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Date:         Wed, 27 Jan 1993 16:05:40 EST
Reply-To:     "Davis A. Foulger (914) 945-2077 (t. 862-2077)"
              <Foulger@WATSON.BITNET>
From:         FOLGER@WATSON.BITNET
Subject:      Interactions

>In a recent discussion with a Professor of Marketing, a difference of
>opinion arose regarding the interpretation of main effects when there
>are interactions present.
>
>To quote Geoffrey Keppel, "When an interaction is present, an investigator
>will not be interested in the main effects anyway - anything that
>might be said about the effects of one independent variable must be
>qualified by a consideration of the levels of the other."
>(Design and Analysis: A Researcher's Handbook, 2nd Ed, 1982, p.173)
>
>How strictly is this to be held true? Should it depend upon the original
>hypothesis and intention of the study? Any comments on the above quote?
>Should interpretation always proceed sequentially from the highest level
>interaction down to the main effects?

Mr. Keppel is quite wrong, or, more to the point, he is engaging in classic
ANOVA interaction thinking.  An interaction variable is the product of two
other variables and it is easy to constuct cases where its variance is
entirely uncorrelated with the main effects (indeed, this is the case by
definition in prototypic ANOVAs).  Even where it is not, however, it is
still a distinct variable from the main effects, and should be interpreted
as such.  Clearly any and all variables associated with the interaction need
to be taken into account when interpreting the interaction, but this in no
way makes separate consideration of the main effects valuable.

Consider, as an example, the expected effects of seatbelts and airbags on
the survivability of a car accident (I don't have any data at hand -- just a
fair memory from what I've read about the effects of each).  We know that
there is a main effect for airbag use.  We know that there is a somewhat
lower main effect for seatbelt use.  The use of both together, however,
can be cast as an interaction term with a very specific shape corresponding
to what I have called an AND interaction (each condition coded as 1 for use
or 0 for non-use before the interaction product is computed).

Assuming both main effects are significant with with positve betas and the
much larger effect associated with airbag use, there are three major
distinct possibilities for what will happen with this interaction term:

   -- The interaction will be non-significant.  This can be interpreted as
      an indication that the effect of seat belt and air bag use is
      additive (e.g. the beta increment for survivability associated with
      each main effect remains constant even when both are used together.

   -- The interaction is significant with a negative beta.  This means that
      the effect of using both is less than additive; that the seat belt
      already does some of the work of the airbag or the airbag negates some
      of the value of the seatbelt.  There may still be value in using both,
      but the value is reduced.

   -- The interaction is significant with a positive beta.  This means that
      the OVERALL effect of using both is greater than the ADDITIVE effect
      of using both (e.g. the whole is greater than the sum of the parts).
      In this case the beta term represents the increment in survivability
      above and beyond the sum of the main effects.  Interestingly, it is
      my recollection that this outcome is the one that actually obtains in
      real world testing.

One notes that, even in the presence of a significant interaction effect,
one would not want to ignore the main effects.  They are obviously very
meaningful, especially to folks who have air bags but don't use seat belts
or people who have seatbelts but lack air bags.

Clearly the original hypothesis matters here.  Using a classic ANOVA XOR
interaction term would totally confuse the interpretability of what I have
just laid out.  Using an AND interaction, on the other hand, makes
interpretability rather straightforward.

As for the method of interpretation.  I don't think it matters which term
you interpret first.  What matters is that you consider them all and, if
possible, shape the interactions to match the hypotheses.

Davis

Snailmail..........................Davis A. Foulger
Internet: FOLGER@WATSON.IBM.COM    IBM T.J. Watson Research Center
Prodigy:  XFRR20A                  P O Box 218, Yorktown Ht, NY 10598
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