Nonlinear Measurement Error Model Research Articles

A nonlinear measurement error model and its application to describing the dependency of health outcomes on dietary intake

ABSTRACT Many nutritional studies focus on the relationship between individuals' diets and resulting health outcomes. When examining these relationships, researchers are generally interested in individuals' long-term, average intake of nutrients; however, typically only 1–2 days of data are collected. If analyses are performed without accounting for the error in estimating usual intake, estimates will be biased. In this work, we focus on situations where the association between intake and health outcomes is nonlinear. Since we can only obtain noisy measurements of intake, we propose implementing a nonlinear measurement error model which accounts for the nuisance day-to-day variance when estimating long-term average intake. Estimation of the model is performed using maximum likelihood. Properties of the estimators are explored for a model where we assume that the unobservable usual intake is normally distributed. We then propose an extended model where we no longer assume that the distribution for the unobservable predictor is normal, but is instead a finite mixture of discrete distributions. We finish with an application using data from the 2015–2016 National Health and Nutrition Examination Survey (NHANES) where we examine the association between potassium intake and systolic blood pressure.

An approximate method for generalized linear and nonlinear mixed effects models with a mechanistic nonlinear covariate measurement error model

The literature on measurement error for time-dependent covariates has mostly focused on empirical models, such as linear mixed effects models. Motivated by an AIDS study, we propose a joint modeling method in which a mechanistic nonlinear model is used to address the time-varying covariate measurement error for a longitudinal outcome that can be either discrete such as binary and count or continuous. We implement an inference procedure that uses first-order Taylor approximation to linearize both the covariate model and the response model. We study the asymptotic properties of the joint model based estimator and provide proof of consistency and normality. We then evaluate the performance of estimation in finite sample size scenario through simulation. Finally, we apply the new method to real data in an HIV/AIDS study.

Weighted total least squares for solving non-linear problem: GNSS point positioning

In this contribution, two algorithms are developed for parameter estimation in a non-linear measurement error model with errors in both the coefficient matrix and the vector of measurement. They are based on the complete description of the variance–covariance matrices of the observation errors and of the coefficient matrix errors without any restriction. The paper reinvestigates the nonlinear measurement model associated with GNSS point positioning. Various simulation experiments indicate that GNSS point positioning is much better formulated as a non-linear WTLS problem with errors in both the coefficient matrix and measurement variables. The efficacy of the proposed algorithms is verified through the numerical results.

Nonlinear measurement error models subject to additive distortion

We study nonlinear regression models when the response and predictors are unobservable and distorted in a multiplicative fashion by additive models of some observed confounding variables. After approximating the additive nonparametric components via polynomial splines and calibrating the error-prone response and predictors, we develop a semi-parametric profile nonlinear least squares procedure to estimate the parameters of interest. We show that the resulting estimators are asymptotically normal. We further suggest an empirical likelihood-based statistic for statistical inference to improve the accuracy of the associated normal approximation with the aim to avoid estimating the asymptotic covariance matrix that involves infinite-dimensional nuisance of additive distorting functions. We also show that the empirical likelihood statistic is asymptotically chi-squared. Moreover, a test procedure is proposed to check whether the parametric model is adequate or not under this distorted measurement error setting. A wild bootstrap procedure is suggested to compute p-values. Simulation studies are conducted to examine the performance of the proposed procedures. The methods are applied to analyze real data from a low birth infants weight for an illustration.

Iterative algorithm for weighted total least squares adjustment

In this contribution, an iterative algorithm is developed for parameter estimation in a nonlinear measurement error model y−e = (A−EA)x, which is based on the complete description of the variance–covariance matrices of the observation errors e and of the coefficient matrix errors EA without any restriction, e.g. in the case that there are correlations among observations. This paper derives the weighted total least squares solution without applying Lagrange multipliers in a straightforward manner. The algorithm is simple in the concept, easy in the implementation, and fast in the convergence. The final exact solution can be achieved through iteration. Based on the similarity between the proposed algorithm and the ordinary least squares method, the estimate for the covariance matrix of the unknown parameters can be analogously computed by using the error propagation law. The efficacy of the proposed WTLS algorithm is demonstrated by solving three WTLS problems, i.e. a linear regression model, a planar similarity transformation and two-dimensional affine transformation in the case of diagonal and fully populated covariance matrices in both start and transformed coordinate systems.

Intake_epis_food(): An R Function for Fitting a Bivariate Nonlinear Measurement Error Model to Estimate Usual and Energy Intake for Episodically Consumed Foods.

We consider a Bayesian analysis using WinBUGS to estimate the distribution of usual intake for episodically consumed foods and energy (calories). The model uses measures of nutrition and energy intakes via a food frequency questionnaire (FFQ) along with repeated 24 hour recalls and adjusting covariates. In order to estimate the usual intake of the food, we phrase usual intake in terms of person-specific random effects, along with day-to-day variability in food and energy consumption. Three levels are incorporated in the model. The first level incorporates information about whether an individual in fact reported consumption of a particular food item. The second level incorporates the amount of intake from those individuals who reported consumption of the food, and the third level incorporates the energy intake. Estimates of posterior means of parameters and distributions of usual intakes are obtained by using Markov chain Monte Carlo calculations. This R function reports to users point estimates and credible intervals for parameters in the model, samples from their posterior distribution, samples from the distribution of usual intake and usual energy intake, trace plots of parameters and summary statistics of usual intake, usual energy intake and energy adjusted usual intake.

On nonlinear regression estimator with denoised variables

In this paper, a class of denoised nonlinear regression estimators is suggested for a nonlinear measurement error model where the variables in error are observed together with an auxiliary variable. The programming involved in this denoised nonlinear regression estimation is relatively simple and it can be modified with a little effort from the existing programs for nonlinear regression estimation. We establish the consistency and asymptotic normality of such denoised estimators based on the least squares and M-methods. A simulation study is carried out to illustrate the performance of these estimates. An empirical application of the model to production models in economics further demonstrates the potential of the proposed modeling procedures.

Prediction and confidence intervals for nonlinear measurement error models without identifiability information

A major difficulty in applying a measurement error model is that one is required to have additional information in order to identify the model. In this paper, we show that there are cases in nonlinear measurement error models where it is not necessary to have additional information to construct prediction intervals for the future dependent variable Y and confidence intervals for the conditional expectation E(Y | X) where X is the future observable independent variable. In particular, we consider two nonlinear models, the exponential and loglinear models. By applying pseudo-likelihood estimation of variance functions in the weighted least squares method, we construct theoretically justifiable prediction and confidence intervals in these two models. Some simulation results which show that the proposed intervals perform well are also provided.

A UNIVARIATE MEASUREMENT ERROR MODEL FOR LONGITUDINAL CHANGE

Analyses of variables measured with error are often flawed when measurement error is ignored. An extensive literature on linear measurement error models is available by [1], and more recently, for non-linear measurement error models by [2] and [3]. We investigate problems encountered in the analysis of longitudinal change, where the difference in some outcome variable is modeled on a set of regressor variables in a linear regression model. This type of analysis has become quite popular in biomedical research, and the results are often biased when the outcome variable is measured with error and its observed baseline value is included as a covariate in the fitted model. [4] demonstrated the effects of measurement error in the analysis of change in wall thickness of the common carotid artery. They showed the naive analysis led to erroneous findings due to the measurement error bias, even when the regressor variables were assumed to be measured precisely. In this paper, we present a method-of-moments correction for measurement error bias, provided the measurement error variance is known or can be estimated. This work extends the work of [4] to include regressor variables that are measured with error.

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.

A stochastic approximation algorithm with Markov chain Monte-carlo method for incomplete data estimation problems.

We propose a general procedure for solving incomplete data estimation problems. The procedure can be used to find the maximum likelihood estimate or to solve estimating equations in difficult cases such as estimation with the censored or truncated regression model, the nonlinear structural measurement error model, and the random effects model. The procedure is based on the general principle of stochastic approximation and the Markov chain Monte-Carlo method. Applying the theory on adaptive algorithms, we derive conditions under which the proposed procedure converges. Simulation studies also indicate that the proposed procedure consistently converges to the maximum likelihood estimate for the structural measurement error logistic regression model.

REGRESSION MODELS FOR RECURRENT EVENT DATA: PARAMETRIC RANDOM EFFECTS MODELS WITH MEASUREMENT ERROR

Statistical methodology is presented for the statistical analysis of non-linear measurement error models. Our approach is to provide adjustments for the usual maximum likelihood estimators, their standard errors and associated significance tests in order to account for the presence of measurement error in some of the covariates. We illustrate the technique with a mixed effects Poisson regression model for recurrent event data applied to a randomized clinical trial for the prevention of skin tumours.

A Nonlinear Measurement Error Model with Fixed Observed X (Berkson Case)

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Effect of measurement error in testing separate linear models

Measurement error models (or errors-in variables models) have been explored since the latter part of the 19th century. Much activity and results in estimation and statistical inference have been forthcoming in linear and non-linear measurement error models, particularly in the past 10 to 15 years. Likewise, tests of separate families of hypotheses have received much attention sice Cox (1961, 1962) introduced a method of testing separate families of hypotheses. The aim of this paper is to provide results for testing separate linear models using Cox's methodology under the assumption of the functional measurement error model for which the error structure has a normal distribution. Simulations are used to compare the results of the Cox test when using estimates adjusted for measurement error, naive estimats, and estimates derived from true values. The overall conclusions derived from the simulation results are mixed. However, if the measurement error is small and collinearity between explanatory variables fr...

Assessment of influence in non-linear measurement error models

We consider the assessment of outliers and influential observation in non-linear measurement error models. Residuals, leverage measures and case-deletiondiagonostics are examined. The method of local influence is also applied to the models. In particular, the perturbation of measurement error variances has been found useful in assessing the adequancy of the model assumptions. A numerical example is given to illustrate the application of the diagonostics.

A Nonlinear Measurement Error Model with Fixed Observed X (Berkson Case)

A Nonlinear Measurement Error Model with Fixed Observed X (Berkson Case)