We consider local polynomial kernel regression with a single covariate for clustered data using estimating equations. We assume that at most m < ∞ observations are available on each cluster. In the case of random regressors, with no measurement error in the predictor, we show that it is generally the best strategy to ignore entirely the correlation structure within each cluster and instead pretend that all observations are independent. In the further special case of longitudinal data on individuals with fixed common observation times, we show that equivalent to the pooled data approach is the strategy of fitting separate nonparametric regressions at each observation time and constructing an optimal weighted average. We also consider what happens when the predictor is measured with error. Using the SIMEX approach to correct for measurement error, we construct an asymptotic theory for both the pooled and the weighted average estimators. Surprisingly, for the same amount of smoothing, the weighted a...

https://www.jstor.org/stable/pdfplus/2669396.pdf

Nonparametric Function Estimation for Clustered Data When the Predictor is Measured without/with Error

In the presence of covariate measurement error, estimating a regression function nonparametrically is extremely difficult, the problem being related to deconvolution. Various frequentist approaches exist for this problem, but to date there has been no Bayesian treatment. In this article we describe Bayesian approaches to modeling a flexible regression function when the predictor variable is measured with error. The regression function is modeled with smoothing splines and regression P-splines. Two methods are described for exploration of the posterior. The first, called the iterative conditional modes (ICM), is only partially Bayesian. ICM uses a componentwise maximization routine to find the mode of the posterior. It also serves to create starting values for the second method, which is fully Bayesian and uses Markov chain Monte Carlo (MCMC) techniques to generate observations from the joint posterior distribution. Use of the MCMC approach has the advantage that interval estimates that directly model and ...

Bayesian Smoothing and Regression Splines for Measurement Error Problems

In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without it. However, if additional data are available, then it is possible to estimate consistently the unknown error density. Data are seldom available directly on the transformation, but repeated, or replicated, measurements increasingly are becoming available. Such data consist of ``intrinsic'' values that are measured several times, with errors that are generally independent. Working in this setting we treat the nonparametric deconvolution problems of density estimation with observation errors, and regression with errors in variables. We show that, even if the number of repeated measurements is quite small, it is possible for modified kernel estimators to achieve the same level of performance they would if the error distribution were known. Indeed, density and regression estimators can be constructed from replicated data so that they have the same first-order properties as conventional estimators in the known-error case, without any replication, but with sample size equal to the sum of the numbers of replicates. Practical methods for constructing estimators with these properties are suggested, involving empirical rules for smoothing-parameter choice.

On deconvolution with repeated measurements

We have developed a new general approach for handling misclassification in discrete covariates or responses in regression models. The simulation and extrapolation (SIMEX) method, which was originally designed for handling additive covariate measurement error, is applied to the case of misclassification. The statistical model for characterizing misclassification is given by the transition matrix Pi from the true to the observed variable. We exploit the relationship between the size of misclassification and bias in estimating the parameters of interest. Assuming that Pi is known or can be estimated from validation data, we simulate data with higher misclassification and extrapolate back to the case of no misclassification. We show that our method is quite general and applicable to models with misclassified response and/or misclassified discrete regressors. In the case of a binary response with misclassification, we compare our method to the approach of Neuhaus, and to the matrix method of Morrissey and Spiegelman in the case of a misclassified binary regressor. We apply our method to a study on caries with a misclassified longitudinal response.

https://www.jstor.org/stable/info/3695708

A General Method for Dealing with Misclassification in Regression: The Misclassification SIMEX

A method of computerized content analysis that gives “approximately unbiased and statistically consistent estimates” of a distribution of elements of structured, unstructured, and partially structured source data among a set of categories. In one embodiment, this is done by analyzing a distribution of small set of individually-classified elements in a plurality of categories and then using the information determined from the analysis to extrapolate a distribution in a larger population set. This extrapolation is performed without constraining the distribution of the unlabeled elements to be equal to the distribution of labeled elements, nor constraining a content distribution of content of elements in the labeled set (e.g., a distribution of words used by elements in the labeled set) to be equal to a content distribution of elements in the unlabeled set. Not being constrained in these ways allows the estimation techniques described herein to provide distinct advantages over conventional aggregation techniques.

System for estimating a distribution of message content categories in source data

Summary In many regression applications the independent variable is measured with error. When this happens, conventional parametric and nonparametric regression techniques are no longer valid. We consider two diVerent approaches to nonparametric regression. The first uses the simex, simulation-extrapolation, method and makes no assumption about the distribution of the unobserved error-prone predictor. For this approach we derive an asymptotic theory for kernel regression which has some surprising implications. Penalised regression splines are also considered for fixed number of known knots. The second approach assumes that the error-prone predictor has a distribution of a mixture of normals with an unknown number of components, and uses regression splines. Simulations illustrate the results.

Nonparametric regression in the presence of measurement error

In many regression applications both the independent and dependent variables are measured with error. When this happens, conventional parametric and nonparametric regression techniques are no longer valid. We consider two different nonparametric techniques, regression splines and kernel estimation, of which both can be used in the presence of measurement error. Within the kernel regression context, we derive the limit distribution of the SIMEX estimate. With the regression spline technique, two different methods of estimations are used. The first method is the SIMEX algorithm which attempts to estimate the bias, and remove it. The second method is a structural approach, where one hypothesizes a distribution for the independent variable which depends on estimable parameters. A series of examples and simulations illustrate the methods.

Nonparametric Kernel and Regression Spline Estimation in the Presence of Measurement Error

In many regression applications both the independent and dependent variables are measured with error. When this happens, conventional parametric and nonparametric regression techniques are no longer valid. This is further complicated when one instead wants to fit a generalized linear model to the collected data. We consider two different estimation techniques. The first method is the SIMEX (SIMulation Extrapolation) algorithm which attempts to estimate the bias, and remove it. The second method is a structural approach, where one hypothesizes a distribution for the independent variable which depends on estimable parameters. For both methods, two different knot selection methods are developed.

Jeffrey D. Maca

Papers

Nonparametric regression in the presence of measurement error

Nonparametric Kernel and Regression Spline Estimation in the Presence of Measurement Error

Nonparameteric Regression Splines for Generalized Linear Measurement Error Models