3 Identifiability
What repeated measurements constrain.
This chapter gives the degrees-of-freedom accounting that governs which models can be estimated from a given design. It is the basis for the regime chapters.
3.1 Within-subject degrees of freedom
From \(T\) observations of one subject and one region, finite differencing yields \(T - 1\) first differences and \(T - 2\) second differences. A first-order rate law is constrained by the first differences; curvature, and therefore any distinction between linear, exponential, and sigmoidal decline, is constrained only by the second differences.
| \(T\) | First differences | Second differences | Distinguishable within subject |
|---|---|---|---|
| 1 | 0 | 0 | Nothing; the observation is cross-sectional |
| 2 | 1 | 0 | Rate only. No functional form is identified |
| 3 | 2 | 1 | Rate and a single, poorly conditioned curvature estimate |
| 4–6 | 3–5 | 2–4 | Low-order shape; monotonicity; approximate inflection |
| 7–15 | 6–14 | 5–13 | Subject-specific parameters of a low-dimensional rate law |
| \(>20\) | — | — | Nonparametric rate estimation; system identification |
3.2 Noise amplification in differences
Let the measurement error be independent with variance \(\sigma^2\). Then
\[ \operatorname{Var}(y_{j+1} - y_j) = 2\sigma^2, \qquad \operatorname{Var}(y_{j+1} - 2y_j + y_{j-1}) = 6\sigma^2 . \]
Curvature is therefore estimated with three times the error variance of a first difference, while the curvature signal is typically an order of magnitude smaller than the slope signal over a short follow-up. The signal-to-noise ratio for curvature at \(T = 3\) is consequently poor even when the slope is estimated well. Any claim that a nonlinear rate law fits better than a linear one at \(T = 3\) must be supported by an explicit noise-floor calculation (Chapter 4), not by a likelihood ratio alone.
3.3 Subject-level versus population-level identifiability
The table above concerns a single subject in isolation. Hierarchical models change the accounting substantially: a population model with \(p\) fixed-effect parameters can be identified from \(N\) subjects with \(T = 2\) each, provided the design contains between-subject variation in the directions the parameters describe. Total information scales with \(NT\), not \(T\).
Two conditions must hold for this to be legitimate.
- Coverage. The union of individual observation windows must span the time or stage range over which the population trajectory is claimed. Extrapolating a population curve beyond the observed range is unsupported regardless of \(N\).
- Exchangeability of stage. Subjects observed at different points must be comparable after conditioning on the model’s covariates and random effects. Violations — for example, cohort effects confounded with age — bias the inferred shape.
A useful diagnostic is the shrinkage factor for each subject-level random effect. If posterior subject-specific parameters are shrunk almost entirely to the population mean, that parameter is determined by the prior and not by the subject’s data. Fitting such a model is possible; interpreting its subject-level parameters is not.
3.4 Equivalence of arms at \(T = 2\)
Consider any model whose prediction for a subject depends on the observed data only through \((y_{i1}, \Delta_{i1}, \mathbf{x}_i)\), where \(\mathbf{x}_i\) are covariates. At \(T = 2\) this includes the fitted values of a first-order ordinary differential equation, a linear mixed model with a random intercept and slope, and a regression of change on baseline and covariates. These produce identical in-sample fits under matched parametrization; they differ only in
- the extrapolation function applied outside the observed interval,
- the pooling structure that ties subjects, regions, or parameters together,
- the uncertainty model and therefore the calibration of prediction intervals.
A three-arm comparison at \(T = 2\) is informative only if it is designed to exercise at least one of these three. Otherwise it compares a model against a reparametrization of itself.
3.5 Latent time as a degrees-of-freedom multiplier
Disease progression models introduce a subject-specific time shift \(\tau_i\) and rate \(\alpha_i\), so that subject \(i\) observed at \(t_{ij}\) is modeled as occupying population time \(\alpha_i (t_{ij} - \tau_i)\) (Oxtoby 2023). This assembles short individual arcs into a long population trajectory and is the principal mechanism by which sparse designs support nonlinear shape estimation.
The cost is \(2N\) additional parameters and an identifiability condition: the model is determined only up to an affine reparametrization of the latent time axis unless an anchor is imposed. Standard anchors are fixing the population mean and variance of \(\tau\), fixing the trajectory of a reference variable, or imposing monotonicity of a designated marker. Results are sensitive to this choice and it should be reported.
Latent time also imposes a strong assumption: that all subjects follow a common trajectory shape differing only in onset and speed. Where subtypes exist, this assumption is violated and mixture extensions are required.