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For binary $A$, conditional covariance satisfiesBy conditional exchangeability and consistency of potential outcomes, the difference in braces is $\mathbb{E}[Y(1)-Y(0)\mid X]$. Consequentlyso the required weight isThis overlap weight emphasizes covariate strata with treatment probabilities near one half and downweights strata near a violation of positivity in causal inference.

By consistency of potential outcomes and conditional exchangeability,Taking conditional expectations given $X_1=x_1$ and subtracting the cases $z=1$ and $z=0$ proves the outcome-regression identification formula.

The raw difference is generally not a causal effect. Existing UTCs may differ systematically from non-UTCs in location, admissions, pupil composition, prior attainment, resources, and the local apprenticeship market. These are possible confounders of school type and outcome. The comparison also weights the 43 UTCs and roughly 3500 controls very differently from a clearly defined target population. Without conditional exchangeability, positivity, and consistency, the difference mixes the effect of UTC status with selection bias.

The propensity score for binary treatment $A$ and pretreatment covariates $X$ is $e(X)=P(A=1\mid X)$. It is a balancing score:This distributional fact does not require conditional exchangeability of the potential outcomes. It follows from the definition of the true assignment probability.^[ref]

For a bounded function $g$, condition successively on $X$ and on $e=e(X)$:Also $P(A=1\mid e)=e$, so division by $e$ gives the same conditional covariate distribution among treated units as in the full score stratum; the analogous argument uses $1-e$ for controls. Comparison of both arms requires $0<e<1$.

Identifying a causal effect is an additional step. If $(Y(0),Y(1))\perp A\mid X$, then the balancing property lets the scalar propensity score replace the full measured $X$ in that exchangeability statement. Without this causal assumption, balancing measured covariates leaves possible unmeasured confounding. An estimated score also need not deliver exact finite-sample covariate balance; its estimation and the subsequent matching in causal inference or weighting procedure must be checked.

A randomized controlled trial assigns experimental units to competing treatments by a specified random mechanism and compares their outcomes. Under complete randomization of $N$ units, exactly $n_1$ receive assignment one and $n_0=N-n_1$ assignment zero. With fixed potential outcomes, each unit has assignment probability $n_1/N$. ThereforeThis proves design-unbiasedness of the mean difference for the finite-population average treatment effect, assuming consistency in causal inference and no interference in causal inference.

Randomization controls the assignment distribution; it does not force every realized baseline covariate difference to be zero. Equal assignment probabilities yield unconditional exchangeability in the design, whereas covariate-dependent randomization can require conditional exchangeability and an analysis respecting that design.^[ref]

When assigned and received treatments differ, an intention-to-treat analysis estimates an assignment-policy effect. A treatment-receipt effect needs additional assumptions. Allocation concealment, blinded outcome assessment and handling of missing follow-up address other sources of distortion beyond random assignment itself.

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.