A necessary and sufficient condition is given for a specified factorial effect to be orthogonal to every other factorial effect, after adjustment is made for blocks. The results are extended to the case of regular disconnected designs. The structure of a generalized inverse of the intrablock matrix is investigated when certain pairs of factorial spaces are orthogonal. A useful class of designs exhibiting partial orthogonal factorial structure is identified and examples are given.
"Orthogonality of Factorial Effects." Ann. Statist. 14 (2) 743 - 752, June, 1986. https://doi.org/10.1214/aos/1176349951