Class Of Semiparametric Transformation Models Research Articles

Imputation methods for the semiparametric transformation models with doubly-truncated and interval-censored data

Interval sampling is often used in epidemiological studies. With interval sampling, data are collected only from individuals who fail within a certain calendar time interval. Such interval sampling induces doubly truncated (DT) data if the calendar time of the failure event can be observed exactly. In practice, the failure time may not be directly observed and it is only recorded within time intervals, leading to doubly truncated and interval censored (DTIC) data. In this article, we analyze DTIC data using a class of semiparametric transformation models. By iterating between an imputation step and an optimization step for DT data, we propose two imputation methods for obtaining parameter estimates for regression parameters and cumulative hazard function of models. Simulation studies indicate that the proposed estimators perform adequately.

Regression Analysis of Multivariate Interval-Censored Failure Time Data under Transformation Model with Informative Censoring

We consider a regression analysis of multivariate interval-censored failure time data where the censoring may be informative. To address this, an approximated maximum likelihood estimation approach is proposed under a general class of semiparametric transformation models, and in the method, the frailty approach is employed to characterize the informative interval censoring. For the implementation of the proposed method, we develop a novel EM algorithm and show that the resulting estimators of the regression parameters are consistent and asymptotically normal. To evaluate the empirical performance of the proposed estimation procedure, we conduct a simulation study, and the results indicate that it performs well for the situations considered. In addition, we apply the proposed approach to a set of real data arising from an AIDS study.

Joint semiparametric models for case-cohort designs.

Two-phase studies such as case-cohort and nested case-control studies are widely used cost-effective sampling strategies. In the first phase, the observed failure/censoring time and inexpensive exposures are collected. In the second phase, a subgroup of subjects is selected for measurements of expensive exposures based on the information from the first phase. One challenging issue is how to utilize all the available information to conduct efficient regression analyses of the two-phase study data. This paper proposes a joint semiparametric modeling of the survival outcome and the expensive exposures. Specifically, we assume a class of semiparametric transformation models and a semiparametric density ratio model for the survival outcome and the expensive exposures, respectively. The class of semiparametric transformation models includes the proportional hazards model and the proportional odds model as special cases. The density ratio model is flexible in modeling multivariate mixed-type data. We develop efficient likelihood-based estimation and inference procedures and establish the large sample properties of the nonparametric maximum likelihood estimators. Extensive numerical studies reveal that the proposed methods perform well under practical settings. The proposed methods also appear to be reasonably robust under various model mis-specifications. An application to the National Wilms Tumor Study isprovided.

Semiparametric regression analysis of partly interval-censored failure time data with application to an AIDS clinical trial.

Failure time data subject to various types of censoring commonly arise in epidemiological and biomedical studies. Motivated by an AIDS clinical trial, we consider regression analysis of failure time data that include exact and left-, interval-, and/or right-censored observations, which are often referred to as partly interval-censored failure time data. We study the effects of potentially time-dependent covariates on partly interval-censored failure time via a class of semiparametric transformation models that includes the widely used proportional hazards model and the proportional odds model as special cases. We propose an EM algorithm for the nonparametric maximum likelihood estimation and show that it unifies some existing approaches developed for traditional right-censored data or purely interval-censored data. In particular, the proposed method reduces to the partial likelihood approach in the case of right-censored data under the proportional hazards model. We establish that the resulting estimator is consistent and asymptotically normal. In addition, we investigate the proposed method via simulation studies and apply it to the motivating AIDS clinical trial.

Bi-level variable selection in semiparametric transformation mixture cure models for right-censored data

We investigate the bi-level variable selection problem in semiparametric transformation mixture cure models (STMCM). In this type of mixture cure models, we consider a class of semiparametric transformation models for the conditional survival function and a logistic regression for the incidence component, then conduct group variable selection. The group bridge penalty is adopted for bi-level variable selection on both parts of the mixture cure models. Through simulation studies and real data analyses, we show that the proposed method can identify the important variables and groups that contribute to the cure proportion and the survival function for the uncured subjects, respectively.

Penalized estimation of semiparametric transformation models with interval-censored data and application to Alzheimer's disease.

Variable selection or feature extraction is fundamental to identify important risk factors from a large number of covariates and has applications in many fields. In particular, its applications in failure time data analysis have been recognized and many methods have been proposed for right-censored data. However, developing relevant methods for variable selection becomes more challenging when one confronts interval censoring that often occurs in practice. In this article, motivated by an Alzheimer's disease study, we develop a variable selection method for interval-censored data with a general class of semiparametric transformation models. Specifically, a novel penalized expectation-maximization algorithm is developed to maximize the complex penalized likelihood function, which is shown to perform well in the finite-sample situation through a simulation study. The proposed methodology is then applied to the interval-censored data arising from the Alzheimer's disease study mentioned above.

Semiparametric regression analysis for left-truncated and interval-censored data without or with a cure fraction

Interval censoring and truncation arise often in cohort studies, longitudinal and sociological research. In this article, we formulate the effects of covariates on left-truncated and mixed case interval-censored (LTIC) data without or with a cure fraction through a general class of semiparametric transformation models. We propose the conditional likelihood approach for statistical inference. For data without a cure fraction, we propose a computationally efficient EM algorithm, facilitated by a novel data augmentation method, to obtain the conditional maximum likelihood estimator (cMLE). For data with a cure fraction, we consider a semiparametric mixture cure model, which combines a logistic regression formula for the uncured probability with the class of transformation models for the failure time of uncured individuals. To overcome the computational complexity due to the presence of a cure fraction, by reparameterization of cure rate in the conditional likelihood function, we propose a computationally stable EM algorithm for obtaining the cMLE. We show that the cMLEs for the regression parameters are consistent and asymptotically normal. Based on the profile likelihood, we apply an EM-aided numerical differentiation method to compute the asymptotic variance estimates. We demonstrate the performance of our procedures through intensive simulation studies and application to the datasets from the Cardiovascular Disease Risk Factors Two-Township Study.

Semiparametric regression analysis of doubly censored failure time data from cohort studies

Doubly censored failure time data occur when the failure time of interest represents the elapsed time between two events, an initial event and a subsequent event, and the observations on both events may suffer censoring. A well-known example of such data is given by the acquired immune deficiency syndrome (AIDS) cohort study in which the two events are HIV infection and AIDS diagnosis, and several inference methods have been developed in the literature for their regression analysis. However, all of them only apply to limited situations or focus on a single model. In this paper, we propose a marginal likelihood approach based on a general class of semiparametric transformation models, which can be applied to much more general situations. For the implementation, we develop a two-step procedure that makes use of both the multiple imputation technique and a novel EM algorithm. The asymptotic properties of the resulting estimators are established by using the modern empirical process theory, and the simulation study conducted suggests that the method works well in practical situations. An application is also provided.

A Simulation Study on Comparing General Class of Semiparametric Transformation Models for Survival Outcome with Time-Varying Coefficients and Covariates

The consideration of the time-varying covariate and time-varying coefficient effect in survival models are plausible and robust techniques. Such kind of analysis can be carried out with a general class of semiparametric transformation models. The aim of this article is to develop modified estimating equations under semiparametric transformation models of survival time with time-varying coefficient effect and time-varying continuous covariates. For this, it is important to organize the data in a counting process style and transform the time with standard transformation classes which shall be applied in this article. In the situation when the effect of coefficient and covariates change over time, the widely used maximum likelihood estimation method becomes more complex and burdensome in estimating consistent estimates. To overcome this problem, alternatively, the modified estimating equations were applied to estimate the unknown parameters and unspecified monotone transformation functions. The estimating equations were modified to incorporate the time-varying effect in both coefficient and covariates. The performance of the proposed methods is tested through a simulation study. To sum up the study, the effect of possibly time-varying covariates and time-varying coefficients was evaluated in some special cases of semiparametric transformation models. Finally, the results have shown that the role of the time-varying covariate in the semiparametric transformation models was plausible and credible.

SEMIPARAMETRIC TRANSFORMATION MODELS WITH MULTILEVEL RANDOM EFFECTS FOR CORRELATED DISEASE ONSET IN FAMILIES

Large cohort studies are commonly launched to study risk of genetic variants or other risk factors on age at onset (AAO) of a chronic disorder. In these studies, family history data including AAO of disease in family members are collected to provide additional information and can be used to improve efficiency. Statistical analysis of these data is challenging due to missing genotypes in family members and the heterogeneous dependence attributed to both shared genetic back-ground and shared environmental factors (e.g., life style). In this paper, we propose a class of semiparametric transformation models with multilevel random effects to tackle these challenges. The proposed models include both proportional hazards model and proportional odds model as special cases. The multilevel random effects contain individual-specific random effects including kinship correlation structure dependent on the family pedigree, and a shared random effect to account for unobserved environment exposure. We use nonparametric maximum likelihood approach for inference and propose an expectation-maximization algorithm for computation in the presence of missing genotypes among family members. The obtained estimators are shown to be consistent, asymptotically normal, and semiparametrically efficient. Simulation studies demonstrate that the proposed method performs well with finite sample sizes. Finally, the proposed method is applied to study genetic risks in an Alzheimer's disease study.

Semiparametric transformation joint models for longitudinal covariates and interval-censored failure time

In many clinical trials and epidemiology research, subjects are followed-up repeatedly, and repeated measurements on longitudinal covariates as well as an observation on a possibly censored time-to-event are collected on each subject. The longitudinal covariates are often measured intermittently with measurement errors, and the measurement process is terminated by a correlated event process, leading to informative missing data. Methods for joint modelling of longitudinal and time to-event data have received much attention in the statistics literature in recent years. Most research has focused on right-censoring mechanism for the event time. In practice, the event time is often examined at the pre-scheduled times, at which the longitudinal covariates are also measured, resulting in interval-censored survival data. To take the interval censoring into account and to provide a more general framework for studying the effects of covariates on survival time, a new class of joint models is proposed. The joint model comprises a linear mixed-effects model for the longitudinal biomarkers and a class of semiparametric transformation models for the failure time, which incorporates the underlying longitudinal biomarkers as time-dependent covariates. The likelihood approach and an EM algorithm for obtaining the semiparametric maximum likelihood estimator (SPMLE) are developed. In M-step, a hybrid algorithm combining the Newton–Raphson and self-consistency algorithms is used to compute the finite-dimensional and infinite-dimensional parameters. The existence and consistency of the SPMLE are established. The proposed method is investigated through simulation studies and illustrated using a real dataset from a Taiwanese HIV/AIDS cohort study.

A Class of Semiparametric Transformation Models for Doubly Censored Failure Time Data

AbstractDoubly censored failure time data occur in many areas including demographical studies, epidemiology studies, medical studies and tumorigenicity experiments, and correspondingly some inference procedures have been developed in the literature (Biometrika, 91, 2004, 277; Comput. Statist. Data Anal., 57, 2013, 41; J. Comput. Graph. Statist., 13, 2004, 123). In this paper, we discuss regression analysis of such data under a class of flexible semiparametric transformation models, which includes some commonly used models for doubly censored data as special cases. For inference, the non‐parametric maximum likelihood estimation will be developed and in particular, we will present a novel expectation–maximization algorithm with the use of subject‐specific independent Poisson variables. In addition, the asymptotic properties of the proposed estimators are established and an extensive simulation study suggests that the proposed methodology works well for practical situations. The method is applied to an AIDS study.

Estimation and variable selection for semiparametric transformation models under a more efficient cohort sampling design

Two-phase cohort sampling designs, or sometimes known as retrospective sampling designs, are often used in large cohort studies for saving sampling time and cost. Commonly used designs include case–cohort design, case–control design, nested case–control design, and so on. Efforts had been taken to improve the estimation efficiency under these commonly used designs. We propose a different retrospective sampling design, called end-point design, under the class of semiparametric transformation models. An inverse probability weighting likelihood approach is designed for estimating the model parameters, and the proposed design shows higher efficiency than the case–cohort and case–control design with comparable size of covariates ascertainment. We also consider variable selection under the proposed design. A specially designed objective function with adaptive lasso penalty is proposed. The large sample properties of the proposed estimation and variable selection procedure are developed. Extensive simulation studies are carried out to show favorable evidence for the proposed approaches. A real data set is analyzed for illustration.

Maximum likelihood estimation for semiparametric regression models with multivariate interval-censored data.

SummaryInterval-censored multivariate failure time data arise when there are multiple types of failure or there is clustering of study subjects and each failure time is known only to lie in a certain interval. We investigate the effects of possibly time-dependent covariates on multivariate failure times by considering a broad class of semiparametric transformation models with random effects, and we study nonparametric maximum likelihood estimation under general interval-censoring schemes. We show that the proposed estimators for the finite-dimensional parameters are consistent and asymptotically normal, with a limiting covariance matrix that attains the semiparametric efficiency bound and can be consistently estimated through profile likelihood. In addition, we develop an EM algorithm that converges stably for arbitrary datasets. Finally, we assess the performance of the proposed methods in extensive simulation studies and illustrate their application using data derived from the Atherosclerosis Risk in Communities Study.

A Sieve Semiparametric Maximum Likelihood Approach for Regression Analysis of Bivariate Interval-Censored Failure Time Data

ABSTRACT Interval-censored failure time data arise in a number of fields and many authors have discussed various issues related to their analysis. However, most of the existing methods are for univariate data and there exists only limited research on bivariate data, especially on regression analysis of bivariate interval-censored data. We present a class of semiparametric transformation models for the problem and for inference, a sieve maximum likelihood approach is developed. The model provides a great flexibility, in particular including the commonly used proportional hazards model as a special case, and in the approach, Bernstein polynomials are employed. The strong consistency and asymptotic normality of the resulting estimators of regression parameters are established and furthermore, the estimators are shown to be asymptotically efficient. Extensive simulation studies are conducted and indicate that the proposed method works well for practical situations. Supplementary materials for this article are available online.

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.

Analysis of transformation models with right-truncated data

ABSTRACTIn many applications, statistical data are frequently observed subject to a retrospective sampling criterion resulting in right-truncated data. 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 right-truncated data. We proposed two estimators for regression coefficients. The first estimator is based on martingale estimating equations. The second estimator is based on the conditional likelihood function given the truncation times. The asymptotic properties of both estimators are derived. The finite sample performance is examined through a simulation study.

Maximum likelihood estimation for semiparametric transformation models with interval-censored data.

Interval censoring arises frequently in clinical, epidemiological, financial and sociological studies, where the event or failure of interest is known only to occur within an interval induced by periodic monitoring. We formulate the effects of potentially time-dependent covariates on the interval-censored failure time through a broad class of semiparametric transformation models that encompasses proportional hazards and proportional odds models. We consider nonparametric maximum likelihood estimation for this class of models with an arbitrary number of monitoring times for each subject. We devise an EM-type algorithm that converges stably, even in the presence of time-dependent covariates, and show that the estimators for the regression parameters are consistent, asymptotically normal, and asymptotically efficient with an easily estimated covariance matrix. Finally, we demonstrate the performance of our procedures through simulation studies and application to an HIV/AIDS study conducted in Thailand.

Efficient Estimation of Semiparametric Transformation Models for the Cumulative Incidence of Competing Risks.

The cumulative incidence is the probability of failure from the cause of interest over a certain time period in the presence of other risks. A semiparametric regression model proposed by Fine and Gray (1999) has become the method of choice for formulating the effects of covariates on the cumulative incidence. Its estimation, however, requires modeling of the censoring distribution and is not statistically efficient. In this paper, we present a broad class of semiparametric transformation models which extends the Fine and Gray model, and we allow for unknown causes of failure. We derive the nonparametric maximum likelihood estimators (NPMLEs) and develop simple and fast numerical algorithms using the profile likelihood. We establish the consistency, asymptotic normality, and semiparametric efficiency of the NPMLEs. In addition, we construct graphical and numerical procedures to evaluate and select models. Finally, we demonstrate the advantages of the proposed methods over the existing ones through extensive simulation studies and an application to a major study on bone marrow transplantation.

Maximum likelihood estimation for semiparametric transformation models with interval-censored data

Interval censoring arises frequently in clinical, epidemiological, financial, and sociological studies, where the event or failure of interest is known only to occur within an interval induced by periodic monitoring. We formulate the effects of potentially time-dependent covariates on the interval-censored failure time through a broad class of semiparametric transformation models that encompasses proportional hazards and proportional odds models. We consider nonparametric maximum likelihood estimation for this class of models with an arbitrary number of monitoring times for each subject. We devise an EM-type algorithm that converges stably, even in the presence of time-dependent covariates, and show that the estimators for the regression parameters are consistent, asymptotically normal, and asymptotically efficient with an easily estimated covariance matrix. Finally, we demonstrate the performance of our procedures through extensive simulation studies and application to an HIV/AIDS study conducted in Thailand.