On the use of repeated measurement errors in linear regression models
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On the use of repeated measurement errors in linear regression models Mengli Zhang1 · Yang Bai1 Received: 19 February 2019 / Accepted: 14 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In a linear mean regression setting with repeated measurement errors, we develop asymptotic properties of a naive estimator to better clarify the effects of these errors. We then construct a group of unbiased estimating equations with independent repetitions and make use of these equations in two ways to obtain two estimators: a weighted averaging estimator and an estimator based on the generalized method of moments. The proposed estimators do not require any additional information about the measurement errors. We also prove the consistency and asymptotic normality of the two estimators. Our theoretical results are verified by simulation studies and a real data analysis. Keywords Measurement error · Estimating equation · GMM
1 Introduction Measurement errors generally occur in data collection because of inaccuracy of experimental instruments, instability of the surrounding environment, and omissions by investigators, among other things. Alzheimer’s disease is one of the many examples. Campbell (2002) found that the level of aluminum deposits in the brain had an effect on an evaluation score for Alzheimer’s disease. However, analysis of the association between Alzheimer’s disease and the level of aluminum deposits may be unreliable because accurate measurements of aluminum levels are not available. Goldstein (1979) pointed out that the reverse conclusion could be drawn if the measurement error in data analysis on education of children from different classes was taken into account. It is also known that if measurement error is simply ignored, then standard estimation methods for model parameters give biased and inconsistent estimators. We will demonstrate this in detail with a simple linear regression model in Sect. 3.
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Yang Bai [email protected] School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai 200433, People’s Republic of China
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M. Zhang, Y. Bai
To tackle the above problem, Fuller (1987) systematically constructed a measurement error model that considers additive errors in explained variables. There are two kinds of measurement error models. One is the structural measurement error model, which regards the true values of independent variables as random. When fitting a structural measurement error model, common problems that arise are lack of identifiability and biased least squares estimation, and additional information is needed for the identifiability. The other kind of model is the functional measurement error model, which takes the true values as unknown constants and deals mainly with parameter redundancy and inefficient maximum likelihood estimation (MLE) estimators. For the structural error model with a known error variance, Cook and Stefanski (1994) introduced a method called simulation–extrapolation (SIMEX). In this method,
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