Component Analysis When Testing for Fixed Effects in Unbalanced ANOVAs

In possibly unbalanced fixed effects in ANOVAs, we examine both parametric and nonparametric tests for main and two-way interaction effects when the levels of each factor may be ordered or unordered. For main effects, we decompose the factor sum of squares into one degree of freedom components involving contrasts, albeit not necessarily orthogonal contrasts. For interactions, we develop what we call coefficients. These are an extension of part of the interaction sum of squares in potentially unbalanced designs. They may be used to test nonparametrically for focused interaction effects. The tests developed here provide focused and objective assessments of main and interaction effects and augment traditional methods.