# How do we know to which sources of variation B x W interactions are orthogonal?

The covariance term in least squares regressions is of the explicit form $$(X^{text{T}X)}text{-1}$$. It follows that two terms in an analysis of variance are orthogonal (ie the inclusion of the terms does not effect the sums of squares involved in computing the F ratio of the other terms) if the cross product of their predictor values, X, is zero.

If there are two centred between subject covariates B1 and B2 and 2 within subject factors W1 and W2 then B1 x W1 and B2 x W1 are orthogonal to each other but not to any other terms in the anova. To see this suppose we have three subjects taking values 2, -1 and -1 and 3, -6 and -3 respectively on B1 and B2. Suppose W1 and W2 have two levels defined by contrast coefficients (1, 1, -1, -1) and (1, -1, 1, -1) respectively. This gives 12 predictor combinations (4 for each subject) for each Wi with Bi (i=1,2).

The cross-product term

W1 x B1 has predictor coefficients (2,2,-2,-2,-1,-1,1,1,-1,-1,1,1),

W1 x B2 has predictor coefficients (3,3,-3,-3,-6,-6,6,6,3,3,-3,-3),

W2 x B1 has predictor coefficients (2,-2,2,-2,-1,1,-1,1,-1,1,-1,1) and

W2 x B2 has predictor coefficients (3,-3,3,-3,-6,6,-6,6,3,-3,3,-3).

The sum of the cross-products of (W1 x B1, W2 x B1), (W1 x B1, W1 x B2), (W2 x B1, W2 x B2), (W1 x B2, W2 x B2) are zero so these pairs of terms are orthogonal. It can also be shown that the error terms, Wi x subjects, are both orthogonal to their respective B x Wi interactions as these are just extensions of W1 and W2, as described above.

The pairs (W1 x B1, W1 x B2) and (W2 x B1, W2 x B2) contain elements which are not orthogonal to each other which means that Wi x B1 and Wi x B2 need to be included together in the anova with Wi and Wi x subject terms and the Wi x B1 terms removed, as necessary, in a stepwise manner, assuming there is no interaction involving W1 x W2.

Since no term involving a between subjects covariate interaction with W1 influences any term involving W2 and vice-versa two separate anovas may be computed involving W1, B1 and B2 and W2, B1 and B2 respectively with the comparative importance of W1 x B1 and W1 x B2 being assessed by the former and W2 x B1 and W2 x B2 by the latter.