Develop a sensitivity analysis for the unrecoverable confounder component

Develop a sensitivity analysis, under a probit treatment model, for the residual bias term associated with the unrecoverable confounder component in the recoverU+ estimator, using its closed-form dependence on the correlation and scale of that component.

Background

The recoverU+ estimator recovers only the component of the spatial confounder orthogonal to the treatment field. The treatment-correlated component remains unidentified and produces the residual bias term b_UA, so the estimator is only partially doubly robust. The paper notes that, under a probit treatment model, this residual bias has a closed form in terms of the correlation and scale of the unrecoverable component. Developing a sensitivity analysis based on this relationship would provide a practical way to assess how strongly the unrecoverable confounding component could affect causal conclusions.

References

Third, under a probit treatment model $b_{U_A}$ has a closed form in the correlation and scale of $U_A$ (by standard inverse-Mills-ratio algebra for a jointly Gaussian confounder and latent treatment index), anchoring a sensitivity analysis in the spirit of the E-value \citep{vanderweele2017}; developing that sensitivity analysis is beyond the scope of this paper and is left to future work.

— A unified framework for estimating direct causal effect under spatial confounding and interference, with the R package spaci  (2609.28799 - Ogunsola et al., 23 Sep 2026) in Section 4.2, subsection “Partial double robustness”