Variances of higher-order vs. lower-order random terms

If I understand your question correctly, I think there is one exception.
Whenever you are dealing with balanced 2 x 2 x 2 x ... design all main
effects and interactions have equal statistical power.

Say what? A main effect in that case is a difference between two averages. A (1st order) interaction effect is a difference between two differences of means, each based on half as many observations. So the standard error in the latter case is twice that of the former. Of course, you can scale the interaction so that the s.e. is the same as that of the main effects, but it is then not clear that similarly sized effects are equal in importance. At the very least, you need to be very specific about what you mean by "equal power".

Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com