Characterizing Unsystematic Dispersion Of Misfit In Structural Equation Models

Abstract

Structural equation modeling relies on fit evaluation to assess how well models approximate theoretical relationships, with the goal of providing evidence for the validity of constituent hypotheses. While global fit indices offer convenient summaries of overall model adequacy (Jackson, 2009; McNeish & Wolf, 2023), they cannot guarantee the validity of individual hypotheses, potentially masking substantial localized misspecifications. This limitation becomes particularly problematic in larger models, where acceptable global fit may conceal severe localized misspecification. Current practice supplements global fit with local fit evaluation, but this approach lacks a formalized tolerance for approximation error, unlike global indices that explicitly define acceptable thresholds (Browne & Cudeck, 1993; Hu & Bentler, 1999; Steiger, 2016; Steiger & Lind, 1980). To address this, the current study makes three key contributions to SEM methodology. First, it introduces misfit dispersion as a new dimension of model fit evaluation. Second, it operationalizes the concept of evenly dispersed misfit through analysis of the statistical properties of local fit, yielding the misfit proportion test for assessing disproportionate local misfit. Third, through comprehensive simulation studies, it evaluates the empirical false positive rate and power of the misfit proportion test in finite samples, establishing guidelines for use in practice. The novel framework presented here extends the principles of approximate global fit to local fit evaluation, unifying them within a consistent paradigm. By considering both the size of global misfit and its dispersion, this refined criteria better ensure that models and their constituent hypotheses provide acceptable approximations of the substantive processes they represent.