Newsgroups: bit.listserv.stat-l
Date:         Fri, 31 Jul 1992 14:37:46 EDT
From:         Tim Dorcey <TCD@CORNELLC.BITNET>
Subject:      Interaction effects in regression

A few more comments to add to the discussion on whether "main effects"
should be deleted when the interaction in a regression model is
significant...

I find it helpful to keep in mind that, fundamentally, regression
models are about prediction, i.e., how much can I expect the response
variable to change as changes are made in the predictors.  Typically,
a regression model is parameterized in such a way that the parameter
estimates have an immediate connection to this question.  For example,
suppose we have the linear model:
E[y|x,z] = a + b*x + c*z
If I then ask, what would happen if I increased x by 1 unit while holding
z constant, I can compute:
E[y|x+1,z] = a + b*(x+1) + c*z
and then subtract to get:
E[y|x+1,z] - E[y|x,z] = b
This is very nice because a change in x has the same effect regardless of
the actual values of x & z.  Thus, my question has a simple answer, and a
test of the hypothesis that b=0 has a direct connection to the issue of
prediction.  But, now suppose I ask this same question when the model is:
E[y|x,z] = a + b*x + c*z + d*x*z
I compute:
E[y|x+1,z] = a + b*(x+1) + c*z + d*(x+1)*z
and then subtract to get:
E[y|x+1,z] - E[y|x,z] = b + d*z
Evidentally, the answer now depends upon the value of z.  E.g., when
z = -b/d, x has no effect on the response.  Furthermore, a test of the
hypothesis that b = 0, now has only narrow implications for prediction.
In particular, if b = 0, it simply means that a change in x has no
effect on y when z = 0.  I can imagine situations where this conclusion
might be interesting, but that would be the exception rather than the
rule (clearly, it would have to be a situation where the origin of z
was theoretically meaningful).
      So, to get back to the original question, if b is not significantly
different from 0, should we force it to be zero (i.e., delete x from
the model)?  The logic in forcing a coefficient to be 0 (or any other
fixed value) is that the coefficients of a model with fewer parameters can
be estimated with greater precision.  The main drawback is that if the
true coefficient is different from the value that we fixed it at, then
the other estimates will be biased.  Furthermore, if the decision to
omit a variable is based upon the same set of data that the reduced model
is then fit to, none of the distributional results (e.g., t-tests) are
valid.  I.e., these tests are based on the assumption that we chose our
predictors without looking at the data.  It is interesting to consider
the "test-and-refit" approach in the context of forcing parameters to
be some other value than 0.  Suppose we adopted the following strategy:
1) fit an initial model   E[y|x,z] = a + b*x + c*z
2) test the hypothesis that:  b = 2
3) if b is not significantly different than 2, force it to be 2 and
   refit the model to get better estimates of a and c.
I suspect that many who are quite comfortable omitting non-significant
variables from regression models would be skeptical of this approach,
even though, in the context of linear regression theory, it is absolutely
equivalent.  So, what is it about 0 that is special and how does that
relate to the original question about main effects and interactions?

1)  Fixing a coefficient to 0 means that we don't even need to know the
value of the corresponding variable, so we end up with a more parsimonious
model.  In the case, E[y|x,z] = a + b*x + c*z + d*x*z,
however, regardless of whether we set b and/or c to 0, we still need to
know the values of x and z.  The resulting model is no more parsimonious.

2)  On an a priori basis, it seems that "this variable has no effect"
is a more plausible conclusion than "this variable has a regression
coefficient of 2".  As discussed previously, however, in the interaction
model, "b = 0" is equivalent to "this variable has no effect when z=0".
Unless there were prior reasons to expect that particular result, "b=0"
remains on the same footing as "b=2".

Therefore, my conclusion would be to leave the main effects in the model,
except perhaps under the special circumstance where it was expected that
"the effect of x when z=0" might be 0.  Even then, I would personally
keep the full model, because I don't think the increased precision of
parameter estimates is worth the risk of introducing bias.  The exercise
above was only meant to show that even if you buy the general idea of
omitting non-significant variables from regression models, the
interaction model is different.

Tim Dorcey                        BITNET:   TCD@CORNELLC
Statistical Software Consultant   Internet: TCD@CORNELLC.CIT.CORNELL.EDU
Cornell Information Technologies  Phone:    (607) 255-5715
Cornell University
Ithaca, NY  14853
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