A number of-Group Evaluation in Structural Equation Modeling | by Laura Castro-Schilo | Jun, 2023

Testing Results throughout Subpopulations

A number of-group evaluation (MGA) is a statistical approach that permits researchers to research variations throughout subpopulations, or demographic segments, by enabling specification of structural equations models (SEMs) with group-specific estimates or with equal estimates throughout teams.

Variations in means, regressions, loadings, variances, and covariances of variables might be investigated utilizing MGA, as all these parameters might be modeled in SEM. Thus, regardless that different modeling strategies (e.g., evaluation of variance or regression with interplay results) make it doable to research the function of a grouping variable, these strategies are much less versatile than MGA in SEM.

Determine 1. Common overview of multiple-group evaluation and the technique for making inferences. Picture by creator.

Anytime there’s curiosity in exploring group variations, MGA could be a useful instrument. When knowledge are gathered on people, teams are most frequently outlined based mostly on components with few ranges (e.g., gender, ethnicity, occupation, household standing, well being standing, and so forth.) however can be outlined based mostly on quite a lot of different components relying on the sector, knowledge, and analytic context. Some examples of questions that may be answered with MGA in a number of totally different fields are,

Shopper analysis

  • Is product satisfaction (or high quality) totally different throughout demographic segments?

Folks analytics

  • Is worker efficiency (or motivation) equal throughout firm branches or divisions?

Well being care

  • Do affected person reported outcomes differ based mostly on drug producer?


  • Is new advertising marketing campaign efficient at rising model status in numerous geographical areas?


  • Are there cross-cultural variations in emotional expertise?


  • Is development in tutorial achievement equal throughout females and males?

All questions listed above contain variables which are unobserved (e.g., satisfaction, efficiency, and so forth.), also called latent variables. As a result of these variables can’t be noticed straight, they’re troublesome to measure.

Determine 2. Evaluating measurement of unobserved (latent) versus noticed variables. Picture by creator.

One such issue is that totally different teams can have totally different conceptualizations of those variables. Ask your self:

What is satisfaction?

What is good efficiency?

Is it probably that your responses to those questions can be totally different than these with totally different life experiences?

Many instances, the reply is sure.

Fortunately, we will check empirically whether or not totally different teams conceptualize latent variables in an identical approach. This check is carried out with MGA within the SEM framework and is called factorial invariance (aka measurement invariance). Factorial invariance checks are important to make sure comparisons throughout teams are legitimate; subsequently, these checks should be performed previous to evaluating regressions or means throughout teams (aka structural parameters) if latent variables are current.

Determine 3. The problem of modeling unobserved variables is that they won’t be measuring the identical factor throughout subpopulations. Picture by creator.

To check for variations in parameters throughout teams, researchers normally match SEMs with and with out equality constraints throughout teams. Then, the 2 ensuing fashions are in contrast utilizing a chance ratio check (equivalently, a chi-square difference test) and variations in different match statistics (e.g., the comparative match index and root imply sq. error of approximation) to evaluate whether or not imposing constraints produce statistically important worsening of mannequin match. If the match of the mannequin doesn’t worsen considerably, then the mannequin with equality constraints is retained, and one concludes that the populations into consideration don’t differ considerably on the parameter(s) examined. In distinction, if the match of the mannequin worsens considerably, the mannequin with out constraints (i.e., the place every group is allowed to have its personal estimate(s)) is retained, and one concludes that the populations into consideration differ considerably on the parameter(s) examined.

The determine beneath illustrates the technique behind MGA in a two-group instance the place a easy linear regression is match. This determine reveals equality constraints positioned on one parameter. Mannequin 1 has zero levels of freedom (i.e., it’s absolutely saturated), whereas Mannequin 2 has one diploma of freedom ensuing from the equality constraint. These fashions are in contrast based mostly on the distinction of their chi-squares, which can also be chi-square distributed with levels of freedom equal to 1 (the distinction between levels of freedom throughout fashions). A much less particular check might be carried out by inserting equality constraints on a number of parameters at a time.

Determine 4. Technique behind MGA in a two-group instance with a easy linear regression. Picture by creator.

SEMs had been developed as confirmatory fashions. That’s, one devises hypotheses, interprets them right into a testable statistical mannequin, and inferences are used to find out if the info assist the hypotheses. This method can also be utilized in MGA and is important to keep away from massive kind I error charges, which result in discovering statistical results that aren’t actually current within the inhabitants(s) of examine. Because of this, conducting all-possible comparisons throughout teams is not really useful.

Disclaimer: The paragraphs beneath are for methodologists that want to deepen their understanding of MGA. This part assumes readers perceive the full-information most chance estimator. Furthermore, the steps outlined listed here are just for explaining the logic behind MGA. In actuality, conducting MGA with these steps can be inefficient as a result of statistical software program ought to leverage algorithms that simplify this course of.

The estimation of MGA will not be totally different from that of a easy SEM with lacking knowledge. In a typical implementation of MGA-SEM, customers submit the info they wish to analyze together with a grouping variable, which signifies the group that every statement belongs to. A easy knowledge manipulation step — utilizing the grouping variable — is required to arrange the evaluation for a number of teams. The determine beneath illustrates the info which are provided for evaluation and the restructuring of knowledge for MGA.

Determine 5. Information inputted by customers and knowledge after restructuring for doing multiple-group evaluation. Picture by creator.

The ensuing knowledge can now be used with full data most chance because the estimator to make sure all rows within the knowledge are submitted for evaluation regardless of there being lacking knowledge. Just a few handy outcomes from the restructured knowledge are:

  • The log chance of any given row is barely influenced by the non-missing cells, such that including the log chance of all of the ‘Group 0’ rows yields the log chance for that group. Equally, including the log chance of all ‘Group 1’ rows yields the log chance for group 1. Every group’s log chances are used to estimate a chi-square statistic for the general mannequin, which quantifies the misfit for every group.
  • The sample of lacking values prohibits estimation of any parameter throughout the teams’ variables (e.g., the covariance of Var1_0 and Var1_1 will not be estimable), which is inconsequential as a result of MGA is worried with comparability of results throughout teams relatively than cross-group estimates.
  • ‘Vanilla SEM’ permits one to set equality constraints on parameters. Thus, utilizing the restructured knowledge in SEM, one can specify two an identical fashions with every group’s subset of variables, and equality constraints might be positioned on equal parameters throughout the teams. To reiterate, all of this may be performed in customary SEM with out asking the software program to conduct MGA explicitly.

Fortunately, these steps don’t must be carried out by customers who wish to do MGA-SEM! SEM software program makes becoming multiple-group fashions quite simple by permitting customers to specify a grouping variable. Nevertheless, doing the info manipulation (Determine 5) and utilizing customary SEM to conduct MGA-SEM will deepen your understanding of this subject. To study much more, try the sources cited beneath.

Step-by-step example of utilized multiple-group evaluation in JMP.

Book chapter on multiple-group evaluation for factorial (measurement) invariance:

Widaman, Ok. F., & Olivera-Aguilar, M. (2022). Investigating measurement invariance utilizing confirmatory issue evaluation. Handbook of Structural Equation Modeling, 367.

Journal article on utilizing various match indices to check for invariance:

Chen, F. F. (2007). Sensitivity of goodness of match indexes to lack of measurement invariance. Structural equation modeling: a multidisciplinary journal, 14(3), 464–504.

This text was initially printed within the JMP user community on February 27, 2023.

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