# What is the effect of dropping a between by within 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). The mixed anova can be thought of as an amalgamation 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. It then 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. The above also applies equally if B is a factor or a covariate. Van Breukelen and Van Dijk (2007) and Aiken and West (1991) suggest always centering covariates before using them in an interaction. Van Breukelen and Van Dijk (2007) show that without centering covariates the main effects W1 and W2 are correlated with W1 x B and W2 x B respectively however, in any case, care should be taken when interpreting main effects in the presence of an interaction. See also here.

SPSS recognizes this partition by having separate syntax subcommands for specifying patterns of within subjects factors and covariates (/WSDESIGN) and between subjects factors and covariates (/DESIGN).

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

References

Aiken LS and West SG (1991) Multiple Regression: Testing and Interpreting Interactions. Sage:London.

Van Breukelen GJP and Van Dijk KRA (2007) Use of covariates in randomized controlled trials. *Journal of the International Neuropsychological Society* **13** 903-904.