In article
<[EMAIL PROTECTED]>,
[EMAIL PROTECTED] (William B. Ware) wrote:
> Keeping in mind that it's a textbook, I suspect that the authors were
just
> trying to keep the numbers of numbers small. All replicates within a
cell
> having the same value is rather rare in practice.
>
> However, the greater question appears to be that of violating the
> assumption of homogeneity of variance. ANOVA is robust against such
> violations when the cell sizes are equal...
>
> WBW
Thanks to all for your responses. I wasn't aware that ANOVA was robust
to violations of homoscedasticity, but after reading your note I looked
in my stats collections. Most books mention this, but some only in a
phrase. In Winer et al. 1991 (Statistical principles in exptl design
3rd ed), p. 100-110, there is a good summary of the issues with the main
citation being Glass et al (1972). Winer et al. reproduce a table from
Glass describing potential problems with violations of different ANOVA
assummptions in balanced and unbalanced designs. With equal sample
sizes, the effect on alpha is apparently very minor indeed. Zar
(1999,p.185) also cites Glass et al. (1972) on ANOVA robustness to
heteroscedasticity. Underwood (1989, p. 182-183 cites Box (1953) and
his own work to state that with balanced samples, there is no problem in
balanced designs UNLESS one group has a very large variance compared to
the other groups. In the textbook problem that I assigned my class
(Devore & Farnum 1999, problem 10.2.13, taken from a published study
with citation), two of the 9 groups have identical zero variances. I'm
taking this to mean that this is probably not a problem.
This problem is not at all rare in ecology. Often, a treatment will
have ZERO individuals in all replicate samples. I'd always assumed that
parametric ANOVA should not be used on such data. I may have been too
hasty. In environmental chemistry, a group of samples may all be below
the level of detection. In effect, they would all be zeros. I'd
assumed that parametric ANOVA shouldn't be used for such data, but maybe
there are cases where parametric methods could be used.
I'm still going to recommend to my students that they test for
homoscedasticity, and after inspecting boxplots or variance to mean
plots, use appropriate transformations where appropriate. The zeros
problem isn't amenable to transformation though.
In many years of reading ecological papers, I have never seen a
statement similar to: "Our data violated the homogeneity of variance
assumption, but because we used a balanced design, we don't anticipate a
problem." It sounds like such a statement would be consistent with the
studies in this area.
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