A New Estimator for Cox Proportional Hazard Regression Model in Presence of Collinearity

We propose a new approach to estimate the parameters of the Cox proportional hazards model in the presence of collinearity. Generally, a maximum partial likelihood estimator is used to estimate parameters for the Cox proportional hazards model. However, the maximum partial likelihood estimators can be seriously affected by the presence of collinearity since the parameter estimates result in large variances.

In this study, we develop a Liu-type estimator for Cox proportional hazards model parameters and compare it with a ridge regression estimator based on the scalar mean squared error (MSE). Finally, we evaluate its performance through a simulation study.     

A Note on the Estimate of Treatment Effect from a Cox Regression Model When the Proportionality Assumption Is Violated

ABSTRACT

Cox proportional hazards regression model has been widely used to estimate the effect of a prognostic factor on a time-to-event outcome. In a survey of survival analyses in cancer journals, it was found that only 5% of studies using Cox proportional hazards model attempted to verify the underlying assumption. Usually an estimate of the treatment effect from fitting a Cox model was reported without validation of the proportionality assumption. It is not clear how such an estimate should be interpreted if the proportionality assumption is violated. In this article, we show that the estimate of treatment effect from a Cox regression model can be interpreted as a weighted average of the log-scaled hazard ratio over the duration of study. A hypothetic example is used to explain the weights.     

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.     

Hazards Regression for Cancer Studies

The complication in analyzing tumor data is that the tumors detected in a screening program tend to be slowly progressive tumors, which is the so-called length-biased sampling that is inherent in screening studies. Under the assumption that all subjects have the same tumor growth function, Ghosh (2008 Ghosh , D. ( 2008 ). Proportional hazards regression for cancer studies . Biometrics 64 : 141 – 148 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) developed estimation procedures for proportional hazards model. In this article, by modeling growth function as a function of covariates, we demonstrate that Ghosh (2008 Ghosh , D. ( 2008 ). Proportional hazards regression for cancer studies . Biometrics 64 : 141 – 148 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar])'s approach can be extended to the case when each subject has a specific growth function. A simulation study is conducted to demonstrate the potential usefulness of the proposed estimators for the regression parameters in the proportional and additive hazards model.     

Maximum Likelihood Estimation for Cox's Regression Model Under Case–Cohort Sampling

Abstract.  Case–cohort sampling aims at reducing the data sampling and costs of large cohort studies. It is therefore important to estimate the parameters of interest as efficiently as possible. We present a maximum likelihood estimator (MLE) for a case–cohort study based on the proportional hazards assumption. The estimator shows finite sample properties that improve on those by the Self & Prentice [Ann. Statist. 16 (1988)] estimator. The size of the gain by the MLE varies with the level of the disease incidence and the variability of the relative risk over the considered population. The gain tends to be small when the disease incidence is low. The MLE is found by a simple EM algorithm that is easy to implement. Standard errors are estimated by a profile likelihood approach based on EM-aided differentiation.     

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.     

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.     

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.     

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.     

A Proportional Hazards Regression Model for the Subdistribution with Covariates‐adjusted Censoring Weight for Competing Risks Data

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate‐dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate‐dependent censoring. We consider a covariate‐adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate‐adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate‐adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research. Here, cancer relapse and death in complete remission are two competing risks.     

A Study of Suicide Risk Using a Cox Cure Model via a Retrospective Sampling and Multiple Imputation

The incidence of suicide is an example of rare event. A Cox cure model is adopted to examine the suicide risk of a cohort study of a sample of 65,000 Personal Emergency Link users over a 10-year observation period. Our objective is to investigate the effect of personal covariates on the failure time to suicide as well as the long-term survival from suicide. Based on the Cox cure model, a new and efficient estimation approach using retrospective sampling and multiple imputation was used. Compared with the standard Cox proportional hazards model, the Cox cure model provides a new perspective on analyzing the suicide data.     

A multicollinearity diagnostic for models fit to censored data

A multicollinearity diagnostic is discussed for parametric models fit to censored data. The models considered include the Weibull, exponential and lognormal models as well as the Cox proportional hazards model. This diagnostic is an extension of the diagnostic proposed by Belsley, Kuh, and Welsch (1980). The diagnostic is based on the condition indicies and variance proportions of the variance covariance matrix. Its use and properties are studied through a series of examples. The effect of centering variables included in model is also discussed.     

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.     

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。     

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.     

The Dantzig Selector in Cox's Proportional Hazards Model

Abstract. The Dantzig selector (DS) is a recent approach of estimation in high‐dimensional linear regression models with a large number of explanatory variables and a relatively small number of observations. As in the least absolute shrinkage and selection operator (LASSO), this approach sets certain regression coefficients exactly to zero, thus performing variable selection. However, such a framework, contrary to the LASSO, has never been used in regression models for survival data with censoring. A key motivation of this article is to study the estimation problem for Cox's proportional hazards (PH) function regression models using a framework that extends the theory, the computational advantages and the optimal asymptotic rate properties of the DS to the class of Cox's PH under appropriate sparsity scenarios. We perform a detailed simulation study to compare our approach with other methods and illustrate it on a well‐known microarray gene expression data set for predicting survival from gene expressions.     

Heteroscedastic Regression Models and Applications to Off-line Quality Control

We discuss in the present paper the analysis of heteroscedastic regression models and their applications to off-line quality control problems. It is well known that the method of pseudo-likelihood is usually preferred to full maximum likelihood since estimators of the parameters in the regression function obtained are more robust to misspecification of the variance function. Despite its popularity, however, existing theoretical results are difficult to apply and are of limited use in many applications. Using more recent results in estimating equations, we obtain an efficient algorithm for computing the pseudo-likelihood estimator with desirable convergence properties and also derive simple, explicit and easy to apply asymptotic results. These results are used to look in detail at variance minimization in off-line quality control, yielding techniques of inferences for the optimized design parameter. In application of some existing approaches to off-line quality control, such as the dual response methodology, rigorous statistical inference techniques are scarce and difficult to obtain. An example of off-line quality control is presented to discuss the practical aspects involved in the application of the results obtained and to address issues such as data transformation, model building and the optimization of design parameters. The analysis shows very encouraging results, and is seen to be able to unveil some important information not found in previous analyses.     

Hypotheses Tests of Strain-specific Vaccine Efficacy Adjusted for Covariate Effects

In the evaluation of efficacy of a vaccine to protect against disease caused by finitely many diverse infectious pathogens, it is often important to assess if vaccine protection depends on variations of the exposing pathogen. This problem can be formulated under a competing risks model where the endpoint event is the infection and the cause of failure is the infecting strain type determined after the infection is diagnosed. The strain-specific vaccine efficacy is defined as one minus the cause-specific hazard ratio (vaccine/placebo). This paper develops some simple procedures for testing if the vaccine affords protection against various strains and if and how the strain-specific vaccine efficacy depends on the type of exposing strain, adjusting for covariate effects. The Cox proportional hazards model is used to relate the cause-specific outcomes to explanatory variables. The finite sample properties of proposed tests are studied through simulations and are shown to have good performances. The tests developed are applied to the data collected from an oral cholera vaccine trial.     

Relative Curvature Measure for Heteroscedastic or Non Normal Nonlinear Regression

The Bates–Watts relative curvature measure can assess the validity of the linearized approximation in nonlinear regression models. However, it is developed based on an ordinary nonlinear regression in which the observation is assumed to be homoscedastically and normally distributed. In this article, we extend the original Bates–Watts relative curvature measure to one that can be applicable to nonlinear regression with heteroscedastic or non normal data, based on the transformation-both-sides (TBS) approach. In pharmacokinetic models, a diagnostic use of their measures is illustrated. By means of a simulation experiment, the performance of the relative curvature measure for the TBS approach is evaluated.     

Regression Analysis of Left-truncated and Case I Interval-censored Data with the Additive Hazards Model

In recent years the analysis of interval-censored failure time data has attracted a great deal of attention and such data arise in many fields including demographical studies, economic and financial studies, epidemiological studies, social sciences, and tumorigenicity experiments. This is especially the case in medical studies such as clinical trials. In this article, we discuss regression analysis of one type of such data, Case I interval-censored data, in the presence of left-truncation. For the problem, the additive hazards model is employed and the maximum likelihood method is applied for estimations of unknown parameters. In particular, we adopt the sieve estimation approach that approximates the baseline cumulative hazard function by linear functions. The resulting estimates of regression parameters are shown to be consistent and efficient and have an asymptotic normal distribution. An illustrative example is provided.