Penalized empirical likelihood for quantile regression with missing covariates and auxiliary information

Based 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.     

A resampling method by perturbing the estimating functions for quantile regression with missing data

In this article, we propose a resampling method based on perturbing the estimating functions to compute the asymptotic variances of quantile regression estimators under missing at random condition. We prove that the conditional distributions of the resampling estimators are asymptotically equivalent to the distributions of quantile regression estimators. Our method can deal with complex situations, where the response and part of covariates are missing. Numerical results based on simulated and real data are provided under several designs.     

Robust nonparametric estimation with missing data

In this paper, under a nonparametric regression model, we introduce two families of robust procedures to estimate the regression function when missing data occur in the response. The first proposal is based on a local M$M$-functional applied to the conditional distribution function estimate adapted to the presence of missing data. The second proposal imputes the missing responses using the local M$M$-smoother based on the observed sample and then estimates the regression function with the completed sample. We show that the robust procedures considered are consistent and asymptotically normally distributed. A robust procedure to select the smoothing parameter is also discussed.     

Smoothed empirical likelihood confidence intervals for quantile regression parameters with auxiliary information

This paper develops a smoothed empirical likelihood (SEL)-based method to construct confidence intervals for quantile regression parameters with auxiliary information. First, we define the SEL ratio and show that it follows a Chi-square distribution. We then construct confidence intervals according to this ratio. Finally, Monte Carlo experiments are employed to evaluate the proposed method.     

Nonparametric curve estimation with missing data: A general empirical process approach

A general nonparametric imputation procedure, based on kernel regression, is proposed to estimate points as well as set- and function-indexed parameters when the data are missing at random (MAR). The proposed method works by imputing a specific function of a missing value (and not the missing value itself), where the form of this specific function is dictated by the parameter of interest. Both single and multiple imputations are considered. The associated empirical processes provide the right tool to study the uniform convergence properties of the resulting estimators. Our estimators include, as special cases, the imputation estimator of the mean, the estimator of the distribution function proposed by Cheng and Chu [1996. Kernel estimation of distribution functions and quantiles with missing data. Statist. Sinica 6, 63–78], imputation estimators of a marginal density, and imputation estimators of regression functions.     

Semi-empirical likelihood inference for the ROC curve with missing data

The receiver operating characteristic (ROC) curve is one of the most commonly used methods to compare the diagnostic performance of two or more laboratory or diagnostic tests. In this paper, we propose semi-empirical likelihood based confidence intervals for ROC curves of two populations, where one population is parametric and the other one is non-parametric and both have missing data. After imputing missing values, we derive the semi-empirical likelihood ratio statistic and the corresponding likelihood equations. It is shown that the log-semi-empirical likelihood ratio statistic is asymptotically scaled chi-squared. The estimating equations are solved simultaneously to obtain the estimated lower and upper bounds of semi-empirical likelihood confidence intervals. We conduct extensive simulation studies to evaluate the finite sample performance of the proposed empirical likelihood confidence intervals with various sample sizes and different missing probabilities.     

Using auxiliary data for parameter estimation with non-ignorably missing outcomes

We propose a method for estimating parameters in generalized linear models when the outcome variable is missing for some subjects and the missing data mechanism is non-ignorable. We assume throughout that the covariates are fully observed. One possible method for estimating the parameters is maximum likelihood with a non-ignorable missing data model. However, caution must be used when fitting non-ignorable missing data models because certain parameters may be inestimable for some models. Instead of fitting a non-ignorable model, we propose the use of auxiliary information in a likelihood approach to reduce the bias, without having to specify a non-ignorable model. The method is applied to a mental health study.     

Inferences on missing information under multiple imputation and two-stage multiple imputation

In the presence of missing values, researchers may be interested in the rates of missing information. The rates of missing information are (a) important for assessing how the missing information contributes to inferential uncertainty about, Q, the population quantity of interest, (b) are an important component in the decision of the number of imputations, and (c) can be used to test model uncertainty and model fitting. In this article I will derive the asymptotic distribution of the rates of missing information in two scenarios: the conventional multiple imputation (MI), and the two-stage MI. Numerically I will show that the proposed asymptotic distribution agrees with the simulated one. I will also suggest the number of imputations needed to obtain reliable missing information rate estimates for each method, based on the asymptotic distribution.     

Efficient estimation for incomplete multivariate data

We review the Fisher scoring and EM algorithms for incomplete multivariate data from an estimating function point of view, and examine the corresponding quasi-score functions under second-moment assumptions. A bias-corrected REML-type estimator for the covariance matrix is derived, and the Fisher, Godambe and empirical sandwich information matrices are compared. We make a numerical investigation of the two algorithms, and compare with a hybrid algorithm, where Fisher scoring is used for the mean vector and the EM algorithm for the covariance matrix.     

Empirical Likelihood in a Semi-Parametric Model for Missing Response Data

Let Y be a response and, given covariate X,Y has a conditional density f(y | x, θ), where θ is a unknown p-dimensional vector of parameters and the marginal distribution of X is unknown. When responses are missing at random, with auxiliary information and imputation, we define an adjusted empirical log-likelihood ratio for the mean of Y and obtain its asymptotic distribution. A simulation study is conducted to compare the adjusted empirical log-likelihood and the normal approximation method in terms of coverage accuracies.     

A doubly robust goodness-of-fit test in general linear models with missing covariates

In this article, we utilize a form of general linear model where missing data occurred randomly on the covariates. We propose a test function based on the doubly robust method to investigate goodness of fit of the model. For this aim, kernel method is used to estimate unknown functions under estimating equation method. Doubly robustness and asymptotic properties of the test function are obtained under local and alternative hypotheses. Furthermore, we investigate the power of the proposed test function by means of some simulation studies and finally we apply this method on analyzing a real dataset.     

Analysing survey data with incomplete responses by using a method based on empirical likelihood

Summary.  In many surveys, missing response is a common problem. As an example, Zahner, Jacobs, Freeman and Trainor analysed data from a study of child psychopathology in the State of Connecticut, USA. In that study, the response variable, psychopathology, was inferred from questions that were addressed to teachers of the children and was subject to a high level of missingness. However, the missing responses were supplemented by surrogate information that was provided by the parents and/or the primary care providers of the children. In such a situation, it is conceivable that the supplemental information can be used to recover some of the information that has been lost in the cases with missing response. This paper considers a method using empirical likelihood. Empirical likelihood is well known in providing nonparametric inference. But its application has largely been confined to complete-data situations. The method proposed exploits the semiparametric nature of empirical likelihood. The method gives consistent estimates if the cases with non-missing responses form a random sample of the population. In large samples, the method behaves similarly to a regression estimate that is applied to estimating equations. The method is easy to implement with standard statistical packages. In a small sample study, the method was found to give favourable results, when compared with existing methods.     

Pseudolikelihood estimation in a class of problems with response-related missing covariates

Many practical situations involve a response variable Y and covariates X , where data on (Y, X ) are incomplete for some portion of a sample of individuals. We consider two general types of pseudolikelihood estimation for problems in which missingness may be response-related. These are typically simpler to implement than ordinary maximum likelihood, which in this context is semiparametric. Asymptotics for the pseudolikelihood methods are presented, and simulations conducted to investigate the methods for an important class of problems involving lifetime data. Our results indicate that for these problems the two methods are effective and comparable with respect to efficiency.     

Multiple imputation of missing data with ante-dependence covariance structure

A controlled clinical trial was conducted to investigate the efficacy effect of a chemical compound in the treatment of Premenstrual Dysphoric Disorder (PMDD). The data from the trial showed a non-monotone pattern of missing data and an ante-dependence covariance structure. A new analytical method for imputing the missing data with the ante-dependence covariance is proposed. The PMDD data are analysed by the non-imputation method and two imputation methods: the proposed method and the MCMC method.     

Semiparametric estimation of regression quantiles with application to standardizing weight for height and age in US children

The appropriate interpretation of measurements often requires standardization for concomitant factors. For example, standardization of weight for both height and age is important in obesity research and in failure-to-thrive research in children. Regression quantiles from a reference population afford one intuitive and popular approach to standardization. Current methods for the estimation of regression quantiles can be classified as nonparametric with respect to distributional assumptions or as fully parametric. We propose a semiparametric method where we model the mean and variance as flexible regression spline functions and allow the unspecified distribution to vary smoothly as a function of covariates. Similarly to Cole and Green, our approach provides separate estimates and summaries for location, scale and distribution. However, similarly to Koenker and Bassett, we do not assume any parametric form for the distribution. Estimation for either cross-sectional or longitudinal samples is obtained by using estimating equations for the location and scale functions and through local kernel smoothing of the empirical distribution function for standardized residuals. Using this technique with data on weight, height and age for females under 3 years of age, we find that there is a close relationship between quantiles of weight for height and age and quantiles of body mass index (BMI=weight/height²) for age in this cohort.     

Empirical likelihood for nonlinear regression models with nonignorable missing responses

This article develops three empirical likelihood (EL) approaches to estimate parameters in nonlinear regression models in the presence of nonignorable missing responses. These are based on the inverse probability weighted (IPW) method, the augmented IPW (AIPW) method and the imputation technique. A logistic regression model is adopted to specify the propensity score. Maximum likelihood estimation is used to estimate parameters in the propensity score by combining the idea of importance sampling and imputing estimating equations. Under some regularity conditions, we obtain the asymptotic properties of the maximum EL estimators of these unknown parameters. Simulation studies are conducted to investigate the finite sample performance of our proposed estimation procedures. Empirical results provide evidence that the AIPW procedure exhibits better performance than the other two procedures. Data from a survey conducted in 2002 are used to illustrate the proposed estimation procedure. The Canadian Journal of Statistics 48: 386–416; 2020 © 2020 Statistical Society of Canada     

Statistical inference under imputation for proportional hazard model with missing covariates

Missing covariate data are common in biomedical studies. In this article, by using the non parametric kernel regression technique, a new imputation approach is developed for the Cox-proportional hazard regression model with missing covariates. This method achieves the same efficiency as the fully augmented weighted estimators (Qi et al. 2005. Journal of the American Statistical Association, 100:1250) and has a simpler form. The asymptotic properties of the proposed estimator are derived and analyzed. The comparisons between the proposed imputation method and several other existing methods are conducted via a number of simulation studies and a mouse leukemia data.     

Dimension reduced kernel estimation for distribution function with incomplete data

This work focuses on the estimation of distribution functions with incomplete data, where the variable of interest Y has ignorable missingness but the covariate X is always observed. When X is high dimensional, parametric approaches to incorporate X—information is encumbered by the risk of model misspecification and nonparametric approaches by the curse of dimensionality. We propose a semiparametric approach, which is developed under a nonparametric kernel regression framework, but with a parametric working index to condense the high dimensional X—information for reduced dimension. This kernel dimension reduction estimator has double robustness to model misspecification and is most efficient if the working index adequately conveys the X—information about the distribution of Y. Numerical studies indicate better performance of the semiparametric estimator over its parametric and nonparametric counterparts. We apply the kernel dimension reduction estimation to an HIV study for the effect of antiretroviral therapy on HIV virologic suppression.     

A nonparametric inverse probability weighted estimation for functional data with missing response data at random

This paper considers the nonparametric inverse probability weighted estimation for functional data with missing response data at random. Under mild conditions, the asymptotic properties of the proposed estimation method are established. Based on the resampling method, the estimation of the asymptotic variance of the proposed estimator is obtained. Finally, the finite sample properties of the proposed estimation method are investigated via Monte Carlo simulation studies. A real data analysis is given to illustrate the use of the proposed method.