Evaluating Local Structural-After-Measurement (LSAM) and Traditional Approaches for the Estimation of Complex Nonlinear Effects Among Latent Variables

Various methods exist to include quadratic or interaction terms involving latentvariables in structural equation models (SEM). Most use system-wide estimation,where all parameters are estimated simultaneously, and have been investigated inmodels with few nonlinear effects. Recently, structural-after-measurement (SAM)approaches have been proposed, where estimation proceeds in two stages:measurement model first, then structural model. Rosseel et al. (2025) extended localSAM (LSAM) to handle second-order nonlinear effects among latent variables. In thisarticle, we present a formula for two-step standard errors (SEs) not previously availablefor LSAM. We also conducted two simulation studies varying latent exogenouspredictor distributions (normal, right-skewed, uniform), reliabilities (0.4, 0.6, 0.8),sample sizes (400, 1000), and measurement error and structural disturbancedistributions (normal, right-skewed). The first study examined a model with threenonlinear effects. LSAM produced largely unbiased estimates with stable coverage,SEs, and Type I error rates, though power fluctuated and performance was adverselyaffected under low reliability, particularly with uniform latent exogenous distributions.Larger sample sizes mitigated these issues. Traditional methods (LMS, QML, UPI)showed results consistent with prior literature. The second study, excluding LMS,tested a more complex model with eight nonlinear effects. LSAM maintained adequateperformance despite greater complexity, though similar limitations emerged and powerdecreased under certain conditions. Traditional methods yielded more variable results,with generally poorer performance relative to Simulation 1 and LSAM. Right-skewedmeasurement errors affected the abovementioned results more than right-skewedstructural disturbances, particularly under low and medium reliability.