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Radka Hanzlová In which way do the results differ? Do you get different chi-square values and/or df? Do you get different parameter estimates? If you could post the Mplus inputs/outputs, it would be easier to figure out what's going on.

Dear Christian Geiser Thank you very much much. I posted my outputs from Mplus. As you can see, the results differs in everything from chi-square, df, values and numbers of estimated paramaters. I would be very grateful from you comments to my output.

Just my two cents: I am not a Mplus user, therefore I do not know how it is handled there, but I had a problem in lavaan with DWLS and ULS estimators for ordinal data in measurement invariance assessment, if in some subgroups not all categories of the items have been used. E.g. an item had possible values from 1 to 5 and in group A all possible values were realized, but in group B nobody used the value 5. Lavaan provided an error message. Maybe Mplus circumvents this and estimates it nonetheless? Just my idea for your problem, but maybe it is unrelated.

Rainer Duesing Thank you very much for your answer. Yes, this is the problem also in Mplus and I also encoutered with that, but this problem is solved.

Radka Hanzlová how did you solve it?

Rainer Duesing We found that the problem was with one item in one country, so we had to exclude that country from the cross-country comparison.

Radka Hanzlová One issue that I noticed in your scalar and metric input files is that you fixed both one loading per factor to 1 (per the Mplus default, the first loading for each factor/BY statement is automatically constrained to 1) and the factor variances to 1 (e.g., f1@1). This leads to a much more restrictive (and typically unrealistic/implausible) specification. Typically, we would only do either or--that is, fix one loading per factor or the factor variance of that factor to scale the latent variables--but not both. This could explain why the Mplus output "all.out" shows much lower chi-squares and fewer degrees of freedom. You can also see that in the "all.out" output, the factor variances were freely estimated in all groups, whereas in your individual output files at least some factor variances are fixed to 1.

To replicate the (correct-looking) Mplus "all.out" output file, try deleting all statements that say f1@1, f2@1, and so on in your individual input files to free up all factor variances. That should do the trick (unless there are other problems that I may have missed of course.).

Radka Hanzlová Rainer Duesing I'm not sure you'd have to exclude the entire country only because of a single item threshold. I would tend to think that freeing up this single item threshold(s) in the country in question should suffice to account for the issue of empty categories. You may still have partial measurement equivalence (probably) and would not loose the information on an entire country just because of one item category.

Christian Geiser I am not sure if Radka Hanzlová solution with the item thresholds was also the solution for her original question here (at least I did not understand it that way, but that the thresholds have been a completely other problem she solved before...), but maybe I am wrong. I was just curious to hear a solution to it. In my case, I did not exclude a group, but rescaled this particular item so that I was able to include all groups. But maybe others have different/better solutions, which i could use next time.

Rainer Duesing It was my understanding too that the "empty category" issue was a separate topic. I just wanted to give my two cents since Radka Hanzlová mentioned she had excluded an entire country from the analysis because of a single item (at least, that was my understanding). I don't think it is necessary to exclude an entire country with thousands of cases, as partial measurement equivalence is often sufficient for latent variable comparisons across groups. A single "non-invariant" item (with freely estimated thresholds) should not do much harm unless it is a key (marker) item for that factor. As long as there are other items for the same factor that do show measurement equivalence, you may have a sufficient level of partial measurement equivalence that would permit meaningful group comparisons of, for example, latent factor means.

How did you "rescale" the item(s)? My preliminary reaction to the rescaling approach would be that it probably does not solve the problem. When there are no responses in a certain category, the information is simply not there. Rescaling does not change that, but may obscure the fact that there is missing information.

Hi Radka Hanzlová,

In addition to the helpful comments from Christian and Rainer, I would recommend that you:

For discussions around measurement invariance with ordered-categorigcal variables using WLSMV have a look at:

1. Millsap et al., 2004. Assessing factorial invariance in ordered-categorical measures

2. Liu et al., 2017. Testing measurement invariance in longitudinal data with ordered-categorical measures

For discussions around empty response options (zero cells) have a look at:

1. Rutkowski et al., 2019. Collapsing categorical variables and measurement invariance.

2. Savalei, Victoria 2011. What to do about zero frequency cells when estimating polychoric correlations.

Hope this helps!

Best,

Darun