Measurement validation in the behavioral sciences is generally carried out in a psychometric modeling framework that assumes unobservable traits/constructs (ie, latent factors) created from the observed variables (often items measuring that construct) are the variables of interest. Unfortunately, many researchers compare these constructs across populations/groups (eg, males & females) assuming that they have the same psychometric properties and association between unobserved and observed across groups of interest. In other words, researchers will often make the premature and untested assumption that the theoretical constructs they are interested in are invariant (or equivalent) from one group to another. When researchers assume a measure is invariant, they are failing to investigate whether the construct has factorial invariance (Byrne, Shavelson, & Muthén, 1989; Millsap, 1998). In particular, they fail to test whether the latent factor scores were generated in a similar fashion across groups, thus producing the same metric (unstandardized factor loadings) and scalar (intercept or threshold) parameters.

With the advent of computers and accessible latent modeling software, invariance testing within multi-group confirmatory factor analysis (MCFA) and multi-group structural equation modeling (MSEM) literature has increased considerably over the past 20 years (Meade, Johnson, & Braddy, 2008; Vandenberg & Lance, 2000). This has enabled researchers to more easily explore whether latent factors, along with the relationship between latent factors, are invariant across populations. Despite methodological guidelines, statistical procedures, and widely available software, researchers continue to struggle with the numerous decisions that need to be made when testing for invariance. As delineated below, several authors have assessed the utility and applicability of invariance procedures to provide guidelines including: 1) setting the measurement scale, 2) evaluating model fit and statistical power, and 3) estimating the appropriate model depending on the data characteristics. The present chapter aims to provide an up-to-date review of important considerations when conducting MCFA and MSEM analyses; especially as they relate to assessing whether or not the latent factors (ie, metric and scalar parameters), latent factor mean scores, and structural coefficients are equal across populations. A demonstration