Experimental functional regression

Use this route when the response is a continuous functional outcome and the scientific question is how declared scalar experimental predictors change that response over time.

1. Define the response and predictor design

Construct a complete common-grid functional response and an aligned scalar design table. Encode interactions explicitly. The package does not choose categorical coding, interactions, centering or scaling automatically.

2. Choose the correct repeated-measures route

For independent curves, fit

fit = fit_function_on_scalar_regression(
    trajectories,
    design,
    predictors=("condition",),
    dimension="response",
)

If repeated participant trials exist, participant averaging is only defensible when the predictor is participant-constant and that aggregation matches the scientific estimand. Trial-varying predictors should move to the repeated-trial mixed-effects workflow.

3. Use whole-function inference

For simultaneous coefficient inference, use the fixed-design wild bootstrap provided for function-on-scalar regression rather than reading pointwise intervals as whole-function evidence.

bootstrap = bootstrap_function_on_scalar_coefficients(
    fit,
    n_bootstrap=1000,
    random_state=55,
)

4. Interpret coefficients as functions

A coefficient \(\beta_k(t)\) is the expected change in the functional response at time (t) for a one-unit change in the declared predictor, conditional on the fitted scalar design. Report the time domain over which simultaneous coverage is claimed.

5. Report the complete contract

Use function_on_scalar_reporting_text(). Report predictor coding, basis or grid contract, uncertainty method, bootstrap seed/size, simultaneous scope and any aggregation performed upstream.

Advanced branches

Scalar-on-function FPCR and heteroscedastic fixed-target wild-bootstrap inference answer different questions. They are not interchangeable with this canonical function-on-scalar route.