11 Dense designs

Seven or more observations per subject.

11.1 What becomes available

With seven or more observations per subject, the subject becomes a unit of dynamical analysis rather than a contributor to a pooled population curve.

Subject-specific rate laws can be estimated and compared across subjects, and heterogeneity in dynamics can be characterized directly rather than as a random effect around a common form.
Coupling matrices are constrained by within-subject information. The distinction between cross-sectional covariance and dynamical coupling becomes empirically accessible.
Nonparametric rate estimation becomes possible: \(dv/dt\) can be estimated as a function of \(v\) without committing to a parametric family.
Latent-variable dynamical models (H5) acquire enough within-subject signal to constrain the latent dynamics rather than only the encoder.

Beyond roughly twenty observations, data-driven system identification — sparse regression on a candidate library, and its relatives — becomes feasible, and the governing question shifts from identifiability to model selection and to controlling false discovery over the candidate library.

11.2 Design considerations

Dense longitudinal morphometry is uncommon because of cost and because of the measurement floor: at short inter-scan intervals, the biological change between consecutive scans may fall entirely below the pipeline’s error. Increasing \(T\) by shortening intervals does not necessarily increase information, and can reduce it by adding observations that are pure noise while inflating apparent sample size.

The relevant quantity is total observed change relative to measurement error over the full study duration, not the number of scans. A design with seven scans over one year may carry less dynamical information than three scans over six years. This should be evaluated explicitly at the design stage using the floor estimate from Chapter 4.

11.3 Evaluation

Rolling-origin evaluation as in Chapter 10, extended to long horizons, with the addition of:

Held-out-subject evaluation of learned dynamics. Where subject-specific dynamics are estimated, the relevant generalization question is whether the learned form transfers to subjects not seen in training.
Recovery experiments on simulated data. With a dense design, the ability of each arm to recover known dynamics from realistic noise can be tested directly. This is the most informative validation available and is not possible in sparse regimes.

11.4 Relationship to the sparse regime

A dense sub-study embedded in a larger sparse cohort is a strong design. The dense subset supports identification of the functional form and the coupling structure; the sparse majority supports estimation of population-level parameters and of between-subject variation with adequate power. Methods developed on the dense subset can then be applied, with the form fixed, to the full cohort.

Where a study has the option, this is preferable to either a uniformly sparse or a uniformly dense design at equal total cost.