The κ curvature diagnostic

The delta method is a first-order Taylor approximation. The approximation breaks when the estimand is too curved in β for the local linearization to track the true sampling distribution. The κ diagnostic measures that curvature and, when it crosses a threshold, auto-falls-back to simulation.

Definition

For an estimand h(β) with gradient g = ∂h/∂β |_{β̂} and Hessian H = ∂²h/∂β² |_{β̂}, define the whitened gradient and Hessian

\[ \tilde g = L^\top g, \qquad \tilde H = L^\top H L, \]

where L L^⊤ = V̂ is the Cholesky factor of the parameter covariance. Skovgaard’s relative curvature in this metric is

\[ \kappa = \frac{\lVert \tilde H \rVert}{\lVert \tilde g \rVert^2}. \]

The whitening transform is critical: without it κ is parameterization-dependent and uninterpretable. With it, κ has the property that an affine change of variables leaves it invariant.

Calibration

The default thresholds are taken from the nonlinear-regression literature:

κ value

Interpretation

κ < 0.1

delta method is highly reliable

0.1 ≤ κ < 0.3

borderline; delta usable but report κ

κ ≥ 0.3

delta unsafe; auto-fall-back to simulation

These are configurable via kappa_threshold= on the session.

What the fallback does

When κ exceeds the threshold and the session was constructed with method="delta", the call recomputes inference via Krinsky–Robb simulation. The result records:

  • the realized inference method ("simulation", not the requested "delta");

  • the fallback reason ("kappa exceeded");

  • the κ value itself, surfaced on result.summary().

This is meant to be loud. pymargins’ position is that silent delta-method use on highly curved estimands is the most common inference bug in published applied work, and that a tool that just computes the delta number — as Stata’s margins and R’s marginaleffects both do — gives the analyst no way to know they should have used something else.

Disabling

  • kappa_threshold=float("inf") — keep the requested method, never fall back.

  • diagnostics=False — skip κ entirely (useful in tight loops where the second derivative is expensive).

See Inference scale (phi / phi_inv) for why picking the right scale is the first move before tightening κ tolerance.