  In this paper, we develop a quantile functional regression modeling framework
that models the distribution of a set of common repeated observations from a
subject through the quantile function, which is regressed on a set of
covariates to determine how these factors affect various aspects of the
underlying subject-specific distribution. To account for smoothness in the
quantile functions, we introduce custom basis functions we call
\textit{quantlets} that are sparse, regularized, near-lossless, and empirically
defined, adapting to the features of a given data set and containing a Gaussian
subspace so {non-Gaussianness} can be assessed. While these quantlets could be
used within various functional regression frameworks, we build a Bayesian
framework that uses nonlinear shrinkage of quantlet coefficients to regularize
the functional regression coefficients and allows fully Bayesian inferences
after fitting a Markov chain Monte Carlo. Specifically, we apply global tests
to assess which covariates have any effect on the distribution at all, followed
by local tests to identify at which specific quantiles the differences lie
while adjusting for multiple testing, and to assess whether the covariate
affects certain major aspects of the distribution, including location, scale,
skewness, Gaussianness, or tails. If the difference lies in these commonly-used
summaries, our approach can still detect them, but our systematic modeling
strategy can also detect effects on other aspects of the distribution that
might be missed if one restricted attention to pre-chosen summaries. We
demonstrate the benefit of the basis space modeling through simulation studies,
and illustrate the method using a biomedical imaging data set in which we
relate the distribution of pixel intensities from a tumor image to various
demographic, clinical, and genetic characteristics.
