Page 370 - Contributed Paper Session (CPS) - Volume 4

CPS2292 Roger S. Zoh, PhD et al.
series data. Rather, we consider the functional covariate as a single function
that is used to estimate a latent variable such as true energy expenditure.
Under our newly developed methods, estimation of the measurement error
covariance is not required for parameter estimation. To the best of our
knowledge, the use of function-valued instrumental variables in the functional
linear regression model is novel. We illustrate the impacts of measurement
error and covariance structures on the estimated parameters through
simulation studies. With the increasing use of wearable or activity monitoring
devices to study biological phenomenon in biomedical research, it is critical
that statistical methods that allow their accurate and unbiased assessments be
developed.

2. Methodology
We propose a generalized method of moments based estimator to
estimate the function-valued coefficient of the functional linear regression
model. In this setting, the outcome is scalar-valued, while the covariate, X(t) is
a function. Our proposed method requires no distributional assumptions for
the measurement errors. However, the estimation of the function-valued
coefficient depends on the assumption that an instrumental variable exists in
the data. Additionally, estimation of the covariance matrix for the
measurement error is not required for the successful implementation of our
proposed methodology. Under current functional data methodology, a naive
estimator of the coefficient would be based on the observed measures and
the outcomes, where the observed measures are treated as the true measures
for the unobservable latent covariate. The strength of our proposed estimator
is that while the function-value covariate might not be directly observed,
estimation of its effect on the response is based on its unbiased measure as
well as additional information provided in the data in the form of the
instrumental variable.

3. Result
In this section, we describe the application of our methods to the
motivating example. Students enrolled in the study were followed over an 18-
month period. The study design was a cluster randomized trial where teachers
within three schools in the College Station Independent School District were
randomly assigned to receive either the treatment (stand-biased desks) or
control (traditional desks) (Benden, 2011). The data contain measurements
obtained at baseline and at the beginning of each semester over two academic
years. An objective of the study was to investigate the relationship between
energy expenditure behaviour at baseline and the 18-month change in body
mass index (BMI) from baseline among the students. Thus, an outcome of
interest was the difference or change in BMI values from baseline to 18 months

359 | I S I W S C 2 0 1 9