Balanced Hypotheses and Unbalanced Data

Abstract

In this article we consider the fixed-effects analysis of variance model with unbalanced data. We study the standard main-effects and interaction hypotheses normally tested with balanced data and determine simple, necessary, and sufficient conditions for testing the equivalent hypotheses with unbalanced data involving missing cells. These conditions depend on the pattern of missing cells. They are expressed in terms of the ranks of the design matrices for the full model and reduced models related to the hypothesis under test. A simple iterative method is given to determine these ranks without storing the design matrices.

Key Words:

• Analysis of variance
• Hypothesis testing
• Unbalanced data
• Rank computations