Second order factors
Message/Author
 Tim Powers posted on Tuesday, June 16, 2015 - 7:14 pm
I have a question on handling 2nd order factors in a structural model within MPlus (and perhaps SEM in general). Even though I identify a 2nd order factor with “BY” statements, I notice that the factor loadings appear in the Beta array (Tech 1 output). I expected it to show up in the Lambda array to distinguish the measurement part of the model. If I regress a separate factor on the 2nd order factor, does this not change the nature of the 2nd order factor? How does Mplus distinguish between a 2nd order indicator versus a regression in the structural part of the model when factors are used as unobserved measures?
Here is an example:

f1 by x1 x2 x3;
f2 by x4 x5 x6;
f3 by x7 x8 x9;
f4 by f1 f2 f3;
f5 by y1 y2 y3;
f5 on f4;

Tim
 Linda K. Muthen posted on Wednesday, June 17, 2015 - 6:49 am
Where the parameters appear in the arrays is a matter of convenience and does not define the model. The rows of the lambda matrix are observed variables and the columns are continuous latent variables. So a second-order factor does not go there. It goes in beta for which rows and columns are continuous latent variables.
 Tim Powers posted on Wednesday, June 17, 2015 - 5:47 pm
Thank you.
So now I am trying to understand how a 2nd order factor is handled in Mplus modelling. How does Mplus distinguish between 2nd order factor indicators (using BY) versus a regression path onto some other endogenous factor (using "ON"; example above is f5 on f4). Apart from not covarying the indicator residuals, is there more happening behind the scenes?

Question:
Does the 2nd order factor change in meaning if it has indicator factors AND has a regression path (using "ON") to other (possibly many) endogenous factors? Stated another way, will the first order indicator loadings change in this scenario? And why?

I have colleagues who suggest that I cannot use 2nd order factors within a model whereby it is used to explain multiple other factors directly (and through other intervening factors). Their argument is that SEM cannot distinguish between an indicator and a regression path, and therfore, the meaning of the factor changes substantially from what is proposed. They argue that I need to replace the first order factors (i.e., factors with their own observed indicators) with factor scores (i.e., making the first order factors into observed variables).

Appreciatively,
Tim
 Bengt O. Muthen posted on Wednesday, June 17, 2015 - 6:28 pm
In general

y1-y5 on f;

is different from

f by y1-y5;

because the former correlates the residuals of y1-y5.

Your questions are akin to asking if f1 changes meaning in SEM where you have

Case 1: f1 by y1-y5;
Case 2: f1 by y1-y5;
f2 by y6-y10;
f2 on f1;

The f1 loadings will change from Case 1 to Case 2 because there is more information in Case 2 for estimating them. But does f1 change meaning? That's an interpretational matter. If the loadings change a lot that may be because the larger model does not fit. If they don't change a lot, the meaning remains the same.
 Tim Powers posted on Wednesday, June 17, 2015 - 6:58 pm
Many thanks.
Tim
 Tim Powers posted on Sunday, June 28, 2015 - 1:22 am
As a late follow-up to this discussion … I am intrigued by Linda’s comment ‘Where the parameters appear in the arrays is a matter of convenience and does not define the model.’ Do the different arrays perform different functions in defining or measuring fit in a model? Is the lambda array used for the measurement part of the model, and the beta for the structural part? Or does the covariance from all observed and unobserved variables get used in the model without real distinction?

Are the arrays simply produced for convenience and interpretational use?

Tim
 Linda K. Muthen posted on Sunday, June 28, 2015 - 9:47 am
Where the parameters appear does not impact the value that the parameter is estimated at. It is bookkeeping and has no interpretation. See Technical Appendix 2 on the website in particular the section All y Continuous Variables. The definition of the parameters comes from the MODEL command.