Orthogonal Contrasts for both Balanced and Unbalanced Designs and both Ordered and Unordered Treatments

We consider designs with t treatments, the ith level of which has ni observations. Four cases are examined: treatment levels both ordered and not, and the design balanced, with all ni equal, and not. A general construction is given that takes observations, typically treatment sums or treatment rank sums, constructs a simple quadratic form and expresses it as a sum of squares of orthogonal contrasts. For the case of ordered treatment levels, the Kruskal–Wallis, Friedman and Durbin tests are recovered by this construction. A dataset where the design is the supplemented balanced, which is an unbalanced design in our terminology, is analyzed. When treatment levels are not ordered the construction also applies. We then focus on Helmert contrasts.