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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.

One way to see this is using the following [:FAQ/hmeg:example.]

Reference

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