The unweighted ``harmonic mean'' solution for unbalanced anova designs : A detailed argument - Laurencelle, Louis (Tutorials in Quantitative Methods for Psychology, 2017)
The treatment of unbalanced designs in analysis of variance (anova) has a long and still controversial history, an issue being the choice between the so-called harmonic mean or unweighted solution and the classical weighted solution. We here argue in favour of the unweighted, i.e. equally weighted solution, based on the following reasons. The classical solution gives more weight to the means obtained from a more numerous group of data, thus inducing a positive bias in the computation of the between-group mean square, irrespective of the groups' effect sizes. Indeed, this differential weighing is at variance with the determination and handling of effect sizes, whose values are kept free of the various group sizes implied, so that the final weighted' $F$ statistic cannot stand for a truthful reflection of those. Besides, the oft-quoted argument around the demographic representativeness of the various groups compared is specious in the context of most anova applications, the purpose of anova being to compare groups/conditions one to the other, whatever their sample sizes. Finally, in the cases of two- or multi-way designs, the weighted solution precludes the calculation of truly orthogonal and additive variance components, the linear regression' alternatives for this problem being complex and essentially arbitrary. The harmonic mean solution preserves orthogonality and additivity in the variance decomposition for multi-dimensional designs, is congruent with effect sizes and entails no differential bias in the calculation of the $F$ test whatever the sample sizes. On the other hand, it suffers from a positive bias in the $F$'s significance, a bias negligible for mildly unbalanced group sizes and aptly corrected by Rankin (1974) modified degrees of freedom.