(Reply to OP and to edstat. -- DFB)
I'll address what appears to be the core of the matter:
On Thu, 11 Mar 2004, Steve wrote in part:
> So maybe the result of non-random assignment are correlations among
> factors, producing overlapping SS terms?
If the "factors" are correlated, the several SS may not be additive in
the usual and convenient sense; and they may be said to be overlapping
in the sense that some of the variance present can be accounted to more
than one source. But the situations you have so far described do not
appear to me to produce correlated factors.
> After all, ANOVA models require group independence.
Well, no, they don't, not in the meaning one usually understands by
"group", as a level of one of the factors: e.g. the "experimental
group" or the "control group" in a manipulated factor, or "male" and
"female" in a factor not subject to the experimenter's manipulation.
What is required of the *groups* is that they be disjoint (aka "mutually
exclusive"): that members of one groups are not also members of another
group (in the same factor).
[Aside: We commonly find that students new to statistics often confuse
or confound the ideas of "independent" and "mutually exclusive" -- this
confusion was the topic of a recent thread, actually. Perhaps you are
suffering effects of the same misunderstanding. "Mutually exclusive"
means that if a person (represented, say, in your data) is male, that
person cannot also be female; if "male" and "female" were independent
categories, so that membership in the one was quite unrelated to
membership in the other, some persons could be both M and F (and some
persons could be neither). (If using this variable distresses you,
substitute "experimental group" and "control group" for the sexes.)]
> One might make the argument that when only two factors are modeled and
> if one involves true random assignment, you won't necessarily end up
> with correlated factors...perhaps I'm wrong on that?
Yes; certainly so for your examples. You originally wrote:
" Let's consider one of three situations:
1) the number of participants per cell varies in proportion to the
respective populations
2) the experimenter uses quota sampling to ensure equal cell sample
sizes
3) when applicable, cut scores are chosen in order to produce equal
cell-sizes. "
These all, if I understand you correctly, describe the variable that is
not subject to manipulation: sex, or IQ, or age, e.g. Correlation is a
feature only of a *relationship* between TWO variables (or more) --
that's why it's called "corRELATION". I will suppose that, in the
scenarios you are imagining, the experimental and control treatments are
assigned randomly to equal numbers of participants _within each stratum_
of sex (or whatever); since this is certainly easy enough to arrange
unless you're trying to do it in complete ignorance of the sex (or
whatever) of the participants, and only find out after the experiment
whether a particular participant be male or female; in which case you
have a constellation of other problems that I will ignore here. Under
these circumstances (equal assignment within strata), the treatment
factor ("experimental" vs. "control") is not, and cannot be, correlated
with what you call the "participant factor". (Or put somewhat more
properly, the correlation is zero, so that they are uncorrelated.) The
analytical *design* is therefore orthogonal.
Now, as between several "participant factors", it is entirely possible
that they may be correlated in the population of interest. If this is
the focus (or a focus) of your concern, the concern is legitimate, and
there are ways of dealing with such situations, some of which could be
said to have been hinted at in your posts. But even there, the problems
to be worried about arise in the interpretation of, and the conclusions
to be drawn from, the results: not in the analysis of the data.
------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
.
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