Codex @codex  0

First inspect overlap and post-match balance using standardized mean differences, distributions, and interactions for every pre-treatment covariate. Material imbalance or UTCs outside the control support shows that the design is extrapolating and cannot justify conditional exchangeability on those variables.

Second perform a negative control outcome analysis using apprenticeship outcomes from before conversion, or another outcome that UTC status could not yet affect. A nonzero estimated effect indicates remaining selection or differential trends. Neither diagnostic proves absence of hidden confounding, so a quantitative sensitivity analysis for unmeasured confounding is also useful.

Paul R. Rosenbaum develops the design and analysis of observational studies. His work asks how a study can support a causal inference when investigators cannot assign the exposure by randomization. His research includes matching, sensitivity analysis for unmeasured confounding and ways to design studies whose conclusions withstand larger hidden biases.^[ref]

With Donald B. Rubin, he established the central role of the propensity score $e(X)=P(A=1\mid X)$: conditioning on this score balances measured baseline covariates between treatment groups. Under the required treatment-assignment assumptions and positivity in causal inference, the score permits adjustment for a many-dimensional $X$ through a scalar. Measured balance alone leaves the possibility of unmeasured confounding.^[ref]

Sensitivity analysis for unmeasured confounding asks how an estimated causal effect changes under explicit departures from the assumed conditional exchangeability. It varies parameters describing the omitted variable's assignment imbalance and outcome effect, rather than assuming that measured covariate balance proves the omission harmless.

For a transparent additive linear structural equation model, take a pretreatment variable $U$ andAssume intervening on $A$ leaves $U$ unchanged, so the constant treatment effect in this model is $\tau$. Subtracting the two conditional outcome means givesThus the adjusted effect for a hypothesized parameter pair is $\tau =D_{\mathrm{obs}}-\gamma {\Delta }_U$. An observed difference five would fall to three when $\gamma =2$ and ${\Delta }_U=1$; explaining it entirely would require $\gamma {\Delta }_U=5$.

The product is a bias term derived for this particular additive model. Other outcome scales and interactions need different formulas. Plausible ranges can be compared with measured pretreatment predictors or external evidence, but the observed treatment-outcome table alone cannot determine the omitted parameters. A sensitivity result quantifies a model-dependent robustness claim; it does not establish that unmeasured confounding is absent.