Variable selection for semiparametric proportional hazards model under progressive Type-II censoring

Variable selection is an effective methodology for dealing with models with numerous covariates. We consider the methods of variable selection for semiparametric Cox proportional hazards model under the progressive Type-II censoring scheme. The Cox proportional hazards model is used to model the influence coefficients of the environmental covariates. By applying Breslow’s “least information” idea, we obtain a profile likelihood function to estimate the coefficients. Lasso-type penalized profile likelihood estimation as well as stepwise variable selection method are explored as means to find the important covariates. Numerical simulations are conducted and Veteran’s Administration Lung Cancer data are exploited to evaluate the performance of the proposed method.     

Smoothed empirical likelihood confidence intervals for the relative distribution with left‐truncated and right‐censored data

The study of differences among groups is an interesting statistical topic in many applied fields. It is very common in this context to have data that are subject to mechanisms of loss of information, such as censoring and truncation. In the setting of a two‐sample problem with data subject to left truncation and right censoring, we develop an empirical likelihood method to do inference for the relative distribution. We obtain a nonparametric generalization of Wilks' theorem and construct nonparametric pointwise confidence intervals for the relative distribution. Finally, we analyse the coverage probability and length of these confidence intervals through a simulation study and illustrate their use with a real data set on gastric cancer. The Canadian Journal of Statistics 38: 453–473; 2010 © 2010 Statistical Society of Canada     

Imputation for statistical inference with coarse data

Coarse data is a general type of incomplete data that includes grouped data, censored data, and missing data. The likelihood‐based estimation approach with coarse data is challenging because the likelihood function is in integral form. The Monte Carlo EM algorithm of Wei & Tanner [Wei & Tanner (1990). Journal of the American Statistical Association, 85, 699–704] is adapted to compute the maximum likelihood estimator in the presence of coarse data. Stochastic coarse data is also covered and the computation can be implemented using the parametric fractional imputation method proposed by Kim [Kim (2011). Biometrika, 98, 119–132]. Results from a limited simulation study are presented. The proposed method is also applied to the Korean Longitudinal Study of Aging (KLoSA). The Canadian Journal of Statistics 40: 604–618; 2012 © 2012 Statistical Society of Canada     

Consistency of The MMGLE under the piecewise proportional hazards models with interval-censored data

Wong et al. [(2018), ‘Piece-wise Proportional Hazards Models with Interval-censored Data’, Journal of Statistical Computation and Simulation, 88, 140–155] studied the piecewise proportional hazards (PWPH) model with interval-censored (IC) data under the distribution-free set-up. It is well known that the partial likelihood approach is not applicable for IC data, and Wong et al. (2018) showed that the standard generalised likelihood approach does not work either. They proposed the maximum modified generalised likelihood estimator (MMGLE) and the simulation results suggest that the MMGLE is consistent. We establish the consistency and asymptotically normality of the MMGLE.     

An Estimator of the Survival Function Basedon the Semi-Markov Model Under Dependent Censorship

Lee and Wolfe (Biometrics vol. 54 pp. 1176–1178, 1998) proposed the two-stage sampling design for testing the assumption of independent censoring, which involves further follow-up of a subset of lost-to-follow-up censored subjects. They also proposed an adjusted estimator for the survivor function for a proportional hazards model under the dependent censoring model. In this paper, a new estimator for the survivor function is proposed for the semi-Markov model under the dependent censorship on the basis of the two-stage sampling data. The consistency and the asymptotic distribution of the proposed estimator are derived. The estimation procedure is illustrated with an example of lung cancer clinical trial and simulation results are reported of the mean squared errors of estimators under a proportional hazards and two different nonproportional hazards models.     

Adjusting regression attenuation in the Cox proportional hazards model

In the measurement error Cox proportional hazards model, the naive maximum partial likelihood estimator (MPLE) is asymptotically biased. In this paper, we give the formula of the asymptotic bias for the additive measurement error Cox model. By adjusting for this error, we derive an adjusted MPLE that is less biased. The bias can be further reduced by adjusting for the estimator second and even third time. This estimator has the advantage of being easy to apply. The performance of the proposed estimator is evaluated through a simulation study.     

Estimation of treatment effects based on possibly misspecified Cox regression

In randomized clinical trials, a treatment effect on a time-to-event endpoint is often estimated by the Cox proportional hazards model. The maximum partial likelihood estimator does not make sense if the proportional hazard assumption is violated. Xu and O'Quigley (Biostatistics 1:423-439, 2000) proposed an estimating equation, which provides an interpretable estimator for the treatment effect under model misspecification. Namely it provides a consistent estimator for the log-hazard ratio among the treatment groups if the model is correctly specified, and it is interpreted as an average log-hazard ratio over time even if misspecified. However, the method requires the assumption that censoring is independent of treatment group, which is more restricted than that for the maximum partial likelihood estimator and is often violated in practice. In this paper, we propose an alternative estimating equation. Our method provides an estimator of the same property as that of Xu and O'Quigley under the usual assumption for the maximum partial likelihood estimation. We show that our estimator is consistent and asymptotically normal, and derive a consistent estimator of the asymptotic variance. If the proportional hazards assumption holds, the efficiency of the estimator can be improved by applying the covariate adjustment method based on the semiparametric theory proposed by Lu and Tsiatis (Biometrika 95:679-694, 2008).     

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.     

Proportional cause-specific reversed hazards model

The proportional reversed hazards model explains the multiplicative effect of covariates on the baseline reversed hazard rate function of lifetimes. In the present study, we introduce a proportional cause-specific reversed hazards model. The proposed regression model facilitates the analysis of failure time data with multiple causes of failure under left censoring. We estimate the regression parameters using a partial likelihood approach. We provide Breslow's type estimators for the cumulative cause-specific reversed hazard rate functions. Asymptotic properties of the estimators are discussed. Simulation studies are conducted to assess their performance. We illustrate the applicability of the proposed model using a real data set.     

Regression analysis for a summed missing data problem under an outcome‐dependent sampling scheme

In this paper, we consider a regression analysis for a missing data problem in which the variables of primary interest are unobserved under a general biased sampling scheme, an outcome‐dependent sampling (ODS) design. We propose a semiparametric empirical likelihood method for accessing the association between a continuous outcome response and unobservable interesting factors. Simulation study results show that ODS design can produce more efficient estimators than the simple random design of the same sample size. We demonstrate the proposed approach with a data set from an environmental study for the genetic effects on human lung function in COPD smokers. The Canadian Journal of Statistics 40: 282–303; 2012 © 2012 Statistical Society of Canada     

Local composite partial likelihood estimation for length-biased and right-censored data

Length-biased data, which are often encountered in engineering, economics and epidemiology studies, are generally subject to right censoring caused by the research ending or the follow-up loss. The structure of length-biased data is distinct from conventional survival data, since the independent censoring assumption is often violated due to the biased sampling. In this paper, a proportional hazard model with varying coefficients is considered for the length-biased and right-censored data. A local composite likelihood procedure is put forward for the estimation of unknown coefficient functions in the model, and large sample properties of the proposed estimators are also obtained. Additionally, an extensive simulation studies are conducted to assess the finite sample performance of the proposed method and a data set from the Academy Awards is analyzed.     

Evaluating functional covariate‐environment interactions in the Cox regression model

Children exposed to mixtures of endocrine disrupting compounds such as phthalates are at high risk of experiencing significant friction in their growth and sexual maturation. This article is primarily motivated by a study that aims to assess the toxicants‐modified effects of risk factors related to the hazards of early or delayed onset of puberty among children living in Mexico City. To address the hypothesis of potential nonlinear modification of covariate effects, we propose a new Cox regression model with multiple functional covariate‐environment interactions, which allows covariate effects to be altered nonlinearly by mixtures of exposed toxicants. This new class of models is rather flexible and includes many existing semiparametric Cox models as special cases. To achieve efficient estimation, we develop the global partial likelihood method of inference, in which we establish key large‐sample results, including estimation consistency, asymptotic normality, semiparametric efficiency and the generalized likelihood ratio test for both parameters and nonparametric functions. The proposed methodology is examined via simulation studies and applied to the analysis of the motivating data, where maternal exposures to phthalates during the third trimester of pregnancy are found to be important risk modifiers for the age of attaining the first stage of puberty. The Canadian Journal of Statistics 47: 204–221; 2019 © 2019 Statistical Society of Canada     

Proportional hazards regression with interval censored data using an inverse probability weight

The prevalence of interval censored data is increasing in medical studies due to the growing use of biomarkers to define a disease progression endpoint. Interval censoring results from periodic monitoring of the progression status. For example, disease progression is established in the interval between the clinic visit where progression is recorded and the prior clinic visit where there was no evidence of disease progression. A methodology is proposed for estimation and inference on the regression coefficients in the Cox proportional hazards model with interval censored data. The methodology is based on estimating equations and uses an inverse probability weight to select event time pairs where the ordering is unambiguous. Simulations are performed to examine the finite sample properties of the estimate and a colon cancer data set is used to demonstrate its performance relative to the conventional partial likelihood estimate that ignores the interval censoring.     

Generalized Supremum Tests for the Equality of Cause Specific Hazard Rates

In this paper we propose two new classes of asymptotically distribution-free Renyi-type tests for testing the equality of two risks in a competing risk model with possible censoring. This work extends the work of Aly, Kochar and McKeague [1994, Journal of American Statistical Association, 89, 994–999] and many of the existing tests for this problem belong to these newly proposed classes. The asymptotic properties of the proposed tests are investigated. Simulation studies are done to compare the performance with existing tests. A competing risks data set is analyzed to demonstrate the usefulness of the procedure.     

SIMEX method for censored quantile regression with measurement error

Censored quantile regression serves as an important supplement to the Cox proportional hazards model in survival analysis. In addition to being exposed to censoring, some covariates may subject to measurement error. This leads to substantially biased estimate without taking this error into account. The SIMulation-EXtrapolation (SIMEX) method is an effective tool to handle the measurement error issue. We extend the SIMEX approach to the censored quantile regression with covariate measurement error. The algorithm is assessed via extensive simulations. A lung cancer study is analyzed to verify the validation of the proposed method.     

Partial least squares Cox regression for genome-wide data

Most methods for survival prediction from high-dimensional genomic data combine the Cox proportional hazards model with some technique of dimension reduction, such as partial least squares regression (PLS). Applying PLS to the Cox model is not entirely straightforward, and multiple approaches have been proposed. The method of Park et al. (Bioinformatics 18(Suppl. 1):S120–S127, 2002) uses a reformulation of the Cox likelihood to a Poisson type likelihood, thereby enabling estimation by iteratively reweighted partial least squares for generalized linear models. We propose a modification of the method of park et al. (2002) such that estimates of the baseline hazard and the gene effects are obtained in separate steps. The resulting method has several advantages over the method of park et al. (2002) and other existing Cox PLS approaches, as it allows for estimation of survival probabilities for new patients, enables a less memory-demanding estimation procedure, and allows for incorporation of lower-dimensional non-genomic variables like disease grade and tumor thickness. We also propose to combine our Cox PLS method with an initial gene selection step in which genes are ordered by their Cox score and only the highest-ranking k% of the genes are retained, obtaining a so-called supervised partial least squares regression method. In simulations, both the unsupervised and the supervised version outperform other Cox PLS methods.     

Additive hazards regression with censoring indicators missing at random

In this article, the authors consider a semiparametric additive hazards regression model for right‐censored data that allows some censoring indicators to be missing at random. They develop a class of estimating equations and use an inverse probability weighted approach to estimate the regression parameters. Nonparametric smoothing techniques are employed to estimate the probability of non‐missingness and the conditional probability of an uncensored observation. The asymptotic properties of the resulting estimators are derived. Simulation studies show that the proposed estimators perform well. They motivate and illustrate their methods with data from a brain cancer clinical trial. The Canadian Journal of Statistics 38: 333–351; 2010 © 2010 Statistical Society of Canada     

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.     

Effects of model misspecification on tests of no randomized treatment effect arising from Cox's proportional hazards model

We examine the asymptotic and small sample properties of model-based and robust tests of the null hypothesis of no randomized treatment effect based on the partial likelihood arising from an arbitrarily misspecified Cox proportional hazards model. When the distribution of the censoring variable is either conditionally independent of the treatment group given covariates or conditionally independent of covariates given the treatment group, the numerators of the partial likelihood treatment score and Wald tests have asymptotic mean equal to 0 under the null hypothesis, regardless of whether or how the Cox model is misspecified. We show that the model-based variance estimators used in the calculation of the model-based tests are not, in general, consistent under model misspecification, yet using analytic considerations and simulations we show that their true sizes can be as close to the nominal value as tests calculated with robust variance estimators. As a special case, we show that the model-based log-rank test is asymptotically valid. When the Cox model is misspecified and the distribution of censoring depends on both treatment group and covariates, the asymptotic distributions of the resulting partial likelihood treatment score statistic and maximum partial likelihood estimator do not, in general, have a zero mean under the null hypothesis. Here neither the fully model-based tests, including the log-rank test, nor the robust tests will be asymptotically valid, and we show through simulations that the distortion to test size can be substantial.     

Conditional maximum likelihood estimation in semiparametric transformation model with LTRC data

Left-truncated data often arise in epidemiology and individual follow-up studies due to a biased sampling plan since subjects with shorter survival times tend to be excluded from the sample. Moreover, the survival time of recruited subjects are often subject to right censoring. In this article, a general class of semiparametric transformation models that include proportional hazards model and proportional odds model as special cases is studied for the analysis of left-truncated and right-censored data. We propose a conditional likelihood approach and develop the conditional maximum likelihood estimators (cMLE) for the regression parameters and cumulative hazard function of these models. The derived score equations for regression parameter and infinite-dimensional function suggest an iterative algorithm for cMLE. The cMLE is shown to be consistent and asymptotically normal. The limiting variances for the estimators can be consistently estimated using the inverse of negative Hessian matrix. Intensive simulation studies are conducted to investigate the performance of the cMLE. An application to the Channing House data is given to illustrate the methodology.