Received August 1, 2010; revised October 1, 2010; accepted for publication
November 1, 2010

# Scalable regression calibration approaches to correcting measurement error
in multi-level generalized functional linear regression models with
heteroscedastic measurement errors

ROGER S. ZOH1 YUANYUAN LUAN1 ERJIA CUI2 XIAOXIN YU1 HEYANG JI1 SNEHA JADHAV3
CARMEN D. TEKWE∗,1
1 Department of Epidemiology and Biostatistics, Indiana University,
Bloomington, IN 47405
2 Department of Biostatistics, Johns Hopkins University, Baltimore, MD 21205
3 Department of Mathematics & Statistics, Wake Forest University, NC, 27109
<EMAIL_ADDRESS>
###### Abstract

Wearable devices permit the continuous monitoring of biological processes,
such as blood glucose metabolism, and behavior, such as sleep quality and
physical activity. The continuous monitoring often occurs in epochs of 60
seconds over multiple days, resulting in high dimensional longitudinal curves
that are best described and analyzed as functional data. From this
perspective, the functional data are smooth, latent functions obtained at
discrete time intervals and prone to homoscedastic white noise. However, the
assumption of homoscedastic errors might not be appropriate in this setting
because the devices collect the data serially. While researchers have
previously addressed measurement error in scalar covariates prone to errors,
less work has been done on correcting measurement error in high dimensional
longitudinal curves prone to heteroscedastic errors. We present two new
methods for correcting measurement error in longitudinal functional curves
prone to complex measurement error structures in multi-level generalized
functional linear regression models. These methods are based on two-stage
scalable regression calibration. We assume that the distribution of the scalar
responses and the surrogate measures prone to heteroscedastic errors both
belong in the exponential family and that the measurement errors follow
Gaussian processes. In simulations and sensitivity analyses, we established
some finite sample properties of these methods. In our simulations, both
regression calibration methods for correcting measurement error performed
better than estimators based on averaging the longitudinal functional data and
using observations from a single day. We also applied the methods to assess
the relationship between physical activity and type 2 diabetes in community
dwelling adults in the United States who participated in the National Health
and Nutrition Examination Survey.

Basis splines; Diabetes; Functional data; Physical activity; Splines; Wearable
accelerometer

00footnotetext: To whom correspondence should be addressed.

##  1 INTRODUCTION

Wearable monitoring devices permit the continuous monitoring of biological
processes, such as blood glucose metabolism (Tsai _and others_ , 2019;
Gaynanova _and others_ , 2022), and behavior, such as sleep quality (Kuo _and
others_ , 2016; Tuominen _and others_ , 2019) and physical activity (Troiano
_and others_ , 2008). The continuous monitoring often occurs in epochs of 60
seconds over multiple days resulting in high dimensional longitudinal curves
that are best described and analyzed as functional data.

While researchers have previously addressed measurement error in scalar
covariates prone to error (Carroll _and others_ , 2006), less work has been
done on correcting measurement error in high dimensional longitudinal curves
prone to complex heteroscedastic errors (Cardot _and others_ , 2007; Crambes
_and others_ , 2009; Goldsmith _and others_ , 2011). When correcting for
measurement error, functional data analysts have previously assumed that the
serially observed functional data represent a curve of a latent process
contaminated by noise with an independent error structure (Silverman and
Ramsay, 2005). Because observation periods for functional data often extend
over multiple days, it is necessary to develop methods for correcting
measurement error in serially observed data prone to complex heteroscedastic
error structures. Prior approaches to correcting measurement error in serially
observed functional data prone to complex errors involve the assumption that
the observed curves are from the normal distribution (Tekwe _and others_ ,
2022; Jadhav _and others_ , 2022). However, sometimes the curves might be best
described as belonging in more flexible distributions such as those in the
exponential family (Crainiceanu _and others_ , 2009). To address these
limitations, we propose two regression calibration approaches for correcting
measurement error in longitudinal functional data from the exponential family
of distributions and prone to complex heteroscedastic measurement error.

In our current work, we seek to assess the role of physical activity in type 2
diabetes (T2D). The incidence and prevalence of T2D have increased over the
past 20 years. Over one-third of U.S. adults are obese, and $\geq$ 11% of
people over 20 years old have T2D (Eckel _and others_ , 2011). Some
researchers project that the overall prevalence of T2D in the U.S. will be 21%
by 2050 (Boyle _and others_ , 2010). Obesity results from a chronic imbalance
between energy intake and energy expenditure (Spiegelman and Flier, 2001). To
combat the growing epidemic of obesity and its associated health outcomes such
as T2D, clinicians and researchers have proposed treatments to reduce dietary
intake and increase daily physical activity (Balk _and others_ , 2015).
Therefore, it is critical to assess these behaviors accurately and evaluate
how they influence obesity and T2D. Self-reported measures of physical
activity are prone to recall bias (Sallis and Saelens, 2000; Armstrong and
Bull, 2006). Consequently, researchers increasingly use wearable devices to
monitor physical activity (Kozey _and others_ , 2010). Wearable devices for
physical activity are based on motion sensor technology and data processing to
provide measures of physical activity frequency, duration, and intensity
(Kozey _and others_ , 2010; Troiano _and others_ , 2008). While wearable
device measures of physical activity are not prone to recall and researcher
bias, their accuracy may be questionable (Bassett, 2012; Crouter _and others_
, 2006; Jacobi _and others_ , 2007; Warolin _and others_ , 2012; Rothney _and
others_ , 2008; Kozey _and others_ , 2010). The sources of error in wearable
device measures of physical activity are both random and systematic. To
process accelerometer data from wearable devices, the devices must first be
calibrated to record activity counts with another physiological variable, such
as the metabolic equivalent task (MET)(Kozey _and others_ , 2010; Crouter _and
others_ , 2006; Swartz _and others_ , 2000). Next, the device approximates
activity counts from its relationship with the physiological variable with a
regression equation (Kozey _and others_ , 2010). However, there are currently
no standard equations, with over 30 equations available for this estimation
(Plasqui and Westerterp, 2007; Kozey _and others_ , 2010; Crouter _and others_
, 2006; Rothney _and others_ , 2008). The accuracy of the data generated by
the devices thus depends on the estimation equations. The accuracy also
depends on the activity type, sex, and body composition of device users
(Freedson _and others_ , 1998; Valenti _and others_ , 2014). In short,
physical activity is a latent variable not observed directly but represented
by a proxy, the wearable device measure. Such surrogate measurement tends to
introduce measurement error (Carroll _and others_ , 2006).

We present fast, scalable regression calibration methods for correcting
measurement error in functional covariates in multi-level generalized
functional linear regression models with complex heteroscedastic measurement
errors. Our new methods have several advantages. First, we assume the
distributions of the device-based measures of physical activity belong in the
exponential family. This allows for a more general specification of the
distributions of the device-based measures compared to current approaches that
often entail assumptions of normality for these surrogate measures. Second, we
treat the random errors in the surrogate measures as complex heteroscedastic
errors from the Gaussian distribution with covariance error functions. Third,
our methods can be used to evaluate relationships between multi-level
functional covariates with complex measurement error structures and scalar
outcomes with distributions in the exponential family. Fourth, we treat the
functional covariate as a surrogate measure for true functional covariate. We
also report on our application of the new methods to assess the relationship
between physical activity and T2D in data from the National Health and
Nutrition Examination Survey (NHANES).

##  2 GENERALIZED FUNCTIONAL REGRESSION WITH MULTI-LEVEL HETEROSCEDASTIC
MEASUREMENT ERRORS

###  2.1 The model

Let $\mathcal{D}_{i}=\\{Y_{i},{\mathbf{W}}_{i},{\mathbf{Z}}_{i}\\}$ be a
triplet for individual $i(i=1,\dots,n)$ consisting of, respectively, a scalar
outcome $Y_{i}$, a functional covariate measured at multiple time points which
we denote as ${\mathbf{W}}_{i}=\\{W_{ij}(t)\\}^{t\in
T_{i}}_{j=1,\cdots,J}\in\mathbb{R}^{T_{i}\times J}$, where $T_{i}$ is the
number of distinct time points at which the covariates $W_{ij}(t)$ are
measured, and ${\mathbf{Z}}_{i}\in\mathbb{R}^{p}$ is a vector of length $p$
representing the error-free covariates. For example, $Y_{i}$ could represent a
health outcome such as T2D status and ${\mathbf{W}}_{i}$ could represent
physical activity curves obtained from a monitoring device. Our model for the
$i$th subject is as follows:

$\displaystyle g[E\\{Y_{i}\mid X_{i}(t),{\mathbf{Z}}_{i}\\}]$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\beta(t)X_{i}(t)dt+{\mathbf{Z}}_{i}^{\rm
T}\mbox{\boldmath$\alpha$},$ (1) $\displaystyle h[E\\{W_{ij}(t)\\}]$
$\displaystyle=$ $\displaystyle X_{i}(t)+U_{ij}(t);$ (2)

where $g(\bullet)$ and $h(\bullet)$ are both monotone, twice continuously
differentiable functions; and $X_{i}(t)$ is a latent functional covariate with
its associated unknown functional coefficient, $\beta(t)$. The response
variable $Y_{i}$ may be either a continuous or discrete outcome with a
distribution belonging in the exponential family; $\alpha$ is a $p\times 1$
vector of coefficients associated with the $p$ error-free covariates,
${\mathbf{Z}}_{i}$. Further, $W_{ij}(t)$ is the $t$th $t=(1,\ldots,T)$
observation collected at the $j$th $j=(1,\ldots,J)$ day for subject $i$. Let
$X_{i}(t)$ be a functional covariate that is not directly observable but
approximated by $W_{ij}(t)$ with some error. The random terms or intercepts,
$U_{ij}(t)$, in Equation 2 are subject-specific deviations of the observed
data, $W_{ij}(t)$, from their overall mean functions, $X_{i}(t)$. Equations 1
and 2 represent the systematic component of a generalized functional linear
model with the random component left completely unspecified for now. The
exponential family specification of the measurement error component of the
model is a significant departure from current approaches that are based on
Gaussian processes.

##  3 ESTIMATION AND INFERENCE

Estimation and inference about the functional parameter $\beta(t)$ is our
primary goal. However, $X_{i}(t)$ is latent and not directly observed.
Estimating $\beta(t)$ with the observed measure of $X_{i}(t)$, $W_{ij}(t)$,
leads to biased estimates of $\beta(t)$ (Carroll _and others_ , 2006). The
biased estimator of $\beta(t)$ may attenuate or over-estimate the true effects
of $X_{i}(t)$ depending on the regression model. However, the presence of some
additional information in the data, such as validation data, repeated
measurements on $W_{ij}(t)$, or instrumental variables for $X_{i}(t)$, may be
used to correct for the biases due to measurement errors (Carroll _and others_
, 2006). Given additional identifying information, estimation approaches such
as regression calibration may be used to obtain unbiased estimators for
$\beta(t)$. Regression calibration is a bias correction method that
substitutes $E\\{X_{i}(t)\mid W_{ij}(t),{\mathbf{Z}}_{i}\\}$ for $X_{i}(t)$ in
Equation 1. We use replicates for identifying the model and regression
calibration for correcting measurement error.

Most methods for correcting measurement error in functional data (Tekwe _and
others_ , 2019) require approximating the functional parameters and covariates
in Equations 1 and 2 prior to estimation. However, with our new methods, we
first perform point-wise generalized linear mixed effects analysis of
$W_{ij}(t)$ by $t$, the time of observation, a process called massive
univariate analysis (Cui _and others_ , 2021). This step yields predicted
values for $X_{i}(t)$, the fixed component of the measurement error model in
Equation 2. The use of generalized linear mixed effects models to obtain the
predicted values for $X_{i}(t)$ is possible due to the repeated measures on
$W_{ij}(t)$ across the $J$ days. In the second step, we smooth the predicted
values of $X_{i}(t)$, $\hat{X}_{i}(t)\mid{\mathbf{U}}_{ij}(t)$ across $t$ with
any smoother, such as Bsplines or penalized splines. In the third step, we
substitute these values for $X_{i}(t)$ in the regression equations for the
outcome in Equation 1. The last step involves nonparametric bootstraps to
estimate confidence bands.

###  3.1 Likelihood

Given the observed data
$\mathcal{D}_{i}=\\{Y_{i},{\mathbf{W}}_{i},{\mathbf{Z}}_{i}\\}$, the
contribution of individual $i$ to the likelihood, $\ell_{i}$, is equal to

$\displaystyle\ell_{i}\\{{\mathbf{X}}_{i},\beta(t),\mbox{\boldmath$\alpha$},\mbox{\boldmath$\theta$}_{w}\\}$
$\displaystyle=$ $\displaystyle
f_{y}(Y_{i}|\beta(t),\mbox{\boldmath$\alpha$},{\mathbf{X}}_{i},{\mathbf{Z}}_{i})\prod^{J}_{j=1}f_{W}({\mathbf{W}}_{ij}|{\mathbf{X}}_{i},\mbox{\boldmath$\theta$}_{w}),$
(3)

where $f_{y}(\bullet)$ and $f_{w}(\bullet)$ are densities from the exponential
family which depend on the choice of the link function and the random
component. We note that the likelihood specified in Equation 3 is flexible as
it allows for non-normal data. For example, with an identity function that has
Gaussian random components for $h(\bullet)$ and $g(\bullet)$, our model
becomes the usual Gaussian measurement error model. The joint likelihood then
is

$\displaystyle\ell$ $\displaystyle=$
$\displaystyle\prod^{n}_{i=1}\ell_{i}\\{{\mathbf{X}}_{i},\beta(t),\mbox{\boldmath$\alpha$},\mbox{\boldmath$\theta$}_{w}\\}.$
(4)

In likelihoods 3 and 4, the functional covariate
${\mathbf{X}}_{i}=\\{X_{i}(t)\\}^{i\in n}$ is a random function because it is
unknown and unobserved, which is characteristic of the scalar-on-function
regression with measurement error (Tekwe _and others_ , 2019).

###  3.2 Model assumptions

To estimate the model parameters, we assume the following:

  * [A1

] $Y_{i}\sim EF(\bullet)$, where EF refers to an exponential family
distribution.

  * [A2

] The functions, $g(\bullet)$ and $h(\bullet)$, are monotone, twice
continuously differentiable functions.

  * [A3

] $\mathrm{h[{E}}\\{W_{ij}(t)\mid X_{i}(t)\\}]=X_{i}(t)$, and if $h(\bullet)$
is the identity function, then $\mathrm{{E}}\\{W_{ij}(t)\mid
X_{i}(t)\\}=X_{i}(t)$.

  * [A4

] $\mathrm{Cov}\\{W_{ij}(t),X_{i}(t)\\}\neq 0$.

  * [A5

] $\mathrm{Cov}\\{X_{i}(t),X_{i}(t^{\prime})\\}\neq 0$ and
$\mathrm{Cov}\\{W_{ij}(t),W_{ij}(t^{\prime})\\}\neq 0$ for $t\neq t^{\prime}$.

  * [A6

] $W_{ij}(t)|{\cal X}_{i}(t)\sim EF(\bullet)$.

  * [A7

] $f\\{Y_{i}|X_{i}(t),W_{ij}(t)\\}=f\\{Y_{i}|X_{i}(t)\\}$.

  * [A8

] $U_{ij}(t)\sim MVN(0,\Sigma_{uu})$,
$\mathrm{Cov}\\{U_{ij}(t),U_{ij^{\prime}}(t)\\}\neq 0$ for $j\neq j^{\prime}$,
and $\mathrm{Cov}\\{U_{ij}(t),U_{ij}(t^{\prime})\\}\neq 0$ for $t\neq
t^{\prime}$.

A1 indicates that the scalar response may be discrete or continuous with a
distribution belonging in the EF. A2 includes the usual assumptions for link
functions in generalized linear regression models. A3 indicates that nonlinear
functions of $\mathrm{E}\\{W_{ij}(t)\mid X_{i}(t)\\}$ are unbiased measures of
$X_{i}(t)$. A4 states that the surrogate measures and the true unobserved
covariate are correlated, a classical assumption in measurement error models.
A5 and A6 indicate correlated measurement errors for the surrogate measures
and the true covariate for each subject $i$. A7 is the non-differential
measurement error assumption specifying that the surrogate measure,
$W_{ij}(t)$, does not provide any additional information about the response,
$Y_{i}$, beyond the information given by $X_{i}(t)$. A8 indicates that
subject-specific intercept or deviation of the surrogate measures from the
true covariates have Gaussian distributions. Additionally, the covariance
function for $U_{ij}(t)$ is correlated across the $J$ days of observation and
also across the wear times, $t$.

##  4 REGRESSION CALIBRATION

Regression calibration (Carroll _and others_ , 2006) is an approach to
correcting classical measurement error in regression models with additional
identifying data. The method requires a gold standard for the true unobserved
covariate, repeated measurements of the observed covariate, or an instrumental
variable for the true covariate (Carroll _and others_ , 2006). Researchers
have developed maximum likelihood (Spiegelman and Casella, 1997),
semiparametric (Pepe and Fleming, 1991; Ye _and others_ , 2008), Bayesian
(Bartlett and Keogh, 2018; Zoh _and others_ , 2022), and Monte Carlo (Li _and
others_ , 2007; Fearn _and others_ , 2008; Tekwe _and others_ , 2014)
approaches to correcting measurement error with regression calibration.

We describe two approaches to using repeated measures for correcting
measurement error. Under the assumption that
$\hat{E\\{X_{ij}(t)\\}}=\hat{h[E\\{W_{ij}(t)\\}]}$, the predicted values from
regressing $h[E\\{W_{ij}(t)\\}]$ on the intercept may be used as calibrated
values for ${\mathbf{X}}_{i}$ when assessing its association with the outcome
of interest such as in Equation 1. In our approaches, regression calibration
proceeds in four steps:

  1. 1.

Fit the model

$\displaystyle h\\{X_{i}(t)\\}$ $\displaystyle=$
$\displaystyle\gamma_{x}(t)+\varepsilon_{xi}(t),$ (5)

where $\gamma_{x}(t)$ is the fixed intercept or mean function for $X_{i}(t)$
at time $t$ and $\varepsilon_{ix}(t)$ is the random error term in the model.
We assume $\varepsilon_{ix}(t)\sim MVN\\{0,\Sigma_{xx}(t,t^{\prime})\\}$, for
$t\neq t^{\prime}$. However, we cannot fit the model in Equation 5 because
$X_{i}(t)$ is latent and unobserved. We therefore replace $X_{i}(t)$ with
$W_{ij}(t)$, its unbiased but measurement error-prone observed measure.
Following this replacement Equation 5 becomes

$\displaystyle h[E\\{W_{ij}(t)\\}]$ $\displaystyle=$
$\displaystyle\gamma_{w}(t)+U_{ij}(t),$ (6)

where $\gamma_{w}(t)$ is the fixed intercept and $U_{ij}(t)$ is the subject-
specific random intercept. We assume $U_{ij}(t)\sim
MVN\\{0,\Sigma_{uu}(t,t^{\prime})\\}$ for $t\neq t^{\prime}$.

  2. 2.

Replace $X_{i}(t)$ with $\hat{h[\\{W_{ij}(t)\\}]}$ obtained from the predicted
values in Equation 6. Most current approaches to correcting measurement error
in functional data analysis with functional covariates prone to measurement
error require some dimension reduction before correcting measurement error.
However, the regression calibration methods we propose do not require
dimension reduction of $W_{ij}(t)$ to obtain predicted values of $X_{i}(t)$
for fitting the regression model in Equation 1.

  3. 3.

While dimension reduction is not required to obtain the measurement error-
corrected measure of $X_{i}(t)$, we do reduce dimensions in the regression
step. Using $\hat{h[\\{W_{ij}(t)\\}]}$ for $X_{i}(t)$ in Equation 1, we reduce
the dimensions of the functional terms by approximating them with polynomial
splines $\beta(t)\approx\sum_{k=1}^{K_{n}}\omega_{k}b_{k}(t)$ and
$\hat{h[\\{W_{ijk}\\}]}\approx\int_{0}^{1}\hat{h[\\{W_{ij}(t)\\}]}b_{k}(t)dt$,
where $\left\\{\omega_{k}\right\\}_{k=1}^{K_{n}}$ are unknown spline
coefficients and $\left\\{b_{k}(t)\right\\}_{k=1}^{K_{n}}$ are a set of spline
basis functions on $[0,1]$. Equation 1 becomes

$\displaystyle g[E\\{Y_{i}\mid{\mathbf{X}}_{i},{\mathbf{Z}}_{i}\\}]$
$\displaystyle\approx$
$\displaystyle\sum_{k=1}^{K_{n}}\omega_{k}\hat{h[\\{W_{ijk}\\}]}+{\mathbf{Z}}_{i}^{\rm
T}\mbox{\boldmath$\alpha$}.$ (7)

  4. 4.

Maximize the likelihood function in the reduced form model in Equation 7 by
replacing ${\mathbf{X}}_{i}$ with $\hat{h[\\{W_{ijk}\\}]}$ to obtain estimated
model parameters $E\\{\hat{\beta}(t)\\}\approx
E\\{\sum_{k=1}^{K_{n}}\hat{\omega}_{k}b_{k}(t)\\}$ and
$E(\hat{\mbox{\boldmath$\alpha$}})=\mbox{\boldmath$\alpha$}$.

Our methods for correcting measurement error allow arbitrary error structures
for $\Sigma_{uu}(t,t^{\prime})$, such as unstructured, compound symmetry, or
autoregressive 1. We propose two regression calibration approaches. For our
first approach, we adapt the fast univariate inference (FUI) modeling
developed by Cui and colleagues (Cui _and others_ , 2021) to implement the
functional analog of regression calibration. Using the repeated measurements
on $W_{ij}(t)$ as additional identifying information for estimating
$\Sigma_{uu}(t,t^{\prime})$, FUI treats the repeated functional measures
across multiple days as longitudinal high dimensional data and fits a point-
wise generalized linear mixed effects model at each wear time $t,t=1,\ldots,T$
across all the $J$ observational periods or days resulting in $T$ estimated
fixed effects. The term ”univariate” in FUI refers to the point-wise fitting
for each wear time separately. We use the predicted values from these point-
wise generalized linear mixed effects models as replacement values for
$X_{i}(t)$.

A potential drawback of the FUI method is that it may not account for the
serial correlation across days or observation periods adequately due to its
univariate focus. Therefore, we propose a fast short multivariate inference
approach (FSMI). The FSMI method also involves fitting point-wise generalized
linear mixed effects models to the data with Equation 6, but by analyzing
observations from multiple wear times concurrently and proceeding across all
the $T_{i}$ wear times in a moving average. The term ”multivariate” here
refers to the use of multiple wear times concurrently in fitting the point-
wise generalized linear mixed effects model.

For inference, we used $95\%$ nonparametric point-wise bootstrap confidence
intervals.

##  5 SIMULATIONS

We conducted simulations to investigate the finite sample properties of our
methods. We generated 1,000 data sets with sample sizes
$n\in(100,200,500,1,000,2,000,5,000)$ independently from the model in Equation
1 using an identity function for $g(\bullet)$. We simulated the outcomes as
$Y_{i}\sim Binomial(p_{i})$ with $p_{i}=expit(\eta_{i})$ and
$\eta_{i}=\int_{0}^{1}\beta(t)X_{i}(t)dt+\boldsymbol{Z_{i}\alpha}$, where
$\beta(t)=sin(2\pi t)$ and $\boldsymbol{\alpha}=(1,2)$. We simulated the true
functional covariate with $X_{i}(t)=1/[1+\exp\\{8(t-0.5)\\}]+\epsilon_{X}(t)$,
and generated $\epsilon_{X}(t)$’s independently from a Gaussian process (GP)
with mean 0 and an autoregressive one structure for the covariance matrix.
Thus, $\epsilon_{X}(t)\sim GP\\{\boldsymbol{0},\Sigma_{xx}(t,t^{\prime})\\}$
for $t\neq t^{\prime}$. We included a continuous error-free covariate,
$Z_{1i}$, and a binary error-free covariate, $Z_{2i}$, simulated independently
with $Z_{1i}\sim N(2,1)$ and $Z_{2i}\sim Binomial(1,0.6)$, respectively. We
generated the surrogate measure, $W_{ij}(t)$, from a Poisson process (PP) with
mean function $\lambda_{i}(t)$. That is, $W_{ij}(t)\sim
PP\\{\lambda_{i}(t)\\}$, where $\lambda_{i}(t)=\exp\\{X_{i}(t)\\}$ and
$j=1,\dots,J=5$. The measurement error component of our model in Equation 2
was $\hbox{log}[E\\{W_{ij}(t)\\}]=X_{i}(t)+U_{ij}(t)$ for the $i$th subject at
the $j$th replication, and we assumed $U_{ij}(t)\sim
GP\\{\boldsymbol{0},\Sigma_{uu}(t,t^{\prime})\\}$ for $t\neq t^{\prime}$.

We compared five estimators for $\beta(t)$ in Equation 1 in analyzing the
simulated data: FUI, FSMI, average, naive, and oracle. The average estimator
averages the repeatedly observed surrogate for $X_{i}(t)$ across the $J$
replicates to obtain $\bar{W}_{i\bullet}(t)$. The naive estimator is based on
the first replicate $\\{W_{i1}(t)\\}$ of the surrogate measures as a
substitute for $X_{i}(t)$. The oracle or benchmark estimator uses $X_{i}(t)$.
The average method provides an ad hoc method for reducing bias due to
measurement error, while the naive estimator does not correct for any
measurement error.

We performed three sets of simulations to investigate the performance of the
five estimators in response to three different factors, respectively: 1)
increasing sample size, with $n\in(100,200,500,1,000,2,000,5,000)$,
$\sigma_{x}=3.0$, where $\sigma_{x}$ indicates the diagonal elements in the
covariance matrix for $X_{i}(t)$, and $\rho_{x}=0.50$; 2) the increasing
magnitude of the measurement error with $n=5,000$,
$\sigma_{x}\in(1.5,2.0,3.0)$, and $\rho_{x}=0.75$; and 3) increasing magnitude
of the serial correlation across time points for $X_{i}(t)$ with
$\rho_{x}\in(0.05,0.25,0.5,0.75)$, $n=5,000$, and $\sigma_{x}=3.0$. All
simulations were conducted with 1,000 iterations.

We compared the estimators with three measures. Let $\hat{\beta}^{r}(t)$ be
the estimator of $\beta(t)$ at the $r$th $(r=1,\ldots,R)$ simulation replicate
and $\bar{\beta}(t)=\frac{1}{R}\sum_{r=1}^{R}\hat{\beta}^{r}(t)$, the averaged
estimate over a total of $R=1,000$ replications. Let
$\left\\{t_{l}\right\\}_{l=1}^{n_{grid}}$ be a sequence of equally spaced grid
points on $[0,1]$. We define the average squared bias (ABias2) of
$\hat{\beta}(t)$, the average sample variance (AVar), and the average
integrated mean square error (AIMSE) as

$\displaystyle\text{ABias}^{2}\\{\hat{\beta}(t)\\}$ $\displaystyle=$
$\displaystyle\frac{1}{n_{grid}}\sum_{l=1}^{n_{grid}}\\{\bar{\beta}(t_{l})-{{{\beta_{true}(t_{l})}}}\\}^{2},$
$\displaystyle\text{AVar}\\{\hat{\beta}(t)\\}$ $\displaystyle=$
$\displaystyle\frac{1}{R}\sum_{r=1}^{R}\frac{1}{n_{grid}}\sum_{l=1}^{n_{grid}}\left\\{\hat{\beta}^{r}(t_{l})-\bar{{\beta}}(t_{l})\right\\}^{2},\text{and}$
$\displaystyle\text{AIMSE}\\{\hat{\beta}(t)\\}$ $\displaystyle=$
$\displaystyle\text{ABias}^{2}\\{\hat{\beta}(t)\\}+\text{AVar}\\{\hat{\beta}(t)\\},\text{respectively.}$

Small values of these measures indicate good performance and large values
indicate poor performance.

###  5.1 Simulation results

####  5.1.1 Impact of sample size

Table 1 summarizes the results for varying sample size. As the sample size
increased, the ABias2, AVar, and AIMSE consistently decreased for all the
estimators. The FSMI estimator had ABias2 consistently smaller than that of
the FUI, average, and naive estimators and closest to that of the oracle
estimator. The oracle estimator had the smallest Avar, followed in order by
the FSMI, FUI, naive, and the average method. Similarly, the average method
had the highest values for AIMSE while the AIMSE associated with the oracle
estimator was the smallest.

Table 1: The effect of varying sample size on the performance of
$\hat{\beta}(t)$ under the five estimators with
$n\in(100,200,500,1,000,2,000,5,000)$, $\sigma_{x}=3.0$, and $\rho_{x}=0.50$.

| ABias2 | AVar | AIMSE
---|---|---|---
n | Oracle | FSMI | FUI | Average | Naive | Oracle | FSMI | FUI | Average | Naive | Oracle | FSMI | FUI | Average | Naive
100 | 0.0079 | 0.0174 | 0.0297 | 0.2942 | 0.2097 | 3.2800 | 3.8213 | 4.2202 | 9.0955 | 8.4759 | 3.2879 | 3.8387 | 4.2499 | 9.3897 | 8.6856
200 | 0.0016 | 0.0056 | 0.0129 | 0.2132 | 0.1520 | 1.4365 | 1.6621 | 1.8318 | 3.9350 | 3.6675 | 1.4381 | 1.6677 | 1.8447 | 4.1482 | 3.8196
500 | 0.0009 | 0.0023 | 0.0067 | 0.1686 | 0.1185 | 0.6369 | 0.7418 | 0.8155 | 1.7658 | 1.6114 | 0.6378 | 0.7440 | 0.8222 | 1.9344 | 1.7300
1000 | 0.0002 | 0.0019 | 0.0067 | 0.1761 | 0.1247 | 0.3605 | 0.4177 | 0.4571 | 0.9729 | 0.9109 | 0.3607 | 0.4195 | 0.4638 | 1.1490 | 1.0357
2000 | 0.0001 | 0.0015 | 0.0061 | 0.1727 | 0.1220 | 0.2111 | 0.2449 | 0.2661 | 0.5614 | 0.5163 | 0.2112 | 0.2464 | 0.2722 | 0.7341 | 0.6384
5000 | 0.0001 | 0.0014 | 0.0060 | 0.1705 | 0.1220 | 0.1092 | 0.1280 | 0.1378 | 0.2931 | 0.2670 | 0.1093 | 0.1294 | 0.1437 | 0.4636 | 0.3890

####  5.1.2 Impact of the magnitude of measurement error

Table 2 provides the results on varying levels of the magnitude of measurement
error. The oracle estimator consistently had the smallest Abias2 and AVar of
the five estimators. The FSMI estimator had the next smallest Abias2 and AVar
values, followed by, in ascending order, the FUI, naive, and average
estimators. The oracle estimator had the smallest values of Avar and AIMSE
while the FSMI estimator had the second smallest values.

Table 2: The effect of varying magnitude of measurement error on the
performance of $\hat{\beta}(t)$ for the five estimators with $n=5,000$,
$\sigma_{x}\in(1.5,2.0,3.0)$, and $\rho_{x}=0.75$

| ABias2 | AVar | AIMSE
---|---|---|---
$\sigma_{x}$ | Oracle | FSMI | FUI | Average | Naive | Oracle | FSMI | FUI | Average | Naive | Oracle | FSMI | FUI | Average | Naive
1.5 | 0.0001 | 0.0017 | 0.0078 | 0.1791 | 0.1519 | 0.1100 | 0.1280 | 0.1349 | 0.2820 | 0.2564 | 0.1102 | 0.1298 | 0.1426 | 0.4611 | 0.4083
2.0 | 0.0000 | 0.0026 | 0.0079 | 0.1820 | 0.1686 | 0.0648 | 0.0761 | 0.0796 | 0.1653 | 0.1567 | 0.0648 | 0.0787 | 0.0875 | 0.3473 | 0.3253
3.0 | 0.0000 | 0.0048 | 0.0103 | 0.1824 | 0.1772 | 0.0294 | 0.0367 | 0.0383 | 0.0778 | 0.0759 | 0.0294 | 0.0415 | 0.0486 | 0.2602 | 0.2531

####  5.1.3 Impact of the strength of serial correlation across time points

Table 3 summarizes the impacts of the varying strength of the serial
correlation across time points for the functional covariate, $X_{i}(t)$. The
simulation results demonstrate that the biases associated with the oracle,
FSMI, and FUI methods were all smaller than those obtained under the average
and naive approaches. The strength of correlation of the functional covariate
did not affect the estimators obtained under the oracle and FSMI methods as
their ABias2’s remained more stable for increasing values of $\rho_{x}$. As
the correlation coefficient, $\rho_{x}$, increased, the Avar and AIMSE of all
five estimations decreased. The oracle estimator had the smallest values of
Avar and AIMSE and followed in decreasing order by the FSMI, FUI, average, and
naive estimators.

Table 3: The effect of increasing magnitude of the serial correlation cross
time points for $X_{i}(t)$ on the performance of $\hat{\beta}(t)$ for the five
estimators with $\rho_{x}\in(0.05,0.25,0.5,0.75)$, $n=5,000$, and
$\sigma_{x}=3.0$.

| ABias2 | AVar | AIMSE
---|---|---|---
$\rho_{x}$ | Oracle | FSMI | FUI | Average | Naive | Oracle | FSMI | FUI | Average | Naive | Oracle | FSMI | FUI | Average | Naive
0.05 | 0.0001 | 0.0039 | 0.0044 | 0.1146 | 0.0989 | 0.0519 | 0.0667 | 0.0673 | 0.1389 | 0.1349 | 0.0520 | 0.0706 | 0.0716 | 0.2535 | 0.2338
0.25 | 0.0000 | 0.0043 | 0.0068 | 0.1507 | 0.1370 | 0.0378 | 0.0478 | 0.0497 | 0.1041 | 0.1025 | 0.0378 | 0.0521 | 0.0565 | 0.2547 | 0.2395
0.50 | 0.0000 | 0.0052 | 0.0099 | 0.1799 | 0.1703 | 0.0285 | 0.0358 | 0.0378 | 0.0793 | 0.0777 | 0.0285 | 0.0410 | 0.0478 | 0.2592 | 0.2479
0.75 | 0.0000 | 0.0048 | 0.0103 | 0.1824 | 0.1772 | 0.0294 | 0.0367 | 0.0383 | 0.0778 | 0.0759 | 0.0294 | 0.0415 | 0.0486 | 0.2602 | 0.2531

##  6 APPLICATION

###  6.1 Data and Model

We applied our methods to data from the 2005–2006 cycle of the NHANES to
assess the association between wearable device-based physical activity (PA)
and T2D status while adjusting for age, sex, and race/ethnicity in community-
dwelling adults in the U.S. The participants wore ActiGraph uniaxial
accelerometers (model #AM7164) on their hips for at least five days during
waking hours except during water-related activities, as the ActiGraph devices
were not water-proof. The ActiGraph uniaxial accelerometers (model #AM7164)
registered PA intensity and step counts as zero when participants were not
wearing them or were sedentary. In this application, we define the latent
variable, $X_{i}(t)$, as the true unknown activity counts for a given
participant $i$ during wear time $t$. The surrogate measure for $X_{i}(t)$ was
the step count recorded by the device. Because NHANES did not provide any
information to distinguish between non-wear and non-moving time, we defined
non-wear time to be when the device-based PA measures were recorded as zero
for at least 60 consecutive minutes and set the PA measures (step counts) to
be non-applicable values during the non-wear time. We included data on
participants who were older than 20 and had PA data for at least two days. We
applied sample weights to account for the oversampling of racial subgroups,
which were Hispanic and black racial groups, per NHANES guidelines (Johnson
_and others_ , 2013). The final analytic sample included 2,001 participants,
whose average age was 44 ± 14 years. Fifty-one percent were female and
approximately $14\%$ were black, $13\%$ were Hispanic, $68\%$ were white, and
$4\%$ were of another race or ethnicity after adjusting for oversampling from
some racial groups.

We classified participants who reported they had ever been told by a doctor
that they have T2D as having a prior diagnosis of T2D, and classified others
as not having such a diagnosis. Figure 1 illustrates patterns of wearable-
device-based step counts for weekdays and weekends by time (hours) over 24
hours. The patterns of step counts on weekdays and weekends were similar.
Therefore, we analyzed the data for weekdays and weekends together as a whole.

We considered the following functional logistic regression model with
measurement error:

$\displaystyle\hbox{log}\left(\frac{P_{i}}{1-P_{i}}\right)$
$\displaystyle=\int_{0}^{1}\beta(t)X_{i}(t)dt+{\mathbf{Z}}_{i}^{\rm
T}\mbox{\boldmath$\alpha$}\text{ and}$ $\displaystyle h[E\\{W_{ij}(t)\\}]$
$\displaystyle=X_{i}(t)+U_{ij}(t),$

where $P_{i}$ is the probability of being diagnosed with T2D for the $i$th
subject, $\beta(t)$ is the unknown functional regression coefficient, and
$\alpha$ is the unknown scalar regression coefficients of the error-free
covariates. The $X_{i}(t)$ are the unobservable true activity counts, with
surrogate measure, $W_{ij}(t)$, representing device-based step counts prone to
measurement error, and ${\mathbf{Z}}_{i}$ are the error-free covariates,
including age, sex, and race/ethnicity, for the $i$th subject.

Figure 1: Wearable-device-based step counts from the 2005-2006 NHANES plotted
against time (hour) for weekdays and weekends. The grey lines on the
background are the trend of step counts over time for each subject and the
blue line is the average step count across all participants over time for each
day

###  6.2 Application Results

Table 4 shows the estimated exponentiated regression coefficients of the
error-free covariates and their corresponding $95\%$ confidence intervals
(CIs) for the FSMI, FUI, average, and naive estimators of $\beta(t)$. We
obtained the $95\%$ CIs via a nonparametric bootstrap with $1,000$ replicates.
Correcting for measurement error influenced estimates for some error-free
covariates but not others. For example, the statistically significant
estimates of the age coefficient were similar for the four estimators. There
was no statistically significant difference in the odds of T2D between the
male and female participants based on all four estimates. Additionally, the
regression calibration- and naive-based estimators were smaller than that
obtained from the average-based method. There was an increased odds of T2D
among blacks when compared to whites. The estimated odds obtained under the
FSMI and FUI methods were smaller than those obtained under the average and
naive approaches. However, the largest estimated difference in odds of T2D
between blacks and whites was based on the average approach. We also observed
an increased odds of T2D among Hispanics when compared with whites. While the
estimated difference in odds between whites and Hispanics were found to be
equivalent based on the FSMI and FUI methods, the average-based was smallest
among all four methods. Finally, we did not observe any difference in the odds
of T2D between participants belonging in the other racial/ethnic groups when
compared to whites under all four methods of estimation. The FSMI and FUI
approaches were estimated to be equivalent while the average-based method was
the highest.

We graphically illustrate the results for the functional ceofficient in Figure
2. The plots demonstrate pairwise comparisons between the FSMI, FUI, average,
and naive estimators of the exponentiated coefficient for the association
between step count and T2D, [$\exp\\{\hat{\beta}(t)\\}$], after adjusting for
age, sex, and race. The FSMI and FUI estimators were almost identical and
differed from the average and naive estimators. The naive estimator tended to
have different time trends when compared to the FSMI, FUI, and average
estimators. The average and naive estimators were also more attenuated towards
the null compared with the FSMI and FUI estimators. We observed statistically
significant decreases in the odds of T2D and PA when PA was performed between
the $10$th and $15$th and also between the $20$th and $24$th hours based on
the FSMI and FUI. However, this statistical significance was not observed
under the naive-based methods. Furthermore, the plots indicate that the impact
of measurement error in device-based step counts is time-varying.

Table 4: Application of four estimators in estimating the associations of the
error-free covariates with T2D after correcting for measurement error in
device-based step counts in data from the 2005-2006 NHANES.

Covariates |  | FSMI |  $95\%$ C.I. | FUI |  $95\%$ C.I. | Average |  $95\%$ C.I. | Naive |  $95\%$ C.I.
---|---|---|---|---|---|---|---|---|---
Intercept |  | 0.0011 | (0.00042, 0.0024 ) | 0.0011 | (0.00042, 0.0024) | 0.0013 | (0.00042, 0.0024) | 0.0012 | (0.00045, 0.0026)
Age |  | 1.0806 | (1.06632, 1.0979) | 1.0806 | (1.06630, 1.0978) | 1.0790 | (1.06631, 1.0979) | 1.0806 | (1.06458, 1.0974)
Sex |  |  |  |  |  |  |  |  |
| Female | 0.9956 | (0.68115, 1.4172 ) | 0.9956 | (0.68116, 1.4173) | 1.0580 | (0.68129, 1.417) | 0.9936 | (0.68664, 1.4326)
Race |  |  |  |  |  |  |  |  |
| Black | 3.1656 | (2.08932, 4.9656) | 3.1645 | (2.08862, 4.9638) | 3.7987 | (2.08908, 4.9655) | 3.4526 | (2.21831, 5.3201)
| Hispanic | 3.5241 | (2.05753, 5.9562) | 3.5237 | (2.05714, 5.9550) | 3.3188 | (2.05715, 5.9538) | 3.6691 | (1.98040, 5.9763)
| Other | 2.2926 | (0.65906, 5.2862 ) | 2.2922 | (0.65895, 5.2853) | 2.3697 | (0.65920, 5.2870) | 2.3190 | (0.64141, 5.5556)

Figure 2: Plots of pairwise comparisons between FSMI, FUI, average, and naive
estimates of the exponentiated coefficients of the association between step
count and T2D [$\exp\\{\hat{\beta}(t)\\}$] after adjusting for age, sex, and
race in data from the 2005-2006 NHANES. The shaded areas are the nonparametric
$95\%$ point-wise bootstrap confidence intervals (CI) of
[$\exp{\hat{\beta}(t)}$] with the FSMI estimator. The dashed cyan lines
represent the $95\%$ point-wise lower and upper bounds of these CIs.

##  7 DISCUSSION

We developed the multi-level generalized functional linear regression model
with functional covariates prone to heteroscedastic errors. These models are
suited to massive longitudinal functional data assumed to be error-prone, such
as those collected by wearable devices at frequent intervals over multiple
days. Additionally, the measurement error component of the model was also
developed under the multi-level generalized functional linear regression
models framework. To date, most approaches to correcting measurement error in
functional data analysis are based on the assumption that surrogate measures,
$W_{ij}(t)$, follow Gaussian distributions. We assumed an exponential family
distribution for the surrogate measures and Gaussian error processes for the
measurement error. We implemented regression calibration methods to correct
for measurement error and allow arbitrary heteroscedastic covariance matrices
for the measurement errors. To evaluate our new methods, we conducted
simulations that involved Poisson assumptions for the surrogate measures and
Gaussian error processes for the measurement errors. The regression
calibration methods generally had lower $\text{Abias}^{2}$ than estimators
based on averaging and naive methods that do not correct for measurement error
explicitly. While the average estimator reduced bias more than the naive
estimator, it does not provide any formal correction of measurement error in
the estimation. Likewise, the average and naive estimators produced
$\text{Abias}^{2}$ values that were larger than the FSMI and FUI estimators
and not statistically consistent. The FSMI estimator yielded lower
$\text{Abias}^{2}$ and Avar than the FUI estimator because FSMI focuses on
multiple time points concurrently in the estimation while FUI is a univariate
approach. Overall, both regression calibration methods performed better than
the average and naive methods with increasing sample sizes and varying levels
of correlations in the serially observed functional covariate prone to errors.

We illustrated our new methods in analyses of data from the NHANES. We
assessed the association between a device-based measure of step count, a
surrogate for true physical activity counts, and T2D in adults in the U.S. The
FSMI and FUI estimators were equivalent in our analyses and they demonstrated
some correction of measurement error compared to the average and naive
estimators.

##  8 SUPPLEMENTARY MATERIAL

None

## ACKNOWLEDGEMENTS

Conflict of Interest: None declared. The authors would like to thank Dr. Devon
Brewer for his constructive feedback and for editing the manuscript. This
research was supported by National Institute of Diabetes and Digestive and
Kidney Diseases Award 1R01DK132385-01. This research was also supported in
part by Lilly Endowment, Inc., through its support for the Indiana University
Pervasive Technology Institute.

## References

  * Armstrong and Bull (2006) Armstrong, Timothy and Bull, Fiona. (2006).  Development of the World Health Organization Global Physical Activity Questionnaire (GPAQ).  Journal of Public Health 14(2), 66–70.
  * Balk _and others_ (2015) Balk, Ethan M, Earley, Amy, Raman, Gowri, Avendano, Esther A, Pittas, Anastassios G and Remington, Patrick L. (2015).  Combined diet and physical activity promotion programs to prevent type 2 diabetes among persons at increased risk: a systematic review for the community preventive services task force.  Annals of Internal Medicine 163(6), 437–451.
  * Bartlett and Keogh (2018) Bartlett, Jonathan W and Keogh, Ruth H. (2018).  Bayesian correction for covariate measurement error: a frequentist evaluation and comparison with regression calibration.  Statistical Methods in Medical Research 27(6), 1695–1708.
  * Bassett (2012) Bassett, D.R. (2012).  Device-based monitoring in physical activity and public health research.  Physiological Measurement 33(11), 1769.
  * Boyle _and others_ (2010) Boyle, James P, Thompson, Theodore J, Gregg, Edward W, Barker, Lawrence E and Williamson, David F. (2010).  Projection of the year 2050 burden of diabetes in the US adult population: dynamic modeling of incidence, mortality, and prediabetes prevalence.  Population Health Metrics 8(1), 29.
  * Cardot _and others_ (2007) Cardot, H., Crambes, C., Kneip, A. and Sarda, P. (2007).  Smoothing splines estimators in functional linear regression with errors-in-variables.  Computational Statistics & Data Analysis 51(10), 4832–4848.
  * Carroll _and others_ (2006) Carroll, Raymond J, Ruppert, David, Stefanski, Leonard A and Crainiceanu, Ciprian M. (2006).  Measurement error in nonlinear models: a modern perspective. CRC press.
  * Crainiceanu _and others_ (2009) Crainiceanu, Ciprian M, Staicu, Ana-Maria and Di, Chong-Zhi. (2009).  Generalized multilevel functional regression.  Journal of the American Statistical Association 104(488), 1550–1561.
  * Crambes _and others_ (2009) Crambes, Christophe, Kneip, Alois, Sarda, Pascal _and others_. (2009).  Smoothing splines estimators for functional linear regression.  The Annals of Statistics 37(1), 35–72.
  * Crouter _and others_ (2006) Crouter, S.E., Churilla, J.R. and Bassett, D.R. (2006).  Estimating energy expenditure using accelerometers.  European Journal of Applied Physiology 98(6), 601–612.
  * Cui _and others_ (2021) Cui, E., Leroux, A., Smirnova, E. and Crainiceanu, C.M. (2021).  Fast univariate inference for longitudinal functional models.  Journal of Computational and Graphical Statistics, 1–12.
  * Eckel _and others_ (2011) Eckel, Robert H, Kahn, Steven E, Ferrannini, Ele, Goldfine, Allison B, Nathan, David M, Schwartz, Michael W, Smith, Robert J and Smith, Steven R. (2011).  Obesity and type 2 diabetes: what can be unified and what needs to be individualized?  The Journal of Clinical Endocrinology & Metabolism 96(6), 1654–1663.
  * Fearn _and others_ (2008) Fearn, T, Hill, DC and Darby, SC. (2008).  Measurement error in the explanatory variable of a binary regression: regression calibration and integrated conditional likelihood in studies of residential radon and lung cancer.  Statistics in Medicine 27(12), 2159–2176.
  * Freedson _and others_ (1998) Freedson, P.S., Melanson, E. and Sirard, J. (1998).  Calibration of the computer science and applications, inc. accelerometer.  Medicine and Science in Sports and Exercise 30(5), 777–781.
  * Gaynanova _and others_ (2022) Gaynanova, Irina, Punjabi, Naresh and Crainiceanu, Ciprian. (2022).  Modeling continuous glucose monitoring (cgm) data during sleep.  Biostatistics 23(1), 223–239.
  * Goldsmith _and others_ (2011) Goldsmith, Jeff, Bobb, Jennifer, Crainiceanu, Ciprian M, Caffo, Brian and Reich, Daniel. (2011).  Penalized functional regression.  Journal of Computational and Graphical Statistics 20(4), 830–851.
  * Jacobi _and others_ (2007) Jacobi, D., Perrin, A.E., Grosman, N., Doré, M.F., Normand, S., Oppert, J.M. and Simon, C. (2007).  Physical activity-related energy expenditure with the rt3 and tritrac accelerometers in overweight adults.  Obesity 15(4), 950–956.
  * Jadhav _and others_ (2022) Jadhav, Sneha, Tekwe, Carmen D and Luan, Yuanyuan. (2022).  A function-based approach to model the measurement error in wearable devices.  Statistics in Medicine 41(24), 4886–4902.
  * Johnson _and others_ (2013) Johnson, Clifford L, Paulose-Ram, Ryne, Ogden, Cynthia L, Carroll, Margaret D, Kruszan-Moran, Deanna, Dohrmann, Sylvia M and Curtin, Lester R. (2013).  National health and nutrition examination survey. analytic guidelines, 1999-2010.
  * Kozey _and others_ (2010) Kozey, S.L., Lyden, K., Howe, C.A., Staudenmayer, J.W. and Freedson, P.S. (2010).  Accelerometer output and met values of common physical activities.  Medicine and Science in Sports and Exercise 42(9), 1776\.
  * Kuo _and others_ (2016) Kuo, Chih-En, Liu, Yi-Che, Chang, Da-Wei, Young, Chung-Ping, Shaw, Fu-Zen and Liang, Sheng-Fu. (2016).  Development and evaluation of a wearable device for sleep quality assessment.  IEEE Transactions on Biomedical Engineering 64(7), 1547–1557.
  * Li _and others_ (2007) Li, Yehua, Guolo, Annamaria, Hoffman, F Owen and Carroll, Raymond J. (2007).  Shared uncertainty in measurement error problems, with application to nevada test site fallout data.  Biometrics 63(4), 1226–1236.
  * Pepe and Fleming (1991) Pepe, Margaret Sullivan and Fleming, Thomas R. (1991).  A nonparametric method for dealing with mismeasured covariate data.  Journal of the American Statistical Association 86(413), 108–113.
  * Plasqui and Westerterp (2007) Plasqui, G. and Westerterp, K.R. (2007).  Physical activity assessment with accelerometers: an evaluation against doubly labeled water.  Obesity 15(10), 2371–2379.
  * Rothney _and others_ (2008) Rothney, M.P., Schaefer, E.V., Neumann, M.M., Choi, L. and Chen, K.Y. (2008).  Validity of physical activity intensity predictions by actigraph, actical, and rt3 accelerometers.  Obesity 16(8), 1946–1952.
  * Sallis and Saelens (2000) Sallis, J.F. and Saelens, B.E. (2000).  Assessment of physical activity by self-report: status, limitations, and future directions.  Research Quarterly for Eexercise and Sport 71(sup2), 1–14.
  * Silverman and Ramsay (2005) Silverman, BW and Ramsay, JO. (2005).  Functional Data Analysis. Springer.
  * Spiegelman and Flier (2001) Spiegelman, Bruce M and Flier, Jeffrey S. (2001).  Obesity and the regulation of energy balance.  Cell 104(4), 531–543.
  * Spiegelman and Casella (1997) Spiegelman, Donna and Casella, Mario. (1997).  Fully parametric and semi-parametric regression models for common events with covariate measurement error in main study/validation study designs.  Biometrics, 395–409.
  * Swartz _and others_ (2000) Swartz, A.M., Strath, S.J., Bassett, D.R., O’Brien, W.L., King, G.A. and Ainsworth, B.E. (2000).  Estimation of energy expenditure using csa accelerometers at hip and wrist sites.  Medicine and Science in Sports and Exercise 32(9; SUPP/1), S450–S456.
  * Tekwe _and others_ (2019) Tekwe, C.D., Zoh, R.S., Yang, M., Carroll, R.J., Honvoh, G., Allison, D.B., Benden, M. and Xue, L. (2019).  Instrumental variable approach to estimating the scalar-on-function regression model with measurement error with application to energy expenditure assessment in childhood obesity.  Statistics in Medicine 38(20), 3764–3781.
  * Tekwe _and others_ (2014) Tekwe, Carmen D, Carter, Randy L, Cullings, Harry M and Carroll, Raymond J. (2014).  Multiple indicators, multiple causes measurement error models.  Statistics in Medicine 33(25), 4469–4481.
  * Tekwe _and others_ (2022) Tekwe, Carmen D, Zhang, Mengli, Carroll, Raymond J, Luan, Yuanyuan, Xue, Lan, Zoh, Roger S, Carter, Stephen J, Allison, David B and Geraci, Marco. (2022).  Estimation of sparse functional quantile regression with measurement error: a simex approach.  Biostatistics 23(4), 1218–1241.
  * Troiano _and others_ (2008) Troiano, R.P., Berrigan, D., Dodd, K.W., Masse, L.C., Tilert, T. and McDowell, M. (2008).  Physical activity in the united states measured by accelerometer.  Medicine and Science in Sports and Exercise 40(1), 181\.
  * Tsai _and others_ (2019) Tsai, Chih-Wei, Li, Chun-Hung, Lam, Ringo Wai-Kit, Li, Chi-Kwong and Ho, Sam. (2019).  Diabetes care in motion: Blood glucose estimation using wearable devices.  IEEE Consumer Electronics Magazine 9(1), 30–34.
  * Tuominen _and others_ (2019) Tuominen, Jarno, Peltola, Karoliina, Saaresranta, Tarja and Valli, Katja. (2019).  Sleep parameter assessment accuracy of a consumer home sleep monitoring ballistocardiograph beddit sleep tracker: a validation study.  Journal of Clinical Sleep Medicine 15(3), 483–487.
  * Valenti _and others_ (2014) Valenti, G., Camps, S.G.J.A., Verhoef, S.P.M., Bonomi, A.G. and Westerterp, K.R. (2014).  Validating measures of free-living physical activity in overweight and obese subjects using an accelerometer.  International Journal of Obesity 38(7), 1011–1014.
  * Warolin _and others_ (2012) Warolin, Joshua, Carrico, Amanda R, Whitaker, Lauren E, Wang, Li, Chen, Kong Y, Acra, Sari and Buchowski, Maciej S. (2012).  Effect of bmi on prediction of accelerometry-based energy expenditure in youth.  Medicine and Science in Sports and Exercise 44(12), 2428\.
  * Ye _and others_ (2008) Ye, Wen, Lin, Xihong and Taylor, Jeremy MG. (2008).  Semiparametric modeling of longitudinal measurements and time-to-event data–a two-stage regression calibration approach.  Biometrics 64(4), 1238–1246.
  * Zoh _and others_ (2022) Zoh, Roger S, Luan, Yuanyuan and Tekwe, Carmen. (2022).  A fully Bayesian semi-parametric scalar-on-function regression (SoFR) with measurement error using instrumental variables.  arXiv preprint arXiv:2202.00711.