Restricted mean survival time for medical data analysis

Abstract

Restricted mean survival time (RMST), defined as the mean event-free time truncated up to a prespecified time point t, has gained popularity among clinicians. The RMST can be calculated as the area under the survival curve from baseline to time t. Frequentist inference of the RMST with right-censored data has been extensively investigated. In the first part of this thesis, the RMST estimation is extended to accommodate more general censoring types. A nonparametric interval-censored RMST estimator is developed by employing the linear smoothing technique to overcome the ambiguity of the survival function due to interval censoring. Asymptotic properties of the interval-censored RMST estimator are discussed and hypothesis testing procedures are constructed. Extensive simulation studies and two real clinical examples show that the interval-censored RMST estimator can provide accurate estimation and deliver promising performance in detecting survival differences between treatment groups. The second part of this thesis develops a Bayesian nonparametric approach to estimating the RMST. A mixture of Dirichlet processes (MDP) prior is assigned on the distribution function F and a Gibbs sampler is constructed to generate the posterior samples of RMST. The proposed Bayesian MDP estimation of RMST is compared with another Bayesian nonparametric approach using the Dirichlet process mixture model and other frequentist nonparametric counterparts. 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. Two real clinical examples are used to illustrate the application of the proposed Bayesian nonparametric RMST for both right- and interval-censored data. The third part introduces an alternative measure to RMST, called the area above quantile (AAQ). The AAQ can be viewed as the area under the survival curve when projected to the quantile axis compared with the RMST which accumulates survival information along the time axis. The AAQ assesses the average survival time among participants with the survival rate larger than a specified quantile level τ. Asymptotic properties of the AAQ estimator are investigated, for which the asymptotic variance can be approximated by an ϵ-perturbation approach. Hypothesis testing procedures for the AAQ difference at a single quantile or a finite set of quantiles are constructed. Numerical studies and real data analysis show that the proposed AAQ-based test can perform better than the RMST-based and weighted log-rank tests under specific survival patterns. The last part of this thesis proposes a new clinical quantity named reduction in number to treat (RNT), as an alternative to the commonly used number needed to treat (NNT). The RNT is calculated as the difference of the reciprocals of clinical measures of interest between two arms, e.g., response rates for binary endpoints, survival rates and RMSTs for time-to-event endpoints. The RNT can be interpreted as the average reduction in the number of patients to treat for the treatment compared with the control to induce one response. Five real clinical trials are used to illustrate the concept of RNT and compare the performances between RNT and NNT.