2 Formalizing the problem
2.1 Notation
Let \(y_{ijr}\) denote the measurement of region \(r \in \{1,\dots,R\}\) on subject \(i \in \{1,\dots,N\}\) at visit \(j \in \{1,\dots,T_i\}\), acquired at time \(t_{ij}\). Write \(\mathbf{y}_{ij} \in \mathbb{R}^{R}\) for the full regional vector at a visit and \(\Delta_{ij} = t_{i,j+1} - t_{ij}\) for the inter-visit interval. Throughout, \(T\) without subscript refers to the typical or minimum number of visits per subject.
Two structural features distinguish this setting from standard time-series analysis:
- \(T\) is small (often 2 or 3) while \(N\) and \(R\) are large. Information about dynamics comes predominantly from the cross-section, not from within-subject series length.
- \(t_{ij}\) is irregular across subjects, and subjects enter the study at different and unobserved points along whatever underlying process is being modeled.
2.2 Three distinct objectives
The phrase “model the evolution over time” conflates three objectives with different data requirements. Stating which one is intended is a precondition for a valid method comparison.
Description. Estimate the population-average trajectory shape and its between-subject variation. Requires that the aggregate of short individual arcs covers the time range of interest. Feasible with \(T = 2\) given large \(N\) and adequate coverage of the age or stage range.
Prediction. Forecast \(y_{ijr}\) at an unobserved time, either by interpolation within the observed window or extrapolation beyond it. Requires sufficient within-subject information to personalize the forecast; otherwise the prediction reduces to the population mean conditioned on covariates.
Mechanism. Identify the structure governing change — which regions drive which, what the functional form of the rate law is, what exogenous inputs matter. This is the most demanding objective and is not attainable at the subject level with small \(T\). It is partially attainable at the population level under strong structural priors (Chapter 7).
Comparing a differential-equation model against a supervised learner on a prediction task, and then interpreting the result as evidence about mechanism, is a common and invalid inference.
2.3 Choice of modeling target
Several targets are in use and they are not interchangeable.
| Target | Definition | Comment |
|---|---|---|
| Level | \(y_{i,j+1,r}\) | Dominated by baseline value; inflates apparent \(R^2\) |
| Change | \(y_{i,j+1,r} - y_{ijr}\) | Direct, but sensitive to baseline measurement error |
| Rate | \((y_{i,j+1,r} - y_{ijr}) / \Delta_{ij}\) | Assumes local linearity over \(\Delta_{ij}\) |
| Log-ratio | \(\log(y_{i,j+1,r} / y_{ijr})\) | Natural under exponential dynamics; scale-free |
| Latent stage | \(\tau_i\) | Requires a progression model (Chapter 7) |
Reporting model fit on the level target is the single most common source of overstated performance: a model that predicts next-visit hippocampal volume from current hippocampal volume achieves high \(R^2\) while carrying no information about change. Performance should be reported on change or log-ratio, and always against a no-change baseline (Chapter 14).
2.4 Constraints specific to regional morphometry
Compositional structure. Regional volumes sum toward total brain volume. Without explicit handling, a substantial fraction of apparent regional covariation is a single global scaling factor. Options are modeling on the log scale, including total intracranial volume or total brain volume as a covariate, or working in proportions with an isometric log-ratio transform. The choice materially changes which regions appear to be coupled, and it determines the leading component of any decomposition (Chapter 5).
Volume and density are not independent channels. Modulated voxel-based morphometry density and region-of-interest volume are strongly collinear by construction, yet not interchangeable: Gennatas et al. (2017) reported divergent age trajectories for density, volume, mass, and cortical thickness over the same regions. Treating them as two observations of a two-dimensional state will produce a near-singular system. Either select one, or model the pair explicitly as a measurement model over a shared latent state.
Irregular intervals carry information. \(\Delta_{ij}\) must enter the model directly. Binning visits into nominal waves discards real variation and induces attenuation bias.