8  Supervised learning

Models in this family regress a future quantity on present information without specifying a generating process.

8.1 Target and feature construction

The target should be change or log-ratio rather than level, for the reasons given in Chapter 2. The feature set typically comprises:

  • the full baseline regional vector \(\mathbf{y}_{i1}\), not only the target region;
  • age, sex, and normalization covariates;
  • the interval \(\Delta_{ij}\), entered explicitly;
  • site, scanner, and software version;
  • where \(T \geq 3\), the observed first-interval rate, which is generally the single most informative feature.

The finding that prior observed change substantially improves forecasts over baseline-only information is robust. In a three-visit ADNI analysis with elastic net regression, longitudinal models outperformed single-timepoint models for hippocampal volume, ventricular volume, and total gray matter, with correlations of approximately 0.62, 0.51, and 0.41 respectively (Hadji et al. 2026). These values are a reasonable calibration for what is achievable and should inform the power calculation for any new comparison.

8.2 Pooling across regions

Fitting \(R\) independent models discards the strong covariation among regions and divides the effective sample. The extent of that covariation is an empirical question and should be established first (Chapter 5). Preferable alternatives:

  • Pooled model with region as a factor or learned embedding. One model over \(N \times R\) rows, with region identity as an input. Permits partial pooling of regional effects.
  • Multi-task or reduced-rank multi-output regression. Explicitly models the low-rank structure of regional change.

The pooled formulation is usually the stronger baseline and should be reported alongside any per-region result.

8.3 Regression toward the mean

Regressing change on baseline when baseline is measured with error induces a spurious negative association: the estimated coefficient is attenuated toward \(-1\) in proportion to the error fraction. This is a bias, not a nuisance, and it affects every model in this family that uses the same region’s baseline value as a predictor of its change.

Mitigations, in order of preference:

  1. An errors-in-variables or measurement-error model using the reliability estimate from Chapter 4.
  2. Instrumenting baseline with an independent acquisition where available.
  3. Excluding the same-region baseline from the feature set and reporting the result as a sensitivity analysis.

Reporting a model in this family without addressing this point is a substantive omission.

8.4 Forecast structure

For multi-visit prediction, two structures are available:

  • Direct multi-horizon. A separate model per horizon. Robust, no error accumulation, but does not enforce coherence across horizons.
  • Autoregressive. One step-ahead model applied recursively. Coherent and able to extrapolate, but accumulates error and is sensitive to distribution shift as predictions move away from the training distribution.

Autoregressive application of a supervised learner is the closest analogue in this family to a differential equation, and is the appropriate comparator when the evaluation involves extrapolation.

8.5 Model class

Within this family, the choice among regularized linear models, gradient boosting, and neural networks is usually secondary. The TADPOLE result — that feature selection mattered more than algorithm class, and that ensembles ranked highest (Marinescu et al. 2021) — should be taken as the prior. Effort is better spent on target construction, pooling structure, and measurement-error handling than on model class search.