I have a bifactor model with categorical indicators: 4-option Likert-type items. One of my research objectives involves testing measurement invariance between two groups. I have been doing it as follows:
1. A configural model in which all loadings are allowed to vary between groups. Factors' variances are set to 1 in both groups. One threshold per item is constrained to be equal across groups, but the rest of them are freely estimated. (The model does not converge if this constraints are not included).
2. A scalar model in which equality constraints are applied upon all items and factor loadings. (I have read that metric invariance should not be tested alone in nonlinear MG-CFA).
Despite the increasing popularity of bifactor models and categorical CFA, I have not been able to find clear guidelines about how to conduct such an analysis. Am I doing this right? Is there an expert consensus about how to evaluate invariance in categorical bifactor models (e.g. Δχ², ΔCFI, DIFFTEST)?
Any suggestion will be appreciated. I am working with lavaan in R.
One method that I think is particularly good for testing measurement invariance is permutation testing, as implemented in the semTools package. It establishes an empirical null distribution for the change in a given ΔAFI or Δχ² statistic, and then based on the empirical value determines whether the decrement in for us statistically significant. The nicest thing about this method is that it's light on assumptions, and it maintains nominal type 1 error rates even if the configural model is incorrectly specified.
See here for some literature on the method:
https://www.tandfonline.com/doi/full/10.1080/10705511.2017.1421467
https://www.frontiersin.org/articles/10.3389/fpsyg.2017.01455/full
Dear Pablo D Valencia ,
the information below may be useful.
The one-way multivariate analysis of variance (one-way MANOVA) is used to determine whether there are any differences between independent groups on more than one continuous dependent variable. In this regard, it differs from a one-way ANOVA, which only measures one dependent variable.
For example, you could use a one-way MANOVA to understand whether there were differences in the perceptions of attractiveness and intelligence of drug users in movies (i.e., the two dependent variables are "perceptions of attractiveness" and "perceptions of intelligence", whilst the independent variable is "drug users in movies", which has three independent groups: "non-user", "experimenter" and "regular user"). Alternatively, you could use a one-way MANOVA to understand whether there were differences in students' short-term and long-term recall of facts based on three different lengths of lecture (i.e., the two dependent variables are "short-term memory recall" and "long-term memory recall", whilst the independent variable is "lecture duration", which has four independent groups: "30 minutes", "60 minutes", "90 minutes" and "120 minutes").
Note: If you have two independent variables rather than one, you can run a two-way MANOVA instead. Alternatively, if you have one independent variable and a continuous covariate, you can run a one-way MANCOVA. In addition, if your independent variable consists of repeated measures, you can use the one-way repeated measures MANOVA.
It is important to realize that the one-way MANOVA is an omnibus test statistic and cannot tell you which specific groups were significantly different from each other; it only tells you that at least two groups were different. Since you may have three, four, five or more groups in your study design, determining which of these groups differ from each other is important. You can do this using a post-hoc test (N.B., we discuss post-hoc tests later in this guide).
In this "quick start" guide, we show you how to carry out a one-way MANOVA using SPSS Statistics, as well as interpret and report the results from this test. Since the one-way MANOVA is often followed up with post-hoc tests, we also show you how to carry these out using SPSS Statistics. However, before we introduce you to this procedure, you need to understand the different assumptions that your data must meet in order for a one-way MANOVA to give you a valid result. We discuss these assumptions next.
SPSS Statistics
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