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.     

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.     

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.     

A Bayesian model for measurement and misclassification errors alongside missing data,with an application to higher education participation in Australia

In this paper we consider the impact of both missing data and measurement errors on a longitudinal analysis of participation in higher education in Australia. We develop a general method for handling both discrete and continuous measurement errors that also allows for the incorporation of missing values and random effects in both binary and continuous response multilevel models. Measurement errors are allowed to be mutually dependent and their distribution may depend on further covariates. We show that our methodology works via two simple simulation studies. We then consider the impact of our measurement error assumptions on the analysis of the real data set.     

Robust replicated heteroscedastic measurement error model using heavy-tailed distribution

Heteroscedastic measurement error models are widely used in epidemiological, analytical chemistry, and other research areas. In this article, we propose a heteroscedastic measurement error model for replicated data under scale mixtures of normal distributions with/without equation error, which covers unpair and/or unequal replication cases. We obtain iterative formulas of maximum likelihood estimations via EM algorithm, and provide closed forms of asymptotic variances of the estimators. Simulation studies and a real data application are reported to investigate the effective and robust performances of the model and estimates.     

Exact sampling of the unobserved covariates in Bayesian spline models for measurement error problems

In truncated polynomial spline or B-spline models where the covariates are measured with error, a fully Bayesian approach to model fitting requires the covariates and model parameters to be sampled at every Markov chain Monte Carlo iteration. Sampling the unobserved covariates poses a major computational problem and usually Gibbs sampling is not possible. This forces the practitioner to use a Metropolis–Hastings step which might suffer from unacceptable performance due to poor mixing and might require careful tuning. In this article we show for the cases of truncated polynomial spline or B-spline models of degree equal to one, the complete conditional distribution of the covariates measured with error is available explicitly as a mixture of double-truncated normals, thereby enabling a Gibbs sampling scheme. We demonstrate via a simulation study that our technique performs favorably in terms of computational efficiency and statistical performance. Our results indicate up to 62 and 54 % increase in mean integrated squared error efficiency when compared to existing alternatives while using truncated polynomial splines and B-splines respectively. Furthermore, there is evidence that the gain in efficiency increases with the measurement error variance, indicating the proposed method is a particularly valuable tool for challenging applications that present high measurement error. We conclude with a demonstration on a nutritional epidemiology data set from the NIH-AARP study and by pointing out some possible extensions of the current work.     

Multivariate measurement error models based on Student-t distribution under censored responses

Measurement error models constitute a wide class of models that include linear and nonlinear regression models. They are very useful to model many real-life phenomena, particularly in the medical and biological areas. The great advantage of these models is that, in some sense, they can be represented as mixed effects models, allowing us to implement well-known techniques, like the EM-algorithm for the parameter estimation. In this paper, we consider a class of multivariate measurement error models where the observed response and/or covariate are not fully observed, i.e., the observations are subject to certain threshold values below or above which the measurements are not quantifiable. Consequently, these observations are considered censored. We assume a Student-t distribution for the unobserved true values of the mismeasured covariate and the error term of the model, providing a robust alternative for parameter estimation. Our approach relies on a likelihood-based inference using an EM-type algorithm. The proposed method is illustrated through some simulation studies and the analysis of an AIDS clinical trial dataset.     

Functional measurement error in functional regression

Measurement error is an important problem that has not been studied very well in the context of functional data analysis. To the best of our knowledge, there are no existing methods that address the presence of functional measurement errors in generalized functional linear models. In this article, a novel approach is proposed to estimate the slope function in the presence of measurement error in the generalized functional linear model with a scalar response. This work significantly advances the existing conditional score method to accommodate the case where both the measurement error and independent variables lie in infinite dimensional spaces. Asymptotic results are established for the proposed estimate, and its behaviour is studied via simulations, where the response is continuous or binary. Analysis of Canadian Weather data highlights the practical utility of our method. The Canadian Journal of Statistics 48: 238–258; 2020 © 2020 Statistical Society of Canada     

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.     

Generalized estimator of population mean by using conventional and non-conventional measures in the presence of measurement errors

In survey research, it is assumed that reported response by the individual is correct. However, given the issues of prestige bias, self-respect, respondent's reported data often produces estimated values which are highly deviated from the true values. This causes measurement error (ME) to be present in the sample estimates. In this article, the estimation of population mean in the presence of measurement error using information on a single auxiliary variable is studied. A generalized estimator of population mean is proposed. The class of estimators is obtained by using some conventional and non-conventional measures. Simulation and numerical study is also conducted to assess the performance of estimators in the presence and absence of measurement error.     

Errors-in-variables with systematic biases

A structural regression model is considered in which some of the variables are measured with error. Instead of additive measurement errors, systematic biases are allowed by relating true and observed values via simple linear regressions. Additional data is available, based on standards, which allows for “calibration” of the measuring methods involved. Using only moment assumptions, some simple estimators are proposed and their asymptotic properties are developed. The results parallel and extend those given by Fuller (1987) in which the errors are additive and the error covariance is estimated. Maximum likelihood estimation is also discussed and the problem is illustrated using data from an acid rain study in which the relationship between pH and alkalinity is of interest but neither variable is observed exactly.     

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.     

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.     

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.     

A joint-modeling approach to assess the impact of biomarker variability on the risk of developing clinical outcome

In some clinical trials and epidemiologic studies, investigators are interested in knowing whether the variability of a biomarker is independently predictive of clinical outcomes. This question is often addressed via a naïve approach where a sample-based estimate (e.g., standard deviation) is calculated as a surrogate for the “true” variability and then used in regression models as a covariate assumed to be free of measurement error. However, it is well known that the measurement error in covariates causes underestimation of the true association. The issue of underestimation can be substantial when the precision is low because of limited number of measures per subject. The joint analysis of survival data and longitudinal data enables one to account for the measurement error in longitudinal data and has received substantial attention in recent years. In this paper we propose a joint model to assess the predictive effect of biomarker variability. The joint model consists of two linked sub-models, a linear mixed model with patient-specific variance for longitudinal data and a full parametric Weibull distribution for survival data, and the association between two models is induced by a latent Gaussian process. Parameters in the joint model are estimated under Bayesian framework and implemented using Markov chain Monte Carlo (MCMC) methods with WinBUGS software. The method is illustrated in the Ocular Hypertension Treatment Study to assess whether the variability of intraocular pressure is an independent risk of primary open-angle glaucoma. The performance of the method is also assessed by simulation studies.     

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.     

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.     

Cases for the nugget in modeling computer experiments

Most surrogate models for computer experiments are interpolators, and the most common interpolator is a Gaussian process (GP) that deliberately omits a small-scale (measurement) error term called the nugget. The explanation is that computer experiments are, by definition, “deterministic”, and so there is no measurement error. We think this is too narrow a focus for a computer experiment and a statistically inefficient way to model them. We show that estimating a (non-zero) nugget can lead to surrogate models with better statistical properties, such as predictive accuracy and coverage, in a variety of common situations.     

Bayesian hierarchical models for food frequency assessment

The aim of this study is to assess the biases of a Food Frequency Questionnaire (FFQ) by comparing total energy intake (TEI) with total energy expenditure (TEE) obtained from doubly labelled water(DLW) biomarker after adjusting measurement errors in DLW. We develop several Bayesian hierarchical measurement error models of DLW with different distributional assumptions on TEI to obtain precise bias estimates of TEI. Inference is carried out by using MCMC simulation techniques in a fully Bayesian framework, and model comparisons are done via the mean square predictive error. Our results showed that the joint model with random effects under the Gamma distribution is the best fit model in terms of the MSPE and residual diagnostics, in which bias in TEI is not significant based on the 95% credible interval. The Canadian Journal of Statistics 38: 506–516; 2010 © 2010 Statistical Society of Canada