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Date:         Thu, 28 Jan 1993 17:00:31 EST
Reply-To:     "Davis A. Foulger (914) 945-2077 (t. 862-2077)"
              <Foulger@WATSON.BITNET>
From:         FOLGER@WATSON.BITNET
Subject:      Interactions

I was asked, in a private note, if, perhaps:

>Keppel was speaking only of what Campbell and Stanley refer to as
>"nonmonotonic" interactions -- where the slopes for the simple main effects
>change direction across levels of the other factor -- for example, were the
>effect of airbags to reduce mortality if you did not wear belts but to
>increase mortality if you did wear a belt.  In such a case it the main
>effects are less likely to be of interest, yes?

After drafting an answer I thought it might be of general interest.
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What you describe, I think, is what I usually call an XOR interaction (the
classic shape of an ANOVA interaction).  We are into fuzzy tirritory here.
If the effects of airbag use varied as you propose (e.g.  they reduced
mortality if you did not wear belts but increased it if you did), there is a
reasonably good chance that the effects would cancel each other out in the
main effect such that there was would be significant main effect for
airbags (and perhaps no significant main effect for seat belts either).
In such a case the main effects would be very uninteresting.

So long as the effects of the seat belt or the airbag alone are different,
however, it seems unlikely that this result would obtain.  It is more likely
that there would be a strong interaction effect and a weaker main effect for
either airbags or seatbelts.  The interpretation of the interaction effect
would be important, for it would indicate that the use of either seat belts
or airbags was a good thing, but that the use of either or neither was a bad
thing.  Interpretation of the main effect would still matter, however, for
it would indicate which of the technologies was to be preferred if you had
the option of using one or the other.

I remain of the opinion that significant main effects are important and
should be interpreted, even when the interaction effect is much more
significant.

Restating things slightly so they will be more intuitive, let us consider
the likelihood that two randomly selected people people will go out together
given the gender of the person who asks and the gender of the person who is
asked.  Without getting into the likelihood of any given combination
happening or the very real differences in preferences that exist in the real
world, the following table probably summarizes existing stereotypic
assumptions of probable behavior.  I believe this table describes what you
are asking about (e.g.  "nonmonotonic" interactions in which the slope of a
simple main effect changes direction across levels of the other factor):

                                   askee
                       male                   female
                male   low likelihood         moderate likelihood
         asker
               female  high likelihood        moderate likelihood

If I were to do a study that actually got results paralleling what is shown
in the table, an ANOVA would most likely describe the results more or less
as follows:

     -- No significant main effect for askee (e.g. the low and high on the
        male side would balance each other against the moderate effect in
        both female askee conditions).

     -- A significant main effect for asker (e.g. females are more likely to
        be accepted when they ask than are males when they ask).

     -- A significant interaction effect (e.g. opposite gender couples are
        more likely to go out than same gender couples).

The significance of the interaction leads to one important and highly
meaningful interpretation (opposite sex couples are more likely to go out
than same sex couples), but the significant main effect also leads to a
significant and highly meaningful interpretation (females are more likely to
succeed in asking someone out regardless of the gender of the other person).
I don't think I would want to ignore the main effect interpretation simply
because the interaction was also interpretable.

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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