A multiple imputation approach to Cox regression with interval-censored data

We propose a general semiparametric method based on multiple imputation for Cox regression with interval-censored data. The method consists of iterating the following two steps. First, from finite-interval-censored (but not right-censored) data, exact failure times are imputed using Tanner and Wei's poor man's or asymptotic normal data augmentation scheme based on the current estimates of the regression coefficient and the baseline survival curve. Second, a standard statistical procedure for right-censored data, such as the Cox partial likelihood method, is applied to imputed data to update the estimates. Through simulation, we demonstrate that the resulting estimate of the regression coefficient and its associated standard error provide a promising alternative to the nonparametric maximum likelihood estimate. Our proposal is easily implemented by taking advantage of existing computer programs for right-censored data.     

Median Regression Model with Interval Censored Data

Quantile regression methods have been used to estimate upper and lower quantile reference curves as the function of several covariates. Especially, in survival analysis, median regression models to the right‐censored data are suggested with several assumptions. In this article, we consider a median regression model for interval‐censored data and construct an estimating equation based on weights derived from interval‐censored data. In a simulation study, the performances of the proposed method are evaluated for both symmetric and right‐skewed distributed failure times. A well‐known breast cancer data are analyzed to illustrate the proposed method.     

A comparison of methods for determining compartmental analysis parameters

The traditional method for determining compartmental analysis parameters relies on a visual selection of data points to be used for regression of data from each cellular compartment. This method is appropriate when the compartments are kinetically discrete and are easily discernible. However, where treatment effects on compartment parameters are being evaluated, a more objective method for determining initial parameters is desirable.

Three methods were examined for determining initial isotopic contents and half-times of ⁸⁶Rb elution from cellular compartments using theoretical data with known parameters. Experimental data from roots of Douglas fir (Pseudotsuga menziesii [Mirb.] Franco) and barley (Hordeum vulgare L.) intact seedlings were also used. The three methods were a visually assisted, linear regression on data of semilog plot of isotope elution versus time, a microcomputer-assisted, linear regression on semilog plot where maximization of the square of the correlation coefficient (r²) was the criterion to determine data points needed for each regression and a mainframe computer-assisted, direct nonlinear regression on elution data using a model of the sum of three exponential decay functions. The visual method resulted in the least accurate estimates of compartmental analysis parameters. The microcomputer-assisted and nonlinear regression methods calculated the parameters equally well.

     

Use of nonlinear regression to analyze enzyme kinetic data: application to situations of substrate contamination and background subtraction

In a recent publication, A. Lundin, P. Arner, and J. Hellmér [Anal. Biochem. 177, 125-131 (1989)] describe a method whereby kinetic substrate assays can be performed when the assay mixture includes a significant contaminating levels of substrate. Their method requires various rearrangements of the data, and involves three separate linear regression calculations. We show how the same data may be analyzed directly, and far more simply, by nonlinear regression. Unlike the linear regression method, nonlinear regression allows direct calculation of the actual values for Km, Vmax, and the concentration of contaminating substrate (as well as estimates of their standard errors); the former method gives only apparent values. The nonlinear regression technique is also statistically a more valid means of analysis, as the rearrangements required to give linearized equations will considerably distort the error distribution and render simple unweighted linear regression inappropriate. The ease of incorporating extra parameters into standard equations when nonlinear regression is used is further illustrated by fitting enzyme reaction data which describe a first-order process when a significant nonspecific background is present. For this equation no simple rearranged linear plot is possible, but nonlinear regression is easily applied to determine the kinetic parameters.     

Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information

This article presents a new method for maximum likelihood estimation of logistic regression models with incomplete covariate data where auxiliary information is available. This auxiliary information is extraneous to the regression model of interest but predictive of the covariate with missing data. Ibrahim (1990, Journal of the American Statistical Association 85, 765-769) provides a general method for estimating generalized linear regression models with missing covariates using the EM algorithm that is easily implemented when there is no auxiliary data. Vach (1997, Statistics in Medicine 16, 57-72) describes how the method can be extended when the outcome and auxiliary data are conditionally independent given the covariates in the model. The method allows the incorporation of auxiliary data without making the conditional independence assumption. We suggest tests of conditional independence and compare the performance of several estimators in an example concerning mental health service utilization in children. Using an artificial dataset, we compare the performance of several estimators when auxiliary data are available.     

Grouped generalized estimating equations for longitudinal data analysis

Generalized estimating equation (GEE) is widely adopted for regression modeling for longitudinal data, taking account of potential correlations within the same subjects. Although the standard GEE assumes common regression coefficients among all the subjects, such an assumption may not be realistic when there is potential heterogeneity in regression coefficients among subjects. In this paper, we develop a flexible and interpretable approach, called grouped GEE analysis, to modeling longitudinal data with allowing heterogeneity in regression coefficients. The proposed method assumes that the subjects are divided into a finite number of groups and subjects within the same group share the same regression coefficient. We provide a simple algorithm for grouping subjects and estimating the regression coefficients simultaneously, and show the asymptotic properties of the proposed estimator. The number of groups can be determined by the cross validation with averaging method. We demonstrate the proposed method through simulation studies and an application to a real data set.     

Distribution-free regression analysis of grouped survival data

Methods based on regression models for logarithmic hazard functions, Cox models, are given for analysis of grouped and censored survival data. By making an approximation it is possible to obtain explicitly a maximum likelihood function involving only the regression parameters. This likelihood function is a convenient analog to Cox's partial likelihood for ungrouped data. The method is applied to data from a toxicological experiment.     

Survival analysis with time-varying regression effects using a tree-based approach

Nonproportional hazards often arise in survival analysis, as is evident in the data from the International Non-Hodgkin's Lymphoma Prognostic Factors Project. A tree-based method to handle such survival data is developed for the assessment and estimation of time-dependent regression effects under a Cox-type model. The tree method approximates the time-varying regression effects as piecewise constants and is designed to estimate change points in the regression parameters. A fast algorithm that relies on maximized score statistics is used in recursive segmentation of the time axis. Following the segmentation, a pruning algorithm with optimal properties similar to those of classification and regression trees (CART) is used to determine a sparse segmentation. Bootstrap resampling is used in correcting for overoptimism due to split point optimization. The piecewise constant model is often more suitable for clinical interpretation of the regression parameters than the more flexible spline models. The utility of the algorithm is shown on the lymphoma data, where we further develop the published International Risk Index into a time-varying risk index for non-Hodgkin's lymphoma.     

Integrating external biological knowledge in the construction of regulatory networks from time-series expression data

ABSTRACT: BACKGROUND: Inference about regulatory networks from high-throughput genomics data is of great interest in systems biology. We present a Bayesian approach to infer gene regulatory networks from time series expression data by integrating various types of biological knowledge. RESULTS: We formulate network construction as a series of variable selection problems and use linear regression to model the data. Our method summarizes additional data sources with an informative prior probability distribution over candidate regression models. We extend the Bayesian model averaging (BMA) variable selection method to select regulators in the regression framework. We summarize the external biological knowledge by an informative prior probability distribution over the candidate regression models. CONCLUSIONS: We demonstrate our method on simulated data and a set of time-series microarray experiments measuring the effect of a drug perturbation on gene expression levels, and show that it outperforms leading regression-based methods in the literature.     

Median regression models for longitudinal data with dropouts

Summary .  Recently, median regression models have received increasing attention. When continuous responses follow a distribution that is quite different from a normal distribution, usual mean regression models may fail to produce efficient estimators whereas median regression models may perform satisfactorily. In this article, we discuss using median regression models to deal with longitudinal data with dropouts. Weighted estimating equations are proposed to estimate the median regression parameters for incomplete longitudinal data, where the weights are determined by modeling the dropout process. Consistency and the asymptotic distribution of the resultant estimators are established. The proposed method is used to analyze a longitudinal data set arising from a controlled trial of HIV disease ( Volberding et al., 1990 , The New England Journal of Medicine 322, 941–949). Simulation studies are conducted to assess the performance of the proposed method under various situations. An extension to estimation of the association parameters is outlined.     

The clustering of regression models method with applications in gene expression data

Identification of differentially expressed genes and clustering of genes are two important and complementary objectives addressed with gene expression data. For the differential expression question, many "per-gene" analytic methods have been proposed. These methods can generally be characterized as using a regression function to independently model the observations for each gene; various adjustments for multiplicity are then used to interpret the statistical significance of these per-gene regression models over the collection of genes analyzed. Motivated by this common structure of per-gene models, we proposed a new model-based clustering method--the clustering of regression models method, which groups genes that share a similar relationship to the covariate(s). This method provides a unified approach for a family of clustering procedures and can be applied for data collected with various experimental designs. In addition, when combined with per-gene methods for assessing differential expression that employ the same regression modeling structure, an integrated framework for the analysis of microarray data is obtained. The proposed methodology was applied to two microarray data sets, one from a breast cancer study and the other from a yeast cell cycle study.     

Quasi Likelihood/Moment Method for Generalized and Restricted Generalized Poisson Regression Models and Its Application

This paper reviews the generalized Poisson regression model, the restricted generalized Poisson regression model and the mixed Poisson regression (negative binomial regression and Poisson inverse Gaussian regression) models which can be used for regression analysis of counts. The aim of this study is to demonstrate the quasi likelihood/moment method, which is used for estimation of the parameters of mixed Poisson regression models, also applicable to obtain the estimates of the parameters of the generalized Poisson regression and the restricted generalized Poisson regression models. Besides, at the end of this study an application related to this method for zoological data is given.     

应用神经网络和多元回归技术预测森林产量

应用传统统计技术常会因样本小和测量数据不符某种分布而受到限制。本文评价一种前馈型神经网络算法以预测落叶阔叶林产量。另外,还介绍一种由定性变为定量的数据变换方法,以用相对小的样本建立多元回归预测模型。数据变换方法有助于改善多元回归模型的预测效果。在本实验的条件下,研究结果表明神经网络技术能够产生最好的预测效果.     

Tree-Structured Exponential Regression Modeling

A method for fitting piecewise exponential regression models to censored survival data is described. Stratification is performed recursively, using a combination of statistical tests and residual analysis. The splitting criterion employed in cross-validation is the average squared error of the residuals. The bootstrap is employed to keep the probability of a type I error (the error of discovering two or more strata when there is only one) of the method close to a predetermined value. The proposed method can thus also serve as a formal goodness-of-fit test for the exponential regression model. Real and simulated data are used for illustration.     

Analysis of failure time data with dependent interval censoring

This article develops a method for the analysis of screening data for which the chance of being screened is dependent on the event of interest (informative censoring). Because not all subjects make all screening visits, the data on the failure of interest is interval censored. We propose a model that will properly adjust for the dependence to obtain an unbiased estimate of the nonparametric failure time function, and we provide an extension for applying the method for estimation of the regression parameters from a (discrete time) proportional hazards regression model. The method is applied on a data set from an observational study of cytomegalovirus shedding in a population of HIV-infected subjects who participated in a trial conducted by the AIDS Clinical Trials Group.     

Adaptive regression splines in the Cox model

We develop a method for constructing adaptive regression spline models for the exploration of survival data. The method combines Cox's (1972, Journal of the Royal Statistical Society, Series B 34, 187-200) regression model with a weighted least-squares version of the multivariate adaptive regressi on spline (MARS) technique of Friedman (1991, Annals of Statistics 19, 1-141) to adaptively select the knots and covariates. The new technique can automatically fit models with terms that represent nonlinear effects and interactions among covariates. Applications based on simulated data and data from a clinical trial for myeloma are presented. Results from the myeloma application identified several important prognostic variables, including a possible nonmonotone relationship with survival in one laboratory variable. Results are compared to those from the adaptive hazard regression (HARE) method of Kooperberg, Stone, and Truong (1995, Journal of the American Statistical Association 90, 78-94).     

A simple linear regression method for quantitative trait loci linkage analysis with censored observations

Standard quantitative trait loci (QTL) mapping techniques commonly assume that the trait is both fully observed and normally distributed. When considering survival or age-at-onset traits these assumptions are often incorrect. Methods have been developed to map QTL for survival traits; however, they are both computationally intensive and not available in standard genome analysis software packages. We propose a grouped linear regression method for the analysis of continuous survival data. Using simulation we compare this method to both the Cox and Weibull proportional hazards models and a standard linear regression method that ignores censoring. The grouped linear regression method is of equivalent power to both the Cox and Weibull proportional hazards methods and is significantly better than the standard linear regression method when censored observations are present. The method is also robust to the proportion of censored individuals and the underlying distribution of the trait. On the basis of linear regression methodology, the grouped linear regression model is computationally simple and fast and can be implemented readily in freely available statistical software.     

Combining regression trees and radial basis function networks

We describe a method for non-parametric regression which combines regression trees with radial basis function networks. The method is similar to that of Kubat, who was first to suggest such a combination, but has some significant improvements. We demonstrate the features of the new method, compare its performance with other methods on DELVE data sets and apply it to a real world problem involving the classification of soybean plants from digital images.     

Response-adaptive regression for longitudinal data

We propose a response-adaptive model for functional linear regression, which is adapted to sparsely sampled longitudinal responses. Our method aims at predicting response trajectories and models the regression relationship by directly conditioning the sparse and irregular observations of the response on the predictor, which can be of scalar, vector, or functional type. This obliterates the need to model the response trajectories, a task that is challenging for sparse longitudinal data and was previously required for functional regression implementations for longitudinal data. The proposed approach turns out to be superior compared to previous functional regression approaches in terms of prediction error. It encompasses a variety of regression settings that are relevant for the functional modeling of longitudinal data in the life sciences. The improved prediction of response trajectories with the proposed response-adaptive approach is illustrated for a longitudinal study of Kiwi weight growth and by an analysis of the dynamic relationship between viral load and CD4 cell counts observed in AIDS clinical trials.     

Linking gene expression data with patient survival times using partial least squares

There is an increasing need to link the large amount of genotypic data, gathered using microarrays for example, with various phenotypic data from patients. The classification problem in which gene expression data serve as predictors and a class label phenotype as the binary outcome variable has been examined extensively, but there has been less emphasis in dealing with other types of phenotypic data. In particular, patient survival times with censoring are often not used directly as a response variable due to the complications that arise from censoring. We show that the issues involving censored data can be circumvented by reformulating the problem as a standard Poisson regression problem. The procedure for solving the transformed problem is a combination of two approaches: partial least squares, a regression technique that is especially effective when there is severe collinearity due to a large number of predictors, and generalized linear regression, which extends standard linear regression to deal with various types of response variables. The linear combinations of the original variables identified by the method are highly correlated with the patient survival times and at the same time account for the variability in the covariates. The algorithm is fast, as it does not involve any matrix decompositions in the iterations. We apply our method to data sets from lung carcinoma and diffuse large B-cell lymphoma studies to verify its effectiveness.