Annals of Applied Statistics

Function-on-scalar quantile regression with application to mass spectrometry proteomics data

Yusha Liu, Meng Li, and Jeffrey S. Morris

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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 interpatient 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 Material for this article is available online.

Article information

Ann. Appl. Stat., Volume 14, Number 2 (2020), 521-541.

Received: March 2019
Revised: October 2019
First available in Project Euclid: 29 June 2020

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Digital Object Identifier

Bayesian hierarchical model functional data analysis functional response regression global-local shrinkage proteomic biomarker quantile regression


Liu, Yusha; Li, Meng; Morris, Jeffrey S. Function-on-scalar quantile regression with application to mass spectrometry proteomics data. Ann. Appl. Stat. 14 (2020), no. 2, 521--541. doi:10.1214/19-AOAS1319.

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Supplemental materials

  • Supplement A to “Function-on-scalar quantile regression with application to mass spectrometry proteomics data”. We provided the pancreatic cancer mass spectrometry dataset and the related code, which are also available at
  • Supplement B to “Function-on-scalar quantile regression with application to mass spectrometry proteomics data”. We provided details of the MCMC sampling procedure, additional results of data application and implementation details of the “FDboost” package.