Stabilizer Variables for Measurement Invariance-Induced Heterogeneity: Identification Theory and Testing in Multi-Group Models

Abstract

When measurement invariance (MI) is violated in multi-group structural equation models, group-specific measurement artifacts inflate the between-group variance of structural parameters beyond their true values. Existing remedies-partial invariance, group-specific estimation, or moderation analysis-address the consequences of inflation but not its mechanism. This article introduces the stabilizer variable, a covariate that absorbs measurement-induced parameter heterogeneity while maintaining structural independence from the focal relationship. Two theoretical results are established: a variance decomposition theorem showing that MI violations inflate dispersion through an identifiable artifactual component, and a purification theorem proving that a stabilizer reduces this dispersion via Frisch-Waugh-Lovell projection. Two stabilization mechanisms are identified: variance purification (Type A) and directional alignment (Type B). We then develop the stabilizer variable test, a dual-criterion procedure combining nonparametric bootstrap testing for stabilization magnitude with binomial testing for directional consistency, incorporating adaptive MI severity scoring with calibrated fit-index weights. Simulations comprising 949,100 replications across varying group counts, sample sizes, and MI severity levels demonstrate 80-99% power with false-positive rates below 2%. Practical guidelines recommend K >= 10 groups and n >= 100 per group for conservative applications. The framework generalizes to any multi-group regression context where systematic measurement error induces spurious parameter heterogeneity.