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A note about unequal group sizes in ANOVA

Howell (1992) pp 410-417 and 542-549 illustrates and recommends the use of the (default) Type III sums of squares (termed Method I by Howell) for looking at sources of variation in an anova where the group sizes differ. This is provided the differences in group sizes are independent of the groups.

The computations for these may be done using either a least squares approach based upon formulating the anova in terms of a linear regression (to generate Type III sums of squarres) or by using the formulae associated with equal groups as given, for example, [:FAQ/ss: here] but replacing the i-th group sample size ($$n_text{i}$$) with its harmonic mean, $$\bar{n}_text{i}$$:

$$\bar{n}_text{i}$$ = Harmonic mean of k means = $$ \frac{k}{\frac{1}{n_text{1}} + ... \frac{1}{n_text{k}}} $$

Using harmonic means, or Type III sums of squares, in an anova gives an unweighted means analysis which is an equally weighted analysis of group means because group means are treated equally irrespective of the size of the group upon which they are based.

Tabachnick and Fidell pp 217-218 als discuss sums of squares. They recommend Type III sums of squares for use for equally weighted groups and type II sums of squares when the sample sizes reflect the importance of the groups, e.g. when it is desired to consider the relative frequency of their occurrence in the general population (giving rise to a weighted means approach).

Harmonic means are also useful for averaging rates - see this [:FAQ/hmeg:example.]

Reference

Howell DC (1992) Statistical Methods for Psychology. Third Edition. Duxbury Press:Belmont, CA.