What is the effect of dropping a between subjects interaction on other terms in a mixed anova?
Suppose we have a mixed anova with one between subjects factor (B) and two within subject factors (W1 and W2). Assuming there is no three-way interaction B x W1 x W2 o the outputted mixed anova can be thought of as an amalgammation of four separate anovas each with their own error sum of squares. These comprise B and the error across subjects; W1, W1 x B, W1 x subjects error, W2, W2 x B, W2 x subjects error and W1 x W2, B X W1 x W2 and the W1 x W2 x subjects error. The error terms are not independent - for example the presence of B influence the variation across subjects which inputs into all four error terms.
Let's assume the two interactions involving W1 x W2 are not statistically significant then it follows that W1 and W2 are independent of each other as are the B x W1 and B x W2 terms. This follows since within subjects factors influence variation within subject and not between subject so neither W1 nor W2 influence the (between) subjects error term, and since we are assuming there is no W1 x W2 interaction, each other.
It follows that the removal of the W2 x B interaction from the model does not influence either the W1 or W1 x B sources of variation and vice-versa.