Bifactor models allow users to model both a general factor and several specific (or group) factors. The focus is usually on the general factor, which is what the measure was designed for in the first place. However, we usually don't model CFAs every time we test a hypothesis; instead, we tend to rely on bifactor CFAs to support the creation of composite measures (i.e. the average or sum of all the items).
My question is: What's the best reliability estimate of such composite measures? Or, alternatively, what should I take as measurement error in a bifactor model? All the variance not explained by all the factors together (i.e. omega total reliability) or only the variance that cannot be attributed to the general factor (i.e. omega hierarchical reliability)?
As an example of the practical implications of this issue, consider the following side question: What reliability estimate (omega total or omega hierarchical) should we use when performing correction for attenuation?
the hierarchical omega, being recommended a value greater than .70
Hello Pablo,
Some of this depends on your future use of the bifactor output (and its meaning, but I will limit myself to the statistical issues). If you are solely focused on the general factor (as seems to be the case), this will lead you to focus on Omega hierarchical … or possibly Cronbach. A couple of issues.
(1) The model fit for a “CFA” on a bifactor model is almost always excellent as it tends to allow multiple pathways for co/variance. So, the issue for you (if you are focused on the general factor) is the reliability of the general factor. This may depend on your field, but generally you are looking for reliability of 0.7 or greater; anything less means the item and factor have a correlation < than 50% (i.e. the square root of the reliability).
(2) If you go with a composite measure that uses the average or sum of all items, you may want to consider your options. A Cronbach alpha may better reflect your reliability measure for such a composite variable, as this measure of reliability assumes all items equally load on (contribute to) the general factor. However, this is often an underestimation of reliability and now moves you away from the bifactor definition/meaning of the “general” factor, which accounts for the group/specific factors. In my experience, I have found the loadings on the general factor are congeneric (dissimilar) across the items. Thus, rather than using an average or sum, I suggest using the loadings to fashion your composite measure (some items measure the general factor better than others) or use the program to save factor scores for your general factor.
Food for thought … and wishing you well in your research.
Tim
Hi Pablo,
I agree with Tim that it depends on the use of your instrument and your research question. If you intend to prove that the instrument is unidimensional and that you can use a single composite score for the entire scale, you could first fit a unidimensional CFA, determine if the fit is sufficient. If it is, it will generally lead to your bi-factor model very clearly displaying unidimensionality as well, so you could skip the bi-factor model, which could save you time if time is an issue. If it is not, you could still apply a bi-factor model.
For the bi-factor model itself I will refer you to Rodriguez, Reise & Haviland (2018):
Article Evaluating Bifactor Models: Calculating and Interpreting Sta...
and Reise, Moore & Haviland (2010)
Article Bifactor Models and Rotations: Exploring the Extent to Which...
They provide cut-off scores for unidimensionality using bi-factor analyses and explain how these measures of reliability (including alpha, omega, ECV) should be interpreted.
As for your final (side)question, omegaH should be used, as mentioned by Cristian.
Best,
Michiel
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