I found Art's question ambiguous in a number of senses: see below, in my
comments on his post.
Karl's example does not, it seems to me, entirely address the question
being asked. For one thing, it depends much on how the interaction
predictor (which I'll call d*g) is actually constructed. If <d*g> is
obtained by multiplying <d> by <g> and no adjustments are made to the
product, <d*g> may (but need not) end up with four different values;
and if it is used alone in an ANOVA context, of course it will take 3 df
to account for its effects.
(E.g., if <d> = 1 or 2 and <g> = 1 or 3, <d*g> = 1, 2, 3, or 6.)
But <d*g> might have only two values (e.g., if both <d> and <g> are
coded {+1, -1}, then <d*g> = +1 or -1), and in that case an ANOVA using
only <d*g> would find only 1 df for the <d*g> effect. (I suspect this
would have happened in Karl's example had he used data with balanced
frequencies; but whether that suspicion be correct depends on what SAS
actually does in computing <d*g>.)
Possibly somewhat more to the point (or at least differently...), Art's
question might have been asked in a multiple-regression (MR) context;
in which case, however <d*g> were constructed, it would only be
represented with one d.f. in an analysis predicting a dependent variable
(Y?) from <d*g>.
Whether such an analysis be useful or not depends on a whole lot of
context: things that Art hasn't told us about his problem, e.g. why (or
perhaps if?) he wants to ignore any main effects that may (but need not)
be present, and whether the interaction variable <d*g>, _as_used_
in_the_analysis_, be correlated with either main effect.
(One could have constructed <d*g> to be orthogonal to both <d> and <g>
-- see my White Paper on interactions in MR, on the MINITAB web site
www.minitab.com for an example and description of how to do that -- so
that <d*g> represents what one might call the "pure interaction" effect,
uncontaminated with main effects; for this variable in Karl's example,
used as a predictor all by itself (in MR, not ANOVA), the SS would be
rather more like 15.9 than 1312.3.)
On Mon, 16 Feb 2004, Karl L. Wuensch wrote:
> Here is SAS GLM output for a 2x2 ANOVA, unequal cell sizes, type
> III sums of squares. First the full factorial design:
> Source DF Type III SS
> deattr 1 1286.325649
> gender 1 3.955131
> deattr*gender 1 15.873457
>
> Now I drop the two main effects and run the analysis again:
> Source DF Type III SS
> deattr*gender 3 1312.292621
>
> As you can, both the sums of squares and the df previously assigned
> to the main effects are now assigned to the interaction term.
>
> ----- Original Message -----
>
> From: "Arthur Tabachneck" <[EMAIL PROTECTED]>, Fri Feb 13:
>
> Is it legal to test for an interaction without testing for the main
> effects of the variables included in those interactions. And, in the
> case it is (which I recall it isn't), does one still have to account
> for the degrees of freedom used by the non-tested main effects?
Depends on what you mean by "legal". Can one do such a thing? Yes.
But it _also_ depends on what you mean by "interaction". One meaning is
"an effect attributable to a combination of two (?) variables, above and
beyond the effects attributable to the two variables separately" -- that
is, what I've called "pure interaction" above and elsewhere. Another
meaning is "an effect attributable to the product of two (?) variables",
and this effect does depend heavily on (1) how the "product" is
constructed (see my notes above) and (2) whether the "main effects" are
also being separately accounted in this analysis. As Karl's analyses
showed, the "interaction" might only contain the "pure interaction"
information (as in his first analysis), or it might contain ALL the
effects due to interaction AND main effects (as in his second analysis).
You ask about "accounting for the degrees of freedom used by the
non-tested main effects"; you didn't ask about "accounting for the sums
of squares explained by the non-tested main effects". But I cannot
think of a context in which one wouldn't want so to account (which
doesn't mean you don't have such a context -- I may not be sufficiently
imaginative, is all), because if you don't as it were extract the SS due
to the main effects, you'll have an "error" SS that includes all those
systematic effects; so your analysis will be relatively insensitive to
the interaction effect (and any other effects that may be retained in
your model), due to the unnecessary inflation of "error".
> Hoping that was sufficiently clear,
> Art
Well, no, I didn't think it was. As I tried to illustrate. There are
lots of things more I can imagine saying about your question, but I
can't tell which of those things may be relevant to your context.
Good luck! -- DFB.
------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
.
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