Estimation in the cox proportional hazards model with left-truncated and interval-censored data

We show that the nonparametric maximum likelihood estimate (NPMLE) of the regression coefficient from the joint likelihood (of the regression coefficient and the baseline survival) works well for the Cox proportional hazards model with left-truncated and interval-censored data, but the NPMLE may underestimate the baseline survival. Two alternatives are also considered: first, the marginal likelihood approach by extending Satten (1996, Biometrika 83, 355-370) to truncated data, where the baseline distribution is eliminated as a nuisance parameter; and second, the monotone maximum likelihood estimate that maximizes the joint likelihood by assuming that the baseline distribution has a nondecreasing hazard function, which was originally proposed to overcome the underestimation of the survival from the NPMLE for left-truncated data without covariates (Tsai, 1988, Biometrika 75, 319-324). The bootstrap is proposed to draw inference. Simulations were conducted to assess their performance. The methods are applied to the Massachusetts Health Care Panel Study data set to compare the probabilities of losing functional independence for male and female seniors.     

Improved semiparametric estimation of the proportional rate model with recurrent event data

Owing to its robustness properties, marginal interpretations, and ease of implementation, the pseudo-partial likelihood method proposed in the seminal papers of Pepe and Cai and Lin et al. has become the default approach for analyzing recurrent event data with Cox-type proportional rate models. However, the construction of the pseudo-partial score function ignores the dependency among recurrent events and thus can be inefficient. An attempt to investigate the asymptotic efficiency of weighted pseudo-partial likelihood estimation found that the optimal weight function involves the unknown variance–covariance process of the recurrent event process and may not have closed-form expression. Thus, instead of deriving the optimal weights, we propose to combine a system of pre-specified weighted pseudo-partial score equations via the generalized method of moments and empirical likelihood estimation. We show that a substantial efficiency gain can be easily achieved without imposing additional model assumptions. More importantly, the proposed estimation procedures can be implemented with existing software. Theoretical and numerical analyses show that the empirical likelihood estimator is more appealing than the generalized method of moments estimator when the sample size is sufficiently large. An analysis of readmission risk in colorectal cancer patients is presented to illustrate the proposed methodology.     

A mixture model for occupational exposure mean testing with a limit of detection

Information from detectable exposure measurements randomly sampled from a left-truncated log-normal distribution may be used to evaluate the distribution of nondetectable values that fall below an analytic limit of detection. If the proportion of nondetects is larger than expected under log normality, alternative models to account for these unobserved data should be considered. We discuss one such model that incorporates a mixture of true zero exposures and a log-normal distribution with possible left censoring, previously considered in a different context by Moulton and Halsey (1995, Biometrics 51, 1570-1578). A particular relationship is demonstrated between maximum likelihood parameter estimates based on this mixture model and those assuming either left-truncated or left-censored data. These results emphasize the need for caution when choosing a model to fit data involving nondetectable values. A one-sided likelihood ratio test for comparing mean exposure under the mixture model to an occupational exposure limit is then developed and evaluated via simulations. An example demonstrates the potential impact of specifying an incorrect model for the nondetectable values.     

Penalized partial likelihood regression for right-censored data with bootstrap selection of the penalty parameter

The Cox proportional hazards model is often used for estimating the association between covariates and a potentially censored failure time, and the corresponding partial likelihood estimators are used for the estimation and prediction of relative risk of failure. However, partial likelihood estimators are unstable and have large variance when collinearity exists among the explanatory variables or when the number of failures is not much greater than the number of covariates of interest. A penalized (log) partial likelihood is proposed to give more accurate relative risk estimators. We show that asymptotically there always exists a penalty parameter for the penalized partial likelihood that reduces mean squared estimation error for log relative risk, and we propose a resampling method to choose the penalty parameter. Simulations and an example show that the bootstrap-selected penalized partial likelihood estimators can, in some instances, have smaller bias than the partial likelihood estimators and have smaller mean squared estimation and prediction errors of log relative risk. These methods are illustrated with a data set in multiple myeloma from the Eastern Cooperative Oncology Group.     

Semiparametric regression analysis of interval-censored data

We propose a semiparametric approach to the proportional hazards regression analysis of interval-censored data. An EM algorithm based on an approximate likelihood leads to an M-step that involves maximizing a standard Cox partial likelihood to estimate regression coefficients and then using the Breslow estimator for the unknown baseline hazards. The E-step takes a particularly simple form because all incomplete data appear as linear terms in the complete-data log likelihood. The algorithm of Turnbull (1976, Journal of the Royal Statistical Society, Series B 38, 290-295) is used to determine times at which the hazard can take positive mass. We found multiple imputation to yield an easily computed variance estimate that appears to be more reliable than asymptotic methods with small to moderately sized data sets. In the right-censored survival setting, the approach reduces to the standard Cox proportional hazards analysis, while the algorithm reduces to the one suggested by Clayton and Cuzick (1985, Applied Statistics 34, 148-156). The method is illustrated on data from the breast cancer cosmetics trial, previously analyzed by Finkelstein (1986, Biometrics 42, 845-854) and several subsequent authors.     

Boosting method for nonlinear transformation models with censored survival data

We propose a general class of nonlinear transformation models for analyzing censored survival data, of which the nonlinear proportional hazards and proportional odds models are special cases. A cubic smoothing spline-based component-wise boosting algorithm is derived to estimate covariate effects nonparametrically using the gradient of the marginal likelihood, that is computed using importance sampling. The proposed method can be applied to survival data with high-dimensional covariates, including the case when the sample size is smaller than the number of predictors. Empirical performance of the proposed method is evaluated via simulations and analysis of a microarray survival data.     

Parametric and Parametrically Smoothed Distribution-Free Proportional Hazard Models with Discrete Data

This paper discusses discrete time proportional hazard models and suggests a new class of flexible hazard functions. Explicitly modeling the discreteness of data is important since standard continuous models are biased; allowing for flexibility in the hazard estimation is desirable since strong parametric restrictions are likely to be similarly misleading. Simulation compare continuous and discrete models when data are generated by grouping and demonstrate that simple approximations recover underlying hazards well and outperform nonparametric maximum likelihood estimates in term of mean squared error.     

Bayesian cure rate frailty models with application to a root canal therapy study

Due to natural or artificial clustering, multivariate survival data often arise in biomedical studies, for example, a dental study involving multiple teeth from each subject. A certain proportion of subjects in the population who are not expected to experience the event of interest are considered to be "cured" or insusceptible. To model correlated or clustered failure time data incorporating a surviving fraction, we propose two forms of cure rate frailty models. One model naturally introduces frailty based on biological considerations while the other is motivated from the Cox proportional hazards frailty model. We formulate the likelihood functions based on piecewise constant hazards and derive the full conditional distributions for Gibbs sampling in the Bayesian paradigm. As opposed to the Cox frailty model, the proposed methods demonstrate great potential in modeling multivariate survival data with a cure fraction. We illustrate the cure rate frailty models with a root canal therapy data set.     

Semiparametric modelling and estimation of covariate-adjusted dependence between bivariate recurrent events

A time-dependent measure, termed the rate ratio, was proposed to assess the local dependence between two types of recurrent event processes in one-sample settings. However, the one-sample work does not consider modeling the dependence by covariates such as subject characteristics and treatments received. The focus of this paper is to understand how and in what magnitude the covariates influence the dependence strength for bivariate recurrent events. We propose the covariate-adjusted rate ratio, a measure of covariate-adjusted dependence. We propose a semiparametric regression model for jointly modeling the frequency and dependence of bivariate recurrent events: the first level is a proportional rates model for the marginal rates and the second level is a proportional rate ratio model for the dependence structure. We develop a pseudo-partial likelihood to estimate the parameters in the proportional rate ratio model. We establish the asymptotic properties of the estimators and evaluate the finite sample performance via simulation studies. We illustrate the proposed models and methods using a soft tissue sarcoma study that examines the effects of initial treatments on the marginal frequencies of local/distant sarcoma recurrence and the dependence structure between the two types of cancer recurrence.     

A nonparametric mixture model for cure rate estimation

Nonparametric methods have attracted less attention than their parametric counterparts for cure rate analysis. In this paper, we study a general nonparametric mixture model. The proportional hazards assumption is employed in modeling the effect of covariates on the failure time of patients who are not cured. The EM algorithm, the marginal likelihood approach, and multiple imputations are employed to estimate parameters of interest in the model. This model extends models and improves estimation methods proposed by other researchers. It also extends Cox's proportional hazards regression model by allowing a proportion of event-free patients and investigating covariate effects on that proportion. The model and its estimation method are investigated by simulations. An application to breast cancer data, including comparisons with previous analyses using a parametric model and an existing nonparametric model by other researchers, confirms the conclusions from the parametric model but not those from the existing nonparametric model.     

Semiparametric transformation models with random effects for joint analysis of recurrent and terminal events

Summary .  We propose a broad class of semiparametric transformation models with random effects for the joint analysis of recurrent events and a terminal event. The transformation models include proportional hazards/intensity and proportional odds models. We estimate the model parameters by the nonparametric maximum likelihood approach. The estimators are shown to be consistent, asymptotically normal, and asymptotically efficient. Simple and stable numerical algorithms are provided to calculate the parameter estimators and to estimate their variances. Extensive simulation studies demonstrate that the proposed inference procedures perform well in realistic settings. Applications to two HIV/AIDS studies are presented.     

Analysis of contingency table data on torus mandibularis using a log linear model

The hypothesized sexual difference in the incidence of torus mandibularis in Eskimoid groups, as well as age and group differences, was studied in two Northwest Territories Eskimo groups and in an Aleut group (examined by Moorrees). The data were analyzed using statistical methods new to the field of anthropology. The method analyzes data sampled from a multinomial distribution. A hierarchy of log linear models are fitted to the cell counts of a contingency table. An iterative proportional fitting procedure is used to obtain expected cell counts under each log linear model. The “goodness of fit” of each model is tested by the log likelihood ratio. This statistic can be partitioned into additive components such that differences between models can be tested. In this way a “best” model, from the hierarchy of models, is chosen. Among these three groups of Eskimos, the incidence of torus mandibularis was not affected by sex but was affected by age and was different between the three groups.     

Joint Models of Longitudinal Data and Recurrent Events with Informative Terminal Event

This article presents semiparametric joint models to analyze longitudinal data with recurrent events (e.g. multiple tumors, repeated hospital admissions) and a terminal event such as death. A broad class of transformation models for the cumulative intensity of the recurrent events and the cumulative hazard of the terminal event is considered, which includes the proportional hazards model and the proportional odds model as special cases. We propose to estimate all the parameters using the nonparametric maximum likelihood estimators (NPMLE). We provide the simple and efficient EM algorithms to implement the proposed inference procedure. Asymptotic properties of the estimators are shown to be asymptotically normal and semiparametrically efficient. Finally, we evaluate the performance of the method through extensive simulation studies and a real-data application.     

Generalized additive models with interval-censored data and time-varying covariates: application to human immunodeficiency virus infection in hemophiliacs

We describe a method for extending smooth nonparametric modeling methods to time-to-event data where the event may be known only to lie within a window of time. Maximum penalized likelihood is used to fit a discrete proportional hazards model that also models the baseline hazard, and left-truncation and time-varying covariates are accommodated. The implementation follows generalized additive modeling conventions, allowing both parametric and smooth terms and specifying the amount of smoothness in terms of the effective degrees of freedom. We illustrate the method on a well-known interval-censored data set on time of human immunodeficiency virus infection in a multicenter study of hemophiliacs. The ability to examine time-varying covariates, not available with previous methods, allows detection and modeling of nonproportional hazards and use of a time-varying covariate that fits the data better and is more plausible than a fixed alternative.     

Proportional hazards model with covariates subject to measurement error.

When covariates of a proportional hazards model are subject to measurement error, the maximum likelihood estimates of regression coefficients based on the partial likelihood are asymptotically biased. Prentice (1982, Biometrika 69, 331-342) presents an example of such bias and suggests a modified partial likelihood. This paper applies the corrected score function method (Nakamura, 1990, Biometrika 77, 127-137) to the proportional hazards model when measurement errors are additive and normally distributed. The result allows a simple correction to the ordinary partial likelihood that yields asymptotically unbiased estimates; the validity of the correction is confirmed via a limited simulation study.     

Modeling late entry bias in survival analysis

In a failure time analysis, we sometimes observe additional study subjects who enter during the study period. These late entries are treated as left-truncated data in the statistical literature. However, with real data, there is a substantial possibility that the delayed entries may have extremely different hazards compared to the other standard subjects. We focus on a situation in which such entry bias might arise in the analysis of survival data. The purpose of the present article is to develop an appropriate methodology for making inference about data including late entries. We construct a model that includes parameters for the effect of delayed entry bias having no specification for the distribution of entry time. We also discuss likelihood inference based on this model and derive the asymptotic behavior of estimates. A simulation study is conducted for a finite sample size in order to compare the analysis results using our method with those using the standard method, where independence between entry time and failure time is assumed. We apply this method to mortality analysis among atomic bomb survivors defined in a geographical study region.     

A Semiparametric Joint Model for Longitudinal and Survival Data with Application to Hemodialysis Study

Summary .  In many longitudinal clinical studies, the level and progression rate of repeatedly measured biomarkers on each subject quantify the severity of the disease and that subject's susceptibility to progression of the disease. It is of scientific and clinical interest to relate such quantities to a later time-to-event clinical endpoint such as patient survival. This is usually done with a shared parameter model. In such models, the longitudinal biomarker data and the survival outcome of each subject are assumed to be conditionally independent given subject-level severity or susceptibility (also called frailty in statistical terms). In this article, we study the case where the conditional distribution of longitudinal data is modeled by a linear mixed-effect model, and the conditional distribution of the survival data is given by a Cox proportional hazard model. We allow unknown regression coefficients and time-dependent covariates in both models. The proposed estimators are maximizers of an exact correction to the joint log likelihood with the frailties eliminated as nuisance parameters, an idea that originated from correction of covariate measurement error in measurement error models. The corrected joint log likelihood is shown to be asymptotically concave and leads to consistent and asymptotically normal estimators. Unlike most published methods for joint modeling, the proposed estimation procedure does not rely on distributional assumptions of the frailties. The proposed method was studied in simulations and applied to a data set from the Hemodialysis Study.     

Flexible maximum likelihood methods for bivariate proportional hazards models

This article presents methodology for multivariate proportional hazards (PH) regression models. The methods employ flexible piecewise constant or spline specifications for baseline hazard functions in either marginal or conditional PH models, along with assumptions about the association among lifetimes. Because the models are parametric, ordinary maximum likelihood can be applied; it is able to deal easily with such data features as interval censoring or sequentially observed lifetimes, unlike existing semiparametric methods. A bivariate Clayton model (1978, Biometrika 65, 141-151) is used to illustrate the approach taken. Because a parametric assumption about association is made, efficiency and robustness comparisons are made between estimation based on the bivariate Clayton model and "working independence" methods that specify only marginal distributions for each lifetime variable.     

Shared frailty models for recurrent events and a terminal event

There has been an increasing interest in the analysis of recurrent event data (Cook and Lawless, 2002, Statistical Methods in Medical Research 11, 141-166). In many situations, a terminating event such as death can happen during the follow-up period to preclude further occurrence of the recurrent events. Furthermore, the death time may be dependent on the recurrent event history. In this article we consider frailty proportional hazards models for the recurrent and terminal event processes. The dependence is modeled by conditioning on a shared frailty that is included in both hazard functions. Covariate effects can be taken into account in the model as well. Maximum likelihood estimation and inference are carried out through a Monte Carlo EM algorithm with Metropolis-Hastings sampler in the E-step. An analysis of hospitalization and death data for waitlisted dialysis patients is presented to illustrate the proposed methods. Methods to check the validity of the proposed model are also demonstrated. This model avoids the difficulties encountered in alternative approaches which attempt to specify a dependent joint distribution with marginal proportional hazards and yields an estimate of the degree of dependence.     

A semiparametric estimate of treatment effects with censored data

A semiparametric estimate of an average regression effect with right-censored failure time data has recently been proposed under the Cox-type model where the regression effect beta(t) is allowed to vary with time. In this article, we derive a simple algebraic relationship between this average regression effect and a measurement of group differences in k-sample transformation models when the random error belongs to the G(rho) family of Harrington and Fleming (1982, Biometrika 69, 553-566), the latter being equivalent to the conditional regression effect in a gamma frailty model. The models considered here are suitable for the attenuating hazard ratios that often arise in practice. The results reveal an interesting connection among the above three classes of models as alternatives to the proportional hazards assumption and add to our understanding of the behavior of the partial likelihood estimate under nonproportional hazards. The algebraic relationship provides a simple estimator under the transformation model. We develop a variance estimator based on the empirical influence function that is much easier to compute than the previously suggested resampling methods. When there is truncation in the right tail of the failure times, we propose a method of bias correction to improve the coverage properties of the confidence intervals. The estimate, its estimated variance, and the bias correction term can all be calculated with minor modifications to standard software for proportional hazards regression.