Sensitivity of the average treatment effect to unmeasured confounding

The \(\Gamma\)-model sensitivity bound of Theorem 6.1 in Ospina (2026): a sharp, Hölder-duality bound on how far an unmeasured confounder of odds-ratio strength at most \(\Gamma\) can move the estimate, calibrated by the per-unit causal influence vector CIC (Theorem 3.6). The bound pairs the box constraint the \(\Gamma\)-model places on confounding with the \(\ell_1\) norm of CIC, not the \(\ell_2\) norm; see the paper for why the \(\ell_2\) pairing, including one built from the curvature BdT itself, is not sharp and was a genuine error in an earlier draft of this method.

Usage

bic_sensitivity(cv, fit, treat, data, tau_hat, eps = 1e-04)

Arguments

cv

A "bic_curvature" object from bic_curvature().

fit, treat, data

As in bic_gcomp(); used to compute the gradient of the estimated effect with respect to the model parameters.

tau_hat

The estimated average treatment effect (e.g. bic_gcomp(fit, treat, data)["tau"]), used to solve for Gamma_star.

eps

Step size for the numerical gradient.

Value

A list with CIC (the per-unit causal influence vector), L1 (\(\|\mathrm{CIC}\|_1\)), Gamma_star (the smallest \(\Gamma\) at which the bound reaches |tau_hat|, or NA if the bound already exceeds it at Gamma = 1), and bound (a function of Gamma returning the bound itself).

Details

Gamma_star answers one question: how strong would an unmeasured confounder need to be, on an odds-ratio scale, before it could plausibly explain away the estimated effect? Gamma_star far above 1 means the conclusion survives all but an implausibly strong confounder; Gamma_star close to 1 (or NA, meaning the bound already exceeds |tau_hat| with no confounding at all) means the bound cannot rule out even a weak one. A small Gamma_star is not necessarily a defect of the data or the estimate: at large sample sizes, the worst case this bound allows requires every unit's confounding to align adversarially with the sign of that unit's own contribution, a configuration no single realistic confounder produces, so the bound can be sharp for the box constraint on confounding it assumes while still not being informative in practice; see the vignette and Ospina (2026), Section 8, for a worked example where this happens.