Sub-Balanced Data and the Mixed Analysis of Variance

Abstract

In the mixed model, it is well known that balanced data are a sufficient condition for an ANOVA with terms that are independent multiples of chi-squared variables (called a proper ANOVA). In this article a weaker condition, which is both necessary and sufficient, is determined and shown to be equivalent to what is defined here as sub-balanced random effects, a condition easily checked in practice. The usual method of generating an ANOVA by fitting sums of squares for a given ordering of the effects in the model is sometimes inadequate with sub-balanced data, requiring instead a spectral decomposition of the covariance matrix. An example is given.

Key Words:

• Variance components
• Balanced data
• Sub-balanced data
• Analysis of variance
• Mixed model