Confirmation of the non-orthogonality of the W and W x B interaction when covariate B is not centred
The covariance between two terms is a function of their cross-product and it follows if the sum of these cross-products equals zero the two terms are orthogonal. The sums of squares of two orthogonal sources of variation are not changed by the presence of either source of variation in the anova.
Van Breukelen GJP and Van Dijk KRA (2007) state that if a between subjects factor, W1, is orthogonal to the W1 x B interaction, for a between subjects covariate, B, only if B is centred about its mean. The example below shows how this is shown using cross-products.
Suppose W1 = (1,-1,1,-1) and B = (1,6,2) [for three subjects] then
Centred B = (-2,3,-1) and
W1 x B = (1,-1,1,-1,6,-6,6,-6,2,-2,2,-2)
W1 x B centred = (-2,2,-2,2,3,-3,3,-3,-1,1,-1,1)
Sum of terms in W1 x ( W1 x B ) = (4 x (1+6+2)) > 0.
Sum of terms in W1 x (W1 x B centred) = 4 x (-2+3-1) = 0.
Therefore the W1 sum of squares is only orthogonal to the W1 x B sum of squares if the covariate, B is centred since the sum of deviation about the mean always sums to zero.
Van Breukelen GJP and Van Dijk KRA (2007) Use of covariates in randomized controlled trials. Journal of the International Neuropsychological Society 13 903-904.