Local polynomial regression and simulation–extrapolation

Summary.  The paper introduces a new local polynomial estimator and develops supporting asymptotic theory for nonparametric regression in the presence of covariate measurement error. We address the measurement error with Cook and Stefanski's simulation–extrapolation (SIMEX) algorithm. Our method improves on previous local polynomial estimators for this problem by using a bandwidth selection procedure that addresses SIMEX's particular estimation method and considers higher degree local polynomial estimators. We illustrate the accuracy of our asymptotic expressions with a Monte Carlo study, compare our method with other estimators with a second set of Monte Carlo simulations and apply our method to a data set from nutritional epidemiology. SIMEX was originally developed for parametric models. Although SIMEX is, in principle, applicable to nonparametric models, a serious problem arises with SIMEX in nonparametric situations. The problem is that smoothing parameter selectors that are developed for data without measurement error are no longer appropriate and can result in considerable undersmoothing. We believe that this is the first paper to address this difficulty.     

SIMEX method for censored quantile regression with measurement error

Censored quantile regression serves as an important supplement to the Cox proportional hazards model in survival analysis. In addition to being exposed to censoring, some covariates may subject to measurement error. This leads to substantially biased estimate without taking this error into account. The SIMulation-EXtrapolation (SIMEX) method is an effective tool to handle the measurement error issue. We extend the SIMEX approach to the censored quantile regression with covariate measurement error. The algorithm is assessed via extensive simulations. A lung cancer study is analyzed to verify the validation of the proposed method.     

Simex approaches to measurement error in roc studies

This paper explores the estimation of the area under the ROC curve when test scores are subject to errors. The naive approach that ignores measurement errors generally yields inconsistent estimates. Finding the asymptotic bias of the naive estimator, Coffin and Sukhatme (1995, 1997) proposed bias-corrected estimators for parametric and nonparametric cases. However, the asymptotic distributions of these estimators have not been developed because of their complexity. We propose several alternative approaches, including the SIMEX procedure of Cook and Stefanski (1994). We also provide the asymptotic distributions of the SIMEX estimators for use in statistical inference. Small simulation studies illustrate that the SIMEX estimators perform reasonably well when compared to the bias-corrected estimators.     

An Empirical Comparison of Estimating Methods for the Ramberg–Osgood EIV Model

The standard tensile test is one of the most frequent tools performed for the evaluation of mechanical properties of metals. An empirical model proposed by Ramberg and Osgood fits the tensile test data using a nonlinear model for the strain in terms of the stress. It is an Error-In-Variables (EIV) model because of the uncertainty affecting both strain and stress measurement instruments. The SIMEX, a simulation-based method for the estimation of model parameters, is powerful in order to reduce bias due to the measurement error in EIV models. The plan of this article is the following. In Sec. 2, we introduce the Ramberg–Osgood model and another reparametrization according to different assumptions on the independent variable. In Sec. 3, there is a summary of SIMEX method for the case at hand. Section 4 is a comparison between SIMEX and others estimating methods in order to highlight the peculiarities of the different approaches. In the last section, there are some concluding remarks.     

Mixtures of Linear Regression with Measurement Errors

Existing research on mixtures of regression models are limited to directly observed predictors. The estimation of mixtures of regression for measurement error data imposes challenges for statisticians. For linear regression models with measurement error data, the naive ordinary least squares method, which directly substitutes the observed surrogates for the unobserved error-prone variables, yields an inconsistent estimate for the regression coefficients. The same inconsistency also happens to the naive mixtures of regression estimate, which is based on the traditional maximum likelihood estimator and simply ignores the measurement error. To solve this inconsistency, we propose to use the deconvolution method to estimate the mixture likelihood of the observed surrogates. Then our proposed estimate is found by maximizing the estimated mixture likelihood. In addition, a generalized EM algorithm is also developed to find the estimate. The simulation results demonstrate that the proposed estimation procedures work well and perform much better than the naive estimates.     

Bias corrected estimation for generalized probit regression with covariate measurement error and censored responses

In this paper, we propose a bias corrected estimate of the regression coefficient for the generalized probit regression model when the covariates are subject to measurement error and the responses are subject to interval censoring. The main improvement of our method is that it reduces most of the bias that the naive estimates have. The great advantage of our method is that it is baseline and censoring distribution free, in a sense that the investigator does not need to calculate the baseline or the censoring distribution to obtain the estimator of the regression coefficient, an important property of Cox regression model. A sandwich estimator for the variance is also proposed. Our procedure can be generalized to general measurement error distribution as long as the first four moments of the measurement error are known. The results of extensive simulations show that our approach is very effective in eliminating the bias when the measurement error is not too large relative to the error term of the regression model.     

SIMEX estimation in case of correlated measurement errors

The simulation-extrapolation (SIMEX) approach of Cook and Stefanski (J. Am. Stat. Assoc. 89:1314–1328, 1994) has proved to be successful in obtaining reliable estimates if variables are measured with (additive) errors. In particular for nonlinear models, this approach has advantages compared to other procedures such as the instrumental variable approach if only variables measured with error are available. However, it has always been assumed that measurement errors for the dependent variable are not correlated with those related to the explanatory variables although such scenario is quite likely. In such a case the (standard) SIMEX suffers from misspecification even for the simple linear regression model. Our paper reports first results from a generalized SIMEX (GSIMEX) approach which takes account of this correlation. We also demonstrate in our simulation study that neglect of the correlation will lead to estimates which may be worse than those from the naive estimator which completely disregards measurement errors.     

Parametrically Guided Non-parametric Regression

We present a new approach to regression function estimation in which a non-parametric regression estimator is guided by a parametric pilot estimate with the aim of reducing the bias. New classes of parametrically guided kernel weighted local polynomial estimators are introduced and formulae for asymptotic expectation and variance, hence approximated mean squared error and mean integrated squared error, are derived. It is shown that the new classes of estimators have the very same large sample variance as the estimators in the standard non-parametric setting, while there is substantial room for reducing the bias if the chosen parametric pilot function belongs to a wide neighbourhood around the true regression line. Bias reduction is discussed in light of examples and simulations.     

Modal non‐linear regression in the presence of Laplace measurement error

In this paper, we propose a robust estimation procedure for a class of non‐linear regression models when the covariates are contaminated with Laplace measurement error, aiming at constructing an estimation procedure for the regression parameters which are less affected by the possible outliers, and heavy‐tailed underlying distribution, as well as reducing the bias introduced by the measurement error. Starting with the modal regression procedure developed for the measurement error‐free case, a non‐trivial modification is made so that the modified version can effectively correct the potential bias caused by measurement error. Large sample properties of the proposed estimate, such as the convergence rate and the asymptotic normality, are thoroughly investigated. A simulation study and real data application are conducted to illustrate the satisfying finite sample performance of the proposed estimation procedure.     

Perturbation by multiplicative noise and the Simulation Extrapolation method

While most of the literature on measurement error focuses on additive measurement error, we consider in this paper the multiplicative case. We apply the Simulation Extrapolation method (SIMEX)—a procedure which was originally proposed by Cook and Stefanski (J. Am. Stat. Assoc. 89:1314–1328, 1994) in order to correct the bias due to additive measurement error—to the case where data are perturbed by multiplicative noise and present several approaches to account for multiplicative noise in the SIMEX procedure. Furthermore, we analyze how well these approaches reduce the bias caused by multiplicative perturbation. Using a binary probit model, we produce Monte Carlo evidence on how the reduction of data quality can be minimized. For helpful comments, we would like to thank Helmut Küchenhoff, Winfried Pohlmeier, and Gerd Ronning. Sandra Nolte gratefully acknowledges financial support by the DFG. Elena Biewen and Martin Rosemann gratefully acknowledge the financial support by the Federal Ministry of Education and Research (BMBF). The usual disclaimer applies.     

Efficient Estimation in Marginal Partially Linear Models for Longitudinal/Clustered Data Using Splines

Abstract.  We consider marginal semiparametric partially linear models for longitudinal/clustered data and propose an estimation procedure based on a spline approximation of the non-parametric part of the model and an extension of the parametric marginal generalized estimating equations (GEE). Our estimates of both parametric part and non-parametric part of the model have properties parallel to those of parametric GEE, that is, the estimates are efficient if the covariance structure is correctly specified and they are still consistent and asymptotically normal even if the covariance structure is misspecified. By showing that our estimate achieves the semiparametric information bound, we actually establish the efficiency of estimating the parametric part of the model in a stronger sense than what is typically considered for GEE. The semiparametric efficiency of our estimate is obtained by assuming only conditional moment restrictions instead of the strict multivariate Gaussian error assumption.     

Improved Monte Carlo inference for models with additive error

Some statistical models defined in terms of a generating stochastic mechanism have intractable distribution theory, which renders parameter estimation difficult. However, a Monte Carlo estimate of the log-likelihood surface for such a model can be obtained via computation of nonparametric density estimates from simulated realizations of the model. Unfortunately, the bias inherent in density estimation can cause bias in the resulting log-likelihood estimate that alters the location of its maximizer. In this paper a methodology for radically reducing this bias is developed for models with an additive error component. An illustrative example involving a stochastic model of molecular fragmentation and measurement is given.     

Improved estimation procedures for multilevel models with binary response: a case-study

During recent years, analysts have been relying on approximate methods of inference to estimate multilevel models for binary or count data. In an earlier study of random-intercept models for binary outcomes we used simulated data to demonstrate that one such approximation, known as marginal quasi-likelihood, leads to a substantial attenuation bias in the estimates of both fixed and random effects whenever the random effects are non-trivial. In this paper, we fit three-level random-intercept models to actual data for two binary outcomes, to assess whether refined approximation procedures, namely penalized quasi-likelihood and second-order improvements to marginal and penalized quasi-likelihood, also underestimate the underlying parameters. The extent of the bias is assessed by two standards of comparison: exact maximum likelihood estimates, based on a Gauss–Hermite numerical quadrature procedure, and a set of Bayesian estimates, obtained from Gibbs sampling with diffuse priors. We also examine the effectiveness of a parametric bootstrap procedure for reducing the bias. The results indicate that second-order penalized quasi-likelihood estimates provide a considerable improvement over the other approximations, but all the methods of approximate inference result in a substantial underestimation of the fixed and random effects when the random effects are sizable. We also find that the parametric bootstrap method can eliminate the bias but is computationally very intensive.     

A Semiparametric Approach for Accelerated Failure Time Models with Covariates Subject to Measurement Error

There are relatively few discussions about measurement error in the accelerated failure time (AFT) model, particularly for the semiparametric AFT model. In this article, we propose an adjusted estimation procedure for the semiparametric AFT model with covariates subject to measurement error, based on the profile likelihood approach and simulation and exploration (SIMEX) method. The simulation studies show that the proposed semiparametric SIMEX approach performs well. The proposed approach is applied to a coronary heart disease dataset from the Busselton Health study for illustration.     

On correcting measurement error in the persistence rate estimator

In the field of education, it is often of great interest to estimate the percentage of students who start out in the top test quantile at time 1 and who remain there at time 2, which is termed as “persistence rate,” to measure the students’ academic growth. One common difficulty is that students’ performance may be subject to measurement errors. We therefore considered a correlation calibration method and the simulation–extrapolation (SIMEX) method for correcting the measurement errors. Simulation studies are presented to compare various measurement error correction methods in estimating the persistence rate.     

A Bezier curve method in interval-censored data

In this paper we propose a Bezier curve method to estimate the survival function and the median survival time in interval-censored data. We compare the proposed estimator with other existing methods such as the parametric method, the single point imputation method, and the nonparametric maximum likelihood estimator through extensive numerical studies, and it is shown that the proposed estimator performs better than others in the sense of mean squared error and mean integrated squared error. An illustrative example based on a real data set is given.     

Effect of assay measurement error on parameter estimation in concentration–QTc interval modeling

Linear mixed‐effects models (LMEMs) of concentration–double‐delta QTc intervals (QTc intervals corrected for placebo and baseline effects) assume that the concentration measurement error is negligible, which is an incorrect assumption. Previous studies have shown in linear models that independent variable error can attenuate the slope estimate with a corresponding increase in the intercept. Monte Carlo simulation was used to examine the impact of assay measurement error (AME) on the parameter estimates of an LMEM and nonlinear MEM (NMEM) concentration–ddQTc interval model from a ‘typical’ thorough QT study. For the LMEM, the type I error rate was unaffected by assay measurement error. Significant slope attenuation ( > 10%) occurred when the AME exceeded > 40% independent of the sample size. Increasing AME also decreased the between‐subject variance of the slope, increased the residual variance, and had no effect on the between‐subject variance of the intercept. For a typical analytical assay having an assay measurement error of less than 15%, the relative bias in the estimates of the model parameters and variance components was less than 15% in all cases. The NMEM appeared to be more robust to AME error as most parameters were unaffected by measurement error. Monte Carlo simulation was then used to determine whether the simulation–extrapolation method of parameter bias correction could be applied to cases of large AME in LMEMs. For analytical assays with large AME ( > 30%), the simulation–extrapolation method could correct biased model parameter estimates to near‐unbiased levels. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimation and variable selection for partially functional linear models

In this paper, a new estimation procedure based on composite quantile regression and functional principal component analysis (PCA) method is proposed for the partially functional linear regression models (PFLRMs). The proposed estimation method can simultaneously estimate both the parametric regression coefficients and functional coefficient components without specification of the error distributions. The proposed estimation method is shown to be more efficient empirically for non-normal random error, especially for Cauchy error, and almost as efficient for normal random errors. Furthermore, based on the proposed estimation procedure, we use the penalized composite quantile regression method to study variable selection for parametric part in the PFLRMs. Under certain regularity conditions, consistency, asymptotic normality, and Oracle property of the resulting estimators are derived. Simulation studies and a real data analysis are conducted to assess the finite sample performance of the proposed methods.     

Estimation of regression parameters in generalized linear models for cluster correlated data with measurement error

Liang and Zeger (1986) introduced a class of estimating equations that gives consistent estimates of regression parameters and of their asymptotic variances in the class of generalized linear models for cluster correlated data. When the independent variables or covariates in such models are subject to measurement errors, the parameter estimates obtained from these estimating equations are no longer consistent. To correct for the effect of measurement errors, an estimator with smaller asymptotic bias is constructed along the lines of Stefanski (1985), assuming that the measurement error variance is either known or estimable. The asymptotic distribution of the bias-corrected estimator and a consistent estimator of its asymptotic variance are also given. The special case of a binary logistic regression model is studied in detail. For this case, methods based on conditional scores and quasilikelihood are also extended to cluster correlated data. Results of a small simulation study on the performance of the proposed estimators and associated tests of hypotheses are reported.     

Estimation of the Variance Function in Heteroscedastic Linear Regression Models

Heteroscedasticity generally exists when a linear regression model is applied to analyzing some real-world problems. Therefore, how to accurately estimate the variance functions of the error term in a heteroscedastic linear regression model is of great importance for obtaining efficient estimates of the regression parameters and making valid statistical inferences. A method for estimating the variance function of heteroscedastic linear regression models is proposed in this article based on the variance-reduced local linear smoothing technique. Some simulations and comparisons with other method are conducted to assess the performance of the proposed method. The results demonstrate that the proposed method can accurately estimate the variance functions and therefore produce more efficient estimates of the regression parameters.