# Analysis of variance with unbalanced data: an update for ecology & evolution

Hector, A; von Felten, S; Schmid, B (2010). Analysis of variance with unbalanced data: an update for ecology & evolution. Journal of Animal Ecology, 79(2):308-316.

## 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.

## 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.

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