| Summary: | Regression models with a scalar response and
a functional predictor have been extensively
studied. One approach is to approximate the
functional predictor using basis function or
eigenfunction expansions. In the expansion,
the coefficient vector can either be fixed or
random. The random coefficient vector
is also known as random effects and thus the
regression models are in a mixed effects
framework.
The random effects provide a model for the
within individual covariance of the
observations. But it also introduces an
additional parameter into the model, the
covariance matrix of the random effects.
This additional parameter complicates the
covariance matrix of the observations.
Possibly, the covariance parameters of the
model are not identifiable.
We study identifiability in normal linear
mixed effects models. We derive necessary and
sufficient conditions of identifiability,
particularly, conditions of identifiability
for the regression models with a scalar
response and a functional predictor using
random effects.
We study the regression model using the
eigenfunction expansion approach with random
effects. We assume the random effects have a
general covariance matrix
and the observed values of the predictor are
contaminated with measurement error.
We propose methods of inference for the
regression model's functional coefficient.
As an application of the model, we analyze a
biological data set to investigate the
dependence of a mouse's wheel running
distance on its body mass trajectory.
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