Proportional Hazards Model for Multivariate Failure Time Data

Multivariate failure time data also referred to as correlated or clustered failure time data, often arise in survival studies when each study subject may experience multiple events. Statistical analysis of such data needs to account for intracluster dependence. In this article, we consider a bivariate proportional hazards model using vector hazard rate, in which the covariates under study have different effect on two components of the vector hazard rate function. Estimation of the parameters as well as base line hazard function are discussed. Properties of the estimators are investigated. We illustrated the method using two real life data. A simulation study is reported to assess the performance of the estimator.     

Estimating Survival Curves Under Proportional Hazards Model with Covariate Measurement Errors

For the Cox proportional hazards model with additive covariate measurement errors, we propose a corrected cumulative baseline hazard estimator that reduces the bias of the na]ve Breslow estimator. We also derive corresponding modified estimators for the hazard functions and the survival functions of individuals with particular covariate values. Using a Monte Carlo technique developed by Lin et al . (1994), we construct confidence bands for such hazard and survival functions.     

An Exact Corrected Log-Likelihood Function for Cox's Proportional Hazards Model under Measurement Error and Some Extensions

Abstract.  This paper studies Cox's proportional hazards model under covariate measurement error. Nakamura's [ Biometrika 77 (1990) 127] methodology of corrected log-likelihood will be applied to the so-called Breslow likelihood, which is, in the absence of measurement error, equivalent to partial likelihood. For a general error model with possibly heteroscedastic and non-normal additive measurement error, corrected estimators of the regression parameter as well as of the baseline hazard rate are obtained. The estimators proposed by Nakamura [Biometrics 48 (1992) 829], Kong et al. [ Scand. J. Statist. 25 (1998) 573] and Kong & Gu [ Statistica Sinica 9 (1999) 953] are re-established in the special cases considered there. This sheds new light on these estimators and justifies them as exact corrected score estimators. Finally, the method will be extended to some variants of the Cox model.     

Tuning Parameter Selection in Cox Proportional Hazards Model with a Diverging Number of Parameters

Regularized variable selection is a powerful tool for identifying the true regression model from a large number of candidates by applying penalties to the objective functions. The penalty functions typically involve a tuning parameter that controls the complexity of the selected model. The ability of the regularized variable selection methods to identify the true model critically depends on the correct choice of the tuning parameter. In this study, we develop a consistent tuning parameter selection method for regularized Cox's proportional hazards model with a diverging number of parameters. The tuning parameter is selected by minimizing the generalized information criterion. We prove that, for any penalty that possesses the oracle property, the proposed tuning parameter selection method identifies the true model with probability approaching one as sample size increases. Its finite sample performance is evaluated by simulations. Its practical use is demonstrated in The Cancer Genome Atlas breast cancer data.     

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Cox's proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Cox's proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.     

Bayesian Model Selection and Averaging in Additive and Proportional Hazards Models

Although Cox proportional hazards regression is the default analysis for time to event data, there is typically uncertainty about whether the effects of a predictor are more appropriately characterized by a multiplicative or additive model. To accommodate this uncertainty, we place a model selection prior on the coefficients in an additive-multiplicative hazards model. This prior assigns positive probability, not only to the model that has both additive and multiplicative effects for each predictor, but also to sub-models corresponding to no association, to only additive effects, and to only proportional effects. The additive component of the model is constrained to ensure non-negative hazards, a condition often violated by current methods. After augmenting the data with Poisson latent variables, the prior is conditionally conjugate, and posterior computation can proceed via an efficient Gibbs sampling algorithm. Simulation study results are presented, and the methodology is illustrated using data from the Framingham heart study.     

A real survival analysis application via variable selection methods for Cox's proportional hazards model

Variable selection is fundamental to high-dimensional statistical modeling in diverse fields of sciences. In our health study, different statistical methods are applied to analyze trauma annual data, collected by 30 General Hospitals in Greece. The dataset consists of 6334 observations and 111 factors that include demographic, transport, and clinical data. The statistical methods employed in this work are the nonconcave penalized likelihood methods, Smoothly Clipped Absolute Deviation, Least Absolute Shrinkage and Selection Operator, and Hard, the maximum partial likelihood estimation method, and the best subset variable selection, adjusted to Cox's proportional hazards model and used to detect possible risk factors, which affect the length of stay in a hospital. A variety of different statistical models are considered, with respect to the combinations of factors while censored observations are present. A comparative survey reveals several differences between results and execution times of each method. Finally, we provide useful biological justification of our results.     

Penalized and Shrinkage Estimation in the Cox Proportional Hazards Model

This article considers the shrinkage estimation procedure in the Cox's proportional hazards regression model when it is suspected that some of the parameters may be restricted to a subspace. We have developed the statistical properties of the shrinkage estimators including asymptotic distributional biases and risks. The shrinkage estimators have much higher relative efficiency than the classical estimator, furthermore, we consider two penalty estimators—the LASSO and adaptive LASSO—and compare their relative performance with that of the shrinkage estimators numerically. A Monte Carlo simulation experiment is conducted for different combinations of irrelevant predictors and the performance of each estimator is evaluated in terms of simulated mean squared error. Simulation study shows that the shrinkage estimators are comparable to the penalty estimators when the number of irrelevant predictors in the model is relatively large. The shrinkage and penalty methods are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

A geometric optimally of Cox's partial likelihood

The paper proves geometric optimality of Cox's partial likelihood score functions via estimating functions. As an illustration, Cox's proportional-hazards model is considered. Les auteurs démontrent l'optimalité g?ométrique des fonctions scores de vraisemblance partielle de Cox au moyen des fonctions d'estimation. Le modéle des risques proportionnels de Cox permet d'illustrer leur résultat.     

A Likelihood Based Estimating Equation for the Clayton–Oakes Model with Marginal Proportional Hazards

Multivariate failure time data arise when data consist of clusters in which the failure times may be dependent. A popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption. In this paper, we consider the Clayton–Oakes model with marginal proportional hazards and use the full model structure to improve on efficiency compared with the independence analysis. We derive a likelihood based estimating equation for the regression parameters as well as for the correlation parameter of the model. We give the large sample properties of the estimators arising from this estimating equation. Finally, we investigate the small sample properties of the estimators through Monte Carlo simulations.     

Local Linear Estimation for Time-Dependent Coefficients in Cox's Regression Models

This article develops a local partial likelihood technique to estimate the time-dependent coefficients in Cox's regression model. The basic idea is a simple extension of the local linear fitting technique used in the scatterplot smoothing. The coefficients are estimated locally based on the partial likelihood in a window around each time point. Multiple time-dependent covariates are incorporated in the local partial likelihood procedure. The procedure is useful as a diagnostic tool and can be used in uncovering time-dependencies or departure from the proportional hazards model. The programming involved in the local partial likelihood estimation is relatively simple and it can be modified with few efforts from the existing programs for the proportional hazards model. The asymptotic properties of the resulting estimator are established and compared with those from the local constant fitting. A consistent estimator of the asymptotic variance is also proposed. The approach is illustrated by a real data set from the study of gastric cancer patients and a simulation study is also presented.     

Variable Selection in Linear Mixed Models Using an Extended Class of Penalties

There is an emerging need to advance linear mixed model technology to include variable selection methods that can simultaneously choose and estimate important effects from a potentially large number of covariates. However, the complex nature of variable selection has made it difficult for it to be incorporated into mixed models. In this paper we extend the well known class of penalties and show that they can be integrated succinctly into a linear mixed model setting. Under mild conditions, the estimator obtained from this mixed model penalised likelihood is shown to be consistent and asymptotically normally distributed. A simulation study reveals that the extended family of penalties achieves varying degrees of estimator shrinkage depending on the value of one of its parameters. The simulation study also shows there is a link between the number of false positives detected and the number of true coefficients when using the same penalty. This new mixed model variable selection (MMVS) technology was applied to a complex wheat quality data set to determine significant quantitative trait loci (QTL).     

Cox模型和寿险退保研究中的比例危险性

利用生存分析研究寿险退保问题是一个很好的工具，因为可以将寿险保单的持续期（persistency duration）视为生存期长，而将保单的退保或失效看作一个“保单生命”的结束，这其中的保单退保或失效就成为生存研究的目标事件。而导致保单失效的因素会有很多，只有通过利用Cox比例危险模型拟合寿险退保数据以分析影响客户退保的原因，并在对Cox模型的比例危险假设进行检验时，发现部分影响因素并不遵守此前提条件，从而推理得到这些影响因素在不同的时间段对客户退保的影响方式不同。也就是说，其影响有短期效应和长期效应之分。     

Covariate Selection for the Semiparametric Additive Risk Model

Abstract.  This paper considers covariate selection for the additive hazards model. This model is particularly simple to study theoretically and its practical implementation has several major advantages to the similar methodology for the proportional hazards model. One complication compared with the proportional model is, however, that there is no simple likelihood to work with. We here study a least squares criterion with desirable properties and show how this criterion can be interpreted as a prediction error. Given this criterion, we define ridge and Lasso estimators as well as an adaptive Lasso and study their large sample properties for the situation where the number of covariates p is smaller than the number of observations. We also show that the adaptive Lasso has the oracle property. In many practical situations, it is more relevant to tackle the situation with large p compared with the number of observations. We do this by studying the properties of the so-called Dantzig selector in the setting of the additive risk model. Specifically, we establish a bound on how close the solution is to a true sparse signal in the case where the number of covariates is large. In a simulation study, we also compare the Dantzig and adaptive Lasso for a moderate to small number of covariates. The methods are applied to a breast cancer data set with gene expression recordings and to the primary biliary cirrhosis clinical data.     

On Estimation and Tests of Time-Varying Effects in the Proportional Hazards Model

Abstract.  Cox's proportional hazards model is routinely used in many applied fields, some times, however, with too little emphasis on the fit of the model. In this paper, we suggest some new tests for investigating whether or not covariate effects vary with time. These tests are a natural and integrated part of an extended version of the Cox model. An important new feature of the suggested test is that time constancy for a specific covariate is examined in a model, where some effects of other covariates are allowed to vary with time and some are constant; thus making successive testing of time-dependency possible. The proposed techniques are illustrated with the well-known Mayo liver disease data, and a small simulation study investigates the finite sample properties of the tests.     

Generalized Proportional Hazards Model Based on Modified Partial Likelihood

The proportional hazards (Cox) model is generalized by assuming that at any moment the ratio of hazard rates is depending not only on values of covariates but also on resources used until this moment. Relations with generalized multiplicative, frailty and linear transformation models are considered. A modified partial likelihood function is proposed, and properties of the estimators are investigated.     

Properties of Added Variable Plots in Cox's Regression Model

The added variable plot is useful for examining the effect of a covariate in regression models. The plot provides information regarding the inclusion of a covariate, and is useful in identifying influential observations on the parameter estimates. Hall et al. (1996) proposed a plot for Cox's proportional hazards model derived by regarding the Cox model as a generalized linear model. This paper proves and discusses properties of this plot. These properties make the plot a valuable tool in model evaluation. Quantities considered include parameter estimates, residuals, leverage, case influence measures and correspondence to previously proposed residuals and diagnostics.     

Validation of A Heteroscedastic Hazards Regression Model

A Cox-type regression model accommodating heteroscedasticity, with a power factor of the baseline cumulative hazard, is investigated for analyzing data with crossing hazards behavior. Since the approach of partial likelihood cannot eliminate the baseline hazard, an overidentified estimating equation (OEE) approach is introduced in the estimation procedure. Its by-product, a model checking statistic, is presented to test for the overall adequacy of the heteroscedastic model. Further, under the heteroscedastic model setting, we propose two statistics to test the proportional hazards assumption. Implementation of this model is illustrated in a data analysis of a cancer clinical trial.     

The Influence of Random Effects on Univariate and Bivariate Discrete Proportional Hazards Models

The random effects survival model has been widely used in the recent literature as a generalization of the continuous proportional hazards model. When a random effect is present, it is known that the hazard rates are generally underestimated in the context of continuous proportional hazards models. This article establishes theorems for the influence of random effects on both univariate and bivariate discrete proportional hazards models.     

The Partial Linear Model in High Dimensions

Partial linear models have been widely used as flexible method for modelling linear components in conjunction with non‐parametric ones. Despite the presence of the non‐parametric part, the linear, parametric part can under certain conditions be estimated with parametric rate. In this paper, we consider a high‐dimensional linear part. We show that it can be estimated with oracle rates, using the least absolute shrinkage and selection operator penalty for the linear part and a smoothness penalty for the nonparametric part.