 Lack-of-fit F-test

For $n$ independent normal responses of common variance at $g$ distinct predictor settings, a rank-$p$ normal linear model has residual degrees $n-p$. The saturated setting-mean model has $g$ parameters and pure error degrees $n-g$. Splitting orthogonal residual spaces gives lack of fit degrees $g-p$. Under the fitted mean model, the two squared lengths divided by the common variance are independent chi-squared variables; their ratio of mean squares has the F-distribution shown. This test is unavailable without repeated settings or positive lack-of-fit degrees.