Protocol Update: The Normative Modelling Paradigm for Computational Psychiatry

Normative Modelling (’brain growth charting’) is now a well-established method for computational psychiatry and involves charting centiles of variation across a population in terms of mappings between biology and behavior, providing statistical inferences at the level of the individual. These models have helped the field to move away from case-control analysis toward individual-level analy- sis. Correspondingly, normative modelling has now been applied to chart brain development and ageing in many populations and has been used to quantify indi- vidual deviations across various neurological and psychiatric conditions. This has been supported by large-scale models that are openly accessible for diverse brain imaging modalities. As normative modelling continues to grow, several recent methodological developments, such as non-Gaussian models, longitudinal models, and federated learning, have been implemented in different software tools, includ- ing the Predictive Clinical Neuroscience toolkit (PCNtoolkit). In this protocol update, we provide: (i) a revised overview of this methodological landscape; (ii) an update to our 2022 standardised analytical protocol for normative modelling of neuroimaging data, including options for federated and longitudinal normative models; (iii) practical guidance suited to both novice and experienced practi- tioners supported by open-source code examples implemented in the refactored version of PCNtoolkit; and (iv) updated models for cortical thickness, volumetric data, diffusion-weighted imaging and longitudinal data for use by the community. Keywords:Normative Modeling, Computational Psychiatry, Precision Medicine, Machine Learning 1 Introduction 1.1 Development of the protocol A major goal in the field of computational psychiatry is to develop reliable biomarkers for psychiatric disorders[1]. Traditional case-control approaches largely fail to provide meaningful insight into psychiatric disorders because the rigid and categorical assump- tions of the methodology do not match the continuous and heterogeneous nature of the biology [2]. Normative Modelling (NM) has emerged as a mature statistical method for analyz- ing heterogeneity in clinical cohorts[3]. The NM paradigm moves away from traditional case-control approaches by mapping trajectories of healthy development – anatomical, neurological, behavioral – from a large reference dataset, and quantifies how individ- uals deviate from these norms. This is typically achieved by modelling biological or other response variables (e.g., regional brain volume) as a function of certain covari- ates (e.g., age, sex, total brain volume), thereby rendering the deviation space mostly independent of these variables. The approach has been widely adopted in recent years. Large-scale normative models have been deployed [4–6] and applied to parse heterogeneity in several clinical applications [7, 8]. Currently, various methodological innovations have been proposed to improve the modelling of non-Gaussian data [9], longitudinal data [10–12], model 2 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint comparison [13], and federated learning [14]. These innovations allow researchers to more accurately estimate the mean and centiles of the population distribution, enable the tracking of deviation scores over time, compare models to find which one gives the most accurate deviation scores and update model parameters without sharing data. The Predictive Clinical Neuroscience toolkit (PCNtoolkit) is an open-source Python package for NM, which was developed to fulfill two critical needs. First, to provide a flexible and user-friendly software package for normative modelling, specif- ically tailored to neuroimaging data, and second, to ease reproducibility efforts. The initial distributions of the PCNtoolkit were published under the name nispat, with the first version of the PCNtoolkit being released as version 0.15 in 2020. A protocol was published for version 0.20 of the PCNtoolkit in 2022 [15], and continued develop- ment led to the last release within this generation, version0.35 in April 2025. Ongoing

development and an increased adoption of the NM paradigm demanded a major refactoring of the PCNtoolkit to keep it usable and maintainable. This led to the PCNtoolkit version 1.x.x, with an easier interface to familiar NM techniques, refinement of model estimation and inference techniques in addition to new features such as longitudinal modelling and federated pipelines. The present protocol provides an update to the 2022 protocol [15] and: (i) pro- vides an overview of the recommended workflow for normative modelling, illustrated with examples of the main methods implemented in the latest PCNtoolkit version 1.x.x, and (ii) accommodates the updated syntax and also showcases new features, including new estimation algorithms for modelling non-Gaussian distributions [9] and longitudinal data [12]. In more detail, this protocol provides updated code examples for the methods covered in the 2022 protocol, and extends the protocol by covering longitudinal normative modelling, hierarchical Bayesian regression (HBR), model com- parison using the Watanabe–Akaike information criterion (WAIC) [13], data synthesis, data harmonization, post-hoc-analysis suggestions, and federated learning methods; transferring, extending, and merging models. See figure 1 for an overview of the protocol. Rather than a comprehensive PCNtoolkit guide, this protocol provides general recommendations to brain charting through normative modelling, accompanied by code examples written for the PCNtoolkit version 1.x.x. For a complete reference to the PCNtoolkit, see the PCNtoolkit documentation and additional examples in the associated code repository. Moreover, the general workflow is also applicable to other software implementations [5, 6, 16, 17]. The protocol is structured into four workflows. The first workflow presents the basic sequential process that can be followed to fit and use NM with the PCNtoolkit. Next, we provide a set of secondary workflows that can be applied in any order, but require a fitted model (i.e., completed main workflow). The subsections on federated learning

require one or more pre-estimated models, and in some cases an independent training dataset. Finally, the final section focuses on community engagement. In more detail, insection 1, the reader is guided through a complete workflow of creating a set of normative models with PCNtoolkit. First, for completeness, we pro- vide a summary of the NM paradigm, what the goals of NM are, and how the NM paradigm can be applied to neuroimaging data. Then, we walk through the installation 3 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint Fig. 1: Overview of the protocol. and verification of the PCNtoolkit. This is followed by a step-by-step guide through the process of creating a normative model. The user is guided through the selection and filtering of a dataset on the basis of clinical and demographic information, data missingness, research question, and data modality. Then, we discuss Quality Control (QC) procedures and demonstrate how to use the PCNtoolkit to perform some basic data preprocessing steps, such as outlier detection and cleaning, missingness detection and handling, train-test splitting, and data scaling. We select and configure the algo- rithm and model parameters, which depend on the data regime, model requirements, and the research question. We fit the model on the selected and prepared dataset, and discuss the fitting of NMs in parallel on a compute cluster, and the evaluation of models in a K-Fold cross-validation setting. We then verify the output and discuss the interpretation of the resulting plots, results, and model evaluation metrics. In section 2, we cover a variety of post-hoc analysis methods and extensions to normative modelling, including the following topics: longitudinal NM using Z- diff scores and the velocity modelling framework, HBR model comparison using the 4 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint WAIC [13], and how to use this framework for hypothesis testing, data synthesis, data harmonization, and other post-hoc-analysis ideas. In section 3 we cover the three strategies for federated learning that are imple- mented in the PCNtoolkit: i) model transfer, which allows for transferring a reference model to a new local dataset (resulting in a smaller local model which can be used on the local data), ii) model extension, which allows for extending a reference model with data from a new dataset (resulting in a larger reference model which can be used on both the new and the original dataset; and iii) model merging, which allows for merg- ing two or more fitted models into a single model. We discuss the differences between these strategies and when to use each one. In section 4, we cover the community engagement aspect of the PCNtoolkit. We mention our GitHub repository, which contains all the source code, and how to interact with the community. We also mention our contribution guidelines and how possible contributions could look like. 1.2 Applications of Normative Modelling NM is a general framework for deriving subject-level deviation scores from a reference population, and as a consequence, it can be applied to a wide range of (medical) data and research questions. Here we list a few: 1. Derive a normative model from a reference population of healthy individuals, and draw further conclusions about the population based on the estimated model parameters and deviation scores. For example, in Gaiser et al. [18], a large-scale nor- mative model is fit to the cerebellum to understand cerebellar development across the adolescent period. 2. Transfer a reference NM trained on a large, healthy population to a new, smaller local dataset from a different population, and use it to derive biomarkers from the local clinical samples, for example, as is done in the context of psychosis in Worker et al. [19]. 3. Use normative models fit to cross-sectional data to identify centile crossings in longitudinal data, as is done in the context of schizophrenia in Rehak Buckova et al. [11] and in dementia in Bayer et al. [12]. 4. Use z-scores as features in downstream analysis, for example, as features in supervised [20] or unsupervised [21] machine learning models. 1.3 Comparison to other Methods When NM is viewed as a framework for deriving centiles of variation from a refer- ence population, a wide range of statistical methods can serve as its foundation. For example, the original Gaussian processes (GP) approach for NM [3, 22], but also other algorithms can be used, including Bayesian Linear Regression (BLR) [23], Hierarchi- cal Bayesian Regression (HBR) [24, 25] and generalised additive models of location, scale and shape (GAMLSS) [5, 26]. Future developments may expand the range of

that can be seen as forms of NM even further. We will not compare these specific methods here because all methods are highly flexible, depending on different parameterisations (e.g., the specific distributional form for the centiles, underlying 5 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint basis for modelling non-linearity, etc.) and whether the model is probabilistic or not. These modelling choices are more important than the particular method chosen, and also make direct comparison difficult. Rather, we believe that the most important aspect of NM is not the underlying (regression) method itself, but the framework that it provides for analyzing deviation scores from a reference population. The important feature of normative modelling is that it estimates centiles of variation in addition to the mean. In contrast, a regression model that is fit to data but does not pro- vide centiles of variation, such as a simple linear regression model, can be seen as an approximation to NM when viewed in this light. However, such simple methods do not properly account for different variance components [7] and may not provide a good fit of the estimated centiles to the underlying data, which can bias downstream inferences [9, 27]. Moreover, it is increasingly recognised that the choice of reference cohort is of equal importance to the choice of algorithm, because of the potential for introducing demographic or racial bias [28, 29]. 1.4 Experimental Design Several important design decisions should be considered before applying this protocol. The most fundamental choices in designing a normative modelling analytic strategy are the choice of response variables and covariates. Whilst many normative mod- els predict brain-derived measures based on age and sex, whilst accounting for site (e.g., using batch effects), this is not the only way such models can be employed. For example, in Marquand et al. [3] a normative model was constructed to predict reward-related brain activity on the basis of delay discounting scores, which measure individual variability in reward sensitivity and are sometimes taken as a proxy for impulsivity. Another important design consideration is the choice of the reference cohort and data partitioning strategy (i.e., how to divide the available data into training and test sets), which directly influences how the deviations from such a model should be interpreted. Whilst these aspects have been covered elsewhere [7, 15], in brief, the ref- erence cohort should be chosen to be as representative as possible of the population of study interest. If a healthy cohort is chosen, then deviations should be understood with respect to this cohort. Alternatively, a population-based cohort could be used (i.e., including individuals with brain disorders), but then the prevalence of each sub- group in the reference cohort (or training set) should be taken into consideration when interpreting the target cohort (or test set), where the prevalence may be different. In many cases (for example, for rare conditions), it may be desirable to ensure that a sufficient number of individuals having the phenotype of interest are retained in the test set to maximise the possibility of detecting downstream effects. In other words, the training and test sets should be stratified for the clinical labels. 1.5 Regulatory Approvals If the goal is to fit a normative model from scratch, the main regulatory considera- tion is permission to use the data used to fit the model, because in most instances it is necessary to aggregate cohorts to provide good coverage of the lifespan. This 6 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint can either be performed using publicly available datasets or according to appropriate

transfer agreements (MTAs). If the data cannot be shared, federated learning techniques can be employed to estimate normative models without the need to share data [14, 24]. Alternatively, if the goal is to transfer a pre-fit model to a new dataset, such approval is not necessary because this is done via the model coefficients, which are summary statistics that are no longer traceable to any individual participant. 2 Materials 2.1 Hardware Computing infrastructure required to run this protocol: a Mac, Linux or Windows with installed Windows Subsystem for Linux (WSL) computer with enough space to store the derived phenotypes from imaging data of the train and test set. In addition, an HPC infrastructure (Slurm or Torque) can be beneficial to parallelize fitting large numbers of normative models (e.g., for high-resolution data) The code examples in this protocol have been tested on a MacBook Pro with an M3 Chip and 18GB of RAM. The dataset used in these experiments is rather small, so everything should be possible to run on any computer with a similar or lower configuration. Experiments using a larger dataset will require more RAM, and we recommend using a compute cluster for those cases (See section 3.2.9 for more details about fitting models in parallel using the PCNtoolkit). The resources requested for the jobs in these experiments are always 16 GB of RAM and 4 cores. Considering the available hard disk space, we recommend having at least 10 times the size of the dataset in free hard disk space. This is because the PCNtoolkit stores the predicted centiles (5 centiles per subject per feature), the Z-scores and the log- probability of the observed values (both 1 per subject per feature), which all together add up to 7 times the size of the original dataset. The saved model files and the plots, which are generated by the PCNtoolkit, also take up additional space. However, these functionalities can be disabled if the user desires to save disk space. 2.2 Software The protocol requires Python 3.12 or a later installation and a few core scientific packages. Python can be installed from the official Python website (https://www.python.org/downloads/) or via Anaconda (https://www.anaconda.com/download ). We recommend running the code examples within an isolated virtual environment, which can be created either with conda or the built-in venv module, to ensure dependency consistency across platforms. Detailed installation instructions are available from the official documentation for each tool: • conda: https://docs.anaconda.com/anaconda/install/ • venv: https://docs.python.org/3/tutorial/venv.html The main package used in this protocol is PCNtoolkit (current ver- sion 1.x.x), which can be installed via pip following the instructions at https://pcntoolkit.readthedocs.io/en/latest/pages/quickstart.html. 7 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint Software environment In this manuscript, all analyses were performed in Python 3.12 using the follow- ing core packages: PCNtoolkit 1.1.2; PyMC 5.25.1; ArviZ 0.22.0; PyTensor 2.31.7; NumPy 2.2.6; SciPy 1.16.1; scikit–learn 1.7.2; pandas 2.3.2; xarray 2025.9.0; xar- ray–einstants 0.9.1; matplotlib 3.10.6; h5py 3.14.0; and h5netcdf 1.6.4. Optional components used where noted include nutpie 0.15.2 (for accelerated sampling) and seaborn 0.13.2 (for figure styling). Random seeds were fixed globally using np.random.seed(42) and pymc.set tt rng(42)to ensure bitwise-reproducible sampling, data splitting, and posterior predictive simulations. All scripts and environment specifications are available in the accompanying repository, allowing for exact replication of the computational environment. 2.3 Data The current protocol uses the FCON1000 dataset, which can be downloaded from https://fcon 1000.projects.nitrc.org/. 3 Procedure 3.1 Installation and verification of the PCNtoolkit We recommend using a virtual environment to run the PCNtoolkit. This can be done using a Conda environment (conda) or a Python virtual environment (venv). The PCNtoolkit can be installed using pip: pip install pcntoolkit And the installation can be verified by running the following command: python -c "import pcntoolkit; print(pcntoolkit.__version__)" 3.2 Main workflow 3.2.1 Data Selection Data inclusion criteria A large reference – or normative – set is required to train a well-calibrated normative model. A generalizable normative model should incorporate two types of uncertainty: epistemic and aleatoric [7]. Epistemic uncertainty refers to the uncertainty in our knowledge, while aleatoric uncertainty arises from the natural variation in the popula- tion. The latter is what we aim to capture with our normative models; to model it as accurately as possible, the former has to be minimized. Because epistemic uncertainty decreases with the size of the training data, it is desirable to have a large reference data set. It is also important to consider the range of the covariates [30]. For example, since many datasets have a limited age range, which can introduce collinearity with 8 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint batch or site effects, it is desirable to have overlap (i.e., multiple datasets spanning the same age range) to assist in disentangling these effects. What can be considered a reference set depends on the research question. If the goal is to parse deviations from a healthy population, the reference set should include only healthy participants. If the goal is to parse heterogeneity within a clinical cohort, one could choose to use a set of participants with that particular condition as the

set. For instance, if one is interested in parsing heterogeneity within a clinical cohort of patients with major depressive disorder (MDD), the reference set could include MDD patients who have responded well to medication, and contrast them with non-responders. We will use cortical thickness imaging-derived phenotypes (IDPs) from several different datasets, which we list below. The datasets are all pre-processed and included in this repository. All preprocessing was done using Freesurfer with the recon-all command. We extracted the mean cortical thickness values following the Destrieux parcellation [31]. We use the FCON1000 dataset, which is freely available at theFCON1000 website. In addition, we will use several clinical datasets from the OpenNeuro portal: 1. ds003568 – Contains data from 49 subjects (31F, 18M), aged between 12 and 19 (mean 15.9, std 1.8). Clinical labels are Healthy Control (HC, 20) and Major Depressive Disorder (MDD, 29) [32]. 2. ds003653 – Contains data from 73 subjects (56F, 17M), aged between 19 and 44 (mean 29.1, std 6.6). Clinical labels are HC (39) and MDD (33) [33]. import pandas as pd fcon_df = pd.read_csv( "https://raw.githubusercontent.com/predictive-clinical-neuroscience/ ,→pu25_code/refs/heads/main/data/fcon1000.csv" ) openneuro_df = pd.read_csv( "https://raw.githubusercontent.com/predictive-clinical-neuroscience/ ,→pu25_code/refs/heads/main/data/openneuro.csv" ) data = pcn.load_data("data/fcon1000.csv") Covariates, batch effects, and response variables As we will see, the input data generally consist of several covariates, batch effects, and response variables. The covariates are typically continuous variables, like age and height, but can also be numerically encoded categorical variables (for example, using one-hot-encoding), like sex. The batch effect dimensions are categorical variables, for example, scan site or scanner brand/model. The response variables are continuous variables, like brain volumes or brain activity patterns. All these variables must be available for all subjects in the reference set. Normative models made using PCNtoolkit version1.x.xconsist of one or more linear regressions on the selected covariates, together with a mechanism to handle 9 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint batch effects, the details of which depend on the model type. The choice of covariates and batch-effect variables depends on the research question, but they should gener- ally include all factors known to influence the measurements and be available in all datasets, such as age, sex, sleep quality, or site. Batch effects are used to group sub- jects together in the same way that random effects are used in classical multilevel modelling. Typically, batch effects are used to model site or scanner effects, but they could also be applied to model ancestry or imaging field strength. Whilst batch effects can be encoded as dummy encoded covariates, this is less flexible and is equivalent to a fixed intercept model (i.e., only allowing shifts in the intercept). In contrast, using batch effects allows the model to estimate separate slope, intercept and basis function parameters for each site with partial pooling across levels of the batch effects (see Kia et al. [14, 24]. The primary goal in normative modelling is to compute deviation (Z) scores that are as independent as possible from these covariates and batch effects. For example, if the aim is to study the effect of meditation on brain activity, the normative model should include all relevant covariates—such as age, sex, sleep quality, and scanner site. By modelling and removing the variance explained by these known factors, the resulting deviation scores reflect variability that is most likely related to effects of interest rather than demographic or technical influences. When selecting covariates, care must be taken when including variables that are highly correlated with one another, as multicollinearity can inflate parameter uncer- tainty and make it difficult to interpret the contribution of individual covariates. However, this is principally relevant for applications where parameter interpretation is important. For example, if both sleep quality and stress level are included as predictors, their shared variance may obscure distinct effects on brain activity. subject_id = "sub_id" covariates = ["age"] batch_effects = ["site", "sex"] response_variables = [ col for col in df.columns if col not in covariates + ,→batch_effects + [subject_id]] response_variables = list( filter(lambda x: df[x].var() > 0, response_variables) )# Remove variables with no variance 3.2.2 Data Preparation Preparing the data for modelling involves a sequence of preprocessing steps designed to ensure data quality and reproducibility. Note that standard neuroimaging preprocess- ing steps such as spatial normalisation and/or cortical surface reconstruction should be applied first. All subjects in a normative modelling analysis need to be in the same space. Steps specific to normative modelling include handling missing values and outliers, combining multiple data sources, splitting the data into training and test sets, and rescaling covariates and response variables (e.g., via standardisation). All procedures 10 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint are implemented in the PCNtoolkit, with the NormData class providing dedicated functionality for data cleaning, harmonization, and preparation. PCNtoolkit automatically excludes subjects with missing values and identifies out- liers based on feature-wise Z-scores. The stringency of this filter is controlled via the z thresholdargument. In this tutorial, we setz threshold = 10 to illustrate the proce- dure while avoiding unnecessary data loss; however, for practical analyses, a different value may be selected to balance sensitivity and specificity. However, care needs to be taken not to be overly aggressive with outlier removal as this may result in truncating the empirical distribution, thereby making it difficult to model with a continuous dis- tribution. If a customized approach to missingness or outlier handling is desired, data may also be preprocessed externally before being passed to PCNtoolkit. The train–test split is performed using thetrain test splitfunction from scikit-learn, with automatic stratification on batch-effect variables (e.g.,siteand sex) to maintain balanced distributions and to avoid demographic or acquisition drift between subsets. Data scaling is applied internally, and the corresponding scaling coefficients are stored within the fitted model object to ensure that identical trans- formations are applied to all future data during prediction, or when transferring the model. from pcntoolkit import NormData import numpy as np reference_norm_data = NormData.from_dataframe( name="fcon1000", dataframe=df, covariates=covariates, batch_effects=batch_effects, response_vars=response_variables, subject_ids=subject_id, remove_Nan=True, remove_outliers=True, z_threshold=10, ) 3.2.3 Combining Different Data Sources In many cases, the reference data are supplemented with additional cohorts from other sources. Here, we demonstrate how to combine the FCON1000 dataset with clinical datasets from OpenNeuro. The reference dataset has already been loaded above; we now load the additional data into NormData objects, merge the healthy controls with the reference set, and retain the patient subset for subsequent analysis. # Split into healthy and patient groups healthy_idx = openneuro_df["group"] == "HC" healthy_group = openneuro_df[healthy_idx] patient_groups = openneuro_df[~healthy_idx] # Load the healthy and patient data 11 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint healthy_clinical_data = NormData.from_dataframe( name="openneuro_healthy", dataframe=healthy_group, covariates=covariates, batch_effects=batch_effects, response_vars=response_variables, remove_Nan=True, remove_outliers=True, ) patient_clinical_data = NormData.from_dataframe( name="openneuro_patient", dataframe=patient_groups, covariates=covariates, batch_effects=batch_effects, response_vars=response_variables, remove_Nan=True, remove_outliers=True, ) # Merge the healthy data with the reference data norm_train = reference_norm_data.merge(healthy_clinical_data) # Check for schema compatibility before merging with the patient data if not norm_train.check_compatibility(patient_clinical_data): raise ValueError("The data are not compatible.") else: print("The data are compatible.") 3.2.4 Train–Test Split The reference set is divided into a training set used for model fitting and a test set reserved for model evaluation. Assessing the model on independent data is critical to ensure accurate estimates of generalisabilty. If the goal is to provide a reference model for new data, the model can be refit to the whole dataset once generalisability has been established. A trade-off exists between model accuracy and the reliability of performance estimates: a larger training set improves model precision, whereas a larger test set yields more stable validation metrics. We recommend an 80/20, 70/30 or 50/50 train–test ratio, depending on dataset size. To obtain unbiased evaluation metrics, the test set should be statistically similar to the training set. Stratification of the train–test split ensures that the distributions of key batch-effect variables (e.g., site, sex) remain consistent across subsets. The following command performs this operation using theNormData.train test split method, which internally calls the scikit-learn function of the same name. train, test = norm_train.train_test_split( splits=(0.8, 0.2), split_names=["train", "test"], random_state=42, 12 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint ) 3.2.5 Standardization As described above, PCNtoolkit automatically stores the scaling coefficients used dur- ing feature scaling parameters within the fitted model object. Note that these are derived from the training set only. This ensures that all subsequent predictions and transferred models apply the same transformations to incoming data, eliminating man- ual preprocessing and guaranteeing consistent scaling across datasets during model development and deployment stages. 3.2.6 Fitting Models TheNormativeModelclass The central class in PCNtoolkit is the NormativeModel class. It has all the functions that are required to build and use a normative model. To construct aNormativeModel object, the user needs to provide a template regression model, which is copied and fit for each feature in the reference set. Here we use the default (unparameterized) HBR class as a template, but we will elaborate on this next. from pcntoolkit import NormativeModel, HBR model = NormativeModel( template_regression_model=HBR(), save_dir="../out/models/main_workflow_model_default_HBR", inscaler="standardize", outscaler="standardize", name="main_workflow_model_default_HBR", ) TheRegressionModelclass The current implementation of PCNtoolkit only supports the creation of univariate normative models. This means that each NormativeModel object contains a collection of RegressionModel objects. For each response variable, the NormativeModel class makes a copy of a template regression and fits it to the response variable. This template is what the user should provide when creating a NormativeModel. The RegressionModel type currently has two concrete implementations: Bayesian Linear Regression (BLR) and Hierarchical Bayesian Regression (HBR), and both come with their strengths and weaknesses. For an in-depth coverage of both methods, including their parameters and usage, please refer to the GitHub repository. 3.2.7 Hierarchical Bayesian Regression (HBR) A Hierarchical Bayesian Regression (HBR) model assumes a hierarchical generative process for the data and performs full posterior estimation via Markov Chain Monte Carlo (MCMC). Configuring an HBR model involves specifying a likelihood function and defining priors over its parameters, which may include both fixed and random effects. 13 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint The PCNtoolkit provides several likelihood functions, including Normal, SHASHb, and Beta likelihoods, each suited to different data characteristics [9]. The Normal likelihood is used for approximately Gaussian response variables, the SHASHb likeli- hood for unbounded and asymmetric data, and the Beta likelihood for interval data bounded within a fixed range. Each likelihood has interpretable parameters, such as µ(mean),σ(scale),ϵ(skewness), and δ (kurtosis), which can each be modeled hier- archically to capture structured variation across covariates and batch effects. See [9] for more details. Once a likelihood is chosen, priors for each parameter are specified using the make priorfunction. Parameters can vary linearly with covariates, include random (batch) effects, or remain fixed across all observations. Random-effect priors are imple- mented in a non-centered hierarchical form (reparameterized) to stabilize sampling and avoid funnel pathologies during inference. In this workflow, we use the Normal likelihood, which assumes Gaussian feature distributions after preprocessing. This is appropriate for cortical thickness data, where deviations are approximately symmetric and unbounded. Bothµandσare modeled hierarchically as functions of age (covariate) with random intercepts for site and sex, allowing age effects to vary smoothly while accounting for systematic offsets across acquisition sites and sexes. Example with Normal likelihood configuration For illustration, we define a hierarchical Normal model where each observation yn is generated as: yn ∼ N (µn, σ+2 n) n ∈ {1, ..., N } (1) µn = w⊤ µ ϕµ(xn) + τµn (2) σn = w⊤ σ ϕσ(xn) + τσ (3) σ+ n = ln(1 + exp(σn/3)) ∗ 3 (4) wσ ∼ pwσ(wσ), τ σ ∼ pτσ(τσ), w µ ∼ pwµ(wµ), (5) τµn = µτµ + βX b=1 σ+ τµ b ντµ b[Bn,b] (6) µτµ ∼ pµτµ (µτµ), σ τµ b ∼ p+ στµ b (στµ b) (7) ντµ b ∼ N (ντµ b | 0vb, Ivb) (8) where: • β is the number of batch effect dimensions, i.e., for batch effects (site, sex), β = 2. • vb denotes the number of unique batch effect values of the b’th batch effect dimension. • N denotes the number of observations/datapoints. • B denotes the N × β matrix of batch effect indices. 14 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint Equations 1–8 describe the generative process implemented in the PCNtoolkit’s HBR class, where each response variable is modeled with a Normal likelihood whose mean (µ n) depends on covariates (here, age) and batch effects (here, site and sex), and whose variance (σ 2 n) depends on covariates. The following code block imple- ments this hierarchical Normal model. The mean term varies nonlinearly with age and includes site- and sex-specific intercepts, while the standard deviation term cap- tures heteroskedasticity across age. For brevity, comments have been omitted; a fully annotated version and extended examples are provided in the accompanying Jupyter notebook. from pcntoolkit import make_prior, BsplineBasisFunction, NormalLikelihood, HBR mu = make_prior( linear=True, slope=make_prior(), intercept=make_prior( random=True, mu=make_prior(dist_params=(0.0, 3.0)), sigma=make_prior( dist_params=(0.0, 3.0), mapping="softplus",# 0 mapping_params=(0.0, 3.0), ), ), basis_function=BsplineBasisFunction(basis_column=0, nknots=5, degree=3),# ,→<- Allow nonlinearity ) sigma = make_prior( linear=True, slope=make_prior(dist_params=(0.0, 5.0)), intercept=make_prior(dist_params=(1.0, 3.0)), basis_function=BsplineBasisFunction(basis_column=0, nknots=5, degree=3), mapping="softplus",# 0 mapping_params=(0.0, 3.0), ) likelihood = NormalLikelihood(mu, sigma) template_hbr = HBR( name="template_hbr", likelihood=likelihood, ) # Create a NormativeModel with the template regression model model = NormativeModel( template_regression_model=template_hbr, save_dir="../out/models/main_workflow_model_HBR", ) 15 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint This example demonstrates how hierarchical priors for µ and σ can be defined in the PCNtoolkit using B-spline basis functions and positive-domain mappings (via softplus). The non-centered parameterization and weakly informative pri- ors improve sampling efficiency and numerical stability. Detailed explanations, additional likelihoods (SHASHb, Beta), and alternative parameterizations (linear, random, fixed) are available in the accompanying Jupyter notebook. For full configuration details and extended examples, we refer the readers to the tuto- rial on the main PCNToolkit workflow at https://github.com/predictive-clinical- neuroscience/pu25 code/blob/main/notebooks/1 main workflow.ipynb. 3.2.8 Fitting the normative model to the data With the data prepared and both the template regression model and the normative model configured, we fit on the training set and generate predictions for the test set in a single call. This command estimates the hierarchical parameters, applies the trained model to the test data, and stores predictions, model artifacts, evaluation metrics, and diagnostic plots. model.fit_predict(train, test) An overview of model outputs Running the workflow writes all intermediate and final outputs in the specified output path undermodels/main workflow model HBR/. This directory contains the fitted model specifications, posterior samples, evaluation metrics, and visual diagnostics used in the subsequent sections. • model/— per-feature subfolders containing idata.nc (posterior traces and diag- nostics) and regression model.json(serialized feature-level model specification); plus a top-levelnormative model.jsonsummarizing the full multi-feature model configuration. • plots/— centile and quantile–quantile (QQ) diagnostics for both train and test sets, one set per feature. • results/— CSV outputs with all numerical results: centiles {train,test}.csv,Z {train,test}.csv,logp {train,test}.csv, andstatistics {train,test}.csv. A schematic of the default folder hierarchy: models/ |-- main_workflow_model_HBR/ | |-- model/ | | |-- / | | | |-- idata.nc | | | ‘-- regression_model.json | | |-- / | | | |-- idata.nc | | | ‘-- regression_model.json | | ‘-- normative_model.json 16 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint | | | |-- plots/ | | |-- centiles__{train,test}_*.png | | ‘-- qq__{train,test}.png | | | ‘-- results/ | |-- centiles_{train,test}.csv | |-- Z_{train,test}.csv | |-- logp_{train,test}.csv | ‘-- statistics_{train,test}.csv Each folder is automatically generated by the PCNtoolkit during model fitting and prediction. Together, these outputs contain all information needed to reproduce the diagnostics, metrics, and visualizations, which will be described in Section 3.3.1. A fully commented walkthrough of the saving conventions, file contents, and plot interpretation is provided in the accompanying Jupyter notebook. 3.2.9 Fitting Models in Parallel For large datasets (either in terms of the number of samples and response variables), fitting models sequentially can be quite time-consuming. In case of access to a high- performance computing cluster running standard workload managers such as Torque or Slurm, operations such as ‘fit predict‘ can be parallelized by using the Runner class. The Runner class will allow running the same tasks as in the previous step, but in a distributed fashion. This is a little more involved than in the previous step. For example, you need to specify the Python executable and time and memory limits for each job. You can also specify the number of batches into which to split the workload. Here we will use a Slurm cluster, using one model per batch with 12 hours max run time and 2GB memory per job. This can be achieved in the following way: import os import sys from pcntoolkit import Runner # get path to python executable venv_path = os.path.join(os.path.dirname(os.path.dirname(sys.executable))) # configure runner runner = Runner( cross_validate=False, parallelize=True, n_batches = len(reference_norm_data.response_vars), environment=venv_path, job_type="slurm", time_limit="12:00:00", memory = "2GB", n_cores=1, 17 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint log_dir="../out/models/main_workflow_model_HBR/logs/", temp_dir="../out/models/main_workflow_model_HBR/tmp/", #preamble = "", ) runner.fit_predict(model, train, train, observe=False) 3.3 Secondary workflows In this section, we describe secondary workflows in Figure 1 that can be performed after the primary normative model has been fit. 3.3.1 Model Evaluation and Calibration Model evaluation in normative modelling involves three complementary components: (i) inspection of convergence diagnostics to ensure stable inference, (ii) assessment of quantitative model performance using both mean-based and probabilistic metrics, and (iii) visualization of model calibration through posterior predictive plots. Convergence diagnostics Posterior summaries were inspected for all model parameters using the ArviZ visuali- sation framework, including slope coefficients, site- and sex-specific offsets, and scale parameters. For the Hierarchical Bayesian Regression (HBR) models used in this workflow, convergence and sampling reliability were evaluated using standard MCMC diagnostics: all parameters were required to have potential scale reduction factors ( ˆR) ≤ 1.01, bulk and tail effective sample sizes (ESS) ≥ 1000, and Monte Carlo standard errors (MCSE) below 1% of the corresponding posterior standard deviation. Chains can also be visually inspected to confirm stationary and adequate mixing across draws. These diagnostics apply specifically to HBR models, which rely on Markov Chain Monte Carlo sampling. For Bayesian Linear Regression (BLR) models—estimated via deterministic optimization rather than MCMC—different convergence criteria apply, typically based on gradient norms and optimizer tolerance; however, such models are not used in the present analysis. To illustrate the diagnostic workflow, we performed this analysis for one represen- tative feature (lh G and S frontomargin). This feature was chosen as a typical example with a moderate age effect, heteroskedasticity, and variance across sites. Because con- vergence behavior is generally consistent across features with comparable sample sizes and data characteristics, diagnostics for this representative case serve as a practical indicator of overall model stability. All convergence diagnostics were computed with ArviZ and saved per feature in the corresponding idata.nc file under ../out/models/main workflow model HBR/model//. Each file stores posterior draws and diagnostics, including ˆR, ESS, and MCSE. The aggregated diagnostic table can be reproduced directly from disk: import arviz as az 18 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint Fig. 2: Traceplots for the covariate slope parameters (slope mu) from the Normal likelihood model. Each color represents one of the MCMC chains. The dense, well- mixed traces indicate stable posterior exploration and full chain convergence. import pandas as pd from pathlib import Path model_dir = Path("../out/models/main_workflow_model_HBR/model") rows = [] for d in model_dir.iterdir(): if d.is_dir() and (d / "idata.nc").exists(): idata = az.from_netcdf(d / "idata.nc") s = az.summary(idata)# includes R-hat, ESS_bulk, ESS_tail, MCSE s["feature"] = d.name rows.append(s) diag = pd.concat(rows) diag.to_csv("../out/models/main_workflow_model_HBR/results/diagnostics_summary ,→.csv") Traceplots for the slope parameters of µ (slope mu) are shown in Fig. 2. All four chains overlap tightly and explore the posterior space evenly, with no visual signs of non-stationarity, confirming that the hierarchical regression components have converged well. Table 1: Aggregated posterior diagnostics across all parameters (74 total). Values are shown as median [IQR]. Metric Median IQR ˆR1.00 1.00–1.01 ESSbulk 4513 2176–5502 ESStail 3789 2533–4454 Posterior mean -0.11 [-0.61, 0.35] Posterior SD 0.42 [0.24, 0.92] Monte Carlo SE (mean) 0.006 [0.003, 0.026] 19 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint Quantitative model performance Model performance was then evaluated using both mean-based and probabilistic met- rics exported by the PCNtoolkit. Mean-based metrics—the coefficient of determination (R2), root mean squared error (RMSE), standardized mean squared error (SMSE), and explained variance (EXPV)—quantify how well the predicted mean ˆyapproxi- mates the observed data. Probabilistic metrics assess the full predictive distribution; therefore, they are more suitable for evaluating the quality of estimated centiles in the NM context: the mean standardized log loss (MSLL) compares predictive log- likelihoods to a variance-standardized baseline, while the mean absolute centile error (MACE) [34] measures the average deviation of predicted centiles from their expected uniform targets. Lower RMSE, SMSE, MSLL and MACE values, and higher EXPV andR 2 indicate better overall fit. Because normative models estimate both mean and variance, mean-based metrics alone provide an incomplete view of performance. A model may display modestR 2 yet excellent calibration of uncertainty (and vice versa), as reflected in the centile and QQ plots (Fig. 3). This reflects the purpose of normative mod- elling: to accurately estimate the conditional distribution of each feature rather than only its central tendency. All metric values are stored automatically in ../out/models/main workflow model HBR/results/statistics {train,test}.csv and can be loaded as follows: import pandas as pd table = pd.read_csv( "../out/models/main_workflow_model_HBR/results/statistics_test.csv", index_col="statistic" ).T Visual assessment of calibration Qualitative assessment of calibration is performed using the PCNtoolkit’s built-in plotting utilities.plot centilesvisualizes predicted centile trajectories across covari- ates, whileplot qqcompares empirical and theoretical quantiles of standardized residuals (Z-scores). Together, these provide complementary views of functional and distributional model fit. Figure 3 shows the two plots produced by this code snippet. from pcntoolkit import plot_centiles, plot_qq plot_centiles( model, centiles=[0.05, 0.5, 0.95], covariate="age", covariate_range=(10, 80), batch_effects={"site": ["ICBM", "Leiden_2180"], "sex": ["M", "F"]}, scatter_data=train, hue_data="site", markers_data="sex", show_other_data=True, ) 20 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint (a) Centile plot (b) QQ plot Fig. 3: Visualization of normative model calibration for a representative region of interest.(a)Predicted centiles (5th, 50th, 95th) across age, with observed data over- laid. Smooth median and outer centile alignment indicate a good functional fit across covariates. (b) Quantile–quantile (QQ) plot comparing empirical versus theoretical quantiles of standardized residuals. Alignment with the identity line indicates accu- rate predictive variance; curvature suggests over- or under-dispersion. For a detailed interpretation of QQ plots in normative modelling, see [26]. plot_qq(train, plot_id_line=True) 3.3.2 Longitudinal normative modelling Whilst many normative models are estimated using longitudinal data, the models themselves are fundamentally cross-sectional in nature in that they involve estimating smooth curves to link the group-level centiles across different timepoints. In order to obtain true longitudinal inferences, additional steps are necessary. Most importantly, to provide a statistical estimate to quantify the magnitude of change between two or more consecutive z-score measurements derived from a normative model. In other words, longitudinal normative modelling seeks to determine whether a centile crossing has occurred for a given individual between two or more timepoints. Since individuals do not track centiles exactly as they move through time (e.g., due to measurement noise), it is essential to take longitudinal variance into account both when estimating the normative model and when evaluating change between time points, to obtain a reliable characterization of individual trajectories [12]. This requires longitudinal data. The PCNtoolkit and supporting scripts allow for two ways of obtaining an estimate of the significance of longitudinal change: namely constructing ‘Z-diff’ scores that provide a statistical estimate of centile crossing between two timepoints based on BLR models [11] or using velocity models, which provide the ability to estimate ’thrive lines’ [12] which can quantify centile crossings for larger numbers of timepoints. An alternative method for longitudinal data analysis is presented in Di Biase et al. [35]. 21 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint 3.3.3 Federated Learning A key feature of PCNtoolkit is the ability to support federated learning, as described in detail in [14, 24]. Concrete examples of this workflow are provide a notebook in the associated code repository. In brief, several methods for federated normative mod- eling, i.e., estimating normative models on decentralized data, are supported: model transfer involves deriving a new set of coefficients for data sites that were not included in the reference dataset by distilling the information from an existing ref- erence model. In the context of HBR, this is operationalised by means of applying an informative prior distribution (corresponding to the posterior distribution from the base reference model) to the new data and re-estimating parameters. Whilst this approach is usually the most efficient, please note that a model transfer may be prob- lematic if the data from the new site has a very different distribution from the data in the training set. In contrast, a model extend involves updating the reference model with data from new sites, which can then be passed back to the site initially con- tributing the original model. Whilst this is usually fairly robust, please note that during model extension, if the new sites have poor quality data, this may have a dele- terious impact on the original model (e.g., containing severe outliers or spanning a non-overlapping portion of the covariate range). A model merge can be performed to combine two separately estimated normative models (e.g., from different sites). All three modules operate without requiring access to the reference datasets used to derive the original reference model, which is essential in settings where data privacy is a primary concern. 3.4 Community Aspects In order to maximise value to the community, we provide access to an updated set of lifespan reference models derived from tens of thousands of individuals aggregated from datasets that span the lifespan. These models supersede the cortical thickness and volumetric models presented in Rutherford et al. [4] and fractional anisotropy measures derived from diffusion data [36], with additional models to be brought online in the future, for example, functional connectivity normative models presented in Rutherford et al. [20]. This is necessary because models estimated in early versions of PCNtoolkit are not compatible with PCNtoolkit 1.x.x. These reference models are available for download via a notebook available in the associated code repository. We welcome the contributions of new models from other groups via PCNPortal [16], our online collaborative normative modelling portal. We also welcome contributions from the field, in terms of contributions of new

for normative modeling and different assessment metrics, as well as general improvements to the code. These can be submitted through standard pull requests or GitHub issues to PCNToolkit GitHub repository. This was a major redesign con- sideration in PCNtoolkit 1.x.x and is now much easier due to the object-oriented design. 22 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint 4 Troubleshooting Troubleshooting is best addressed by consulting the additional documentation avail- able online at ReadTheDocs, including additional tutorials for other algorithm implementations, a glossary to clarify the jargon associated with the software, a

guide with links to normative modeling publications and a frequently-asked- questions page where many common errors (and their solutions) are discussed in detail. Additionally, questions can be raised and answered by submitting issues via GitHub. 5 Timing The normative modeling portion of this protocol (including evaluation and visualiza- tion) can be completed in ∼57–72 minutes, although additional time is required to assess additional models. These timing estimates are based on the use of the compute platform described above to run the code. 6 Anticipated Results Together, this protocol describes the process by which normative models can be fit, model convergence assessed and the fit of the centiles to the data to be ascertained. In the example given, these quantitative and visual diagnostics confirm robust model con- vergence, well-calibrated uncertainty estimates, and accurate functional dependence across covariates. Starting from raw, multi-site neuroimaging data, we have demon- strated a reproducible end-to-end workflow covering data selection, preprocessing, model fitting, evaluation, and visualization. The resulting normative model captures both central trends and population variability, allowing individual-level deviations to be expressed in a statistically calibrated manner. With convergence and calibration verified, the model is ready for downstream applications such as longitudinal analyses, data harmonization, and federated normative modeling on decentralized data. This protocol covers the basic normative model estimation workflow (i.e., how to fit a normative model from scratch). We provide additional examples in the accom- panying GitHub repository, including examples showing how to transfer models to a new dataset, to extend an existing model to include new sites, to synthesize data, to perform longitudinal analysis and to harmonise data to a common normative reference model. 7 Key references • De Boer et al. [9] • Bayer et al. [12] • Fraza et al. [37] • Berthet et al. [38] • Cirstian et al. [36] 23 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted February 18, 2026. ; https://doi.org/10.64898/2026.02.17.706268doi: bioRxiv preprint Acknowledgments We greatly acknowledge funding from the European Research Council under a consolidator grant (grant number 101001118), the Wellcome Trust (grant numbers 215698/Z/19/Z and 226706/Z/22/Z) and the Raynor Foundation (Raynor Cerebellum Charts project). Competing Interests CFB is a shareholder and director of SBGNeuro. The other authors report no competing interests. Author Contributions AAAdB, JMMB, CF, AC, BRB: writing software, running experiments, drafting and revising the manuscript; AC: running experiments, revising the manuscript; KT, ES: writing software, drafting and revising the manuscript AB, RC, MZ, SR, AAK: running experiments, drafting and revising the manuscript; TW, CFB, SMK: conceptualisa- tion, supervision, drafting and revising the manuscript; AFM: funding acqusition, conceptualisation, writing software, supervision, drafting and revising the manuscript. Data Availability All data used in this manuscript are publicly available. This includes the FCON1000 dataset, which is freely available at the FCON1000 website and two clinical datasets from the OpenNeuro portal, namely: 1. ds003568, see ref: [32]. 2. ds003653, see ref: [33]. Code Availability The PCNtoolkit software package is available online at : https://github.com/amarquand/PCNtoolkit and all code necessary to run this pro- tocol is available at: https://github.com/predictive-clinical-neuroscience/pu25 code.

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