and using the values ofand in (2) and (3), we can finally express as follows:
In order to check for orthogonality ofto and where g ≠ i, h ≠ j, we employ Equation 10. Consider the product
As was shown previously, for equal or proportional n's
and in general
We only need to expand the first two terms of Equation 2:
A similar proof establishes the result for
Thus we have defined four sets of variables, G,, , and , with the property, in the case of proportional n's, that any member of one set is orthogonal to any member of the other sets. The last three sets can be used as predictors of the dependent variable in a two-way experiment with proportional n's, knowing that they will account for nonoverlapping variance.
Appelbaum, M. I., & Cramer, E. M. ( 1974). "Some problems in the nonorthogonal analysis of variance." Psychological Bulletin, 75, 335-343.
Blair, R. C., & Higgins, J. J. ( 1978). "Tests of hypotheses for unbalanced factorial designs under various regression/coding method combinations." Educational and Psychological Measurement, 38, 621-631.
Bogartz, W. ( 1975). "Coding dummy variables is a waste of time: Reply to Wolf and Cartwright, among others." Psychological Bulletin, 82, 80.