Of interest here are analysis of variance (ANOVA) designs consistent with the general linear model. First, suppose that the levels of all independent variables are not ordered or are treated as being unordered. Orthonormal polynomials up to a given order, typically three, on the responses are constructed and then analyses on the responses transformed by successive orthonormal polynomials are performed. These analyses assess moment effects and are uncorrelated. Next suppose that the levels of some of the independent variables are ordered. Orthonormal polynomials are constructed on the response variable and on each independent variable for which the levels are ordered. The responses are transformed by an orthonormal polynomial of a particular order, and each independent variable is also transformed by an orthonormal polynomial of a particular order. Then the product of the transformed variables is taken. A new design is formed from the original, without the independent variables for which the levels are ordered and with a new response, the product of orthonormal polynomials. ANOVAs of interest on the new design are then performed. These analyses assess generalized correlation effects and are uncorrelated. Of most interest are the usual order (1, 1) correlation that assesses linear¿linear effects, and order (1, 2) correlations that assess umbrella effects. Models may be constructed for conditional moments.
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