Function-on-Scalar Quantile Regression with Application to Mass Spectrometry Proteomics Data

TL;DR

This paper introduces a Bayesian quantile regression framework for functional data, specifically applied to mass spectrometry proteomics, to detect biomarkers that are missed by mean regression due to heterogeneity.

Contribution

It develops a unified Bayesian approach using asymmetric Laplace likelihood, basis representations, and shrinkage priors for functional quantile regression, with scalable inference and multiple testing correction.

Findings

Improved detection of cancer biomarkers missed by mean regression.

Effective regularization and adaptive shrinkage in functional quantile regression.

Successful application to pancreatic cancer proteomics data.

Abstract

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.