Additional Proofs for Modeling Parkinson's Disease Progression from Longitudinal Voice Biomarkers A Comparative Study of Statistical and Neural Mixed-Effects Models

This is an author-posted companion note containing additional mathematical arguments related to the published article “Modeling Parkinson's Disease Progression from Longitudinal Voice Biomarkers: A Comparative Study of Statistical and Neural Mixed-Effects Models,” published in Computer Methods and Programs in Biomedicine Update, Volume 9, 2026, Article 100242. Published version DOI: 10.1016/j.cmpbup.2026.100242. This document is not the journal's official supplementary material and should not be treated as a substitute for the published article. Its purpose is to provide additional mathematical context for the empirical finding that semi-parametric mixed-effects models can be more sample-efficient than neural mixed-effects models in small-cohort longitudinal telemonitoring studies. The note focuses on a bias–variance comparison between low-dimensional statistical mixed-effects models and higher-dimensional neural mixed-effects models, with emphasis on subject-level sample size, repeated observations, generalized additive mixed models, neural mixed-effects models, and prediction risk.