Empirical-Bayes and Bayesian Hierarchical Modelling for Missingness and Differential Expression in Proteomics

Mass spectrometry-based label-free proteomics data often suffer from missing values, especially for low-abundance proteins or when a protein is completely absent in one condition. This makes it challenging to estimate fold changes reliably and perform downstream analyses. Traditional imputation methods often show inconsistent performance across datasets and they typically treat imputed values as fixed rather than uncertain. This can lead to an underestimation of variability in downstream analyses. To address those problems, we present a hierarchical model that accounts for both observed protein intensities and patterns of missing data. Missing values are modelled as left-censored observations below protein-specific detection limits, reflecting the limited sensitivity of the instrument, or as missing with a probability that depends on the latent intensity (intensity-dependent missingness). Our proposed model captures structure at multiple levels: intensity-level measurements, group-level effects (e.g., experimental conditions), and protein-level variation. To estimate model parameters, we employ an empirical Bayes (EB) framework to infer hyperparameters across proteins and use Markov Chain Monte Carlo (MCMC) methods to sample parameters from the posterior distribution. Our pipeline avoids the need for imputing missing values and is designed to produce more reliable fold-change estimates and uncertainty measures. We benchmark our method against existing approaches and demonstrate that it provides accurate, stable, and robust estimates for differential expression analysis.