Regression Algorithms in Normative Modeling for Neuroimaging

Normative modeling constructs population-level reference trajectories of brain descriptive parameters against clinical covariates, enabling individual-level deviation detection through z-scores. The regression algorithm at its core --- estimating both the conditional mean and the conditional variance --- sets a ceiling on the accuracy of every downstream z-score. As neuroimaging enters the big-data era, regression methods face escalating demands in computational scalability, in robustness to outliers, and in geometric fidelity for non-Euclidean descriptors on Riemannian manifolds. This review provides a systematic survey of regression algorithms within the normative-modeling framework, covering parametric, semiparametric, nonparametric, Bayesian, and deep-learning approaches. To enable horizontal comparison rather than family-by-family enumeration, we organize the exposition around three challenge axes --- scalability, robustness, and geometry --- and place four representative normative methods on a common location--scale template that makes explicit how their differing regression mechanics propagate into z-score quality. Building on this framework, a reproducible simulation benchmark on synthetic heteroscedastic data contrasts seven regression algorithms in statistical quality and outlier resilience, complemented by a 2-D real-data case study on the HarMNqEEG Cuba1990 healthy reference, from which practical method-selection guidance emerges. We further detail M-estimation-based robust extensions and intrinsic regression on symmetric positive-definite manifolds, before closing with open questions on adaptive smoothing-parameter selection, higher-order moment estimation, and non-Euclidean scalability. Together, these threads position the regression algorithm as a first-class design choice in normative modeling --- one whose principled treatment is essential for translating population-scale neuroimaging into reliable individual-level clinical inference.