  Mass spectrometry proteomics, characterized by spiky, spatially heterogeneous
functional data, can be used to identify potential cancer biomarkers. Existing
mass spectrometry analyses utilize mean regression to detect spectral regions
that are differentially expressed across groups. However, given the
inter-patient heterogeneity that is a key hallmark of cancer, many biomarkers
are only present at aberrant levels for a subset of, not all, cancer samples.
Differences in these biomarkers can easily be missed by mean regression, but
might be more easily detected by quantile-based approaches. Thus, we propose a
unified Bayesian framework to perform quantile regression on functional
responses. Our approach utilizes an asymmetric Laplace working likelihood,
represents the functional coefficients with basis representations which enable
borrowing of strength from nearby locations, and places a global-local
shrinkage prior on the basis coefficients to achieve adaptive regularization.
Different types of basis transform and continuous shrinkage priors can be used
in our framework. A scalable Gibbs sampler is developed to generate posterior
samples that can be used to perform Bayesian estimation and inference while
accounting for multiple testing. Our framework performs quantile regression and
coefficient regularization in a unified manner, allowing them to inform each
other and leading to improvement in performance over competing methods as
demonstrated by simulation studies. We also introduce an adjustment procedure
to the model to improve its frequentist properties of posterior inference. We
apply our model to identify proteomic biomarkers of pancreatic cancer that are
differentially expressed for a subset of cancer patients compared to the normal
controls, which were missed by previous mean-regression based approaches.
Supplementary materials for this article are available online.
