> The unweighted means method for unbalanced ANOVA designs uses the
> harmonic mean of the cell sample sizes. I wonder if anyone can explain
> why the harmonic mean, as opposed to any other kind of "average" sample
> size. None of the books I've read comment on this--and I just know some
> keen student will ask me this next week!
The other comments get at the effect of using a harmonic mean (i.e.,
weighting toward the smaller sample size), but I believe the "reason" it is
used comes straight from the derivation. I don't have the ANOVA derivation
right at hand, but if you look at the t-test for unequal sample sizes you
will see that the harmonic mean is there too, although we don't really think
of it as such because there are only two numbers being averaged (e.g., the
1/n[a] + 1/n[b] are usually just written out). When you are dividing by the
weighted variance estimates, the harmonic mean of the weights pops out when
you assume homogeneity of variance.
--
Patrick Phillips
Biology Department
University of Oregon
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