Quantile regression methods with varying-coefficient models for censored data

Considerable intellectual progress has been made to the development of various semiparametric varying-coefficient models over the past ten to fifteen years. An important advantage of these models is that they avoid much of the curse of dimensionality problem as the nonparametric functions are restricted only to some variables. More recently, varying-coefficient methods have been applied to quantile regression modeling, but all previous studies assume that the data are fully observed. The main purpose of this paper is to develop a varying-coefficient approach to the estimation of regression quantiles under random data censoring. We use a weighted inverse probability approach to account for censoring, and propose a majorize–minimize type algorithm to optimize the non-smooth objective function. The asymptotic properties of the proposed estimator of the nonparametric functions are studied, and a resampling method is developed for obtaining the estimator of the sampling variance. An important aspect of our method is that it allows the censoring time to depend on the covariates. Additionally, we show that this varying-coefficient procedure can be further improved when implemented within a composite quantile regression framework. Composite quantile regression has recently gained considerable attention due to its ability to combine information across different quantile functions. We assess the finite sample properties of the proposed procedures in simulated studies. A real data application is also considered.     

Influence diagnostics in log-Birnbaum-Saunders regression models with censored data

In this paper we discuss log-Birnbaum-Saunders regression models with censored observations. This kind of model has been largely applied to study material lifetime subject to failure or stress. The score functions and observed Fisher information matrix are given as well as the process for estimating the regression coefficients and shape parameter is discussed. The normal curvatures of local influence are derived under various perturbation schemes and two deviance-type residuals are proposed to assess departures from the log-Birnbaum-Saunders error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed under log-Birnbaum-Saunders regression models. A diagnostic analysis is performed in order to select an appropriate model.     

Influence diagnostics in log-Birnbaum–Saunders regression models with censored data

In this paper we discuss log-Birnbaum–Saunders regression models with censored observations. This kind of model has been largely applied to study material lifetime subject to failure or stress. The score functions and observed Fisher information matrix are given as well as the process for estimating the regression coefficients and shape parameter is discussed. The normal curvatures of local influence are derived under various perturbation schemes and two deviance-type residuals are proposed to assess departures from the log-Birnbaum–Saunders error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed under log-Birnbaum–Saunders regression models. A diagnostic analysis is performed in order to select an appropriate model.     

Quantile regression with doubly censored data

Quantile regression offers a semiparametric approach to modeling data with possible heterogeneity. It is particularly attractive for censored responses, where the conditional mean functions are unidentifiable without parametric assumptions on the distributions. A new algorithm is proposed to estimate the regression quantile process when the response variable is subject to double censoring. The algorithm distributes the probability mass of each censored point to its left or right appropriately, and iterates towards self-consistent solutions. Numerical results on simulated data and an unemployment duration study are given to demonstrate the merits of the proposed method.     

Log-modified Weibull regression models with censored data: Sensitivity and residual analysis

This paper proposes a regression model considering the modified Weibull distribution. This distribution can be used to model bathtub-shaped failure rate functions. Assuming censored data, we consider maximum likelihood and Jackknife estimators for the parameters of the model. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and we also present some ways to perform global influence. Besides, for different parameter settings, sample sizes and censoring percentages, various simulations are performed and the empirical distribution of the modified deviance residual is displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended for a martingale-type residual in log-modified Weibull regression models with censored data. Finally, we analyze a real data set under log-modified Weibull regression models. A diagnostic analysis and a model checking based on the modified deviance residual are performed to select appropriate models.     

Additive models in censored regression

Additive models in censored regression are considered. A randomly weighted version of the backfitting algorithm that allows for the nonparametric estimation of the effects of the covariates on the response is provided. Given the high computational cost involved, binning techniques are used to speed up the computation in the estimation and testing process. Simulation results and the application to real data reveal that the predictor obtained with the additive model performs well, and that it is a convenient alternative to the linear predictor when some nonlinear effects are expected.     

Rank regression for accelerated failure time model with clustered and censored data

For clustered survival data, the traditional Gehan-type estimator is asymptotically equivalent to using only the between-cluster ranks, and the within-cluster ranks are ignored. The contribution of this paper is two fold, (i) incorporating within-cluster ranks in censored data analysis, and (ii) applying the induced smoothing of Brown and Wang (2005, Biometrika) for computational convenience. Asymptotic properties of the resulting estimating functions are given. We also carry out numerical studies to assess the performance of the proposed approach and conclude that the proposed approach can lead to much improved estimators when strong clustering effects exist. A dataset from a litter-matched tumorigenesis experiment is used for illustration.     

BJ: an S-Plus program to fit linear regression models to censored data using the Buckley-James method

Most researchers are familiar with ordinary multiple regression models, most commonly fitted using the method of least squares. The method of Buckley and James (J. Buckley, I. James, Linear regression with censored data, Biometrika 66 (1979) 429-436.) is an extension of least squares for fitting multiple regression models when the response variable is right-censored as in the analysis of survival time data. The Buckley-James method has been shown to have good statistical properties under usual regularity conditions (T.L. Lai, Z. Ying, Large sample theory of a modified Buckley-James estimator for regression analysis with censored data, Ann. Stat. 19 (1991) 1370-1402.). Nevertheless, even after 20 years of its existence, it is almost never used in practice. We believe that this is mainly due to lack of software and we describe here an S-Plus program that through its inclusion in a public domain function library fully exploits the power of the S-Plus programming environment. This environment provides multiple facilities for model specification, diagnostics, statistical inference, and graphical depiction of the model fit.     

SAS and R functions to compute pseudo-values for censored data regression

Recently, in a series of papers, a method based on pseudo-values has been proposed for direct regression modeling of the survival function, the restricted mean and cumulative incidence function with right censored data. The models, once the pseudo-values have been computed, can be fit using standard generalized estimating equation software. Here we present SAS macros and R functions to compute these pseudo-values. We illustrate the use of these routines and show how to obtain regression estimates for a study of bone marrow transplant patients.     

Chebyshev approaches for imbalanced data streams regression models

Data Mining and Knowledge Discovery - In recent years data stream mining and learning from imbalanced data have been active research areas. Even though solutions exist to tackle these two problems,...     

Simple resampling methods of approximating the distribution of LAD estimators for doubly censored regression models

Recently, least absolute deviation (LAD) estimator for median regression models with doubly censored data was proposed and the asymptotic normality of the estimator was established, and the methods based on bootstrap and random weighting were proposed respectively to approximate the distribution of the LAD estimators. But the calculation of the estimators requires solving a non-convex and non-smooth minimization problem, resulting in high computational costs in implementing the bootstrap or random weighting method directly. In this paper, computationally simple resampling methods are proposed to approximate the distribution of the doubly censored LAD estimators. The objective functions in the resampling stage of the new methods are piece-wise linear and convex, and their minimizer can be obtained by the linear programming in the same way as that for the case of uncensored median regression.     

An empirical study of a test for polynomial relationships in randomly right censored regression models

In this paper, a test statistic is constructed to test polynomial relationships in randomly right censored regression models based on the local polynomial smoothing technique. Two bootstrap procedures, namely the residual-based bootstrap and the naive bootstrap procedures, are suggested to derive the p-value of the test. Some simulations are conducted to empirically assess the performance of the two bootstrap procedures. The results demonstrate that the residual-based bootstrap performs much better than the naive bootstrap and the test method with the residual-based bootstrap to derive the p-value works satisfactorily. Although the limiting distribution of the test statistic and the consistency of the bootstrap approximations remain to be investigated, simulation results indicate that the proposed test method may be of some practical use. As a real example, the proposed test is applied to the Stanford heart transplant data.     

Nonparametric regression models for right-censored data using Bernstein polynomials

In some applications of survival analysis with covariates, the commonly used semiparametric assumptions (e.g., proportional hazards) may turn out to be stringent and unrealistic, particularly when there is scientific background to believe that survival curves under different covariate combinations will cross during the study period. We present a new nonparametric regression model for the conditional hazard rate using a suitable sieve of Bernstein polynomials. The proposed nonparametric methodology has three key features: (i) the smooth estimator of the conditional hazard rate is shown to be a unique solution of a strictly convex optimization problem for a wide range of applications; making it computationally attractive, (ii) the model is shown to encompass a proportional hazards structure, and (iii) large sample properties including consistency and convergence rates are established under a set of mild regularity conditions. Empirical results based on several simulated data scenarios indicate that the proposed model has reasonably robust performance compared to other semiparametric models particularly when such semiparametric modeling assumptions are violated. The proposed method is further illustrated on the gastric cancer data and the Veterans Administration lung cancer data.     

Assessing the lifetime performance index of products from progressively type II right censored data using Burr XII model

Process capability analysis has been widely applied in the field of quality control to monitor the performance of industrial processes. In practice, lifetime performance index C_L is a popular means to assess the performance and potential of their processes, where L is the lower specification limit. Nevertheless, many processes possess a non-normal lifetime model, the assumption of normality is often erroneous. Progressively censoring scheme is quite useful in many practical situations where budget constraints are in place or there is a demand for rapid testing. The study will apply data transformation technology to constructs a maximum likelihood estimator (MLE) of C_L under the Burr XII distribution based on the progressively type II right censored sample. The MLE of C_L is then utilized to develop a new hypothesis testing procedure in the condition of known L. Finally, we give two examples to illustrate the use of the testing procedure under given significance level α.     

Local likelihood regression in generalized linear single-index models with applications to microarray data

Searching for an effective dimension reduction space is an important problem in regression, especially for high-dimensional data such as microarray data. A major characteristic of microarray data consists in the small number of observations n and a very large number of genes p. This “large p, small n” paradigm makes the discriminant analysis for classification difficult. In order to offset this dimensionality problem a solution consists in reducing the dimension. Supervised classification is understood as a regression problem with a small number of observations and a large number of covariates. A new approach for dimension reduction is proposed. This is based on a semi-parametric approach which uses local likelihood estimates for single-index generalized linear models. The asymptotic properties of this procedure are considered and its asymptotic performances are illustrated by simulations. Applications of this method when applied to binary and multiclass classification of the three real data sets Colon, Leukemia and SRBCT are presented.     

Obtaining linguistic fuzzy rule-based regression models from imprecise data with multiobjective genetic algorithms

Backfitting of fuzzy rules is an Iterative Rule Learning technique for obtaining the knowledge base of a fuzzy rule-based system in regression problems. It consists in fitting one fuzzy rule to the data, and replacing the whole training set by the residual of the approximation. The obtained rule is added to the knowledge base, and the process is repeated until the residual is zero, or near zero. Such a design has been extended to imprecise data for which the observation error is small. Nevertheless, when this error is moderate or high, the learning can stop early. In this kind of algorithms, the specificity of the residual might decrease when a new rule is added. There may happen that the residual grows so wide that it covers the value zero for all points (thus the algorithm stops), but we have not yet extracted all the information available in the dataset. Focusing on this problem, this paper is about datasets with medium to high discrepancies between the observed and the actual values of the variables, such as those containing missing values and coarsely discretized data. We will show that the quality of the iterative learning degrades in this kind of problems, because it does not make full use of all the available information. As an alternative to sequentially obtaining rules, we propose a new multiobjective Genetic Cooperative Competitive Learning (GCCL) algorithm. In our approach, each individual in the population codifies one rule, which competes in the population in terms of maximum coverage and fitting, while the individuals in the population cooperate to form the knowledge base.

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Support vector censored quantile regression under random censoring

Censored quantile regression models have received a great deal of attention in both the theoretical and applied statistical literature. In this paper, we propose support vector censored quantile regression (SVCQR) under random censoring using iterative reweighted least squares (IRWLS) procedure based on the Newton method instead of usual quadratic programming algorithms. This procedure makes it possible to derive the generalized approximate cross validation (GACV) method for choosing the hyperparameters which affect the performance of SVCQR. Numerical results are then presented which illustrate the performance of SVCQR using the IRWLS procedure.     

多模型数据集的免疫鲁棒回归分析

针对传统多模型数据集回归分析方法计算时间长、模型识别准确率低的问题,提出了一种新的启发式鲁棒回归分析方法。该方法模拟免疫系统聚类学习的原理,采用B细胞网络作为数据集的分类和存储工具,通过判断数据对模型的符合度进行分类,提高了数据分类的准确性,将模型集抽取过程分解成“聚类”“回归”“再聚类”的反复尝试过程,利用并行启发式搜索逼近模型集的解。仿真结果表明,所提方法回归分析时间明显少于传统算法,模型识别准确率明显高于传统算法。根据8模型数据集分析结果,传统算法中,效果最好的是基于RANSAC的逐次提取算法,其平均模型识别准确率为90.37%,需53.3947s;计算时间小于0.5s的传统算法,其准确率不足1%;所提算法仅需0.5094s,其准确率达到了98.25%。