At 11:57 AM 5/26/00 -0400, Martin H. H. Stevens wrote:
>
>I am afraid this may turn out to be an elementary statistics lesson, but
>here goes, and thanks (and/or apologies) in advance...
It may, but you are probably not alone in needing it. Sigh.
>
>When I use Type I sums of squares in a linear model, the sum of squares
>of each factor (from summary.aov()) plus the residual add up to the
>total sum of squares ([n-1]*s^2 of the response variable. When I use
>Type III SS, I get something less than the total sums of squares when I
>add up the partial Sums of Squares.... Why?
It is because the sums of squares in each case, though labelled similarly
are in many cases testing different hypotheses.
The Type I table tests a sequence of nested hypotheses, that is, in each
case that the term presented contributes nothing more to the model in
addition to all *previous* terms. This is why it is called a sequential
analysis of variance table (sometimes).
The Type III table tests a sequence of non-nested hypotheses. Each sum of
squares tests the hypothesis that the term contributes nothing more to the
model in addition to all *other* terms, before and after. This makes it
pretty well the same as what you do when you test all terms in a regression
model, even though you would usually use t-tests rather than present them
in an anova table. In regression analysis most people would not consider
linear terms when a second degree term in the model were present (except in
rather special circumstances).
The objection to Type III sums of squares is that they encourage naive
users to do silly things such as test main effects in the presence of
interactions, without really asking whether the test makes sense or not,
that is, whether it really addresses a question of any interest.
I have an objection to *all* types of sums of squares, actually. I think
looking at linear models in terms of "which types of sums of squares should
I be using?" is just plain muddled. If you have a testing problem, the
real question is "which null hypotheses should I be testing, within which
outer hypothesis?" Once you sort that out the way to do it is completely
clear. The trouble is people do not want to sort that out, because it
requires thinking clearly about the problem and it is much easier to rely
on someone else doing that and providing you with a variety of Types of
sums of squares to choose from.
More often than not, too, the problem that people should be looking at is
an estimation problem and not a testing problem at all, but I digress..
Well, you did ask!
Bill Venables.
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