ESTIMATING ENDOGENOUS INTERACTIONS
(The APA citation for this paper is Ping, R.A. (2008). "Estimating endogenous interactions."
[on-line paper]. http://www.wright.edu/~robert.ping/categorical.doc)
(An earlier version of this paper, Ping, R.A. (2005). "A status report on estimating endogenous
interactions." [on-line paper]. http://www.wright.edu/~robert.ping/Endog_ints.doc, is available
here.)
In
X
U d
Z a
UZ
c
Y
b
(Figure 1)
UZ is an endogenous interaction because it involves the endogenous variable Z. Endogenous
interaction models also include
X
e
c
Z a
XZ
Y
f
(Figure 2).
In both cases, Z is a mediator and a moderator of the X-Y association. These (sub)models could
also be part of a larger model.
An endogenous interaction may have four considerations in theory testing using structural
equation software (e.g., LISREL, EQS, Amos, etc.): theoretical, methodological, estimation
software considerations, and interpretational considerations.
The theoretical considerations will be discussed first, although I suspect there is more to be said
on this matter. Occasionally, a reviewer remarks that a moderator, Z, can not or should not also
be a mediator. Specifically, in Figure 2 Z can not or should not moderate the X --> Y association
with the XZ interaction, and also mediate the X --> Y association.
However, it seems clear that this is theoretically plausible. Consider the following; Relationship
satisfaction (SAT) and the attractiveness of alternative relationships (ALT) are well known in
several social science literatures to be antecedents of relationship exiting (EXIT) (SAT --> EXIT
and ALT --> EXIT) (e.g., Ping 1993). However, Johnson and Rusbult (1989) proposed that
satisfaction "devalues" or reduces the attractiveness of alternatives (i.e., SAT --> ALT). In
addition, Ping (1994) argued that satisfaction and alternative attractiveness interact in their effect
on relationship exiting (i.e., that alternative attractiveness suppresses the satisfaction association
with relationship exiting) (SATxALT --> EXIT). Specifically, with mediation, satisfaction is
likely to reduce alternative attractiveness which then is likely to reduce exiting. With
moderation, the interaction, increased alternative attractiveness decreases the strength of the
(indirect) satisfaction-exiting association. Thus, alternative attractiveness may be both a mediator
and a moderator of satisfaction's effect on relationship exiting, and in at least one plausible
Figure 2-like model, joint mediation and moderation (and an endogenous interaction) are
theoretically plausible. Moreover, I have tested these paths in an unpublished paper using survey
data, and all four paths were significant. Thus, mediation and moderation are both theoretically
and empirically plausible.
Regarding the software consideration, from e-mails I have received, AMOS may have difficulty
estimating an endogenous interaction. For several reasons, I have not used AMOS yet, so I have
not directly observed the alleged problem. However, based on e-mails, specifying the required
correlation paths between X and XZ, and XZ and Z, when Z is endogenous, has produced an
error message in AMOS. If this happens to you, I have a "work-around" but you will have to email me--I suspect there are still some situations it does not handle.
In summary, if you are using AMOS with one or more endogenous interactions, consider
specifying the model as usual with the X-XZ and XZ-Z correlations (no X-Z correlation). If a
specification error message appears, consider removing the X-XZ or the XZ-Z correlation, and
recheck that there is no X-Z correlation, to be certain it's the X-XZ-Z correlations. If that model
runs, please send me an e-mail or consider using LISREL/EQS.
LISREL seems to be OK with Submodel 3B. Ditto for EQS so far.
Regarding methodological considerations, consider reading the remarks in FAQ D under
FREQUENTLY ASKED QUESTIONS on the previous web page, and "Is there an example that
shows all the steps involved in estimating a latent variable interaction/quadratic?" under
"QUESTIONS OF THE MOMENT" also on the previous web page. Interactions require
attention to centering, model specification, starting values, admissible values, etc.
Regarding interpretational considerations, in Figure 1, for example, the association between Z
and Y can be written as Y = dU + aZ + bUZ + error = dU + (a + bU)Z + error. Thus, the
(unstandardized or standardized) structural coefficient of Z's moderated association with Y is the
coefficient (a + bU). From path analysis (see Wright 1934), the (unstandardized or standardized)
structural coefficient of X's association with Y via or mediated by Z is the product of the
(unstandardized or standardized) structural coefficient on the X-Z path, c, with the
(unstandardized or standardized) moderated structural coefficient on the Z-Y path, (a + bU),
which equals c(a + bU) (= ca + cbU). However, structural equation estimation packages will
report only the unmoderated (indirect) X-Y association that has the coefficient ca.
Thus, the coefficient of the (indirect) X-Y associations and their significances (X's indirect
association with Y depends on the level of U in the coefficient ca + CbU) must be computed
manually. (Click here for an EXCEL spreadsheet to expedite that process.)
REFERENCES
Johnson, D. J. and C. E. Rusbult (1989), "Resisting Temptation: Devaluation of Alternative
Partners as a Means of Maintaining Commitment in Close Relationships," J. of Personality
and Social Psychology, 57 (6), 967-980.
Ping, R. A. (1993), "The Effects of Satisfaction and Structural Constraints on Retailer Exiting,
Voice, Loyalty, Opportunism, and Neglect," J. of Retailing, 69 (3), 320-352.
Ping, R. A. (1994), "Does Satisfaction Moderate the Association between Alternative
Attractiveness and Exit Intention in a Marketing Channel?" J. of the Academy of Marketing
Science, 22 (4), 364-371.