[R] type III Sum Sq in ANOVA table - Howto?

Rolf Turner rolf at math.unb.ca
Fri Mar 7 01:31:37 CET 2003

Andy Liaw wrote:

>  The long(er) answer: think harder about what question(s) you want answered
>  (i.e., what hypotheses you really want to test, and test only those).  The
>  model hierarchy says that a model should not have an interaction term
>  involving a factor whose main effect is not present in the model.  Seen in
>  this light, the hypothesis you're trying to test involves a non-sensical
>  model.

Not really.  The hypothesis being tested by Type III sums of square
may be suspected of not being of ``central interest'', but it is NOT
(as is commonly believed) ``non-sensical''.

Let us think about the 2-way ANOVA case, where one can actually
understand what is going on.  Let the population ***cell means*** be
mu_ij (i = 1, ..., m, j = 1, ..., n) and forget about the confusing
and misleading over-parameterized model.

Testing for the significance of the ``row factor'' by Type III
sums of squares (with interaction in the model of course) tests

	H_0: mu_{1.}-bar = mu_{2.}-bar = ... = mu_{m.}-bar

I.e. that the means of the population cell means, over columns, are
all equal.  I.e. that ``when rows are averaged over columns'' there
is no row effect.

This could, at least conceiveably, be of interest.  Note that the
average is not a weighted average, saying that all columns are
equally important.  If all columns are NOT equally important (e.g.
if an item randomly drawn from the population is more likely to
``come from'' column 1 than from column 2 etc.) then this hypothesis
is less likely to be of interest.

But it isn't nonsensical.

It is true, however, that most of the time when people test things
using Type III sums of squares they don't understand what they are
really testing.  But then (said he cynically) people don't understand
what the hell they are really testing in most situations, not just
in the context of Type III sums of squares.


					Rolf Turner

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