Modeling Mean Residual Life Function Using Scale Mixtures

Abstract

The mean residual life function (mrlf) of a subject is defined as the expected remaining (residual) lifetime of the subject given that the subject has survived up to a given time point. It is well known that under mild regularity conditions, an mrlf determines the probability distribution uniquely. Therefore, the mrlf can be used to formulate a statistical model just as it is done with the survival and hazard functions. In practice, the advantage of the mrlf over the more widely used hazard function lies in its interpretation in many applications where the primary goal is often to characterize the remaining life expectancy of a subject instead of the instantaneous failure rate.

In this thesis, we first develop a smooth nonparametric estimator of the mean residual life function based on a set of right censored observations. The proposed smooth estimator is obtained by a scale mixture of the empirical estimate of the mrlf. The large sample properties of the estimator are established. A simulation study shows that the proposed scale mixture mean residual life function is more efficient in terms of having lower mean squared error (MSE) than some of the existing estimators available in the literature. Further, as the scale mixture mean residual life function has a closed analytical form, it is computationally less demanding for data with a very large sample size compared to other smooth estimators of the mrlf. Thus the scale mixture estimator of the mean residual life function turns out to be both statistically and computationally more efficient.

The scale mixture framework is then extended to the regression model that allows of fixed covariates. The commonly used regression models for the mrlf, such as the proportional mean residual life (PMRL) model and the linear mean residual life (LMRL) model, have limited applications due to ad-hoc restriction on the parameter space. The regression model that we propose does not have any constraint. It turns out that the proposed proportional scaled mean residual life (PSMRL) model is equivalent to the accelerated failure time (AFT) model. We use full likelihood by nonparametrically estimating the baseline mrlf using the smooth scale mixture estimator that we developed earlier. The regression parameters are estimated using an iterative procedure. A simulation study is carried out to assess the properties of the estimates of the regression parameters. We illustrate our regression model by applying it to the well-known Veteran's Administration lung cancer data.

Finally, we incorporate time-dependent covariates into our scale mixture framework by extending the AFT model (or the PSMRL model) using a nonparametric mixture of Weibull distributions. A nonparametric Bayesian approach with the Markov Chain Monte Carlo (MCMC) algorithm is used to make the statistical inference for the regression parameter. Unlike the approaches in the literature, our Bayesian approach is not based on the parametric choice of the functions of time for time-dependent covariates and hence does not suffer from the problem of deterministic bias. Our Bayesian approach is also computationally less demanding and more stable compared to the approaches in the literature. A simulation study is carried out to assess the sampling properties of the estimates of the regression parameters. The application of our Bayesian approach to the TUMOR data demonstrates the effectiveness of our approach.