Quadratic Effect in Observed SEM PreviousNext
Mplus Discussion > Structural Equation Modeling >
 Adam Blake posted on Thursday, May 10, 2012 - 8:27 am
I was wondering about the correct way to model a quadratic effect in a mediating variable within a SEM with observed variables. I am unsure as to how to model the correlation between the mediating variable and its quadratic and how to model the relationship between the quadratic and the exogenous variables. I've been trying the two models structures below.

x1-Exog. Var, m1-Mediating Var., m1m1 - m1*m1, y1 - Endo. Var.

m1 m1m1 on x1;
m1 with m1m1;
y1 on m1 m1m1;


m1 on x1;
m1 on m1m1;
y1 on m1 m1m1;

With some of these model structures I am getting a non-positive definite first-order derivative product-matrix. In this case would the bootstrapped confidence intervals for model parameters be reliable?
 Linda K. Muthen posted on Thursday, May 10, 2012 - 1:18 pm
m1 ON x1;
y ON m1 x1 m1m1;
 Adam Blake posted on Friday, May 11, 2012 - 11:45 am
Based on the theory surrounding the model I would like to at least test if the mediating variable could fully account for the effect of x1 on y1. For that reason I would like to keep x1 out of the model statement of y1.

I tried fitting this model and like many of my other models I observe a very large modification index (~59) indicating that I need to add a path (with or on) between m1 and m1m1. Also with no pathway between x1 and m1m1 would I be underestimating the indirect effect of x1 on y1 since the indirect effects would not incorporate the quadratic effect?

I have been having some luck fitting the first model from my last message. Would using bootstapped confidence intervals give me a reliable test of the significance of model parameters even with a non-positive definite first-order derivative product-matrix?

Thank You
 Linda K. Muthen posted on Saturday, May 12, 2012 - 2:45 pm
 Adam Blake posted on Tuesday, May 22, 2012 - 10:54 am
Is there any way in m-plus to test for an indirect effect of an exogenous variable (x1) on an endogenous variable (y1) when the mediating variable (m1) has solely or primarily a quadratic relationship with y1?

From my understanding if I center the m1 variable its quadratic (m1m1) should have no relationship with m1 or x1 and consequently their will be no causal link back to x1. This is despite the fact that changes in x1 would be expected to affect m1m1 and possibly y1.

Is there any relavant literature that I should be aware of?

 Bengt O. Muthen posted on Tuesday, May 22, 2012 - 5:33 pm
It is for such more complex cases that the value of the general definition of causal indirect effects becomes clear - see my interpretation of what the causal effect researchers have developed:

Muthén, B. (2011). Applications of causally defined direct and indirect effects in mediation analysis using SEM in Mplus.

- you find it under Papers, Mediational Modeling.

See the Appendix Section 13.1 and equations 55-58, which work for y = m*m as well. I haven't explored it, however.
 Adam Blake posted on Wednesday, May 23, 2012 - 2:28 pm
Thanks that paper was exactly what I was looking for. If I understood the paper correctly then for my example model above with this model statement below:

m1 on x1 (gamma1);
y1 on m1 (beta1)
y1 on m1m1 (beta2);

The correct model constraints to obtain the total and pure indirect effect (tie & pie) would be:

tie/pie = gamma1*beta1+gamma1*beta2;

Is this correct?
 Adam Blake posted on Wednesday, May 23, 2012 - 2:33 pm
Follow up question is there any way to generate standardized coefficients for the tie/pie parameters?
 Bengt O. Muthen posted on Wednesday, May 23, 2012 - 5:17 pm
No, I don't believe that follows from those formulas. It is more complex than that. I don't have the explicit answer at hand. You might want to consult with a local statistics person.
Back to top
Add Your Message Here
Username: Posting Information:
This is a private posting area. Only registered users and moderators may post messages here.
Options: Enable HTML code in message
Automatically activate URLs in message