QUESTIONS of the MOMENT...
"When theory proposes an X-Y association and it also proposes that Z moderates this
association, but theory is mute about or doesn't propose a Z-Y association, why does one
still include Z in addition to X and XZ in the model to be tested?"
(The APA citation for this paper is Ping, R.A. (2009). "Why must one always include Z
in addition to X and XZ in the model to be tested?" [on-line paper].
http://home.att.net/~rpingjr/MissingZ.doc)
By hypothesizing Z's moderation of the X-Y association but excluding the Z-Y
association in the model to be tested, one is assuming that Z has no relationship with Y.
However, reviewers will likely request support (argumentation) for this assumption, and
validation of these arguments by testing (including) the Z-Y association it in the model.
There are several reasons for this, including that the logic of science requires an argument
(hypothesis) to support an assumption such as the Z-Y association does not exist in the
population. This in turn requires an hypothesis test (i.e., including the Z-Y association in
the model to be tested).
Omitting the Z association also creates the "missing variable" problem (see James, 1980),
which can bias all structural coefficients, and standard errors in the model. This in turn
casts a shadow on the trustworthiness of the test of the proposed model. (The Z-Y
association will never be exactly zero in real-world data, and the fitted covariance
matrices for the structural models with and without Z will be slightly, to very different,
depending on the covariances of Z with the other model variables.) (In addition, missing
variables are accounted for in the error term(s) (structural disturbances) of the dependent
variable(s). Because these missing variables are almost always correlated with other
antecedent variables in real-world data, when Z is missing, error tem(s) are correlated
with antecedent variables in the model, which is a violation of an important structural
equation analysis assumption.)
In addition, if XZ is significant, excluding Z from the model biases the factored ("true"
contingent) association of Z with Y, EVEN IF A Z-Y ASSOCIATION IS
HYPOTHESIZED TO NOT EXIST IN THE POPULATION. In Y = b1X + b2XZ + b3Z,
the factored (true contingent) coefficient of Z is (b2X + b3)Z. Since b3 will never be zero
in real-world data, this factored coefficient of Z and its significance will be biased by
omitting Z. Further, excluding Z misses an opportunity to "explore" (discuss) the novel
fact that Z, even if it is supposed to be nonsignificant, is significantly (conditionally)
associated with Y.
Thus, if theory suggests that the Z-Y association is zero in the population, or the theory is
mute on this matter, one should consider adding a formal argument, including an
hypothesis, that Z has no association with Y to the paper. Then, one should consider
including the Z-Y association in the model to be tested (to test this hypothesis).
Optionally, to gauge the effect (significances) of excluding Z from the model, one could
rerun the model with Z omitted. (This practice, called "trimming," was once popular in
Sociology, but it is now rarely seen in Social Science.) In addition, if XZ is significant,
one could also compute the factored coefficient of Z and its significances at various
levels of Z in the sample, then discuss this "discovery" in the Discussion section of the
paper.
References
James, Lawrence R. (1980), "The Unmeasured Variables Problem in Path Analysis,"
Journal of Applied Psychology, 65 (4), 415-421.