Rayner, J C W; Best, D J.; and Thas, O, Extended analysis of at least partially ordered multi-factor ANOVA, National Institute for Applied Statistics Research Australia, University of Wollongong, Working Paper 06-14, 2014, 21.
For multifactor experimental designs in which the levels of at least one of the factors are ordered we show how to construct components that provide a deep nonparametric scrutiny of the data. The components assess generalised correlations and the resulting tests include and extend the Page and umbrella tests. Application of the tests described is straightforward. Orthonormal polynomials on the ANOVA responses and the factors need to be constructed. Products of at least two of these orthonormal polynomials are then used as inputs into standard ANOVA routines. For example using the first order orthonormal polynomia l on factor A and the first/second order orthonormal polynomial on the ANOVA response will assess, if, for example, with increasing levels of factor A the response increases or increases and then decreases.