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.     

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.     

A Cautionary Note on the Analysis of Extreme Data with Cox Regression

This article analyzes a small censored data set to demonstrate the potential dangers of using statistical computing packages without understanding the details of statistical methods. The data, consisting of censored response times with heavy ties in one time point, were analyzed with a Cox regression model utilizing SAS PHREG and BMDP2L procedures. The p values, reported from both SAS PHREG and BMDP2L procedures, for testing the equality of two treatments vary considerably. This article illustrates that (1) the Breslow likelihood used in both BMDP2L and SAS PHREG procedures is too conservative and can have a critical effect on an extreme data set, (2) Wald's test in the SAS PHREG procedure may yield absurd results from most likelihood models, and (3) BMDP2L needs to include more than just the Breslow likelihood in future development.     

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.     

Smoothed Isotonic Estimators of a Monotone Baseline Hazard in the Cox Model

We consider the smoothed maximum likelihood estimator and the smoothed Grenander‐type estimator for a monotone baseline hazard rate λ ₀ in the Cox model. We analyze their asymptotic behaviour and show that they are asymptotically normal at rate n ^{m /(2m +1)}, when λ ₀ is m ≥2 times continuously differentiable, and that both estimators are asymptotically equivalent. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behaviour of the two methods.     

Shape Constrained Non‐parametric Estimators of the Baseline Distribution in Cox Proportional Hazards Model

Abstract. We investigate non‐parametric estimation of a monotone baseline hazard and a decreasing baseline density within the Cox model. Two estimators of a non‐decreasing baseline hazard function are proposed. We derive the non‐parametric maximum likelihood estimator and consider a Grenander type estimator, defined as the left‐hand slope of the greatest convex minorant of the Breslow estimator. We demonstrate that the two estimators are strongly consistent and asymptotically equivalent and derive their common limit distribution at a fixed point. Both estimators of a non‐increasing baseline hazard and their asymptotic properties are obtained in a similar manner. Furthermore, we introduce a Grenander type estimator for a non‐increasing baseline density, defined as the left‐hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function, derived from the Breslow estimator. We show that this estimator is strongly consistent and derive its asymptotic distribution at a fixed point.     

Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.     

Regression for censored survival data with lag effects

We consider regression modeling of survival data subject to right censoring when the full effect of some covariates (e.g. treatment) may be delayed. Several models are proposed, and methods for computing the maximum likelihood estimates of the parameters are described. Consistency and asymptotic normality properties of the estimators are derived. Some numerical examples are used to illustrate the implementation of the modeling and estimation procedures. Finally we apply the theory to interim data from a large scale randomized clinical trial for the prevention of skin cancer.     

Asymptotic properties of the maximum smoothed partial likelihood estimator in the change-plane Cox model

The change-plane Cox model is a popular tool for the subgroup analysis of survival data. Despite the rich literature on this model, there has been limited investigation into the asymptotic properties of the estimators of the finite-dimensional parameter. Particularly, the convergence rate, not to mention the asymptotic distribution, has not been fully characterized for the general model where classification is based on multiple covariates. To bridge this theoretical gap, this study proposes a maximum smoothed partial likelihood estimator and establishes the following asymptotic properties. First, it shows that the convergence rate for the classification parameter can be arbitrarily close to $n^{-1}$ up to a logarithmic factor under a certain condition on covariates and the choice of tuning parameter. Given this convergence rate result, it also establishes the asymptotic normality for the regression parameter.     

Semiparametric Efficient Estimation in the Generalized Odds-Rate Class of Regression Models for Right-Censored Time-to-Event Data

The generalized odds-rate class of regression models for time to event data is indexed by a non-negative constant and assumes thatg(S(t|Z)) = (t) + Zwhere g(s) = log(^-1(s-) for > 0, g₀(s) = log(- log s), S(t|Z) is the survival function of the time to event for an individual with qx1 covariate vector Z, is a qx1 vector of unknown regression parameters, and (t) is some arbitrary increasing function of t. When =0, this model is equivalent to the proportional hazards model and when =1, this model reduces to the proportional odds model. In the presence of right censoring, we construct estimators for and exp((t)) and show that they are consistent and asymptotically normal. In addition, we show that the estimator for is semiparametric efficient in the sense that it attains the semiparametric variance bound.     

基于Cox比例危险模型的制造业财务困境恢复研究

利用生存分析中的Cox比例危险模型,对上市公司陷入财务困境后影响其成功恢复的因素进行实证研究。研究发现:困境公司能否顺利摆脱困境受到财务和公司治理因素的共同影响,财务因素中盈利能力的影响最为显著,而变更管理层对困境公司的恢复也有积极影响;此外,在困境发生后进行资产重组和关联交易的企业更容易恢复。鉴于资产重组和关联交易的目前发展现状,不断完善此类活动的信息披露机制并加强对此类活动的监管显得十分必要。     

Testing for Covariate Effect in the Cox Proportional Hazards Regression Model

This article presents methods for testing covariate effect in the Cox proportional hazards model based on Kullback–Leibler divergence and Renyi's information measure. Renyi's measure is referred to as the information divergence of order γ (γ ≠ 1) between two distributions. In the limiting case γ → 1, Renyi's measure becomes Kullback–Leibler divergence. In our case, the distributions correspond to the baseline and one possibly due to a covariate effect. Our proposed statistics are simple transformations of the parameter vector in the Cox proportional hazards model, and are compared with the Wald, likelihood ratio and score tests that are widely used in practice. Finally, the methods are illustrated using two real-life data sets.     

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.     

Effect of censoring on adjusting for covariates in comparison of survival times

The increase in the variance of the estimate of treatment effect which results from omitting a dichotomous or continuous covariate is quantified as a function of censoring. The efficiency of not adjusting for a covariate is measured by the ratio of the variance obtained with and without adjustment for the covariate. The variance is derived using the Weibull proportional hazards model. Under random censoring, the efficiency of not adjusting for a continuous covariate is an increasing function of the percentage of censored observations.     

Empirical Likelihood Inference for the Cox Model with Time-dependent Coefficients via Local Partial Likelihood

Abstract.  The Cox model with time-dependent coefficients has been studied by a number of authors recently. In this paper, we develop empirical likelihood (EL) pointwise confidence regions for the time-dependent regression coefficients via local partial likelihood smoothing. The EL simultaneous confidence bands for a linear combination of the coefficients are also derived based on the strong approximation methods. The EL ratio is formulated through the local partial log-likelihood for the regression coefficient functions. Our numerical studies indicate that the EL pointwise/simultaneous confidence regions/bands have satisfactory finite sample performances. Compared with the confidence regions derived directly based on the asymptotic normal distribution of the local constant estimator, the EL confidence regions are overall tighter and can better capture the curvature of the underlying regression coefficient functions. Two data sets, the gastric cancer data and the Mayo Clinic primary biliary cirrhosis data, are analysed using the proposed method.     

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.     

Assessing the prediction accuracy of cure in the Cox proportional hazards cure model: an application to breast cancer data

A cure rate model is a survival model incorporating the cure rate with the assumption that the population contains both uncured and cured individuals. It is a powerful statistical tool for prognostic studies, especially in cancer. The cure rate is important for making treatment decisions in clinical practice. The proportional hazards (PH) cure model can predict the cure rate for each patient. This contains a logistic regression component for the cure rate and a Cox regression component to estimate the hazard for uncured patients. A measure for quantifying the predictive accuracy of the cure rate estimated by the Cox PH cure model is required, as there has been a lack of previous research in this area. We used the Cox PH cure model for the breast cancer data; however, the area under the receiver operating characteristic curve (AUC) could not be estimated because many patients were censored. In this study, we used imputation‐based AUCs to assess the predictive accuracy of the cure rate from the PH cure model. We examined the precision of these AUCs using simulation studies. The results demonstrated that the imputation‐based AUCs were estimable and their biases were negligibly small in many cases, although ordinary AUC could not be estimated. Additionally, we introduced the bias‐correction method of imputation‐based AUCs and found that the bias‐corrected estimate successfully compensated the overestimation in the simulation studies. We also illustrated the estimation of the imputation‐based AUCs using breast cancer data. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Uniqueness of the Maximum Likelihood Estimate of the Weibull Distribution Tampered Failure Rate Model

Abstract

Bhattacharyya and Soejoeti (Bhattacharyya, G. K., Soejoeti, Z. A. (1989 Bhattacharyya, G. K. and Soejoeti, Z. A. 1989. Tampered failure rate model for step-stress accelerated life test. Commun. Statist.—Theory Meth., 18(5): 1627–1643. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Tampered failure rate model for step-stress accelerated life test. Commun. Statist.—Theory Meth. 18(5):1627–1643.) pro- posed the TFR model for step-stress accelerated life tests. Under the TFR model, this article proves that the maximum likelihood estimate of the shape parameters is unique for the Weibull distribution in a multiple step-stress accelerated life test, and investigates the accuracy of the maximum likelihood estimate using the Monte-Carlo simulation.     

Correction of the p-Value after Multiple Tests in a Cox Proportional Hazard Model

We consider a situation which is common in epidemiology, in which several transformations of an explanatory variable are tried in a Cox model and the most significant test is retained. The p-value should then be corrected to take account of the multiplicity of tests. Bonferroni method is often too conservative because the tests may be highly positively correlated. We propose an asymptotically exact correction of the p-value. The method uses the fact that the tests are asymptotically normal to compute numerically the distribution of the maximum of several tests. Counting processes theory is used to derive estimators of the correlations between tests. The method is illustrated by a simulation and an analysis of the relation between concentration of aluminum in drinking water and risk of dementia.