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CPS2292 Roger S. Zoh, PhD et al.

Figure 1: Plots of observed energy expenditure W(t) and mean step counts
M(t) vs. time for all subjects at baseline from our motivating example. The
figure confirms that the relationship between W(t) with time is nonlinear. In
this setting W(t) is assumed to be an unbiased measure of X(t), while M(t) is
an instrumental variable for X(t).

Impact of measurement error on the analyses
In addition to our method of moments-based instrumental variable
estimator, we also obtained naive estimators of the effects of energy
expenditure on BMI see Figure 2. As illustrated in both sets of analyses, the
approaches obtained without accounting for measurement error appeared
notably different from the estimators obtained from the instrumental variable
based approaches. Based on Figure \ref{fig3}, the impacts of measurement
error on both sets of analyses depended on time. While it is well known in
simple linear regression models that the effects of measurement on estimation
is to attenuate its effects towards zero, its impact in this functional linear
regression setting is more complex. For both sets of analyses, we found that
the measurement error adjusted function-valued coefficients tended to be
larger than the naive coefficient. However, the naive estimate of $\beta(t)$ at
baseline was found to be larger than the measurement error adjusted at the
beginning and the end of the observational period.

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