Several key concepts are introduced: the propensity score, ignorable treatment assignment, and the principal unobserved covariate. The data in an observational study are from a population, and these concepts describe that population. The propensity score \(e\left (\mathbf {x}\right )\) is the conditional probability of treatment given the observed covariates, \(e\left (\mathbf {x}\right ) = \mathrm {Pr}\left (Z=1 \, | \, \mathbf {X}=\mathbf {x}\right )\) , and it is determined by the distribution of observable quantities, (R, Z, X); so, \(e\left (\mathbf {x}\right )\) can be estimated from the observable data. In contrast, the principal unobserved covariate is \(\zeta = \zeta (r_{T}, \, r_{C}, \, \mathbf {X}) = \mathrm {Pr}\left (Z=1 \, | \, r_{T}, \, r_{C}, \, \mathbf {X}=\mathbf {x}\right )\) , and it is not a function of observable quantities, (R, Z, X), because (rT, rC) are never jointly observed. Treatment assignment is ignorable given the observed covariates X if \(0 < e\left (\mathbf {x}\right ) = \zeta (r_{T}, \, r_{C}, \, \mathbf {X}) < 1\) , that is, if the propensity score equals the principal unobserved covariate and is never 0 or 1. If treatment assignment was ignorable given the observed covariates X, then causal inference would be comparatively straightforward; so, the central problem in observational studies is the absence of grounds for believing, and the abundance of grounds for doubting, that treatment assignment is ignorable given X. If 0 < ζ < 1, then treatment assignment is always ignorable given (X, ζ); so, the central problem in observational studies can always be expressed in terms of one scalar unobserved covariate, ζ, where 0 ≤ ζ ≤ 1.