I built a model in SEM (only direct paths between 4 predictors and 1 criterion) and then MGCFA for 4 groups of respondents. It performs well. However, I wanted to test if the model works for subgroups (each group can be divided into two subgroups) but they were too small for additional MGCFA (some of them are under 100 cases). I would like to do multiple regression on factor scores obtained from SEM for each subgroup. The problem is, due to the treatment of the measurement error in multiple regression, that I'll get biased results. Is there an alternative or correction for such situation?
Short answer: Yes, you can.
Longer answer: Why? You will probably mislead yourself and your reader.
You'll get a similar answer with old-style multiple regression as you will with the SEM software, especially with the relatively straightforward model that you have described. The difference is that, in multiple regression, we ignore any bias in measurement. We pretend that it's not there because there isn't a straightforward way of measuring and correcting for bias, or multicollinearity, heteroskedasticity. and so many other violations of the underlying assumptions of MR.
There are important philosophical and practical differences between SEM and MR.
In MR, we assume that the model is correct. That is, relationships are all linear, independent variables are IID (independent and identically distributed), and measured with no error, and the dependent variable is the only variable measured with error. These are never true.
In Covariance-Based SEM (e.g. AMOS, LISREL) we ask how well a theoretical model (the path diagram) corresponds to reality (the data), recognising that there may be several alternative models (theories) that correspond well with the data. Yes, there are rules-of-thumb about sample size in SEM, but these affect the usual measures of Goodness of Fit and "statistical significance" where relevant, not the coefficients. If you recognise these issues, then you will have a valid and useful story to tell.
Rather taking a step backwards from SEM to regression, why would you not rely on the path model in the SEM to estimate your "structural" coefficients?
Hello,
the answer is yes, given the constraints you mentionned in your post (but have you tried to estimate a MGCFA model ?)
You could for example use a "structural after measurement" approach proposed by Rosseel & Loh and available in an R package
The following references should be helpful :
-Rosseel Y, Loh WW. A structural after measurement approach to structural equation modeling. Psychol Methods. 2022 Nov 10. doi: 10.1037/met0000503. Epub ahead of print. PMID: 36355708.
-Rosseel, Y. (2020). Small sample solutions for structural equation modeling. In Small sample size solutions: A guide for applied researchers and practitioners (pp. 226-238). Routledge.
Dear all, many thanks for your answers and proposed literature!
I'll try to do "structural after measurement" and inform you if it works well.
Best,
Nebojša
Actually SEM is an advancement of traditional regression. SEM improve the weakness of regression where it cannot model the constructs in the model. I guess you can differentiate between constructs and variables. Now SEM is the solution for the weakness in OLS regression. Then why you want to go backward and take a factor score for each construct and treat the constructs as variables? The result of OLS regression is not as accurate as SEM.
Good day Nebojša Blanuša; in my opinion, since you mentioned MGCFA for four groups of respondents performs well, I think you should build your model in SEM based on the four groups in the first place. However, to test whether the model works for subgroups (4x2=8), you should recalculate your sample size to make your result more convincing in generalizing the findings. If possible, proceed in SEM instead of multiple regression since your MGCFA performs well (this is proof that your measurements are in the correct direction). All the best in your research work!
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