On testing proportionality in the Cox regression model by Andersen's plot

Andersen's plot, a graphical method for testing the proportionality assumption in the Cox Regression Model (Cox, 1972 Cox, D.R. (1972). Regression models and life tables (with discussion). J. Royal Stat. Soc. Ser. B 34:187–220. [Google Scholar]), first proposed by Kay (1977 Kay, R. (1977). Proportional hazard regression models and the analysis of censored survival data. Appl. Stat. 26(3):227–237.[Crossref] , [Google Scholar]) and popularized by Andersen (1982 Andersen, P.K. (1982). Testing goodness of fit of Cox's regression and life model. Biometrics 38:67–77.[Crossref], [Web of Science ®] , [Google Scholar]), has been used widely in biomedical research to check the validity of applying this popular regression model in survival analysis. Our theoretical derivation and examples show that the theoretical basis of this method is flawed. The graphical method should not be used in testing the proportionality. Instead, formal analytical methods based on residuals such as Cox–Snell residual and martingale residual should be used in practice.     

Model-free slice screening for ultrahigh-dimensional survival data

For ultrahigh-dimensional data, independent feature screening has been demonstrated both theoretically and empirically to be an effective dimension reduction method with low computational demanding. Motivated by the Buckley–James method to accommodate censoring, we propose a fused Kolmogorov–Smirnov filter to screen out the irrelevant dependent variables for ultrahigh-dimensional survival data. The proposed model-free screening method can work with many types of covariates (e.g. continuous, discrete and categorical variables) and is shown to enjoy the sure independent screening property under mild regularity conditions without requiring any moment conditions on covariates. In particular, the proposed procedure can still be powerful when covariates are strongly dependent on each other. We further develop an iterative algorithm to enhance the performance of our method while dealing with the practical situations where some covariates may be marginally unrelated but jointly related to the response. We conduct extensive simulations to evaluate the finite-sample performance of the proposed method, showing that it has favourable exhibition over the existing typical methods. As an illustration, we apply the proposed method to the diffuse large-B-cell lymphoma study.     

An additive Cox model for coronary heart disease study

Existing models for coronary heart disease study use a set of common risk factors to predict the survival time of the disease, via the standard Cox regression model. For complex relationships between the survival time and risk factors, the linear regression specification in the existing Cox model is not flexible enough to accounts for such relationships. Also, the risk factors are actually risky only when they fall in some risk ranges. For more flexibility in modelling and characterize the risk factors more accurately, we study a semi-parametric additive Cox model, using basis splines and LASSO technique. The proposed model is evaluated by simulation studies and is used for the analysis of a real data in the Strong Heart Study.     

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.     

Nonparametric Tests for Stratum Effects in the Cox Model

Thispaper considers the stratified proportional hazards model witha focus on the assessment of stratum effects. The assessmentof such effects is often of interest, for example, in clinicaltrials. In this case, two relevant tests are the test of stratuminteraction with covariates and the test of stratum interactionwith baseline hazard functions. For the test of stratum interactionwith covariates, one can use the partial likelihood method (Kalbfleischand Prentice, 1980; Lin, 1994). For the test of stratum interactionwith baseline hazard functions, however, there seems to be noformal test available. We consider this problem and propose aclass of nonparametric tests. The asymptotic distributions ofthe tests are derived using the martingale theory. The proposedtests can also be used for survival comparisons which need tobe adjusted for covariate effects. The method is illustratedwith data from a lung cancer clinical trial.     

Maximum likelihood estimation for tied survival data under Cox regression model via EM-algorithm

We consider tied survival data based on Cox proportional regression model. The standard approaches are the Breslow and Efron approximations and various so called exact methods. All these methods lead to biased estimates when the true underlying model is in fact a Cox model. In this paper we review the methods and suggest a new method based on the missing-data principle using EM-algorithm that leads to a score equation that can be solved directly. This score has mean zero. We also show that all the considered methods have the same asymptotic properties and that there is no loss of asymptotic efficiency when the tie sizes are bounded or even converge to infinity at a given rate. A simulation study is conducted to compare the finite sample properties of the methods.     

Confidence bands for the difference of two survival functions under the additive risk model

In many clinical studies, a commonly encountered problem is to compare the survival probabilities of two treatments for a given patient with a certain set of covariates, and there is often a need to make adjustments for other covariates that may affect outcomes. One approach is to plot the difference between the two subject-specific predicted survival estimates with a simultaneous confidence band. Such a band will provide useful information about when these two treatments differ and which treatment has a better survival probability. In this paper, we show how to construct such a band based on the additive risk model and we use the martingale central limit theorem to derive its asymptotic distribution. The proposed method is evaluated from a simulation study and is illustrated with two real examples.     

On maximum likelihood estimation of the semi-parametric Cox model with time-varying covariates

Including time-varying covariates is a popular extension to the Cox model and a suitable approach for dealing with non-proportional hazards. However, partial likelihood (PL) estimation of this model has three shortcomings: (i) estimated regression coefficients can be less accurate in small samples with heavy censoring; (ii) the baseline hazard is not directly estimated and (iii) a covariance matrix for both the regression coefficients and the baseline hazard is not easily produced.We address these by developing a maximum likelihood (ML) approach to jointly estimate regression coefficients and baseline hazard using a constrained optimisation ensuring the latter''s non-negativity. We demonstrate asymptotic properties of these estimates and show via simulation their increased accuracy compared to PL estimates in small samples and show our method produces smoother baseline hazard estimates than the Breslow estimator.Finally, we apply our method to two examples, including an important real-world financial example to estimate time to default for retail home loans. We demonstrate using our ML estimate for the baseline hazard can give much clearer corroboratory evidence of the ‘humped hazard’, whereby the risk of loan default rises to a peak and then later falls.     

Cox Regression with Covariate Measurement Error

This article deals with parameter estimation in the Cox proportional hazards model when covariates are measured with error. We consider both the classical additive measurement error model and a more general model which represents the mis-measured version of the covariate as an arbitrary linear function of the true covariate plus random noise. Only moment conditions are imposed on the distributions of the covariates and measurement error. Under the assumption that the covariates are measured precisely for a validation set, we develop a class of estimating equations for the vector-valued regression parameter by correcting the partial likelihood score function. The resultant estimators are proven to be consistent and asymptotically normal with easily estimated variances. Furthermore, a corrected version of the Breslow estimator for the cumulative hazard function is developed, which is shown to be uniformly consistent and, upon proper normalization, converges weakly to a zero-mean Gaussian process. Simulation studies indicate that the asymptotic approximations work well for practical sample sizes. The situation in which replicate measurements (instead of a validation set) are available is also studied.     

General partially linear additive transformation model with right-censored data

We propose a class of general partially linear additive transformation models (GPLATM) with right-censored survival data in this work. The class of models are flexible enough to cover many commonly used parametric and nonparametric survival analysis models as its special cases. Based on the B spline interpolation technique, we estimate the unknown regression parameters and functions by the maximum marginal likelihood estimation method. One important feature of the estimation procedure is that it does not need the baseline and censoring cumulative density distributions. Some numerical studies illustrate that this procedure can work very well for the moderate sample size.     

Expected Survival in the Cox Model

The survival in a group of patients is calculated using a Kaplan–Meier curve and the hypothesis that the survival can be predicted by a specified Cox hazard is studied. This Cox hazard is obtained from a previous study of similar patients. A simple test of the hypothesis is constructed by comparing a suitable average of the individual predicted survival curves with the observed survival. Three averaging procedures are presented; "direct adjusted survival curve", "Bonsel's survival curve" and "expected survival curve". Consistency and asymptotic distribution properties of the comparisons are discussed.     

Full likelihood inference in the Cox model with LTRC data when covariates are discrete

For the regression parameter β in the Cox model, there have been several estimates based on different types of approximated likelihood. For right-censored data, Ren and Zhou [Full likelihood inferences in the Cox model: an empirical approach. Ann Inst Statist Math. 2011;63:1005–1018] derive the full likelihood function for (β, F₀), where F₀ is the baseline distribution function in the Cox model. In this article, we extend their results to left-truncated and right-censored data with discrete covariates. Using the empirical likelihood parameterization, we obtain the full-profile likelihood function for β when covariates are discrete. Simulation results indicate that the maximum likelihood estimator outperforms Cox's partial likelihood estimator in finite samples.     

A study of regression calibration in a partially observed stratified Cox model

Regression calibration is a simple method for estimating regression models when covariate data are missing for some study subjects. It consists in replacing an unobserved covariate by an estimator of its conditional expectation given available covariates. Regression calibration has recently been investigated in various regression models such as the linear, generalized linear, and proportional hazards models. The aim of this paper is to investigate the appropriateness of this method for estimating the stratified Cox regression model with missing values of the covariate defining the strata. Despite its practical relevance, this problem has not yet been discussed in the literature. Asymptotic distribution theory is developed for the regression calibration estimator in this setting. A simulation study is also conducted to investigate the properties of this estimator.     

Regression analysis of multivariate current status data under a varying coefficients additive hazards frailty model

This article discusses regression analysis of multivariate current status failure time data for which the observation time may be related to the underlying survival time. A local partial likelihood technique is used to estimate the varying coefficient covariate effect functions under the additive hazards frailty model. The asymptotic properties of the proposed estimators are established. An extensive simulation study is conducted for the evaluation of the proposed procedure, the results of which indicate that the proposed method works well in practice. Also, a real data study is provided to illustrate the performance of the proposed method.     

Cox Regression with Incomplete Covariate Measurements using the EM-algorithm

Ibrahim (1990) used the EM-algorithm to obtain maximum likelihood estimates of the regression parameters in generalized linear models with partially missing covariates. The technique was termed EM by the method of weights. In this paper, we generalize this technique to Cox regression analysis with missing values in the covariates. We specify a full model letting the unobserved covariate values be random and then maximize the observed likelihood. The asymptotic covariance matrix is estimated by the inverse information matrix. The missing data are allowed to be missing at random but also the non-ignorable non-response situation may in principle be considered. Simulation studies indicate that the proposed method is more efficient than the method suggested by Paik & Tsai (1997). We apply the procedure to a clinical trials example with six covariates with three of them having missing values.     

Variable selection by ensembles for the Cox model

Using Cox regression as the main platform, we study the ensemble approach for variable selection. We use a popular real-data example as well as simulated data with various censoring levels to illustrate the usefulness of the ensemble approach, and study the nature of these ensembles in terms of their strength and diversity. By relating these characteristics to the ensemble's selection accuracy, we provide useful insights for how to choose among different ensemble strategies, as well as guidelines for thinking about how to design more effective ensembles.     

Fitting Cox Models with Doubly Censored Data Using Spline‐Based Sieve Marginal Likelihood

In some applications, the failure time of interest is the time from an originating event to a failure event while both event times are interval censored. We propose fitting Cox proportional hazards models to this type of data using a spline‐based sieve maximum marginal likelihood, where the time to the originating event is integrated out in the empirical likelihood function of the failure time of interest. This greatly reduces the complexity of the objective function compared with the fully semiparametric likelihood. The dependence of the time of interest on time to the originating event is induced by including the latter as a covariate in the proportional hazards model for the failure time of interest. The use of splines results in a higher rate of convergence of the estimator of the baseline hazard function compared with the usual non‐parametric estimator. The computation of the estimator is facilitated by a multiple imputation approach. Asymptotic theory is established and a simulation study is conducted to assess its finite sample performance. It is also applied to analyzing a real data set on AIDS incubation time.     

Estimation for generalized partially functional linear additive regression model

In practice, it is not uncommon to encounter the situation that a discrete response is related to both a functional random variable and multiple real-value random variables whose impact on the response is nonlinear. In this paper, we consider the generalized partial functional linear additive models (GPFLAM) and present the estimation procedure. In GPFLAM, the nonparametric functions are approximated by polynomial splines and the infinite slope function is estimated based on the principal component basis function approximations. We obtain the estimator by maximizing the quasi-likelihood function. We investigate the finite sample properties of the estimation procedure via Monte Carlo simulation studies and illustrate our proposed model by a real data analysis.