On Tue, 23 Sep 2003, Rich Ulrich wrote inter alia (concerning
inferences from an ancova in the presence of interaction between the
covariate and the categoircal variables (aka "factors")):
> The inference drawn from the 95% CI on
> the intersection-of-regression lines is *cute* but
> I don't think you can read it that strongly, as a fair point.
> Also, a point about the technicality: Does the technique
> get applied *only* in the case of disordinal interactions,
> or is it also used when the lines do not cross?
If interaction is present, then the regression lines are not uniformly
the same distance apart vertically (that is, in the direction of the
response variable, which is commonly plotted as the ordinate) at
different values of the covariate (customarily the abscissa).
If two lines actually cross, there is clearly a region of the covariate
where the lines are not significantly different from each other; and
there may be a region where line A is significantly higher than line B,
and there may be another region (in the other direction from the
crossing) where line B is significantly higher than line A. (I write
"there may be" because the existence of these regions, in the observed
range of the covariate, depends on (1) how sharply the two lines
actually diverge from each other and (2) where, in the observed range of
the covariate, the crossing is. The existence of a significant
interaction would ordinarily lead one to expect that at least one of
these regions exists, on logical grounds.) In this case, as you
observe, the method Kylie reports is clearly applicable.
If the two lines do not cross, it is possible that in some part of the
observed domain of the covariate the distance between the lines is small
enough not to be significantly different from zero; in any case, the
interaction implies that the distance between the lines is not constant
and that for at least some part of the domain of the covariate that
distance is large enough to be significantly different from zero.
Possibly the difference between the lines ranges from "significant" to
what one may caricature as "REALLY significant"; but you can't really
tell which of these patterns obtains in your data without (a) looking at
the graph and (b) conducting a formal test of the kind Kylie describes.
And of course if there are more than two lines, describing the entire
pattern of non-parallelism can get decidedly more complex.
> - I think that one thing that affects me here is that I
> tend, rather strongly, to regard 'interactions' as being
> a failure to find the proper elements to model.
I can see how one might hold such a view; but I rather suspect it might
too readily lead to oversimplifying one's analyses. If "finding the
proper elements to model" entails something relatively simple, such as a
square-root or logarithmic transformation of the response variable
and/or the covariate, that's one thing; but if no such convenience
appears to exist, the fact of interaction, if found, still needs to be
dealt with. It may still be the case that "we haven't found the proper
elements for a really good model -- yet"; but right now we have these
data to describe, and to TRY to interpret, in the absence of a model
that really satisfies our sense of model-specifying elegance.
> That is, if the definitions were right, we'd see main effects; while
> the definitions are wrong, we should be rather calm and quiet about
> our pronouncements.
With respect, this seems to me to be invoking an oversimplification
("right" vs. "wrong" is nearly always an oversimplification) of the kind
I was worrying about in the last paragraph. [And so far, we hadn't got
to the business of "pronouncements" -- we were still trying to make
sense of some data that show evidence of interaction.]
Rich, you raised this question of "the proper model" in the context of
ANCOVA and interaction involving the covariate. Do you hold the same
opinion, as strongly, where the interaction involves only (some of) the
categorical predictors? If so, I'd like to know your choice of "the
proper model" for the example (the PULSE data set in MINITAB) dealt with
in my White Paper on modelling interactions in multiple regression, on
the Minitab web site (www.minitab.com -- I forget the rest of the
specific URL, but you can get there from the home page by looking for
the section on "white papers"). I rather thought, in that context, that
interaction was a quite reasonable thing to look for, and perhaps even
to expect...
Ciao! -- Don.
-----------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
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