8.2 Two-Way MANOVA

The model above has a single factor. When units are classified by two factors simultaneously the univariate two-way analysis of variance generalises in the same way, and the additional term it brings — interaction — is usually the one of real interest.

For factors \(A\) at \(g\) levels and \(B\) at \(b\) levels, with \(n\) replicates in each cell, the model is \[\underline {X}_{\ell k r} = \underline {\mu } + \underline {\tau }_{\ell } + \underline {\beta }_{k} + \underline {\gamma }_{\ell k} + \underline {e}_{\ell k r},\] where \(\underline {\tau }_\ell \) is the effect of level \(\ell \) of \(A\), \(\underline {\beta }_k\) that of level \(k\) of \(B\), \(\underline {\gamma }_{\ell k}\) their interaction, and the errors are independent \(N_p\left (\underline {0},\Sigma \right )\). The effects are constrained to sum to zero over each index, as in the univariate case.

The total sum of squares and cross-products decomposes exactly as the univariate total sum of squares does, but into matrices rather than scalars: \[SSP_{\text {tot}} = SSP_{A} + SSP_{B} + SSP_{AB} + SSP_{\text {res}} .\]

Each effect is tested by its own Wilks’ lambda, formed against the residual matrix: \[\Lambda _A = \frac {\left |SSP_{\text {res}}\right |} {\left |SSP_A + SSP_{\text {res}}\right |},\] and similarly for \(B\) and for the interaction, each referred to its own degrees of freedom by the approximations of Section 5.5.