Generalized binary and count functional responses

Use this route for repeated non-Gaussian functional responses when the target is a marginal / population-averaged effect of scalar predictors.

1. Declare the observation contract

Choose exactly one of the supported representations:

• Bernoulli: (Y_{ij}(t)in{0,1});
• grouped binomial: integer successes (S_{ij}(t)) plus explicit positive integer denominators (N_{ij}(t));
• Poisson expected count: non-negative integer counts;
• Poisson rate: non-negative integer counts plus explicit positive exposure (E_{ij}(t)).

Do not supply an arbitrary proportion for grouped binomial data and do not use exposure merely to normalize a count trajectory.

2. Fit the marginal GEE

fit = fit_generalized_function_on_scalar_regression(
    trajectories,
    design,
    predictors=("condition",),
    participant_column="participant_id",
    dimension="response",
    family="binomial",
    basis_size=4,
    spline_degree=2,
)

Participants are independent clusters. Working independence is fixed and coefficient uncertainty uses the robust cluster sandwich covariance.

3. Preserve denominator/exposure information

For grouped binomial observations, denominators travel with successes in every participant bootstrap. For Poisson rate models, exposure travels with counts. Neither is inferred or perturbed independently.

4. Use fixed-profile prediction only for declared scientific targets

Bernoulli/grouped-binomial fits predict marginal success probability. Exposure-adjusted Poisson fits distinguish rate from expected count; expected count requires explicit target exposure.

5. Report the marginal estimand

Use generalized_function_on_scalar_reporting_text(). State family/link, response coding, participant cluster, grouped denominator or exposure definition, basis, robust covariance, bootstrap contract and whether prediction targets were extrapolative.

Scope boundary

This is not a generalized functional random-effects model. Negative binomial, zero-inflated, hurdle and Tweedie families are intentionally outside the 0.51-0.54 canonical observation contract.