Estimability from within-block differences

For a linear regression model $Y_{bj}={\alpha }_b+f(x_{bj}{)}^T\beta +{ϵ}_{bj}$ with one unrestricted intercept per two-run experimental block, differencing the responses removes all experimental block effects. The parameter vector $\beta$ is identifiable precisely when the matrix whose rows are $f(x_{b1})-f(x_{b2})$ has full column rank. This criterion applies to arbitrary pairings, including nonregular factorial designs; requiring regular orthogonal confounding is a stronger constraint.