Orthogonality: A cross factored design is orthogonal if each of its factors is independent of each other. Two categorical cross factors are orthogonal by design if each level of one is measured at every level of the other. Two covariates are orthogonal only if they have a correlation coefficient of zero. Orthogonal designs partition total variation in the response straightforwardly into independent hypotheses using sequential sums of squares for each effect in turn. Although a balanced design generally (but not inevitably) ensures orthogonality, this can be difficult to achieve in practice, especially with covariates. Loss of orthogonality can reduce or enhance the power of a design to detect effects, and usually requires analysis with the aid of adjusted sums of squares calculated in a General Linear Model (GLM).

 

Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press.

http://southampton.likn.co/~cpd/anovas/datasets/