Treatment contrast (${\sum }_i{c}_i{\mu }_i,{\sum }_i{c}_i=0$)

A treatment contrast is a linear contrast of cell means comparing treatment responses, with coefficients adding to zero. Pairwise differences and marginal factorial differences are examples. Its variance depends on the true allocation units and their covariance matrix.

At equally replicated quantitative treatment levels, evaluations of orthogonal polynomials generate pairwise orthogonal treatment contrasts, using the replication-weighted inner product. Five equally spaced levels give four contrast directions representing degrees one through four, besides the constant direction. This decomposes treatment variation without assuming the response has low polynomial degree.

For a vector of estimated treatment means $\hat{\mu }$, the variance of a treatment contrast with coefficient vector $c$ is $c^T\mathrm{Cov}(\hat{\mu })c$. Averaging subsamples from a common experimental unit does not make their shared variance component disappear.

An interaction contrast compares a treatment difference across levels of another factor. A nonzero difference of differences shows that an additive representation of those cell means is inadequate. Its standard error must use the error stratum in which that interaction term is randomized.