The identification of logistic regression models with errors in the variables

The simple logistic regression model with normal measurement error and normal regressor is shown to be identifiable without any extra information about the measurement error. The multiple logistic regression model with more than one regressor variable measured with error is not identifiable. If the covariance matrix of the measurement error is known up to a scalar factor, the model is identified. Further we discuss why in spite of the identifiability the models cannot be estimated in a reasonable way without extra information about the measurement error.     

Local and omnibus goodness-of-fit tests in classical measurement error models

We consider functional measurement error models, i.e. models where covariates are measured with error and yet no distributional assumptions are made about the mismeasured variable. We propose and study a score-type local test and an orthogonal series-based, omnibus goodness-of-fit test in this context, where no likelihood function is available or calculated-i.e. all the tests are proposed in the semiparametric model framework. We demonstrate that our tests have optimality properties and computational advantages that are similar to those of the classical score tests in the parametric model framework. The test procedures are applicable to several semiparametric extensions of measurement error models, including when the measurement error distribution is estimated non-parametrically as well as for generalized partially linear models. The performance of the local score-type and omnibus goodness-of-fit tests is demonstrated through simulation studies and analysis of a nutrition data set.     

Parameter estimation approaches to tackling measurement error and multicollinearity in ordinal probit models

Abstract

The regression model with ordinal outcome has been widely used in a lot of fields because of its significant effect. Moreover, predictors measured with error and multicollinearity are long-standing problems and often occur in regression analysis. However there are not many studies on dealing with measurement error models with generally ordinal response, even fewer when they suffer from multicollinearity. The purpose of this article is to estimate parameters of ordinal probit models with measurement error and multicollinearity. First, we propose to use regression calibration and refined regression calibration to estimate parameters in ordinal probit models with measurement error. Second, we develop new methods to obtain estimators of parameters in the presence of multicollinearity and measurement error in ordinal probit model. Furthermore we also extend all the methods to quadratic ordinal probit models and talk about the situation in ordinal logistic models. These estimators are consistent and asymptotically normally distributed under general conditions. They are easy to compute, perform well and are robust against the normality assumption for the predictor variables in our simulation studies. The proposed methods are applied to some real datasets.     

The effects of model mis-specifications in linear measurement error models

In the literature, there are many results on the consequences of mis-specified models for linear models with error in the response only, see, e.g., Seber(1977). There are also discussions of estimation for the model writh errors both in the response and in the predictor variables (called measurement error models; see, e.g., Fuller(1987)). In this paper, we consider the problem of model mis-specification for measurement error models. Only a few special cases have been tackled in the past (Edland, 1996; Carroll and Ruppert, 1996 and Lakshminarayanan Amp; Gunst, 1984); we deal with the situation here in some generality. Results have been obtained as follows: (a) When a model is under-fitted, the estimate of the variance of the measurement error will be asymptotically biased, as will the regression coefficients, and the asymptotic biases in the estimates of the regression coefficients will always exist for under-fitted models. Even orthogonality of the variables in the model will not make the biases vanish. (b)For over-fitting, the estimates of the variances of measurement errors and of the regression coefficients are asymptotically unbiased. However, the variance of the estimated regression coefficients will increase. Over-fitting will cause larger changes in the variances of the estimated parameters in measurement error models than in no measurement error models.     

A simulation study of estimators for generalized linear measurement error models

The purpose of this paper is to examine the properties of several bias-corrected estimators for generalized linear measurement error models, along with the naive estimator, in some special settings. In particular, we consider logistic regression, poisson regression and exponential-gamma models where the covariates are subject to measurement error. Monte Carlo experiments are conducted to compare the relative performance of the estimators in terms of several criteria. The results indicate that the naive estimator of slope is biased towards zero by a factor increasing with the magnitude of slope and measurement error as well as the sample size. It is found that none of the biased-corrected estimators always outperforms the others, and that their small sample properties typically depend on the underlying model assumptions.     

Double sampling for exact values in the normal discriminant model with application to binary regression

Increasing attention is being given to problems involving binary outcomes with covariates subject to measurement error. Here, we consider the two group normal discriminant model where a subset of the continuous variates are subject to error and will typically be replaced by a vector of surrogates, perhaps of different dimension. Correcting for the measurement error is made possible by a double sampling scheme in which the surrogates are collected on all units and true values are obtained on a random subset of units. Such a scheme allows us to consider a rich set of measurement error models which extend the traditional additive error model. Maximum likelihood estimators and their asymptotic properties are derived under a variety of models for the relationship between true values and the surrogates. Specific attention is given to the coefficients in the resulting logistic regression model. Optimal allocations are derived which minimize the variance of the estimated slope subject to cost constraints for the case where there is a univariate covariate but a possibly multivariate surrogate.     

A Bayesian semiparametric accelerated failure time model for arbitrarily censored data with covariates subject to measurement error

A flexible Bayesian semiparametric accelerated failure time (AFT) model is proposed for analyzing arbitrarily censored survival data with covariates subject to measurement error. Specifically, the baseline error distribution in the AFT model is nonparametrically modeled as a Dirichlet process mixture of normals. Classical measurement error models are imposed for covariates subject to measurement error. An efficient and easy-to-implement Gibbs sampler, based on the stick-breaking formulation of the Dirichlet process combined with the techniques of retrospective and slice sampling, is developed for the posterior calculation. An extensive simulation study is conducted to illustrate the advantages of our approach.     

Correcting for nonlinear measurement errors in the dependent variable in the general linear model

Linear models are considered in which measurement error is present in the dependent variable. Observed values are related to true values via nonlinear regression models with the parameters in the measurement error models being estimated with the use of independent, external data, collected using standards. Pseudo-maximum likelihood estimators and their asymptotic properties are developed under normality assumptions and the common approach of simply analyzing imputed values obtained from the nestimated calibration curves is assessed. A small simulation evaluates the procedures. An example is presented in which urinary neopterin (measured via radioimmunoassay) is nbeing compared between two groups of individuals.     

Estimation of dependence between paired correlated failure times in the presence of covariate measurement error

Summary. In many biomedical studies, covariates are subject to measurement error. Although it is well known that the regression coefficients estimators can be substantially biased if the measurement error is not accommodated, there has been little study of the effect of covariate measurement error on the estimation of the dependence between bivariate failure times. We show that the dependence parameter estimator in the Clayton–Oakes model can be considerably biased if the measurement error in the covariate is not accommodated. In contrast with the typical bias towards the null for marginal regression coefficients, the dependence parameter can be biased in either direction. We introduce a bias reduction technique for the bivariate survival function in copula models while assuming an additive measurement error model and replicated measurement for the covariates, and we study the large and small sample properties of the dependence parameter estimator proposed.     

Simulation-Extrapolation via the Bezier Curve in Measurement Error Models

Simulation-extrapolation (SIMEX) is a method for correcting for bias in measurement error models, and parametric SIMEX estimates are often used. In this paper, we propose a nonparametric method for computing the SIMEX estimate via the Bezier curve, which is a popular smoothing technique in the computer graphics area. Comparisons are done for the bias of the limit values of parametric SIMEX estimates and the Bezier estimate in the various nonlinear measurement error models.     

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.     

Correcting data for measurement error in generalized linear models

This paper discusses a general strategy for reducing measurement-error-induced bias in statistical models. It is assumed that the measurement error is unbiased with a known variance although no other distributional assumptions on the measurement-error are employed,

Using a preliminary fit of the model to the observed data, a transformation of the variable measured with error is estimated. The transformation is constructed so that the estimates obtained by refitting the model to the ‘corrected’ data have smaller bias,

Whereas the general strategy can be applied in a number of settings, this paper focuses on the problem of covariate measurement error in generalized linear models, Two estimators are derived and their effectiveness at reducing bias is demonstrated in a Monte Carlo study.     

Prediction and forecasting in linear models with measurement error

We consider measurement error models within the time series unobserved component framework. A variable of interest is observed with some measurement error and modelled as an unobserved component. The forecast and the prediction of this variable given the observed values is given by the Kalman filter and smoother along with their conditional variances. By expressing the forecasts and predictions as weighted averages of the observed values, we investigate the effect of estimation error in the measurement and observation noise variances. We also develop corrected standard errors for prediction and forecasting accounting for the fact that the measurement and observation error variances are estimated by the same sample that is used for forecasting and prediction purposes. We apply the theory to the Yellowstone grizzly bears and US index of production datasets.     

存在测量误差时分层抽样层均值方差的估计

利用模型的方法研究出现测量误差时多变量间的关系是目前国际上的流行方法,但这不利于对单指标的估计。因此,通过在估计量的设计中纳入测量误差信息,推导测量误差方差的定量测度方法,实现了存在测量误差时分层抽样各层均值方差的估计。采用2007年广东省三个市（县）城镇住户调查中的人均消费性支出数据进行实证分析,定量测度了测量误差在层均值方差估计中的大小及其影响,并对不考虑测量误差的估计结果进行了修正。     

A general maximum likelihood analysis of measurement error in generalized linear models

This paper describes an EM algorithm for maximum likelihood estimation in generalized linear models (GLMs) with continuous measurement error in the explanatory variables. The algorithm is an adaptation of that for nonparametric maximum likelihood (NPML) estimation in overdispersed GLMs described in Aitkin (Statistics and Computing 6: 251–262, 1996). The measurement error distribution can be of any specified form, though the implementation described assumes normal measurement error. Neither the reliability nor the distribution of the true score of the variables with measurement error has to be known, nor are instrumental variables or replication required.Standard errors can be obtained by omitting individual variables from the model, as in Aitkin (1996).Several examples are given, of normal and Bernoulli response variables.     

Objective Priors for Parameters in a Normal Linear Regression with Measurement Error

We investigate certain objective priors for the parameters in a normal linear regression models with one of the explanatory variables subject to measurement error. We first show that the use of the standard non informative prior for normal linear regression without measurement error leads to an improper posterior in the measurement error model. We then derive the Jeffreys prior and reference priors, and show that they lead to proper posteriors. We use simulation study to compare the frequentist performance of the estimates derived using these priors, and the MLE.     

Measurement error in the generalised linear model

This paper considers the problem of estimating the linear parameters of a Generalised Linear Model (GLM) when the explanatory variable is subject to measurement error. In this situation the induced model for dependence on the approximate explanatory variable is not usually of GLM form. However, when the distribution of measurement error is known or estimated from replicated measurements, application of the GLIM iteratively reweighted least squares algorithm with transformed data and weighting is shown to produce maximum quasi likelihood estimates in many cases. Details of this approach are given for two particular generalized linear models; simulation results illustrate the usefulness of the theory for these models.     

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.     

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.     

SMALL AREA ESTIMATION USING SURVEY WEIGHTS WITH FUNCTIONAL MEASUREMENT ERROR IN THE COVARIATE

Nested error linear regression models using survey weights have been studied in small area estimation to obtain efficient model‐based and design‐consistent estimators of small area means. The covariates in these nested error linear regression models are not subject to measurement errors. In practical applications, however, there are many situations in which the covariates are subject to measurement errors. In this paper, we develop a nested error linear regression model with an area‐level covariate subject to functional measurement error. In particular, we propose a pseudo‐empirical Bayes (PEB) predictor to estimate small area means. This predictor borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. We also employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator.