Empirical Likelihood for Censored Linear Regression and Variable Selection.

Numerous variable selection techniques have been developed for complete high-dimensional data but very few of them for censored data. The techniques for complete data must be modified if censoring is present. In this paper, we consider the variable selection technique for accelerated failure time (AFT) models by extending the ranking-based variable selection (RBVS) algorithm and its iterative procedure as proposed in the work of Baranowski et al. through the Stute’s weighted least square technique. Simulation studies are conducted to demonstrate the performance of the proposed methods. We further illustrate the performance of this method with a mantle cell lymphoma microarray example. When there is no correlation among the covariates, the proposed method outperforms the iterative sure independence screening and stability selection methods in terms of overall performance for high-dimensional data. Real data analysis also suggests that the proposed method can be chosen for high-dimensional censored data analysis in parallel to other methods in the literature.

Empirical likelihood based on synthetic right censored data

One of the most important endpoints in haematopoietic cell transplant research is survival. A common objective is to interrogate which, if any, co-variates correlate with these endpoints. The most common statistical approach uses the Cox proportional hazards model. However, there are several problems and limitations of using this model including assumptions of proportional hazards and homogenous effects. In contrast, results of transplant studies often show non-proportional hazards because of early transplant-related mortality such that there is a survival disadvantage to transplants early on followed by a benefit. Even when a transplant proves better than a comparator not all transplant recipients benefit equally and some may be disadvantaged. Also, the favourable or unfavourable impact of a co-variate may vary in different time intervals. The accelerated failure time model which directly evaluates the association between survival and co-variates has similar limitations. Also, these models confer only a static view of the treatment effect. Several articles in our statistics series such as that by Zhen-Huan Hu and us (Bone Marrow Transplant. 2021 Aug 19. doi: 10.1038/s41409-021-01435-2), by Zhen-Huan Hu, Hai-Lin Wang and us and forthcoming articles by Megan Othus and by Liesbeth C. de Wreede, Johannes Schetelig and Hein Putter discuss issues in proper analyses of survival data from transplant studies including observational databases and randomized controlled trials. Are there better alternatives? A new popular model is quantile regression. In this typescript Bo Wei concisely introduce the quantile regression model for right censored data. He uses data from a Center for International Blood and Marrow Transplant Research (CIBMTR) registry study to show how to use the quantile regression and interpret the results. He also discusses use of quantile regression in complex survival analyses such as competing risk data or non-compliant data. Quantile regression is a natural, powerful approach for analyzing censored data with heterogenous co-variate effects. It has advantages compared with other survival models in depicting the dynamic association between survival outcome and co-variates. It can be applied to other transplant outcomes such as cumulative incidence of relapse, event-free and relapse-free survivals. There is an equation, but only one. Remember: The only thing to fear is fear itself (FDR). Please stick with it and you will be rewarded.Robert Peter Gale MD, PhD, DSc(hc), FACP, FRCP, FRCPI(hon), LHD, DPS, Mei-Jie Zhang PhD.

Dealing with high-dimensional censored data is very challenging because of the complexities in data structure. This article focuses on developing a variable selection procedure for censored high-dimensional data with the AFT models using the Modified Correlation Adjusted coRrelation (MCAR) scores method. The latter is developed based on CAR scores method that provides a canonical ordering that encourages grouping of correlated predictors and down-weights antagonistic variables. The proposed MCAR scores method is developed as an extension of the CAR scores method using NOVEL integration of the sample and threshold estimator of the correlation matrix as suggested by Huang and Frylewicz. The proposed MCAR exhibits computationally more efficient estimates under model sparsity and can provide a canonical ordering among the predictors. The MCAR method is a greedy method that is also easy to understand and can perform estimation and variable selection simultaneously. Performances of variable selection by the MCAR method have been compared with other existing regularized techniques in literature-such as the lasso, elastic net and with a machine learning technique called boosting and with the censored CAR by a number of simulation studies and a real microarray data set called diffuse large-B-cell lymphoma. Results indicate that when correlation exists among the covariates, the MCAR method outperforms all five techniques while for uncorrelated data, the MCAR performs quite similar to the CAR method but clearly outperforms the other three methods. The empirical study further reveals that the MCAR method exhibits the best predictive performance among the methods.

In this paper, we consider how to yield a robust empirical likelihood estimation for regression models. After introducing modal regression, we propose a novel empirical likelihood method based on modal regression estimation equations, which has the merits of both robustness and high inference efficiency compared with the least square based methods. Under some mild conditions, we show that Wilks’ theorem of the proposed empirical likelihood approach continues to hold. Advantages of empirical likelihood modal regression as a nonparametric approach are illustrated by constructing confidence intervals/regions. Two simulation studies and a real data analysis confirm our theoretical findings.

Censored outcome data are commonly encountered in criminology. Criminologists sometimes use the tobit model to address these censored data. While tobit models make more realistic demands of censored outcome data than ordinary least squares (OLS) regression, they require the researcher to make strong distributional assumptions. When these assumptions are not met, as is often the case in criminological data, tobit models yield biased and inconsistent estimates. We seek to demonstrate this substantial bias in simulation analyses and present easily applied alternative methods. The tobit model and semiparametric alternatives for censored outcome data are applied with simulated data under varying conditions. These simulations are followed with an empirical example using sentencing data. The bias from tobit can be corrected through application of semiparametric alternatives. Criminologists should begin their analyses of censored outcome data with the least restrictive of the available models (CLAD) before progressing to more efficient, but potentially biased, estimators.

Length-biased data arise when the probability of observing a subject in a sample is proportional to its corresponding value. The statistical tools that should be applied to analyse such sets of data differ from those used for random samples. Ignoring this issue in an investigation results in a biased inference. In this paper, we propose an empirical likelihood-based method to draw inference on covariate effects in an accelerated failure time model while the observations are subject to length-bias. The asymptotic distribution of the empirical log-likelihood ratio statistic is derived to be a weighted sum of independent chi-square distributions. Hence, we then extend the results by an exploration into the adjusted empirical likelihood. We derive an asymptotic standard chi-square distribution for the adjusted log-likelihood statistic. The limiting distributions are applied to obtain confidence regions for the regression parameters. A simulation study is carried out to evaluate and compare the performance of the proposed procedures with an existing method based on the normal approximation approach. Finally, the procedures are illustrated by modelling the regression parameter and estimating confidence intervals for a set of real data on widths of shrubs.

Kendall and Gehan estimating functions are commonly used to estimate the regression parameter in accelerated failure time model with censored observations in survival analysis. In this paper, we apply the jackknife empirical likelihood method to overcome the computation difficulty about interval estimation. A Wilks’ theorem of jackknife empirical likelihood for U-statistic type estimating equations is established, which is used to construct the confidence intervals for the regression parameter. We carry out an extensive simulation study to compare the Wald-type procedure, the empirical likelihood method, and the jackknife empirical likelihood method. The proposed jackknife empirical likelihood method has a better performance than the existing methods. We also use a real data set to compare the proposed methods.

ABSTRACTBased on the inverse probability weight method, we, in this article, construct the empirical likelihood (EL) and penalized empirical likelihood (PEL) ratios of the parameter in the linear quantile regression model when the covariates are missing at random, in the presence and absence of auxiliary information, respectively. It is proved that the EL ratio admits a limiting Chi-square distribution. At the same time, the asymptotic normality of the maximum EL and PEL estimators of the parameter is established. Also, the variable selection of the model in the presence and absence of auxiliary information, respectively, is discussed. Simulation study and a real data analysis are done to evaluate the performance of the proposed methods.

This article introduces an original methodology based on empirical likelihood, which aims at combining different food contamination and consumption surveys to provide risk managers with a risk measure, taking into account all the available information. This risk index is defined as the probability that exposure to a contaminant exceeds a safe dose. It is naturally expressed as a nonlinear functional of the different consumption and contamination distributions, more precisely as a generalized U-statistic. This nonlinearity and the huge size of the data sets make direct computation of the problem unfeasible. Using linearization techniques and incomplete versions of the U-statistic, a tractable "approximated" empirical likelihood program is solved yielding asymptotic confidence intervals for the risk index. An alternative "Euclidean likelihood program" is also considered, replacing the Kullback-Leibler distance involved in the empirical likelihood by the Euclidean distance. Both methodologies are tested on simulated data and applied to assess the risk due to the presence of methyl mercury in fish and other seafood.

Practical variable selection for generalized additive models

This study analyzes the performance of control charts using statistics to monitor the stability of the mean in paired data, in the presence of missing or censored data. To determine the control limits, extensive simulations of paired data with a normal distribution are conducted to establish an empirical distribution under different correlation scenarios, sample sizes, and percentages of censored or missing data. After calculating the control limits, additional simulations are carried out to assess the out of control average run length to detect changes in the mean. We employ the paired t-test and the paired Wilcoxon test when the population is normal as they obtain similar results, however if they differ significantly, thus it is an indication of a potential violation of the hypothesis of normality in the data. Additionally, we evaluated some proposals to minimize the negative effect of data displaying censoring and missing values. We found that the results using parametric and non-parametric approaches are practically similar when the data is complete; the non-parametric approach being more affected by missing and censored data. A practical example monitoring the most-streamed songs in the US illustrates the proposed procedure.

Survival models allowing for random effects (e.g., frailty models) have been widely used for analyzing clustered time‐to‐event data. Accelerated failure time (AFT) models with random effects are useful alternatives to frailty models. Because survival times are directly modeled, interpretation of the fixed and random effects is straightforward. Moreover, the fixed effect estimates are robust against various violations of the assumed model. In this paper, we propose a penalized h‐likelihood (HL) procedure for variable selection of fixed effects in the AFT random‐effect models. For the purpose of variable selection, we consider three penalty functions, namely, least absolute shrinkage and selection operator (LASSO), smoothly clipped absolute deviation (SCAD), and HL. We demonstrate via simulation studies that the proposed variable selection procedure is robust against the misspecification of the assumed model. The proposed method is illustrated using data from a bladder cancer clinical trial.

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.

Penalized empirical likelihood for semiparametric models with a diverging number of parameters