Semiparametric Linear Transformation Model Research Articles

Smoothed rank correlation of the linear transformation regression model

The maximum rank correlation (MRC) approach is the most common method used in the literature to estimate the regression coefficients in the semiparametric linear transformation regression model. However, the objective function Gn(β) in the MRC approach is not continuous. The optimization of Gn(β) requires an extensive search for which the computational cost grows in the order of nd, where d is the dimension of X. Given the lack of smoothing, issues related to variable selection, the variance estimate and other inferences by MRC are not well developed in the model. In this paper, we combine the concept underlying the penalized method, rank correlation and smoothing technique and propose a nonconcave penalized smoothed rank correlation method to select variables and estimate parameters for the semiparametric linear transformation model. The proposed estimator is computationally simple, n1/2−consistent and asymptotically normal. A sandwich formula is proposed to estimate the variances of the proposed estimates. We also illustrate the usefulness of the methodology with real data from a body fat prediction study.

On sparse estimation for semiparametric linear transformation models

Semiparametric linear transformation models have received much attention due to their high flexibility in modeling survival data. A useful estimating equation procedure was recently proposed by Chen et al. (2002) [21] for linear transformation models to jointly estimate parametric and nonparametric terms. They showed that this procedure can yield a consistent and robust estimator. However, the problem of variable selection for linear transformation models has been less studied, partially because a convenient loss function is not readily available under this context. In this paper, we propose a simple yet powerful approach to achieve both sparse and consistent estimation for linear transformation models. The main idea is to derive a profiled score from the estimating equation of Chen et al. [21], construct a loss function based on the profile scored and its variance, and then minimize the loss subject to some shrinkage penalty. Under regularity conditions, we have shown that the resulting estimator is consistent for both model estimation and variable selection. Furthermore, the estimated parametric terms are asymptotically normal and can achieve a higher efficiency than that yielded from the estimation equations. For computation, we suggest a one-step approximation algorithm which can take advantage of the LARS and build the entire solution path efficiently. Performance of the new procedure is illustrated through numerous simulations and real examples including one microarray data.

On Sparse Estimation for Semiparametric Linear Transformation Models

Semiparametric linear transformation models have received much attention due to its high flexibility in modeling survival data. A useful estimating equation procedure was recently proposed by Chen et al. (2002) for linear transformation models to jointly estimate parametric and nonparametric terms. They showed that this procedure can yield a consistent and robust estimator. However, the problem of variable selection for linear transformation models is less studied, partially because a convenient loss function is not readily available under this context. In this paper, we propose a simple yet powerful approach to achieve both sparse and consistent estimation for linear transformation models. The main idea is to derive a profiled score from the estimating equation of Chen et al. (2002), construct a loss function based on the profile scored and its variance, and then minimize the loss subject to some shrinkage penalty. Under regularity conditions, we have shown that the resulting estimator is consistent for both model estimation and variable selection. Furthermore, the estimated parametric terms are asymptotically normal and can achieve higher efficiency than that yielded from the estimation equations. For computation, we suggest a one-step approximation algorithm which can take advantage of the LARS and build the entire solution path efficiently. Performance of the new procedure is illustrated through numerous simulations and real examples including one microarray data.

Marginal Regression of Multivariate Event Times Based on Linear Transformation Models

Multivariate event time data are common in medical studies and have received much attention recently. In such data, each study subject may potentially experience several types of events or recurrences of the same type of event, or event times may be clustered. Marginal distributions are specified for the multivariate event times in multiple events and clustered events data, and for the gap times in recurrent events data, using the semiparametric linear transformation models while leaving the dependence structures for related events unspecified. We propose several estimating equations for simultaneous estimation of the regression parameters and the transformation function. It is shown that the resulting regression estimators are asymptotically normal, with variance-covariance matrix that has a closed form and can be consistently estimated by the usual plug-in method. Simulation studies show that the proposed approach is appropriate for practical use. An application to the well-known bladder cancer tumor recurrences data is also given to illustrate the methodology.

Miscellanea. On the linear transformation model for censored data

Recently Cheng, Wei & Ying (1995, 1997) proposed a class of estimation procedures for semiparametric linear transformation models with censored observations. When the support of the censoring variable is shorter than that of the failure time, the estimators are asymptotically biased. In this paper, we present a simple modification of Cheng's estimation procedures for the regression parameters. Through extensive numerical studies with practical sample sizes, we find that the new proposals perform well, but the original interval estimators may not have correct coverage probabilities when censoring is heavy. Prediction procedures for the survival probabilities of future subjects are also modified accordingly.