Structural after measurement (SAM) approaches for accommodating latent quadratic and interaction effects

Established strategies commonly used to address latent quadratic and interaction effects within structural equation models, such as the unconstrained product indicator (UPI) approach or the latent moderated structural equations (LMS) approach, tend to perform effectively in models featuring only a limited number of nonlinear effects. However, as the complexity of the model increases with a higher number of nonlinear terms, the feasibility of joint or one-step methods such as UPI and LMS progressively diminishes. In response to this challenge, this paper advocates the adoption of structural after measurement (SAM) approaches to overcome this limitation. In a SAM approach, estimation proceeds in two stages. In a first stage, we estimate the parameters related to the measurement part of the model, while in a second stage, we estimate the parameters related to the structural part of the model. In this paper, we discuss three SAM approaches already documented in the literature and introduce a novel method based on the local SAM approach. To illustrate the utility of these SAM approaches, we conduct a modest simulation study, demonstrating that SAM approaches for latent quadratic and interaction effects offer a practical and viable alternative to the well-established one-step approaches.