Multiple-Group Invariance with Categorical Outcomes Using Updated Guidelines: An Illustration Using Mplus and the lavaan/semTools Packages

Abstract

Meaningful comparisons of means or relationships between latent constructs across groups require evidence that measurement is equivalent across the studied groups– a property known as measurement equivalence or invariance (ME/I). Methods typically involve an evaluation of increasingly stringent models via confirmatory factor analysis, a typical assumption of which is continuous observed variables. When that assumption is not met – as is often the case in many surveys – alternative methods that directly model the categorical nature of the data exist. Although well established, categorical ME/I models pose a number of complexities and various recommendations for their evaluation. To that end, we describe the current state of categorical ME/I and demonstrate an up-to-date method for model identification and invariance testing. In the tutorial, we exemplify a common approach to establishing ME/I via multiple-group confirmatory factor analysis using Mplus and the lavaan and semTools packages in R.

Keywords:

• Measurement invariance
• model identification
• categorical variables
• Mplus
• lavaan
• semTools in R

Notes

¹ Through an item response theory (IRT) framework, measurement invariance is also known as an absence of differential item functioning (Hambleton & Rogers, 1989; Mellenbergh, 1994; Swaminathan & Rogers, 1990); however, we do not emphasize the IRT perspective here.

² Although some scholars advocate for strict factorial invariance (Meredith, 1993) or equality of residual variances as a condition for comparing latent means (Deshon, 2004; Lubke & Dolan, 2003), in practice, this level of invariance is rarely pursued given that scalar invariance supports cross-group comparisons of manifest (or latent) variable means on the latent variable of interest (Hancock, 1997; Little, 1997; Thompson & Green, 2006).

³ Readers are directed to the original article for complete detail on all conditions posited by Wu and Estabrook (2016).

⁴ As presented in Figure 1 in Wu and Estabrook, there are six different ways that the model can be identified as a baseline model while remaining statistically equivalent. Superscript ^(g) denotes different parameters for different groups. For purposes of our tutorial, we select one path to identify the model and test for threshold followed by loadings invariance.

⁵ This finding was supported by French and Finch (2006) in the multivariate normal context.

⁶ We are not including all aspects of individual studies. For example, Cheung and Rensvold (2002) studied impact of factor variances, strengths of factor loadings, and factor correlations-aspects not necessarily examined in other studies (which included other aspects).

⁷ We note that large number of resources, including discussion sites and groups, as well as supplemental documentation, are available for Mplus (http://www.statmodel.com/) and lavaan package (http://lavaan.ugent.be/; https://groups.google.com/forum/#!forum/lavaan).

⁸ In the Appendix, we provide selected annotated output.

⁹ We note that the previously used standard chi-square difference test is not appropriate for categorical MG-CFA.

¹⁰ Conceptually, this is the same idea as we would have in multiple groups mean comparisons and using a statistical omnibus test to control for multiple pair-wise comparisons.

Additional information

Funding

This work was supported by the Norges Forskningsråd [255246].