Here, we briefly present the original normative model (Fraza et al., 2021), developed to be trained and used on cross-sectional data. In the following subsection, we take this model pre-trained on a large cross-sectional dataset and extend it so that it can be used on longitudinal data.

The original model (Fraza et al., 2021) is pre-trained on a cross-sectional dataset Y=(ynd)∈RN×D, X=(xnm)∈RN×M of N subjects, for whom we observe D IDPs and M covariates (e.g. age or sex). Thus, ynd is the dth IDP of the nth subject and xnm is the mth covariate of the nth subject.

Since each IDP is treated separately, we focus on a fixed IDP d and drop this index for ease of exposition. To simplify notation, we denote y=(y1,…,yN)T the column of observations of this fixed IDP across subjects. The observations are assumed to be independent (across n subjects). To model the relationship between IDP yn and covariates xn=(xn1,…,xnM)T, we want to exploit a normal linear regression model described in Fraza et al., 2021; Rutherford et al., 2022. However, we make a couple of adjustments first:

We also treat the variance of measurements as a hyper-parameter and we denote it by σ2. Thus, we model the distribution of the transformed IDP φ(yn) conditional on covariates xn, vector of parameters w, and hyper-parameters σ2 and γ as follows:(1)φ(yn)=wTϕ(xn)+εn, εn∼N(0,σ2),

where εn are independent from xn and across n. We further denote the design matrix Φ=(ϕ(xn)k)∈RN×K (ϕ(xn)k is the kthe element of vector ϕ(xn)).

The estimation of parameters w is performed by empirical Bayesian methods. In particular, prior about w(2)N(0,ω2I)

is combined with the likelihood function to derive the posterior(3)w|y,Φ;ω2,σ2,γ∼N(w¯,A−1),(4)A=σ−2ΦTΦ+ω−2I,(5)w¯=σ−2A−1ΦTy

The hyper-parameters 𝜔² , 𝜎² , 𝛾 are estimated by maximising the warped marginal log-likelihood.

The predictive distribution of φ(y) for a subject with x is(6)N(w¯Tϕ(x),ϕ(x)TA−1ϕ(x)+σ2).

Hence, the z-score characterising the position of this subject within population is(7)z=φ(y)−w¯Tϕ(x)ϕ(x)TA−1ϕ(x)+σ2,

where φ(y) is the realised warped observation of IDP d for this subject. This score captures how surprising is the actual observation φ(y) relative to what one would expect for an average subject with the same characteristics w¯Tϕ(x), and this deviation has to be compared with (normalised by) the variability stemming from the natural variability in the data (σ2) and the modelling uncertainty (ϕ(x)TA−1ϕ(x)).

In this form, the original models were fit on a large dataset consisting of 58,836 participants scanned across 82 sites. Specifically, cortical thickness and subcortical volumes were modelled, and the models were validated against a subset of 24,000 participants, the quality of which was checked manually (Rutherford et al., 2022).

Note that Equation 6 and Equation 7 implicitly evaluate only (potentially new) subjects measured at sites already present in the original dataset y, Φ. If we want to evaluate subjects measured at a new site, we will have to run an adaptation procedure to account for its effect. This adaptation procedure is described and readily accessible online in Rutherford et al., 2022. In short, a sample of a reference (healthy) cohort measured on the same scanner as the population of interest is needed to accommodate a site-specific effect.

In the following section, we develop a procedure that allows us to extend the original cross-sectional framework pre-trained on dataset y, Φ to evaluate a new longitudinal dataset for assessment of temporal changes.

We adapt the original cross-sectional normative modelling framework (Fraza et al., 2021) (reviewed in the previous section) to the evaluation of intra-subject longitudinal changes. Specifically, we design a score for a longitudinal change between visits (further referred to as z-diff score), based on which we can assess temporal changes in regional brain thickness and potentially detect any unusually pronounced deviations from normative trajectories.

We start by noticing that the original cross-sectional normative modelling framework (Fraza et al., 2021) features an implicit assumption that pertains to the longitudinal view. Specifically, it assumes that had we randomly sampled the population at a different time (e.g. 5 years sooner or later), we would have gotten equivalent picture about the ‘norm’ (up to randomness of the sampling, Figure 1A). In other words, the parameters of the normative model would be the same irrespective of the time we sampled the population, including the case in which we would sample the same people again, just later (while appropriately compensating for the resulting under-representation of younger ages). We further assume comparability of people of any given age irrespective of their birth time (i.e. we assume independence of birth dates and trajectories, Figure 1B). Together, these assumptions imply a form of stationarity (formally discussed in the next paragraph). These are indispensable assumptions for the practical usefulness of normative modelling, albeit one can see that in the real world they are not fulfilled perfectly, e.g., due to evolutionary dynamics, the ever-changing environment, or any changes in the distribution of those demographic variables that are not explicitly accounted for in the normative model.

Formally, we work with the process {(yn,t,xn,t)}, where n indexes a subject and t indexes age (to avoid technicalities, we assume discrete time). The minimal requirement imposed implicitly by the above assumptions is εt∼N(0,σ2) for every age t. We further restrict our focus to the class of stationary Gaussian processes ε, i.e., processes such that εt1,εt2,…,εtk are jointly normal for any finite set of ages t1,t2,…,tk and their joint distribution is invariant to any admissible time shift t1+s,t2+s,…,tk+s. Furthermore, we focus on a specific process in this section for ease of exposition, and we discuss the general case in the next section.

The specific process we focus on in this section reflects the assumption that a healthy subject does not deviate substantially from their position within the population as they get older (Marquand et al., 2019); the observed position change between the visits stems from observation noise (due to technical or physiological factors) and is therefore constrained by the test-retest reliability of the measurement. Formally,εn,t=ηn+ξn,t,

where η is a subject-specific time-independent factor independent of the iid noise process ξ. Note that this does not imply that a healthy subject does not change over time, but rather that the change follows approximately the population centile at which the individual is placed. We generalise our method to other stationary Gaussian processes in the next section.

According to this model, the ith visit of a healthy subject with given covariates x(i) is generated by(8)φ(y(i))=wTϕ(x(i))+η+ξ(i)η∼N(0,ση2)ξ(i)∼N(0,σξ2)σ2=ση2+σξ2

where η, ξ(i), and x(i) are mutually independent for a given i, and the measurement errors ξ(i) and ξ(j) are independent across visits i≠j. Note that we dropped the subject-specific index n (and subsumed the age tn(i) in the visit index i). This should remind the reader that the goal is just to evaluate longitudinal change of a given subject from our new longitudinal data and not to re-estimate the parameters with these additional data. Nevertheless, to properly adapt the cross-sectional model, we will need to estimate one new parameter stemming from the further structure we impose on ε.

In our longitudinal data, we are interested in the change for a given individual across two visits. According to Equation 8, the difference in the transformed IDP between visits 1 and 2, φ(y(2))−φ(y(1)), for a subject with covariates x(1) and x(2) is given by(9)φ(y(2))−φ(y(1))=wT[ϕ(x(2))−ϕ(x(1))]+ξ(2)−ξ(1)

with ξ(2)−ξ(1)∼N(0,2σξ2). We use the posterior distribution of w with hyper-parameters ω2,σ2,γ estimated on the original cross-sectional dataset y,Φ (the estimates are available at https://github.com/predictive-clinical-neuroscience/braincharts; Rutherford and Marquand, 2024). Therefore, the posterior predictive distribution for the difference φ(y(2))−φ(y(1)) for our subject is (for more detailed derivation, please refer to the Appendix 1)(10)N(w¯T[ϕ(x(2))−ϕ(x(1))],[ϕ(x(2))−ϕ(x(1))]TA−1[ϕ(x(2))−ϕ(x(1))]+2σξ2).

Hence, the z-score for the difference in the transformed IDP between visits 1 and 2 is(11)z-diff=[φ(y(2))−φ(y(1))]−w¯T[ϕ(x(2))−ϕ(x(1))][ϕ(x(2))−ϕ(x(1))]TA−1[ϕ(x(2))−ϕ(x(1))]+2σξ2,

where φ(y(2))−φ(y(1)) is the realised temporal change in the warped observations of the IDP for this subject. Since this z-diff score is standard normal for the population of healthy controls, any large deviations may be used to detect unusual temporal changes.

The primary role of adaptation of the (pre-trained) cross-sectional model to (new) longitudinal data is to account for the measurement noise variance σξ2, thus, taking care of the atemporal source of variability η. In other words, having an estimate of σξ2 in hand helps us to use the proper scaling. To arrive at an estimator of σξ2, notice that from the posterior predictive distribution (Equation 10), we have (denoting the set of conditionals Ω={x(1),x(2);y,Φ;ω2,σ2,γ})(12)E[(φ(y(2))−φ(y(1))−E[φ(y(2))−φ(y(1))|Ω])2|Ω]=[ϕ(x(2))−ϕ(x(1))]TA−1[ϕ(x(2))−ϕ(x(1))]+2σξ2.

Hence, by the law of iterated expectations (to integrate out x(1) and x(2)), we obtain(13)E[(φ(y(2))−φ(y(1))−w¯T[ϕ(x(2))−ϕ(x(1))])2−[ϕ(x(2))−ϕ(x(1))]TA−1[ϕ(x(2))−ϕ(x(1))]|y,Φ;ω2,σ2,γγ]=2σξ2.

Therefore, we estimate 2σξ2 by the sample analogue of the left-hand side in Equation 13. Specifically, we devote a subsample C of the controls from our (new) longitudinal data just to this estimation (i.e. the subjects from C will not be used in the evaluation) and we compute(14)2σξ2^=1|C|∑kϵC[(φ(yk(2))−φ(yk(1))−w¯T[ϕ(xk(2))−ϕ(xk(1))])2−[ϕ(xk(2))−ϕ(xk(1))]TA−1[ϕ(xk(2))−ϕ(xk(1))]],

where |C| denotes the number of subjects in subsample C.

Another useful feature of longitudinal data is that [ϕ(xk(2))−ϕ(xk(1))] is negligible (especially with stable covariates, like sex and age). Sex (typically) does not change across the two visits and age relatively little (in our target application) with respect to the full span of ageing. Consequently, [ϕ(xk(2))−ϕ(xk(1))]TA−1[ϕ(xk(2))−ϕ(xk(1))] in Equation 14 is negligible in adult cohorts but must be treated with caution in developmental or ageing groups. Finally, it is apparent from Equation 4 that A scales with the number of subjects, and its inverse will be negligible for substantial training datasets, such as the one that was used for pre-training.

To conclude this subsection, we caution against the naïve use of the difference of the simple z-scores(15)φ(y(2))−w¯Tϕ(x(2))ϕ(x(2))TA−1ϕ(x(2))+σ2−φ(y(1))−w¯Tϕ(x(1))ϕ(x(1))TA−1ϕ(x(1))+σ2

instead of the z-diff score to evaluate the longitudinal change. The problem with such an approach is apparent by comparing it to the z-diff score (Equation 11); it does not properly account for the modelling uncertainty (instead of using the combined term [ϕ(x(2))−ϕ(x(1))]TA−1[ϕ(x(2))−ϕ(x(1))] to scale the difference of the numerators in Equation 15, it scales the individual terms of the difference by their individual model uncertainty). More importantly, even if the modelling uncertainty is negligible, Equation 15 does not properly scale the difference of the ‘residuals’ because it incorrectly includes the common source of subject-level variability η (it uses ση2+σξ2 instead of 2σξ2), as we later demonstrate in the simulation part of the study.

The model we introduced in the previous section is an intuitive extension of the original model introduced in the ‘Model formulation’ section. However, the model operates with a seemingly strong (although reasonable) assumption that healthy subjects inherently follow their centiles. Due to the lack of large longitudinal data testing this assumption, in this section, we investigate the generalisation to other stationary Gaussian processes to illustrate the robustness of our method. As an example, we are able to deal with a stationary Gaussian AR(1) process εt=ζεt−1+ξt with |ζ|<1, ξ iid N(0,(1−ζ2)σ2), and ε0∼N(0,σ2).

Importantly, our framework evaluates change only between two visits. Hence, we do not need to consider the full specification of the process ε, but only the time dependence between the two visits that can arise under it. Formally, since we are in the class of stationary Gaussian processes, we only need to consider the autocorrelation between ε(1) and ε(2)ρ∈[−1,1]. Just as an example, the stationary Gaussian AR(1) process introduced above would produce autocorrelation ρ=ζT2−T1, where T2−T1 is the time between the two visits.

Considering this more general class of processes, this amounts to ε(2)−ε(1)∼N(0,2σ2(1−ρ)). Going through the same derivations as before, we obtain the score for the evaluation of longitudinal change(16)z-diff=[φ(y(2))−φ(y(1))]−w¯T[ϕ(x(2))−ϕ(x(1))][ϕ(x(2))−ϕ(x(1))]TA−1[ϕ(x(2))−ϕ(x(1))]+2σ2(1−ρ)

and the estimator(17)2σ2(1−ρ)^=1|C|∑k∈C[(φ(yk(2))−φ(yk(1))−w¯T[ϕ(xk(2))−ϕ(xk(1))])2−[ϕ(xk(2))−ϕ(xk(1))]TA−1[ϕ(xk(2))−ϕ(xk(1))]].

These take the same form as in the specific case considered in the previous section. Hence, the method developed in the previous section does not depend on that particular assumption about the process ε and will still yield valid inferences even if the seemingly strong assumption of centile tracking is violated. In either case, we need one more free parameter to properly account for the potential non-iid dynamics of ε in the previous section, ρ in this section. The only substantial difference is that while 2σ2(1−ρ) can be larger than 2σ2 (for ρ<0), the process from the previous section leads to 2σξ2 with values only lower than 2σ2. This could provide a test of the assumption about the process from the previous section: if our estimate 2σξ2^ is larger than the cross-sectional estimate 2σ2^, then, the assumption about ε in the previous section is not justified.

To formally evaluate the performance of the proposed method in making accurate inferences about the longitudinal changes, we conduct a simulation study. We imagine a practitioner who would use some lower and upper thresholds for the z-diff score to detect unusual change. It is natural to choose the probability of dubbing a healthy control as unusual θ∈[0,1], and use the θ2 and 1−θ2 quantiles of the standard normal distribution as the thresholds (denote them qθ2 and q1−θ2, respectively). A subject with z-diff <qθ2 or >q1−θ2 is thus flagged as someone with unusual change. We would like to know how successfully this classification detects true changes, i.e., how often it detects a patient with a disrupted trajectory. We capture the disruption by a process δ, i.e., the trajectory of a patient in our model would be φ(y). We treat the realised change in the disruption between the two visits Δ:=δ(2)−δ(1) as a fixed number to be detected.

For the simulation, we fix θ = 0.05. For each combination of Δ∈[−4,4] and ρ∈(−1,1), we generate a large number of patients with various age and gender, disruption Δ, and (ε(2),ε(1)) from the bivariate normal distribution(18)N([00],[σ2ρσ2ρσ2σ2]).

Specifically, we produce φ(y(1))=w¯Tϕ(x(1))+ε(1), φ(y(2))=w¯Tϕ(x(2))+ε(2)+Δ (the remaining parameters are the cross-sectional estimates). For each patient, we calculate the z-diff score and we look at the fraction of patients with a z-diff score surpassing the thresholds. Additionally, we compare the results to the naïve approach based on evaluating the difference of the z-scores between the two visits, while using the same thresholds (the quantiles of the standard normal distribution). The resulting simulation is depicted in Figure 2.

Two intuitive properties arise from this simulation: larger disruptions are easier to detect; positive autocorrelation in ε makes it easier to detect the disruptions, while negative autocorrelation makes it harder. Strong positive autocorrelation reflects a strong common component in ε across the two visits, which cancels out through subtracting the two visits, while strong negative autocorrelation indicates strong switching in ε across the two visits, which can be easily confounded with the true disruption Δ. Finally, if the true process for ε is as assumed in Equation 8 (stable component plus noise), then, higher noise σξ2 corresponds to lower (but positive) ρ. Intuitively, we can see that more noise makes the detection of true disruptions more difficult.

The results of the simulation also clearly caution against the use of the naïve z-score subtraction (Figure 2). First, the z-diff method maintains a consistent false-positive rate (when no disruption is present) of 5%, unlike the z-score subtraction, where the false-positive rate changes with the autocorrelation.

Second, in terms of detection power the z-diff method outperforms the simple z-score subtraction for high autocorrelation, in particular when the autocorrelation is above 0.5, with the difference in the performance being more pronounced with rising autocorrelation. This is particularly relevant in practice—our real-world data imply very high values of ρ (average ρ^ across all IDPs is 0.9; Appendix 2—figure 1).

Finally, the seemingly better performance of the z-score subtraction under autocorrelation below 0.5 is caused only by the subtraction method’s general tendency to label cases as ‘suspicious’ in the absence of change, e.g., by the increased false-positive rate (Figure 2, right). This leads to a small steady ‘improvement’ across the space of disruptions. See more detailed simulation results in Appendix _3—figure 1.

Let us revisit the earlier example of a practitioner using these methods to identify unusual changes in patients. In most real-world scenarios, the majority of people are healthy, with only a few experiencing pronounced changes that might indicate illness. If the detection method does not properly control the false-positive rate, many healthy individuals could be mistakenly flagged as needing further investigation. This could lead to unnecessary stress, costly follow-ups, or even painful procedures for those individuals. The naïve subtraction of z-scores is problematic in this regard. It often misclassifies healthy individuals, particularly when the correlation between measurements (autocorrelation) is low. This inconsistency makes it unreliable in practice. In contrast, the z-diff method consistently maintains a predictable false-positive rate while also improving the ability to detect true changes (when measurements are highly positively correlated). This balance ensures that more patients with real disruptions are identified while minimising unnecessary interventions for healthy individuals.

To implement the method (Figure 3), we used the PCN toolkit. The exact steps of the analysis with detailed explanations are available in the online tutorial at PCNtoolkit-demo (https://github.com/predictive-clinical-neuroscience/PCNtoolkit-demo) in the tutorials section.

The clinical data used for the analysis were part of the Early Stages of Schizophrenia study (Spaniel et al., 2016). We analysed data from 98 patients in the early stages of schizophrenia (38 females) and 67 controls (42 females) (Table 1). The inclusion criteria were as follows: The subjects were over 18 years of age and undergoing their first psychiatric hospitalisation. They were diagnosed with schizophrenia, or acute and transient psychotic disorders, and suffered from untreated psychosis for less than 24 months. Patients were medically treated upon admission, based on the recommendation of their physician. Patients suffering from psychotic mood disorders were excluded from the study.

Healthy controls over 18 years of age were recruited through advertisements unless: They had a personal history of any psychiatric disorder or had a positive family history of psychotic disorders in first- or second-degree relatives.

If a subject in either group (patient or control) had a history of neurological or cerebrovascular disorders or any MRI contraindications, they were excluded from the study.

The study was carried out in accordance with the latest version of the Declaration of Helsinki. The study design was reviewed and approved by the Research Ethics Board. Each participant received a complete description of the study and provided written informed consent.

Data were acquired at the National Centre of Mental Health in Klecany, Czech Republic. The data were acquired at the National Institute of Mental Health using Siemens MAGNETOM Prisma 3 T. The acquisition parameters of T1-weighted images using MPRAGE sequence were: 240 scans; slice thickness: 0.7 mm; repetition time: 2400 ms; echo time: 2.34 ms; inversion time: 1000 ms; flip angle: 8°; and acquisition matrix: 320 mm × 320 mm.