endogenous moderation

[guest]

In the following well-fitted model, "M2" is acting as a moderator on the "M1 -> y" path (see also attached conceptual diagram): 

df$M1_M2 <- df$M1 * df$M2
require(lavaan)

med.model <- '
M1  ~ a1*X1 + a2*X3 + a3*X4
M2  ~ a4*X1 + a5*X2 + a6*X3 + a7*X4 + a8*M1
y   ~ c1*X1 + c2*X3 + c3*X4 + b1*M1 + b2*M2 + b3*M1_M2
#covariances
X1 ~~ X2 + X3
X2 ~~ X3
M1 ~~ M1_M2
M2 ~~ M1_M2
M1_M2 ~~ M1_M2
#intercepts
M1~1
M2~1
M1_M2~1
y~1
X1~1
X2~1
X3~1
X4~1
'
fitmedsem <- sem(med.model, df)

However, since "M2" is an endogenous (measured) variable - and I know that e.g. the process macro only permits of exogenous moderators - I wanted to hear whether the experts of this group see any conceptual obstacles to this model?

Thanks!   

[Terrence Jorgensen]

However, since "M2" is an endogenous (measured) variable - and I know that e.g. the process macro only permits of exogenous moderators - I wanted to hear whether the experts of this group see any conceptual obstacles to this model?

The PROCESS macro does not use latent variable models.  I think moderators can be exogeonous, but try asking on SEMNET:

https://listserv.ua.edu/cgi-bin/wa?SUBED1=semnet&A=1

The issue to be wary of is that you need to treat your product term as exogenous, even if one (or in this case, both) of the variables in the product term is (are) endogenous.  So also correlate M1_M2 with the Xs (your syntax also omits the correlation of X1-3 with X4).  See models 3-5 in Figure 2:

https://doi.org/10.1080/00273170701341316

Terrence D. Jorgensen

Assistant Professor, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

http://www.uva.nl/profile/t.d.jorgensen

[guest]

Hi Terrence,

thanks a lot!

Your advice about treating the product term as exogenous and introducing covariances between the product term and the Xs helped identifying a well-fitting model according to the various fit-indices (e.g. cfi, tli, rmsea, srmr).

In spite of what you said, I did not introduce covariances between X1-3 and X4, however. This has to do with something that I didn't mention in the original problem specification concerning the different status of X1-X3 and X4:

X1-X3 dummy code four experimental conditions in three indicator variables and are thus dependent via the coding.

X4 is, however, an orthogonal experimental condition that was crossed with the four other experimental conditions in the factorial design.

###

On a different level, your advice about how to fit the model made me curious about the relationship between SEM and causal models. In particular, it raised two issues for me that I was hoping that you (or anybody else for that matter) could either shed light on directly or point to further literature on:

1) when authors like Pearl discuss causally interpreted SEM models, the exogenous factors are assumed to be independent to satisfy the Markov condition. But if one introduces co-variances between all exogenous factors in a model, then isn't this the same as allowing that there are unknown common causes affecting the exogenous variables, which in turn would affect the causal interpretation of the SEM model?

2) conceptually what is the difference between, and implications of, introducing the product term representing the interaction of endogenous variables as itself an exogenous variable, as you recommended?

Thanks!

[Terrence Jorgensen]

1) when authors like Pearl discuss causally interpreted SEM models, the exogenous factors are assumed to be independent to satisfy the Markov condition. But if one introduces co-variances between all exogenous factors in a model, then isn't this the same as allowing that there are unknown common causes affecting the exogenous variables, which in turn would affect the causal interpretation of the SEM model?

Yes.

 

2) conceptually what is the difference between, and implications of, introducing the product term representing the interaction of endogenous variables as itself an exogenous variable, as you recommended?

You can't think of the product as another variable.  That is merely a trick to make the model do the math we want it to do.  A product term is 100% determined by the variables from which it is calculated, and it is related to them in our design matrix because it is a function of them.

[Nickname]

Anonymous Guest Poster:
Interesting question.  If it would be helpful, I can expand a little on Terrence's "Yes".

It is important to distinguish two different terminologies.  Consider the following simple example:  X and Z both cause Y.  In common SEM parlance, X and Z are exogenous variables.  In strict Structural Causal Model (SCM) parlance, their disturbances are the exogenous variables. X  and Z are endogenous.  This just pushes the problem back a level to their disturbances.

The Causal Markov Condition (CMC) permits the substitution of a shared latent parent for any covariance because it posits that for any covariance there is a corresponding causal structure.  Suppose we apply this directly to X and Z, not their disturbances.  Once you make this substitution, the covarying variables, X and Z in our example, have a parent and remain orthogonal when conditioning on this parent.  So, there is no violation of the Markov Condition in allowing X and Z to covary.

While it is true that Pearl (2009) relies heavily on examples with just one exogenous variable, somewhat obscuring the issue of how to treat multiple exogenous variables, one can find examples.  See Figure 4.8 on page 126.  X1 and Z are both exogenous but not uncorrelated.

Both the model with the covariance and the model with the latent parent differ both probabilistically and causally from the model with orthogonal exogenous variables.  The latter is a much more restricted model (and most likely will not fit your data).  However, if one accepts the CMC, then the former two do not differ from one another in any important way.  They both represent an unexplained source of covariation among the exogenous variables and among their children.  Controlling for the two exogenous variables (e.g., X and Z) screens off their shared parent with respect to their endogenous children (e.g., Y).  So, there is no need to model it explicitly.

In the case of your dummy variables, the representation of the patterns of covariance among the dummy variables by pairwise common causes renders the underlying assignment mechanism a little obscure to the eye.  However, it is an acceptable representation for purposes of statistical analysis.  You are not violating any assumptions by adding covariances between exogenous variables in a SEM model.

Caveat: What I am describing is the canonical interpretation of SCM.  I am not necessarily endorsing the uncritical acceptance of that interpretation or of the CMC.

I hope that helps,
Keith
------------------------
Keith A. Markus
John Jay College of Criminal Justice, CUNY
http://jjcweb.jjay.cuny.edu/kmarkus
Frontiers of Test Validity Theory: Measurement, Causation and Meaning.
http://www.routledge.com/books/details/9781841692203/