Reframing Case Influence in SEM: Scalable, Parameter-Specific Diagnostics Via Influence Function Approximation

Identifying observations that disproportionately influence model estimates is essential for evaluating the robustness of structural equation models (SEMs), yet existing case-level diagnostics remain computationally intensive and limited in interpretive depth. This paper introduces a scalable influence diagnostic framework for SEM by adapting a curvature-aware influence function approximation from the machine learning literature. The approach combines casewise score contributions with curvature information from the observed information matrix to produce a vector-valued influence estimate for each observation without requiring repeated model fitting. Unlike scalar diagnostics such as generalized Cook's distance, the resulting influence vectors preserve the joint directional structure of each case's effect across all model parameters simultaneously, enabling parameter-level interpretation and multivariate visualization via principal component analysis. A Monte Carlo simulation study demonstrated that the proposed method detected known influential observations with greater accuracy than the approximate diagnostic implemented in the semfindr package across all 12 experimental conditions, with recall advantages ranging from .044 to .137. An empirical application to the Holzinger-Swineford dataset illustrated how the framework identifies not only which observations are influential, but how and where in the parameter space their influence is concentrated. Confirmatory case deletion verified that the influence function correctly predicted the direction of parameter change for all nine factor loadings. The proposed approach offers a theoretically grounded, computationally efficient, and interpretively richer alternative to deletion-based diagnostics for applied SEM researchers.