Multgroup CFA with ordinal variables: strict and scalar invariance; structural invariance

[Quanfa He]

Hi, 

I am testing multigroup CFA with ordinal variables (three levels, 0,1,2) and testing invariance across three groups. 

Based on the Wu and Estabrook article and a few other discussions I have found online, I am using the measEq.syntax to specify configural, metric, scalar and strict invariance models. Configural and metric models had no issue for me. For simplicity, this is the syntax for scalar model that I used: 

         

Model_full_fit_scalar <- measEq.syntax(configural.model = Model_full,
                                       data = dat,
                                       ordered = TRUE,
                                       parameterization = "theta",
                                       ID.fac = "std.lv",  

                                       ID.cat = "Wu.Estabrook.2016",

                                       group = "grouping",
                                       group.equal = c("thresholds","loadings","intercepts"))

And the strict invariance model only differ by adding "residuals" in group.equal=.

One issue came up: the chi-sq statistics for the strict and scalar model were identical, even though the degrees of freedom were not, and there were slight differences in CFI, and TLI. I have checked the syntax to make sure that the residuals of items were constrained across groups. 

How do I interpret the same chi-sq between the two models? Can I interpret the strict model results in this case? Finally, I also intend to test structural invariance (e.g., factor covariances and variances) across groups. Is strict invariance required to test structural invariance? 

Happy to provide additional details if needed

Thanks all!!

[Edward Rigdon]

Strict invariance should include more restrictions, so degrees of freedom should be higher. If chi-square does not increase, this indicates that the test was not able to detect discrepancy between the incremental constraints and the data. That failure may be due to a lack of power or due to the additional constraints actually matching the population covariance matrix. The usual statistical test is a chi-square difference test. But if chi-squares are identical--quite surprising, actually--then the chi-square difference is 0, supporting the additional constraints. CFI incorporates the quantity max ( chi-square - degrees of freedom, 0), so if chi-square does not change but degrees of freedom get larger, this quantity will get smaller, as long as chi-square is greater than degrees of freedom.

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[Michael Schepisi]

Hi, 

my collaborators and I had the same problem when we tried to run a multigroup CFA with an ordinal variable with 3 response cateogories (levels). 
I sent an email to Ryne Estabrook, since he wrote a very nice article on how to conduct multigroup CFA (Wu, H., & Estabrook, R. (2016). Identification of confirmatory factor analysis models of different levels of invariance for ordered categorical outcomes. Psychometrika, 81(4), 1014-1045). 
This is what he texted me back:

"The fact that you’re at three categories (and thus, two thresholds) suggests something to me. Each ordinal item requires two constraints to scale the underlying continuous variable we assume is beneath it. When you constrain your model to have equal thresholds across groups, you’re simply rescaling each variable, which won’t result in any loss of fit. Put differently, there’s no difference in constraining the thresholds for item 1 group 2 to be zero and one or constraining them to be the same as item 1 group 1.
This issue is the broad point of that paper. There’s no such thing as an unconstrained categorical factor model. You make some constraints to make the model identifiable. Sometimes those constraints interact with other constraints in weird ways. You started at something equivalent to the configurally invariant model, so making that “constraint” didn’t change anything". 

Indeed, when I tried another CFA with more than 3 response categories I didn't have this issue anymore. 

Hope this helps. 

Best regards, 

Michael Schepisi

 

[Quanfa He]

Thank you both for your helpful explanation and comments! Apologize for the delayed responses. 

Two questions I now have are: 

1. Can I test structural invariance (i.e., latent variance and covariance) on the basis of scalar invariance? Or do I have to achieve strict invariance before moving forward with structural invariance testing. My understanding so far is that I can test latent variance and covariance after achieving even metric invariance, but getting some clarification and confirmation would be much appreciated!

2. In my metric, scalar and strict invariance models, I found that many of the residual variances have no standard errors, and some of them have "NA" standard errors. I wonder if this should be a cause of concern? In the configural model, residual variances were estimated and look fine. 

Thanks!