Interpretation of standardized intercepts, and standardized effect sizes for measurement invariance testing

[CARLOS GONZALEZ POSES]

Dear all,

I was just working with measurement invariance testing using lavaan and semTools::miPowerFit function and have some doubts about standardized parameters and unstandardized parameters, and how to interpret them. I found resources discussing the issue of how standardized loadings are to be interpreted (e.g., Brown, 2015), but I couldn't find a resource discussing how standardized intercepts are to be interpreted. Thus, my questions are:

1. How are standardized intercepts to be interpreted? How are standardized intercepts transformed back to unstandardized intercepts? Are there any good resources on this?

I think this question is of interest for lavaan users in general who use an standardized solution. However, in the context of invariance testings, it is equality of unstandardized parameters what we are interested in (see this thread for a nice explanation and discussion:  https://groups.google.com/g/lavaan/c/eRRgQwWrkt4/m/r9eK39WNB00J ). But if one engages in measurement invariance testing using miPowerFit, there's the option of introducing sizes of "standardized factor loadings" and "standardized intercepts" that one would like to be detected (i.e., kind of like "effect sizes" that are too big). Therefore:

2. If in measurement invariance testing what we care is about unstandardized loadings and unstandardized intercepts, should we use standardized effect sizes in miPowerFit for MI testing - for instance for detecting partial measurement invariance? Would this mean we are indeed using different unstandardized thresholds for each item and group? Would that make sense?

I hope the question is of interest and somebody is able to answer it.

Best,

Carlos 

[Terrence Jorgensen]

1. How are standardized intercepts to be interpreted?

The distance from zero in units of that variable's  SD.

 

How are standardized intercepts transformed back to unstandardized intercepts?

By multiplying the standardized intercept by its (model-implied, I believe) SD.

 

2. If in measurement invariance testing what we care is about unstandardized loadings and unstandardized intercepts, should we use standardized effect sizes in miPowerFit for MI testing - for instance for detecting partial measurement invariance? Would this mean we are indeed using different unstandardized thresholds for each item and group? Would that make sense?

An invalid equality constrain on unstandardized intercepts implies that their unconstrained values would differ.  The EPCs in the miPowerFit() output tell you how much the unstandardized estimates are expected to differ when the constraint is released.  The standardized EPCs tell you the same, but in units of SD, which can still be informative.  But I agree, it would be more informative to have a (single) standardized difference between a pair of intercepts, which is the effect size we are interested in.  Millsap & Olivera-Aguilar (2012) came up with a nice definition I think is useful.  In a partial-invariance model where the intercept of interest is freely estimated, divide the difference in intercepts by the difference in observed means.  That expresses uniform DIF as a proportion of the observed mean difference (the remainder of the mean difference being attributable to differences in factor means). 

Millsap, R. E., & Olivera-Aguilar, M. (2012). Investigating measurement invariance using confirmatory factor analysis. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp. 380–392). The Guilford Press.

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

[CARLOS GONZALEZ POSES]

Dear Terrence,

Thank you very much for your responses! 

You said: "The distance from zero in units of that variable's  SD". Do I understand correctly by "that variable's SD", you mean the observed indicator standard deviation (not the latent factor standard deviation)?

Also, then, I guess that in the absence of a "a (single) standardized difference between a pair of intercepts", I guess the next best option is to use standardized effect sizes in its current form. Thanks a lot for the reference, it is an interesting read indeed.

Best,

Carlos

El dia divendres, 26 de novembre de 2021 a les 14:25:23 UTC+1, Terrence Jorgensen va escriure:

[Terrence Jorgensen]

Do I understand correctly by "that variable's SD", you mean the observed indicator standard deviation (not the latent factor standard deviation)?

Every variable in the model (indicators and factors) has an intercept.  A standardized intercept for variable X is X's intercept divided by X's SD.

[CARLOS GONZALEZ POSES]

Thank you so much!

El dia dilluns, 29 de novembre de 2021 a les 12:14:01 UTC+1, Terrence Jorgensen va escriure: