8 Correspondence
Which methods are available at each design density.
This chapter summarizes which methods are available at each design density. Entries are:
- Yes — identifiable from within-subject information.
- Pooled — identifiable only through pooling across subjects, with subject-level parameters prior-dominated.
- No — not identifiable at any sample size with this design.
8.1 Correspondence table
| Method | \(T=2\) | \(T=3\) | \(T=4\)–\(6\) | \(T \geq 7\) | Dense (\(>20\)) |
|---|---|---|---|---|---|
| D1 Exponential rate law | Pooled | Pooled | Yes | Yes | Yes |
| D1 Sigmoidal / Gompertz | No | Pooled | Pooled | Yes | Yes |
| D2 Coupled LDS, constrained | Pooled | Pooled | Pooled | Yes | Yes |
| D2 Coupled LDS, general \(\mathbf{A}\) | No | No | Pooled | Pooled | Yes |
| D3 Network diffusion | Pooled | Pooled | Yes | Yes | Yes |
| D4 Latent time | Pooled | Pooled | Yes | Yes | Yes |
| ML, change on baseline | Yes | Yes | Yes | Yes | Yes |
| ML, autoregressive | Yes | Yes | Yes | Yes | Yes |
| H1 Learned residual | Pooled | Pooled | Yes | Yes | Yes |
| H2 Learned parameterization | Pooled | Pooled | Yes | Yes | Yes |
| H3 Universal DE | No | Pooled | Pooled | Yes | Yes |
| H5 Latent ODE / ODE-RNN | No | No | Pooled | Pooled | Yes |
| Per-subject SINDy | No | No | No | Pooled | Yes |
| Per-subject neural ODE | No | No | No | No | Yes |
Entries in the \(T=2\) and \(T=3\) columns assume \(N\) in the hundreds or more and adequate coverage of the age or stage range. A “Pooled” entry is a statement that the model can be fitted and its population-level parameters interpreted; it is not a licence to interpret subject-level parameters, for which the shrinkage diagnostic of Chapter 3 applies.
8.2 What changes across regimes
The transition that matters is not gradual. Three qualitative thresholds separate the regimes.
\(T = 2 \rightarrow T = 3\): curvature becomes observable. One second difference per subject becomes available. It is noisy (Chapter 3) but it is the first within-subject evidence about functional form, and it enables the held-out-visit evaluation design that makes a three-arm comparison interpretable (Chapter 12).
\(T = 3 \rightarrow T = 4\)–\(6\): shape becomes estimable per subject. Subject-level random effects on a curvature parameter stop being prior-dominated. Collocation and gradient-matching estimators become usable. Hybrid models with a learned component begin to be constrained by within-subject data rather than only by the cross-section.
\(T \geq 7\): the subject becomes a unit of dynamical analysis. Subject-specific rate laws can be estimated and compared. Coupling matrices are constrained by within-subject information rather than only by cross-sectional covariance. This is the point at which “modeling each subject’s dynamics” becomes a literal rather than a figurative description.
Beyond roughly twenty observations per subject, data-driven system identification becomes feasible and the methodological question shifts from identifiability to model selection.
8.3 Practical implication for study design
The table implies a concrete design recommendation. Adding a third visit to a two-visit study yields a qualitative rather than incremental gain, because it converts the comparison from one that cannot distinguish the arms (Chapter 3) to one that can. Adding a fourth to a third yields a further qualitative gain by removing the dependence on pooling for shape. Beyond six visits, returns diminish until dense sampling is reached.
Where resources permit only a fixed number of scans, the allocation question is whether to scan more subjects twice or fewer subjects more often. The answer depends on the objective stated in Chapter 2: description and prediction favor more subjects; mechanism favors more visits.