease variable, D. However, many disease outcomes are time dependent, D(t), and ROC curves that vary as a function of time may be more appropriate. A common example of a time-dependent variable is vital status, where D(t) = 1 if a patient has died prior to time t and zero otherwise. We propose summarizing the discrimination potential of a marker X, measured at baseline (t = 0), by calculating ROC curves for cumulative disease or death incidence by time t, which we denote as ROC(t). A typical complexity with survival data is that observations may be censored. Two ROC curve estimators are proposed that can accommodate censored data. A simple estimator is based on using the Kaplan-Meier estimator for each possible subset X > c. However, this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in X. An alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of (X, T), where T represents survival time (Akritas, M. J., 1994, Annals of Statistics 22, 1299-1327). We present an example where ROC(t) is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the ROC(t) curve displays the impact of modifying eligibility criteria for sample size and power in HIV prevention trials.     

Symmetric tests and confidence intervals for survival probabilities and quantiles of censored survival data

We describe existing tests and introduce two new tests concerning the value of a survival function. These tests may be used to construct a confidence interval for the survival probability at a given time or for a quantile of the survival distribution. Simulation studies show that error rates can differ substantially from their nominal values, particularly at survival probabilities close to zero or one. We recommend our new constrained bootstrap test for its good overall performance.     

A group sequential test for survival trials: an alternative to rank-based procedures

A method of interim monitoring is described for survival trials in which the proportional hazards assumption may not hold. This method extends the test statistics based on the cumulative weighted difference in the Kaplan-Meier estimates (Pepe and Fleming, 1989, Biometrics 45, 497-507) to the sequential setting. Therefore, it provides a useful alternative to the group sequential linear rank tests. With an appropriate weight function, the test statistic itself provides an estimator for the cumulative weighted difference in survival probabilities, which is an interpretable measure for the treatment difference, especially when the proportional hazards model fails. The method is illustrated based on the design of a real trial. The operating characteristics are studied through a small simulation.