Not to get off track here, but if you plot the prediction function in 3
dimensions - factor a and b in the x and y planes, response in z, you may
see quickly what's going on with those non-parallel and crossing lines.
Crossing lines occurs when you view the 3-D response surface from one face
directly, so that the setting of the other factor is not visible. The
lines are then the front and back intersections of the surface with the
limit boundary of the non-visible factor.
In such a case of strong interaction one must know the values of both a and
b to understand where z is likely to be. Natch.
Apology if I don't use the proper terminology - my ignorance.
Jay
Donald Burrill wrote:
> 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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