Tree-augmented Cox proportional hazards models

We study a hybrid model that combines Cox proportional hazards regression with tree-structured modeling. The main idea is to use step functions, provided by a tree structure, to 'augment' Cox (1972) proportional hazards models. The proposed model not only provides a natural assessment of the adequacy of the Cox proportional hazards model but also improves its model fitting without loss of interpretability. Both simulations and an empirical example are provided to illustrate the use of the proposed method.     

Bayesian information criterion for censored survival models

We investigate the Bayesian Information Criterion (BIC) for variable selection in models for censored survival data. Kass and Wasserman (1995, Journal of the American Statistical Association 90, 928-934) showed that BIC provides a close approximation to the Bayes factor when a unit-information prior on the parameter space is used. We propose a revision of the penalty term in BIC so that it is defined in terms of the number of uncensored events instead of the number of observations. For a simple censored data model, this revision results in a better approximation to the exact Bayes factor based on a conjugate unit-information prior. In the Cox proportional hazards regression model, we propose defining BIC in terms of the maximized partial likelihood. Using the number of deaths rather than the number of individuals in the BIC penalty term corresponds to a more realistic prior on the parameter space and is shown to improve predictive performance for assessing stroke risk in the Cardiovascular Health Study.     

Goodness-of-fit tests in proportional hazards models with random effects

This paper deals with testing the functional form of the covariate effects in a Cox proportional hazards model with random effects. We assume that the responses are clustered and incomplete due to right censoring. The estimation of the model under the null (parametric covariate effect) and the alternative (nonparametric effect) is performed using the full marginal likelihood. Under the alternative, the nonparametric covariate effects are estimated using orthogonal expansions. The test statistic is the likelihood ratio statistic, and its distribution is approximated using a bootstrap method. The performance of the proposed testing procedure is studied through simulations. The method is also applied on two real data sets one from biomedical research and one from veterinary medicine.     

Adaptive Lasso for Cox's proportional hazards model

We investigate the variable selection problem for Cox's proportionalhazards model, and propose a unified model selection and estimationprocedure with desired theoretical properties and computationalconvenience. The new method is based on a penalized log partiallikelihood with the adaptively weighted L₁ penalty on regressioncoefficients, providing what we call the adaptive Lasso estimator.The method incorporates different penalties for different coefficients:unimportant variables receive larger penalties than importantones, so that important variables tend to be retained in theselection process, whereas unimportant variables are more likelyto be dropped. Theoretical properties, such as consistency andrate of convergence of the estimator, are studied. We also showthat, with proper choice of regularization parameters, the proposedestimator has the oracle properties. The convex optimizationnature of the method leads to an efficient algorithm. Both simulatedand real examples show that the method performs competitively.     

Flexible maximum likelihood methods for bivariate proportional hazards models

This article presents methodology for multivariate proportional hazards (PH) regression models. The methods employ flexible piecewise constant or spline specifications for baseline hazard functions in either marginal or conditional PH models, along with assumptions about the association among lifetimes. Because the models are parametric, ordinary maximum likelihood can be applied; it is able to deal easily with such data features as interval censoring or sequentially observed lifetimes, unlike existing semiparametric methods. A bivariate Clayton model (1978, Biometrika 65, 141-151) is used to illustrate the approach taken. Because a parametric assumption about association is made, efficiency and robustness comparisons are made between estimation based on the bivariate Clayton model and "working independence" methods that specify only marginal distributions for each lifetime variable.     

Smoothing spline-based score tests for proportional hazards models

We propose "score-type" tests for the proportional hazards assumption and for covariate effects in the Cox model using the natural smoothing spline representation of the corresponding nonparametric functions of time or covariate. The tests are based on the penalized partial likelihood and are derived by viewing the inverse of the smoothing parameter as a variance component and testing an equivalent null hypothesis that the variance component is zero. We show that the tests have a size close to the nominal level and good power against general alternatives, and we apply them to data from a cancer clinical trial.     

Polynomial spline estimation and inference of proportional hazards regression models with flexible relative risk form

The Cox proportional hazards model usually assumes an exponential form for the dependence of the hazard function on covariate variables. However, in practice this assumption may be violated and other relative risk forms may be more appropriate. In this article, we consider the proportional hazards model with an unknown relative risk form. Issues in model interpretation are addressed. We propose a method to estimate the relative risk form and the regression parameters simultaneously by first approximating the logarithm of the relative risk form by a spline, and then employing the maximum partial likelihood estimation. An iterative alternating optimization procedure is developed for efficient implementation. Statistical inference of the regression coefficients and of the relative risk form based on parametric asymptotic theory is discussed. The proposed methods are illustrated using simulation and an application to the Veteran's Administration lung cancer data.     

Hierarchical proportional hazards regression models for highly stratified data

In clinical trials conducted over several data collection centers, the most common statistically defensible analytic method, a stratified Cox model analysis, suffers from two important defects. First, identification of units that are outlying with respect to the baseline hazard is awkward since this hazard is implicit (rather than explicit) in the Cox partial likelihood. Second (and more seriously), identification of modest treatment effects is often difficult since the model fails to acknowledge any similarity across the strata. We consider a number of hierarchical modeling approaches that preserve the integrity of the stratified design while offering a middle ground between traditional stratified and unstratified analyses. We investigate both fully parametric (Weibull) and semiparametric models, the latter based not on the Cox model but on an extension of an idea by Gelfand and Mallick (1995, Biometrics 51, 843-852), which models the integrated baseline hazard as a mixture of monotone functions. We illustrate the methods using data from a recent multicenter AIDS clinical trial, comparing their ease of use, interpretation, and degree of robustness with respect to estimates of both the unit-specific baseline hazards and the treatment effect.     

Validation of the alternating conditional estimation algorithm for estimation of flexible extensions of Cox's proportional hazards model with nonlinear constraints on the parameters

Standard optimization algorithms for maximizing likelihood may not be applicable to the estimation of those flexible multivariable models that are nonlinear in their parameters. For applications where the model's structure permits separating estimation of mutually exclusive subsets of parameters into distinct steps, we propose the alternating conditional estimation (ACE) algorithm. We validate the algorithm, in simulations, for estimation of two flexible extensions of Cox's proportional hazards model where the standard maximum partial likelihood estimation does not apply, with simultaneous modeling of (1) nonlinear and time‐dependent effects of continuous covariates on the hazard, and (2) nonlinear interaction and main effects of the same variable. We also apply the algorithm in real‐life analyses to estimate nonlinear and time‐dependent effects of prognostic factors for mortality in colon cancer. Analyses of both simulated and real‐life data illustrate good statistical properties of the ACE algorithm and its ability to yield new potentially useful insights about the data structure.     

Cox regression model with randomly censored covariates

This paper deals with a Cox proportional hazards regression model, where some covariates of interest are randomly right‐censored. While methods for censored outcomes have become ubiquitous in the literature, methods for censored covariates have thus far received little attention and, for the most part, dealt with the issue of limit‐of‐detection. For randomly censored covariates, an often‐used method is the inefficient complete‐case analysis (CCA) which consists in deleting censored observations in the data analysis. When censoring is not completely independent, the CCA leads to biased and spurious results. Methods for missing covariate data, including type I and type II covariate censoring as well as limit‐of‐detection do not readily apply due to the fundamentally different nature of randomly censored covariates. We develop a novel method for censored covariates using a conditional mean imputation based on either Kaplan–Meier estimates or a Cox proportional hazards model to estimate the effects of these covariates on a time‐to‐event outcome. We evaluate the performance of the proposed method through simulation studies and show that it provides good bias reduction and statistical efficiency. Finally, we illustrate the method using data from the Framingham Heart Study to assess the relationship between offspring and parental age of onset of cardiovascular events.     

Cox Regression in Nested Case–Control Studies with Auxiliary Covariates

Summary Nested case–control (NCC) design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas' maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox's model for NCC data. In this article, we consider a situation in which failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas' estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas' estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms' tumor.     

Marginal proportional hazards models for clustered interval-censored data with time-dependent covariates

The Botswana Combination Prevention Project was a cluster-randomized HIV prevention trial whose follow-up period coincided with Botswana's national adoption of a universal test and treat strategy for HIV management. Of interest is whether, and to what extent, this change in policy modified the preventative effects of the study intervention. To address such questions, we adopt a stratified proportional hazards model for clustered interval-censored data with time-dependent covariates and develop a composite expectation maximization algorithm that facilitates estimation of model parameters without placing parametric assumptions on either the baseline hazard functions or the within-cluster dependence structure. We show that the resulting estimators for the regression parameters are consistent and asymptotically normal. We also propose and provide theoretical justification for the use of the profile composite likelihood function to construct a robust sandwich estimator for the variance. We characterize the finite-sample performance and robustness of these estimators through extensive simulation studies. Finally, we conclude by applying this stratified proportional hazards model to a re-analysis of the Botswana Combination Prevention Project, with the national adoption of a universal test and treat strategy now modeled as a time-dependent covariate.