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Normative modeling of neuroimaging data using generalized additive models of location scale and shape

Richard DingaiD, Charlotte FrazaiD, Johanna BayeriD, Seyed Mostafa KiaiD, Christian F. BeckmanniD, André F. MarquandiD

Abstract

Normative modeling aims to quantify the degree to which an individuals brain deviates from a reference sample with respect to one or more variables, which can be used as a potential biomarker of a healthy brain and as a tool to study heterogeneity of psychiatric disorders. The application of normative models is hindered by methodological challenges and lacks standards for the usage and evaluation of normative models. In this paper, we present generalized additive models for location scale and shape (GAMLSS) for normative modeling of neuroimaging data, a flexible modeling framework that can model heteroskedasticity, non-linear effects of variables, and hierarchical structure of the data. It can model non-Gaussian distributions, and it allows for an automatic model order selection, thus improving the accuracy of normative models while mitigating problems of overfitting. Furthermore, we describe measures and diagnostic tools suitable for evaluating normative models and step-by-step examples of normative modeling, including fitting several candidate models, selecting the best models, and transferring them to new scan sites.

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Claude Opus 5.5 · xhigh effort · 7 Nov 2026 · from the full text of version 1 (14 Jun 2021) · CC BY 4.0

Generalized additive models for location, scale and shape (GAMLSS) using SHASH distributions beat Gaussian brain normative models on log score and calibration in most of 150 UK Biobank cortical-thickness regions.

Per the authors, earlier normative-modeling methods each cover only part of the need: the Kia et al. and Bayer et al. models need slow posterior sampling and assume Gaussian residuals, while quantile regression gives discrete, unstable, possibly crossing centiles. GAMLSS combines non-linear effects, heteroskedasticity, site terms and non-Gaussian shapes in one framework, and the paper adds evaluation tools (proper scoring rules, pseudo-z-score calibration) usable for any normative model.

Key results

  1. ShownA non-Gaussian SHASH model beat a Gaussian model in most of 150 FreeSurfer cortical-thickness regions of interest (ROIs) in UK Biobank. It won on both test-set logarithmic score and calibration of pseudo-z-scores, including in measures that look normally distributed.

    Used a 70%/30% train/test split. Each ROI got a Bland-Altman comparison of test logarithmic scores, plus the Shapiro-Wilk W, skewness and kurtosis of randomized residuals (Figure 5). The text gives only the direction ('vast majority' of ROIs), with no counts or effect sizes. · Model fit comparison; Conclusion; Figure legends

  2. ShownThe choice of distribution matters most in the tails. For median cortical thickness, which looks roughly normal, a subject at z = 2.8 under the Gaussian model is at z = 2.2 under the non-Gaussian model.

    The comparison comes from the UK Biobank median cortical thickness models (Figure 4b). The authors say the gap may matter clinically and builds up across imaging-derived phenotypes (IDPs), though it affects only a minority of subjects. · Conclusion; 1. modeling the age effect

  3. ShownIn a multisite collection of public studies, modeling scan-site effects on both location (mu) and scale (sigma) fit held-out subjects better than ignoring site. It also left smaller site differences in test pseudo-z-scores.

    Data: median cortical thickness (FreeSurfer 5.3 and 6.0). Five variants: no site term (M3a); site on mu as a fixed (M3b) or random (M3c) effect; site on mu and sigma as fixed (M3d) or random (M3e) effects. Judged by test deviance and pseudo-z-scores split by site (Table 3). The text gives no numbers and notes that some site effect will remain by chance. · 3. modeling site effects; Tables

  4. ShownFor median cortical thickness in UK Biobank women, the age effect was slightly non-linear, with mild heteroskedasticity and a non-Gaussian shape. Heteroskedastic SHASH models fit best. The simpler M1d was kept because letting shape also depend on age (M1e) barely changed predictions.

    Five candidates of rising complexity: linear Gaussian (M1a), non-linear homoskedastic and heteroskedastic Gaussian (M1b, M1c), then SHASH (M1d, M1e). They were compared on a 70%/30% split by test-set deviance, training Akaike/Bayesian information criteria (AIC/BIC; Table 1) and wormplots (Figure 4). The text gives no table values. · 1. modeling the age effect; Figure legends; Tables

  5. SuggestedSex had little effect on median cortical thickness. Separate sex intercepts, sex-specific age trajectories or fully separate models (M2b to M2d) added little over a model ignoring sex (M2a), so sex was left out of the site analysis.

    Compared with AIC/BIC-derived measures and test-set fit in UK Biobank men and women (Table 2), but the text gives no values. Only one phenotype was tested, and the authors expect larger sex differences for volumetric measures. · 2. modeling the sex effect; Conclusion; Tables

Methods, data and reproducibility

GAMLSS (penalized-likelihood additive models in which location, scale and shape of a SHASH distribution each depend on smooth or random effects of age, sex and scan site, fitted with mgcv) is applied to cortical thickness. It also gives diagnostics (log score, mean standardized log loss, censored likelihood, pseudo-z-score calibration, Akaike weights) and an offset refit for new sites.

  • UK Biobank regional cortical thickness, median thickness (women only for the age models, both sexes for the sex models), 70%/30% train/test split · human (UK Biobank participants; men and women)
  • UK Biobank FreeSurfer cortical thickness in 150 regions of interest, 70%/30% split, used for the Gaussian vs SHASH comparison · human (UK Biobank participants) · n = 150 ROIs
  • Combined multisite dataset from publicly available studies (as in Kia et al. 2020), median cortical thickness from FreeSurfer 5.3 and 6.0 · human (text refers to people scanned at different sites)
  • Simulated data released with the code so the analyses can be run without access to the real datasets
Methods & tools
GAMLSS, mgcv, R (offset function), Sinh-Arcsinh (SHASH) distribution, FreeSurfer 5.3, FreeSurfer 6.0, Shapiro-Wilk test, worm plots, AIC / BIC / Akaike weights
Code
https://github.com/dinga92/gamlss_normative_paper
Software & environment
The multisite thickness data come from FreeSurfer 5.3 and 6.0. SHASH is the version built into the R library mgcv. No package versions or requirement files are given.
Reproducibility gaps
The real datasets are not shared with the code. They must be obtained from their source repositories, some of them open and some only by application. Simulated data are provided so the analyses can still be run.
Limitations (6)
  • Relies on a parametric family for the response distribution, which may not fit the data. Multimodal distributions in particular may be poorly approximated. · stated by the authors
  • Assumes distribution parameters change smoothly with the inputs, so a sudden shift in the distribution is not captured unless the model is adjusted. · stated by the authors
  • The added flexibility can overfit, so the approach is not suited to small datasets. · stated by the authors
  • Evaluation caveats: likelihood-based scores are sensitive to outliers (removing or Winsorizing them underestimates variance and kurtosis). In the authors' experience, AIC, BIC and GAIC do not reliably pick the best of complex GAMLSS models. · stated by the authors
  • The text says GAMLSS is faster than the Kia et al., Bayer et al. and Fraza et al. methods but reports no runtime or accuracy comparison against them. · noted in this summary
  • Empirical tests use only cortical thickness, from UK Biobank and one multisite collection. The new-site transfer method is shown only as a diagram (Figure 3), with no quantitative test of transferred models in the text. · noted in this summary

Who should read:Read this if you build brain normative or centile models and want a practical recipe for non-linear age effects, heteroskedasticity, scan-site effects and non-Gaussian distributions, plus metrics for judging calibration.

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