Inference scale (phi / phi_inv)

The session-level pair (phi, phi_inv) picks the inference scale — the scale on which the delta method and the κ diagnostic are computed, and from whose CI endpoints the reporting scale is back-transformed.

Mathematically, every estimand is the triple (h, phi, phi_inv):

  • h(β) → scalar or vector, evaluated on the inference scale;

  • phi, applied to CI endpoints to lift them to the reporting scale;

  • phi_inv, used to convert user-supplied null values for tests onto the inference scale.

Why is this a session commitment?

Two reasons.

κ has a stable referent only when the scale is fixed. The curvature diagnostic measures how nonlinear the estimand is in β. That number depends entirely on the scale. Fixing (phi, phi_inv) at session level is what makes the session-level κ comparable across calls.

Inter-call composability requires a shared scale. Two results from the same session can be subtracted, added, scaled, and combined through MarginsResult.__sub__ etc., and the combined object inherits the joint covariance. That arithmetic is only meaningful if the operands live on the same inference scale.

Picking a scale

The rule of thumb is that the contrast should be as linear as possible in β. That keeps κ small and the symmetric Wald CI honest.

Quantity

Natural scale

Helper

additive effect on response

identity

Margins.linear_scale

rate ratio, risk ratio, hazard ratio

log

Margins.log_scale

odds ratio, probability contrast

logit

Margins.logit_scale

correlations

Fisher z (tanh)

Margins.correlation_scale

The constructor also accepts arbitrary phi= / phi_inv= JAX callables for custom scales.

What changes when you change scale?

  • The point estimate is computed by h(β̂) on the inference scale, then back-transformed by phi. Different scale, different reported estimate.

  • The CI is symmetric on the inference scale, asymmetric on the reporting scale.

  • κ is recomputed per scale, so the fallback decision may change.

  • The hypothesis test in .test(value=...) interprets value on the reporting scale and lifts it via phi_inv.

See The κ curvature diagnostic for the curvature-fallback link.