Instrumental variable approach to estimating the scalar‐on‐function regression model with measurement error with application to energy expenditure assessment in childhood obesity

Abstract

Summary Wearable device technology allows continuous monitoring of biological markers and thereby enables study of time-dependent relationships. For example, in this paper, we are interested in the impact of daily energy expenditure over a period of time on subsequent progression toward obesity among children. Data from these devices appear as either sparsely or densely observed functional data and methods of functional regression are often used for their statistical analyses. We study the scalar-on-function regression model with imprecisely measured values of the predictor function. In this setting, we have a scalar-valued response and a function-valued covariate that are both collected at a single time period. We propose a generalized method of moments-based approach for estimation while an instrumental variable belonging in the same time space as the imprecisely measured covariate is used for model identification. Additionally, no distributional assumptions regarding the measurement errors are assumed, while complex covariance structures are allowed for the measurement errors in the implementation of our proposed methods. We demonstrate that our proposed estimator is L2 consistent and enjoys the optimal rate of convergence for univariate nonparametric functions. In a simulation study, we illustrate that ignoring measurement error leads to biased estimations of the functional coefficient. The simulation studies also confirm our ability to consistently estimate the function-valued coefficient when compared to approaches that ignore potential measurement errors. Our proposed methods are applied to our motivating example to assess the impact of baseline levels of energy expenditure on BMI among elementary school-aged children.

A function‐based approach to model the measurement error in wearable devices

“…Then, the naive estimate denoted as $$$ {\hat{\beta}}_{\mathrm{naive}}={\left({W}^{\prime }W\right)}^{-1}{W}^{\prime }Y. $$$ Alt.1 is the method suggested in Tekwe et al 19 by assuming that $$$ Z(t)=\delta X(t)+U(t) $$$. The method suggested by Chakraborty and Panaretos 17 which assumes that the range of dependence for the measurement error process is small is referred to as Alt.2.…”

“…For additional details on the use of instruments in measurement error models refer to Reference 20. Reference 19 propose a functional regression where the relation between the functional instrument $$$ Z\left(\cdotp \right) $$$ and the unobserved variable of interest is assumed as $$$ Z(t)=\delta X(t)+U(t), $$$ with $$$ U\left(\cdotp \right) $$$ as the model error. Thus, the relation between function $$$ X\left(\cdotp \right) $$$ and $$$ Z\left(\cdotp \right) $$$ is constant over time.…”

Partially Functional Linear Quantile Regression With Measurement Errors

“…Sources of measurement error associated with wearable-device-based measures of PA include variability in prediction at various PA intensity levels and errors associated with the prediction equations (Bassett, 2012;Crouter et al, 2006;Jacobi et al, 2007;Warolin et al, 2012;Rothney et al, 2008). Failure to account for measurement error in assessing the effects of PA measures on health outcomes often leads to bias and the underestimation of these effects (Tekwe et al, 2019). Third, data collected from wearable devices are typically not discrete vectors but rather a curvilinear functions of time, and require the use of functional data approaches in (Silverman and Ramsay, 2005).…”

A Function-Based Approach to Model the Measurement Error in Wearable Devices

“…Step counts have been used as instruments for the related variable -energy expenditure in (28). For additional details on the use of instruments in measurement error models refer to (4).…”

“…For additional details on the use of instruments in measurement error models refer to (4). (28) propose a functional regression where the relation between the functional instrument Z(•) and the unobserved variable of interest is assumed as Z(t) = δX(t) + U (t), with U (•) as the model error. Thus, the relation between function X(•) and Z(•) is constant over time.…”