Semiparametric Transformation Models Research Articles

Causal estimation using semiparametric transformation models under prevalent sampling.

This article presents methods and inference for causal estimation in semiparametric transformation models for the prevalent survival data. Through the estimation of the transformation models and covariate distribution, we propose a few analytical procedures to estimate the causal survival function. As the data are observational, the unobserved potential outcome (survival time) may be associated with the treatment assignment, and therefore there may exist a systematic imbalance between the data observed from each treatment arm. Further, due to prevalent sampling, subjects are observed only if they have not experienced the failure event when data collection began, causing the prevalent sampling bias. We propose a unified approach, which simultaneously corrects the bias from the prevalent sampling and balances the systematic differences from the observational data. We illustrate in the simulation study that standard analysis without proper adjustment would result in biased causal inference. Large sample properties of the proposed estimation procedures are established by techniques of empirical processes and examined by simulation studies. The proposed methods are applied to the Surveillance, Epidemiology, and End Results (SEER) and Medicare-linked data for women diagnosed with breast cancer.

Calibration weighted estimation of semiparametric transformation models for two-phase sampling.

Two-phase designs are commonly used to subsample subjects from a cohort in order to study covariates that are too expensive to ascertain for everyone in the cohort. This is particularly true for the study of immune response biomarkers in vaccine immunology, where new, elaborate assays are constantly being developed to improve our understanding of the human immune responses to vaccines and how the immune response may protect humans from virus infection. It has long being recognized that if there exist variables that are correlated with expensive variables and can be measured for every subject in the cohort, they can be leveraged to improve the estimation efficiency for the effects of the expensive variables. In this research article, we developed an improved inverse probability weighted estimation approach for semiparametric transformation models with a two-phase study design. Semiparametric transformation models are a class of models that include the Cox PH and proportional odds models. They provide an attractive way to model the effects of immune response biomarkers as human immune responses generally wane over time. Our approach is based on weights calibration, which has its origin in survey statistics and was used by Breslow et al. to improve inverse probability weighted estimation of the Cox regression model. We develop asymptotic theory for our estimator and examine its performance through simulation studies. We illustrate the proposed method with application to two HIV-1 vaccine efficacy trials.

Education, Intelligence, and Well-Being: Evidence from a Semiparametric Latent Variable Transformation Model for Multiple Outcomes of Mixed Types

This paper uses a semiparametric latent variable transformation model for multiple outcomes to examine the effect of education and maternal education on female multidimensional well-being and proposes a procedure to build a well-being index that is less susceptible to functional form misspecification. We model multidimensional well-being as an unobserved common factor underlying the observed well-being outcomes. The semiparametric methodology allows us to alleviate misspecification bias by combining multiple indicators into a latent construct in an unspecified, data-driven way. Using data from female participants of the 1974–2010 waves of the US General Social Survey, we find that education, intelligence, and maternal education contribute positively to multidimensional well-being. However, the effects of education and maternal education on female multidimensional well-being declined steadily between the mid-1970s and the 1990s, and have not rebounded since.

Estimating the error distribution in semiparametric transformation models

In this paper we consider the semiparametric transformation model $\Lambda_{\theta_{o}}(Y)=m(X)+\varepsilon$, where $\theta_{o}$ is an unknown finite dimensional parameter, the function $m(\cdot)=\mathbb{E}(\Lambda_{\theta_{o}}(Y)|X=\cdot)$ is “smooth”, but otherwise unknown, and the covariate $X$ is independent of the error $\varepsilon$. An estimator of the distribution function of $\varepsilon$ is investigated and its weak convergence is proved. The proposed estimator depends on a profile likelihood estimator of $\theta_{o}$ and a nonparametric kernel estimator of $m$. We also evaluate the practical performance of our estimator in a simulation study for several models and sample sizes. Finally, the method is applied to a data set on the scattering of sunlight in the atmosphere.

Partially linear transformation cure models for interval-censored data

There has been considerable progress in the development of semiparametric transformation models for regression analysis of time-to-event data. However, most of the current work focuses on right-censored data. Significantly less work has been done for interval-censored data, especially when the population contains a nonignorable cured subgroup. A broad and flexible class of semiparametric transformation cure models is proposed for analyzing interval-censored data in the presence of a cure fraction. The proposed modeling approach combines a logistic regression formulation for the probability of cure with a partially linear transformation model for event times of susceptible subjects. The estimation is achieved by using a spline-based sieve maximum likelihood method, which is computationally efficient and leads to estimators with appealing properties such as consistency, asymptotic normality and semiparametric efficiency. Furthermore, a goodness-of-fit test can be constructed for the proposed models based on the sieve likelihood ratio. Simulations and a real data analysis are provided for illustration of the methodology.

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.

Semiparametric transformation models for causal inference in time to event studies with all-or-nothing compliance.

We consider causal inference in randomized survival studies with right censored outcomes and all-or-nothing compliance, using semiparametric transformation models to estimate the distribution of survival times in treatment and control groups, conditional on covariates and latent compliance type. Estimands depending on these distributions, for example, the complier average causal effect (CACE), the complier effect on survival beyond time t, and the complier quantile effect are then considered. Maximum likelihood is used to estimate the parameters of the transformation models, using a specially designed expectation-maximization (EM) algorithm to overcome the computational difficulties created by the mixture structure of the problem and the infinite dimensional parameter in the transformation models. The estimators are shown to be consistent, asymptotically normal, and semiparametrically efficient. Inferential procedures for the causal parameters are developed. A simulation study is conducted to evaluate the finite sample performance of the estimated causal parameters. We also apply our methodology to a randomized study conducted by the Health Insurance Plan of Greater New York to assess the reduction in breast cancer mortality due to screening.

Semiparametric transformation models for semicompeting survival data.

Semicompeting risk outcome data (e.g., time to disease progression and time to death) are commonly collected in clinical trials. However, analysis of these data is often hampered by a scarcity of available statistical tools. As such, we propose a novel semiparametric transformation model that improves the existing models in the following two ways. First, it estimates regression coefficients and association parameters simultaneously. Second, the measure of surrogacy, for example, the proportion of the treatment effect that is mediated by the surrogate and the ratio of the overall treatment effect on the true endpoint over that on the surrogate endpoint, can be directly obtained. We propose an estimation procedure for inference and show that the proposed estimator is consistent and asymptotically normal. Extensive simulations demonstrate the valid usage of our method. We apply the method to a multiple myeloma trial to study the impact of several biomarkers on patients' semicompeting outcomes--namely, time to progression and time to death.

Combined estimating equation approaches for semiparametric transformation models with length-biased survival data.

Survival data are subject to length-biased sampling when the survival times are left-truncated and the underlying truncation time random variable is uniformly distributed. Substantial efficiency gains can be achieved by incorporating the information about the truncation time distribution in the estimation procedure [Wang (1989) Journal of the American Statistical Association 84, 742-748; Wang (1996) Biometrika 83, 343-354]. Under the semiparametric transformation models, the maximum likelihood method is expected to be fully efficient, yet it is difficult to implement because the full likelihood depends on the nonparametric component in a complicated way. Moreover, its asymptotic properties have not been established. In this article, we extend the martingale estimating equation approach [Chen et al. (2002) Biometrika 89, 659-668; Kim et al. (2013) Journal of the American Statistical Association 108, 217-227] and the pseudo-partial likelihood approach [Severini and Wong (1992) The Annals of Statistics 4, 1768-1802; Zucker (2005) Journal of the American Statistical Association 100, 1264-1277] for semiparametric transformation models with right-censored data to handle left-truncated and right-censored data. In the same spirit of the composite likelihood method [Huang and Qin (2012) Journal of the American Statistical Association 107, 946-957], we further construct another set of unbiased estimating equations by exploiting the special probability structure of length-biased sampling. Thus the number of estimating equations exceeds the number of parameters, and efficiency gains can be achieved by solving a simple combination of these estimating equations. The proposed methods are easy to implement as they do not require additional programming efforts. Moreover, they are shown to be consistent and asymptotically normally distributed. A data analysis of a dementia study illustrates the methods.

Additive Transformation Models for Clustered Doubly Censored Data

In this article, we propose estimating procedures for additive semiparametric transformation models with clustered doubly-censored data. A simulation study is conducted to investigate the performance of the proposed estimators. We apply the proposed methods to the dataset from the well-known Diabetic Retinopathy Study.

Likelihood approaches for proportional likelihood ratio model with right‐censored data

Regression methods for survival data with right censoring have been extensively studied under semiparametric transformation models such as the Cox regression model and the proportional odds model. However, their practical application could be limited because of possible violation of model assumption or lack of ready interpretation for the regression coefficients in some cases. As an alternative, in this paper, the proportional likelihood ratio model introduced by Luo and Tsai is extended to flexibly model the relationship between survival outcome and covariates. This model has a natural connection with many important semiparametric models such as generalized linear model and density ratio model and is closely related to biased sampling problems. Compared with the semiparametric transformation model, the proportional likelihood ratio model is appealing and practical in many ways because of its model flexibility and quite direct clinical interpretation. We present two likelihood approaches for the estimation and inference on the target regression parameters under independent and dependent censoring assumptions. Based on a conditional likelihood approach using uncensored failure times, a numerically simple estimation procedure is developed by maximizing a pairwise pseudo-likelihood. We also develop a full likelihood approach, and the most efficient maximum likelihood estimator is obtained by a profile likelihood. Simulation studies are conducted to assess the finite-sample properties of the proposed estimators and compare the efficiency of the two likelihood approaches. An application to survival data for bone marrow transplantation patients of acute leukemia is provided to illustrate the proposed method and other approaches for handling non-proportionality. The relative merits of these methods are discussed in concluding remarks.

Semiparametric transformation models with random effects for clustered doubly censored data

For right-censored data, Zeng et al. [Semiparametirc transformation modes with random effects for clustered data. Statist Sin. 2008;18:355–377] proposed a class of semiparametric transformation models with random effects to formulate the effects of possibly time-dependent covariates on clustered failure times. In this article, we demonstrate that the approach of Zeng et al. can be extended to analyse clustered doubly censored data. The asymptotic properties of the nonparametric maximum likelihood estimators of the model parameters are derived. A simulation study is conducted to investigate the performance of the proposed estimators.

Estimation of the error density in a semiparametric transformation model

Consider the semiparametric transformation model $$\Lambda _{\theta _o}(Y)=m(X)+\varepsilon $$ , where $$\theta _o$$ is an unknown finite dimensional parameter, the functions $$\Lambda _{\theta _o}$$ and $$m$$ are smooth, $$\varepsilon $$ is independent of $$X$$ , and $${\mathbb {E}}(\varepsilon )=0$$ . We propose a kernel-type estimator of the density of the error $$\varepsilon $$ , and prove its asymptotic normality. The estimated errors, which lie at the basis of this estimator, are obtained from a profile likelihood estimator of $$\theta _o$$ and a nonparametric kernel estimator of $$m$$ . The practical performance of the proposed density estimator is evaluated in a simulation study.

Efficient Estimation of Semiparametric Transformation Models for Two-Phase Cohort Studies

Under two-phase cohort designs, such as case–cohort and nested case–control sampling, information on observed event times, event indicators, and inexpensive covariates is collected in the first phase, and the first-phase information is used to select subjects for measurements of expensive covariates in the second phase; inexpensive covariates are also used in the data analysis to control for confounding and to evaluate interactions. This article provides efficient estimation of semiparametric transformation models for such designs, accommodating both discrete and continuous covariates, and allowing inexpensive and expensive covariates to be correlated. The estimation is based on the maximization of a modified nonparametric likelihood function through a generalization of the expectation–maximization algorithm. The resulting estimators are shown to be consistent, asymptotically normal and asymptotically efficient with easily estimated variances. Simulation studies demonstrate that the asymptotic approximations are accurate in practical situations. Empirical data from Wilms’ tumor studies and the Atherosclerosis Risk in Communities (ARIC) study are presented. Supplementary materials for this article are available online.

A flexible semiparametric transformation model for recurrent event data

In this article, we propose a class of semiparametric transformation models for recurrent event data, in which the baseline mean function is allowed to depend on covariates through an additive model, and some covariate effects are allowed to be time-varying. For inference on the model parameters, estimating equation approaches are developed, and the asymptotic properties of the resulting estimators are established. In addition, a lack-of-fit test is presented to assess the adequacy of the model. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer study is illustrated.

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 new class of semiparametric transformation models based on first hitting times by latent degradation processes

In many failure mechanisms, most subjects under study deteriorate physically over time, and thus a depreciation in health may precede failure. A latent stochastic process, called degradation process, may be assumed for modeling such depreciation whereby an event occurs when the process first crosses a threshold. A class of survival regression models can be constructed from the first hitting time of a latent accelerating degradation process, which turns out to be a transformation model in the literature. To characterize these models, we propose to use first‐hitting‐time models for the baseline distribution, specifically inverse Gaussian, Birnbaum–Saunders and gamma distributions, among others. The proposed models have many desirable features, such as a wide variety of shapes of hazard rates, analytical tractability and, most of all, its motivation from a plausible stochastic setting for failure. We estimate the model parameters by the non‐parametric 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 estimate their variances. Simulation studies show that the proposed approach is appropriate for practical use. The methodology is illustrated with two real examples. Copyright © 2013 John Wiley & Sons Ltd

Analysing truncated data with semiparametric transformation models

Left-truncation often arises when patient information, such as time of diagnosis, is gathered retrospectively. In some cases, the distribution function, say G(x), of left-truncated variables can be parameterized as G(x; θ), where θ∈Θ⊂Rq and θ is a q-dimensional vector. Under semiparametric transformation models, we demonstrated that the approach of Chen et al. (Semiparametric analysis of transformation models with censored data. Biometrika. 2002;89:659–668) can be used to analyse this type of data. The asymptotic properties of the proposed estimators are derived. A simulation study is conducted to investigate the performance of the proposed estimators.

Variable selection in semiparametric transformation models for right-censored data

We study variable selection in general transformation models for right-censored data. The models studied can incorporate external time-varying covariates, and they include the proportional hazards model and the proportional odds model as special cases. We propose an estimation method that involves minimizing a weighted negative partial loglikelihood function plus an adaptive lasso penalty, with the initial values obtained from nonparametric maximum likelihood estimation. The objective function is parametric and convex, so the minimization is easy to implement. We show that our selection has oracle properties and that the estimator is semiparametrically efficient. We demonstrate the small-sample performance of the proposed method via simulations, and we use the method to analyse data from the Atherosclerosis Risk in Communities Study. Copyright 2013, Oxford University Press.

A class of semiparametric transformation models for survival data with a cured proportion

We propose a new class of semiparametric regression models based on a multiplicative frailty assumption with a discrete frailty, which may account for cured subgroup in population. The cure model framework is then recast as a problem with a transformation model. The proposed models can explain a broad range of nonproportional hazards structures along with a cured proportion. An efficient and simple algorithm based on the martingale process is developed to locate the nonparametric maximum likelihood estimator. Unlike existing expectation-maximization based methods, our approach directly maximizes a nonparametric likelihood function, and the calculation of consistent variance estimates is immediate. The proposed method is useful for resolving identifiability features embedded in semiparametric cure models. Simulation studies are presented to demonstrate the finite sample properties of the proposed method. A case study of stage III soft-tissue sarcoma is given as an illustration.