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Anova for Unbalanced Data: An Overview

Ecological studies typically involve comparison of biological responses among a variety of environmental conditions. When the response variables have continuous distributions and the conditions are discrete, whether inherently or by design, then it is appropriate to analyze the data using analysis of variance (ANOVA). When data conform to a complete, balanced design (equal numbers of observations in each experimental treatment), it is straightforward to conduct an ANOVA, particularly with the aid of the numerous statistical computing packages that are available. Interpretation of an ANOVA of balanced data is also unambiguous. Unfortunately, for a variety of reasons, it is rare that a practicing ecologist embarks on an analysis of data that are completely balanced. Regardless of its cause, lack of balance necessitates care in the analysis and interpretation. In this paper, our aims is to provide an overview of the consequences of lack of balance and to give some guidelines to analyzing unbalanced data for models involving fixed effects. Our treatment is necessarily cursory and will not substitute for training available from a sequence of courses in mathematical statistics and linear models. It is intended to introduce the reader to the main issues and to the extensive statistical literature that deals with them.