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Analysis of variance with unbalanced data: an update for ecology & evolution
Article first published online: 3 DEC 2009
DOI: 10.1111/j.1365-2656.2009.01634.x
© 2009 The Authors. Journal compilation © 2009 British Ecological Society
Additional Information
How to Cite
Hector, A., Von Felten, S. and Schmid, B. (2010), Analysis of variance with unbalanced data: an update for ecology & evolution. Journal of Animal Ecology, 79: 308–316. doi: 10.1111/j.1365-2656.2009.01634.x
Publication History
- Issue published online: 5 FEB 2010
- Article first published online: 3 DEC 2009
- Received 18 July 2009; accepted 8 October 2009Handling Editor: Karl Cottenie
Keywords:
- adjusted sums of squares;
- anova;
- linear models;
- orthogonality;
- type III sums of squares
Abstract
1. Factorial analysis of variance (anova) with unbalanced (non-orthogonal) data is a commonplace but controversial and poorly understood topic in applied statistics.
2. We explain that anova calculates the sum of squares for each term in the model formula sequentially (type I sums of squares) and show how anova tables of adjusted sums of squares are composite tables assembled from multiple sequential analyses. A different anova is performed for each explanatory variable or interaction so that each term is placed last in the model formula in turn and adjusted for the others.
3. The sum of squares for each term in the analysis can be calculated after adjusting only for the main effects of other explanatory variables (type II sums of squares) or, controversially, for both main effects and interactions (type III sums of squares).
4. We summarize the main recent developments and emphasize the shift away from the search for the ‘right’anova table in favour of presenting one or more models that best suit the objectives of the analysis.