Semiparametric Linear Transformation Model Research Articles

Copula-based analysis of dependent current status data with semiparametric linear transformation model.

This paper discusses regression analysis of current status data with dependent censoring, a problem that often occurs in many areas such as cross-sectional studies, epidemiological investigations and tumorigenicity experiments. Copula model-based methods are commonly employed to tackle this issue. However, these methods often face challenges in terms of model and parameter identification. The primary aim of this paper is to propose a copula-based analysis for dependent current status data, where the association parameter is left unspecified. Our method is based on a general class of semiparametric linear transformation models and parametric copulas. We demonstrate that the proposed semiparametric model is identifiable under certain regularity conditions from the distribution of the observed data. For inference, we develop a sieve maximum likelihood estimation method, using Bernstein polynomials to approximate the nonparametric functions involved. The asymptotic consistency and normality of the proposed estimators are established. Finally, to demonstrate the effectiveness and practical applicability of our method, we conduct an extensive simulation study and apply the proposed method to a real data example.

Analysis of Multivariate Survival Data under Semiparametric Copula Models

AbstractModelling multivariate survival data is complicated by the complex association structure among the responses. To balance model flexibility and interpretability, we propose a semiparametric copula model to modulate multivariate survival data, with the marginal distributions of the response components described by semiparametric linear transformation models. To conduct inference about the model parameters, we develop a two‐stage maximum likelihood method and a three‐stage pseudo‐likelihood estimation procedure. We investigate the impact of model misspecification on the estimation of covariate effects and identify a scenario in which consistent estimation of the marginal parameters is retained even when the copula model is misspecified. The proposed methods are justified both theoretically and empirically. An application to a real dataset is provided to demonstrate the utility of the proposed method.

The sparse estimation of the semiparametric linear transformation model with dependent current status data

In this paper, we study the sparse estimation under the semiparametric linear transformation models for the current status data, also called type I interval-censored data. For the problem, the failure time of interest may be dependent on the censoring time and the association parameter between them is left unspecified. To address this, we employ the copula model to describe the dependence between them and a two-stage estimation procedure to estimate both the association parameter and the regression parameter. In addition, we propose a penalized maximum likelihood estimation procedure based on the broken adaptive ridge regression, and Bernstein polynomials are used to approximate the nonparametric functions involved. The oracle property of the proposed method is established and the numerical studies suggest that the method works well for practical situations. Finally, the method is applied to an Alzheimer's disease study that motivated this investigation.

Semiparametric linear transformation models for indirectly observed outcomes.

We propose a regression framework to analyze outcomes that are indirectly observed via one or multiple proxies. Semiparametric transformation models, including Cox proportional hazards regression, turn out to be well suited to model the association between the covariates and the unobserved outcome. By coupling this regression model to a semiparametric measurement model, we can estimate these associations without requiring calibration data and without imposing strong functional assumptions on the relationship between the unobserved outcome and its proxy. When multiple proxies are available, we propose a data-driven aggregation resulting in an improved proxy. We empirically validate the proposed methodology in a simulation study, revealing good finite sample properties, especially when multiple proxies are aggregated. The methods are demonstrated on two case studies.

Regression analysis of censored data with nonignorable missing covariates and application to Alzheimer Disease

In this paper, we discuss regression analysis of censored failure time data when there exist missing covariates and more specifically, we will consider interval-censored data, a general form of censored data, and the nonignorable missing. Although many methods have been proposed in the literature for censored data with missing covariates, they only apply to limited situations and it does not seem to exist an established procedure for the situation discussed here. For the analysis, we employ the semiparametric linear transformation model and develop a two-step estimation procedure. In addition, the asymptotic properties of the resulting estimators are established and a Poisson variable-based EM algorithm is provided for the implementation of the proposed estimation procedure. Finally the proposed approach is applied to an Alzheimer Disease study that motivated this investigation.

Mendelian randomization using semiparametric linear transformation models.

Mendelian randomization (MR) uses genetic information as an instrumental variable (IV) to estimate the causal effect of an exposure of interest on an outcome in the presence of unknown confounding. We are interested in the causal effect of cigarette smoking on lung cancer survival, which is subject to confounding by underlying pulmonary functions. Despite the well-developed IV analyses for the continuous and binary outcomes, the scarcity of methodology for the survival outcome limits its utility for the time-to-event data collected in many observational studies. We propose an IV analysis method in the survival context, estimating causal effects on a transformed survival time and survival probabilities using semiparametric linear transformation models. We study the conditions under which hazard ratio and the effect on survival probability can be approximated. For statistical inference, we construct estimating equations to circumvent the difficulty in deriving joint likelihood of the exposure and the outcome, due to the unknown confounding. Asymptotic properties of the proposed estimators are established without parametric assumptions about confounders. We study the finite sample performance in extensive simulation studies. The MR analysis of a lung cancer study suggests a harmful prognostic effect of smoking pack-years that would have been missed by the crude association.

An empirical comparison of two novel transformation models.

Continuous response variables are often transformed to meet modeling assumptions, but the choice of the transformation can be challenging. Two transformation models have recently been proposed: semiparametric cumulative probability models (CPMs) and parametric most likely transformation models (MLTs). Both approaches model the cumulative distribution function and require specifying a link function, which implicitly assumes that the responses follow a known distribution after some monotonic transformation. However, the two approaches estimate the transformation differently. With CPMs, an ordinal regression model is fit, which essentially treats each continuous response as a unique category and therefore nonparametrically estimates the transformation; CPMs are semiparametric linear transformation models. In contrast, with MLTs, the transformation is parameterized using flexible basis functions. Conditional expectations and quantiles are readily derived from both methods on the response variable's original scale. We compare the two methods with extensive simulations. We find that both methods generally have good performance with moderate and large sample sizes. MLTs slightly outperformed CPMs in small sample sizes under correct models. CPMs tended to be somewhat more robust to model misspecification and outcome rounding. Except in the simplest situations, both methods outperform basic transformation approaches commonly used in practice. We apply both methods to an HIV biomarker study.

Analysis of linear transformation models with covariate measurement error and interval censoring.

Among several semiparametric models, the Cox proportional hazard model is widely used to assess the association between covariates and the time-to-event when the observed time-to-event is interval-censored. Often, covariates are measured with error. To handle this covariate uncertainty in the Cox proportional hazard model with the interval-censored data, flexible approaches have been proposed. To fill a gap and broaden the scope of statistical applications to analyze time-to-event data with different models, in this paper, a general approach is proposed for fitting the semiparametric linear transformation model to interval-censored data when a covariate is measured with error. The semiparametric linear transformation model is a broad class of models that includes the proportional hazard model and the proportional odds model as special cases. The proposed method relies on a set of estimating equations to estimate the regression parameters and the infinite-dimensional parameter. For handling interval censoring and covariate measurement error, a flexible imputation technique is used. Finite sample performance of the proposed method is judged via simulation studies. Finally, the suggested method is applied to analyze a real data set from an AIDS clinical trial.

Regression analysis of informatively interval-censored failure time data with semiparametric linear transformation model

ABSTRACT Regression analysis of interval-censored failure time data with noninformative censoring has been widely investigated and many methods have been proposed. Sometimes the mechanism behind the interval censoring may be informative and several approaches have also been developed for this latter situation. However, all of these existing methods are for single models and it is well known that in many situations, one may prefer more flexible models. Corresponding to this, the linear transformation model is considered and a maximum likelihood estimation method is established. In the proposed method, the association between the failure time of interest and the censoring time is modelled by the copula model, and the involved nonparametric functions are approximated by spline functions. The large sample properties of the proposed estimators are derived. Numerical results show that the proposed method performs well in practical application. Besides, a real data example is presented for the illustration.

Semiparametric linear transformation models: Effect measures, estimators, and applications.

Semiparametric linear transformation models form a versatile class of regression models with the Cox proportional hazards model being the most well-known member. These models are well studied for right censored outcomes and are typically used in survival analysis. We consider transformation models as a tool for situations with uncensored continuous outcomes where linear regression is not appropriate. We introduce the probabilistic index as a uniform effect measure for the class of transformation models. We discuss and compare three estimators using a working Cox regression model: the partial likelihood estimator, an estimator based on binary generalized linear models and one based on probabilistic index model estimating equations. The latter has a superior performance in terms of bias and variance when the working model is misspecified. For the purpose of illustration, we analyze data that were collected at an urban alcohol and drug detoxification unit.

A semiparametric linear transformation model for general biased-sampling and right-censored data

A semiparametric linear transformation model for general biased-sampling and right-censored data

Efficient estimation of a linear transformation model for current status data via penalized splines.

We propose a flexible and computationally efficient penalized estimation method for a semi-parametric linear transformation model with current status data. To facilitate model fitting, the unknown monotone function is approximated by monotone B-splines, and a computationally efficient hybrid algorithm involving the Fisher scoring algorithm and the isotonic regression is developed. A goodness-of-fit test and model diagnostics are also considered. The asymptotic properties of the penalized estimators are established, including the optimal rate of convergence for the function estimator and the semi-parametric efficiency for the regression parameter estimators. An extensive numerical experiment is conducted to evaluate the finite-sample properties of the penalized estimators, and the methodology is further illustrated with two real studies.

Regression analysis of informative current status data with the semiparametric linear transformation model

ABSTRACTMany methods have been developed in the literature for regression analysis of current status data with noninformative censoring and also some approaches have been proposed for semiparametric regression analysis of current status data with informative censoring. However, the existing approaches for the latter situation are mainly on specific models such as the proportional hazards model and the additive hazard model. Corresponding to this, in this paper, we consider a general class of semiparametric linear transformation models and develop a sieve maximum likelihood estimation approach for the inference. In the method, the copula model is employed to describe the informative censoring or relationship between the failure time of interest and the censoring time, and Bernstein polynomials are used to approximate the nonparametric functions involved. The asymptotic consistency and normality of the proposed estimators are established, and an extensive simulation study is conducted and indicates that the proposed approach works well for practical situations. In addition, an illustrative example is provided.

Linear transformation models for censored data under truncation

In many observational cohort studies, a pair of correlated event times are usually observed for each individual. This paper develops a new approach for the semiparametric linear transformation model to handle the bivariate survival data under both truncation and censoring. By incorporating truncation, the potential referral bias in practice is taken into account. A class of generalised estimating equations are proposed to obtain unbiased estimates of the regression parameters. Large sample properties of the proposed estimator are provided. Simulation studies under different scenarios and analyses of real-world datasets are conducted to assess the performance of the proposed estimator.

Integrative genomic testing of cancer survival using semiparametric linear transformation models.

The wide availability of multi-dimensional genomic data has spurred increasing interests in integrating multi-platform genomic data. Integrative analysis of cancer genome landscape can potentially lead to deeper understanding of the biological process of cancer. We integrate epigenetics (DNA methylation and microRNA expression) and gene expression data in tumor genome to delineate the association between different aspects of the biological processes and brain tumor survival. To model the association, we employ a flexible semiparametric linear transformation model that incorporates both the main effects of these genomic measures as well as the possible interactions among them. We develop variance component tests to examine different coordinated effects by testing various subsets of model coefficients for the genomic markers. A Monte Carlo perturbation procedure is constructed to approximate the null distribution of the proposed test statistics. We further propose omnibus testing procedures to synthesize information from fitting various parsimonious sub-models to improve power. Simulation results suggest that our proposed testing procedures maintain proper size under the null and outperform standard score tests. We further illustrate the utility of our procedure in two genomic analyses for survival of glioblastoma multiforme patients. Copyright © 2016 John Wiley & Sons, Ltd.

Semiparametric linear transformation model with differential measurement error and validation sampling

For the semiparametric linear transformation model with covariate measurement error and validation sampling, we propose an estimation method to estimate the covariate coefficient. The method updates the validation set based estimator to get a more efficient estimator using the data information available on the whole cohort. It can be used to deal with both differential and nondifferential measurement errors. Consistency and asymptotic normality are established for the proposed estimator and a closed form formula is derived for the limiting variance–covariance matrix. Simulation studies and a real data analysis are used to illustrate the performances of the proposed method.

Estimation for semiparametric transformation models with length-biased sampling

For length-biased and right-censored data, we propose an estimation method to assess the effects of risk factors under the semiparametric linear transformation model. Unlike the existing method of Shen et al. (2009) based on the ranks of observed failure times, the new estimators are obtained from counting process-based unbiased estimating equations. Consistency and asymptotic normality for the estimators are derived under suitable regularity conditions. We evaluate the finite sample performance of the proposed method and make a comparison with that of Shen et al. (2009) by simulation studies. A real data example is analyzed to illustrate the proposed method.

A semiparametric linear transformation model to estimate causal effects for survival data.

Semiparametric linear transformation models serve as useful alternatives to the Cox proportional hazard model. In this study, we use the semiparametric linear transformation model to analyze survival data with selective compliance. We estimate regression parameters and the transformation function based on pseudo-likelihood and a series of estimating equations. We show that the estimators for the regression parameters and transformation function are consistent and asymptotically normal, and both converge to their true values at the rate of n -1/2, the convergence rate expected for a parametric model. The practical utility of the methods is confirmed via simulations as well as an application of a clinical trial to evaluate the effectiveness of sentinel node biopsy in guiding the treatment of invasive melanoma.

Semiparametric regression analysis for clustered doubly-censored data

This paper considers clustered doubly-censored data that occur when there exist several correlated survival times of interest and only doubly censored data are available for each survival time. In this situation, one approach is to model the marginal distribution of failure times using semiparametric linear transformation models while leaving the dependence structure completely arbitrary. We demonstrate that the approach of Cai et al. (Biometrika 87:867---878, 2000) can be extended to clustered doubly censored data. We propose two estimators by using two different estimated censoring weights. A simulation study is conducted to investigate the proposed estimators.

A Unified Approach to Semiparametric Transformation Models Under General Biased Sampling Schemes

We propose a unified estimation method for semiparametric linear transformation models under general biased sampling schemes. The new estimator is obtained from a set of counting process-based unbiased estimating equations, developed through introducing a general weighting scheme that offsets the sampling bias. The usual asymptotic properties, including consistency and asymptotic normality, are established under suitable regularity conditions. A closed-form formula is derived for the limiting variance and the plug-in estimator is shown to be consistent. We demonstrate the unified approach through the special cases of left truncation, length bias, the case-cohort design, and variants thereof. Simulation studies and applications to real datasets are presented.