The next section details the statistical underpinnings of the measurement equivalence procedure. It starts by introducing the foundations of the model and then details the procedure for testing for measurement equivalence.
Measurement Model
The multiple-group confirmatory factor analysis (MGCFA) was introduced to test measurement equivalence across observable groups (Jöreskog, 1971). Jöreskog's (1971) hierarchical tests for measurement equivalence excluded the mean structure or tests for equality of intercepts and latent means. Sorböm (1974) extended this technique to incorporate means and covariance structures (MACS) or compare intercepts and latent means across groups. In the MACS, equality constraints for factor loadings and intercepts across groups are imposed to detect latent mean differences (Sorböm, 1974).
MGCFA allows researchers to test a priori hypotheses about model parameters across groups. The confirmatory factor analysis model (CFA) provides a framework for evaluating equality constraints across groups. The CFA is defined as:
(1) where
Xg is the vector of observed scores, 𝜏
g is the vector of intercepts or observed means on items, Λ
g is a matrix of factor loadings or the regression of the observed variables on the latent factors ξ, and δ is the vector of measurement errors for each group
g. The mean and variance-covariance matrices of the observed are defined as:
(2)(3) where
E(
Xg) is a vector of observed means, Σ
g is a matrix of observed variances and covariances, 𝜅
g is a vector of factor means, Φ
g is a matrix of factor variances and covariances, and Θ
g is a diagonal matrix of unique variances.