Equal-cell replication variance formula ($\mathrm{Cov}(\hat{\mu })={\sigma }^{2}I/m$)

Suppose a normal linear model gives an unrestricted mean to each cell and has $m$ independent observations per cell, with common error variance ${\sigma }^2$. Each fitted cell mean is its sample mean, with variance ${\sigma }^2/m$; different fitted cell means are independent. Estimate ${\sigma }^2$ by the pooled within-cell residual sum of squares divided by $n-q$, where $q$ is the number of cells. Thus a cell mean has standard error $s/\sqrt{m}$, a difference of two cell means has standard error $s\sqrt{2/m}$, and a two-by-two difference-of-differences has standard error $2s/\sqrt{m}$.

// Target: probability-and-statistics.bigb