ROC-guided survival trees and ensembles

Tree-based methods are popular nonparametric tools in studying time-to-event outcomes. In this article, we introduce a novel framework for survival trees and ensembles, where the trees partition the dynamic survivor population and can handle time-dependent covariates. Using the idea of randomized tests, we develop generalized time-dependent receiver operating characteristic (ROC) curves for evaluating the performance of survival trees. The tree-building algorithm is guided by decision-theoretic criteria based on ROC, targeting specifically for prediction accuracy. To address the instability issue of a single tree, we propose a novel ensemble procedure based on averaging martingale estimating equations, which is different from existing methods that average the predicted survival or cumulative hazard functions from individual trees. Extensive simulation studies are conducted to examine the performance of the proposed methods. We apply the methods to a study on AIDS for illustration.     

The use of frailty hazard models for unrecognized heterogeneity that interacts with treatment: considerations of efficiency and power

Increasingly, genetic studies of tumors of the same histologic diagnosis are elucidating subtypes that are distinct with respect to clinical endpoints such as response to treatment and survival. This raises concerns about the efficiency of using the simple log-rank test for analysis of treatment effect on survival in studies of possibly heterogeneous tumors. Furthermore, such studies, designed under the assumption of homogeneity, may be severely underpowered. We derive analytic approximations for the asymptotic relative efficiency of the simple log-rank test relative to the optimally weighted log-rank test and for the power of the simple log-rank test when applied to subjects with unobserved heterogeneity, as reflected in a continuous frailty, that may interact with treatment. Numerical studies demonstrate that the simple log-rank test may be quite inefficient if the frailty interacts with treatment. Further, there may be a substantial loss of power in the presence of the frailty with or without an interaction with treatment.     

Combining complete multivariate outcomes with incomplete covariate information: a latent class approach

This work was motivated by the need to combine outcome information from a reference population with risk factor information from a screened subpopulation in a setting where the analytic goal was to study the association between risk factors and multiple binary outcomes. To achieve such an analytic goal, this article proposes a two-stage latent class procedure that first summarizes the commonalities among outcomes using a reference population sample, then analyzes the association between outcomes and risk factors. It develops a pseudo-maximum likelihood approach to estimating model parameters. The performance of the proposed method is evaluated in a simulation study and in an illustrative analysis of data from the Women's Health and Aging Study, a recent investigation of the causes and course of disability in older women. Combining information in the proposed way is found to improve both accuracy and precision in summarizing multiple categorical outcomes, which effectively diminishes ambiguity and bias in making risk factor inferences.     

A note on including time-dependent covariate in regression model for competing risks data

Recently, regression analysis of the cumulative incidence function has gained interest in competing risks data analysis, through the model proposed by Fine and Gray (JASA 1999; 94: 496-509). In this note, we point out that inclusion of time-dependent covariates in this model can lead to serious bias. We illustrate the problems arising in such a context, using bone marrow transplant data as a working example and numerical simulations. Practical advices are given, preventing the misuse of this model.     

Parametric survival models for interval-censored data with time-dependent covariates

We present a parametric family of regression models for interval-censored event-time (survival) data that accomodates both fixed (e.g. baseline) and time-dependent covariates. The model employs a three-parameter family of survival distributions that includes the Weibull, negative binomial, and log-logistic distributions as special cases, and can be applied to data with left, right, interval, or non-censored event times. Standard methods, such as Newton-Raphson, can be employed to estimate the model and the resulting estimates have an asymptotically normal distribution about the true values with a covariance matrix that is consistently estimated by the information function. The deviance function is described to assess model fit and a robust sandwich estimate of the covariance may also be employed to provide asymptotically robust inferences when the model assumptions do not apply. Spline functions may also be employed to allow for non-linear covariates. The model is applied to data from a long-term study of type 1 diabetes to describe the effects of longitudinal measures of glycemia (HbA1c) over time (the time-dependent covariate) on the risk of progression of diabetic retinopathy (eye disease), an interval-censored event-time outcome.     

Relative index of inequality: definition, estimation, and inference

The relative index of inequality (RII) is a commonly used measure of the extent to which the occurrence of an outcome such as chronic illness or early death varies with socioeconomic status or some other background variable. The standard RII estimator applies only to linear variation in incidence rates. In this paper a general definition of the RII is introduced, alternative approaches to point estimation are considered, and a parametric bootstrap method is suggested for the construction of approximate confidence intervals. Estimation based on cubic splines fitted by maximum penalized likelihood is developed in some detail, and the proposed approach handles naturally the commonly needed adjustment for a 'standardizing' covariate such as age. Death rates in a large longitudinal study in England and Wales from 1996-2000 are analyzed in order to illustrate the various methods. A small simulation study explores the relative merits of different estimators. The approach based on cubic splines is found to reduce bias substantially, at the expense of some increase in variance, when variation in incidence rates is nonlinear.