The unconstrained product indicator (PI) approach is a simple and popular approach for modeling nonlinear effects among latent variables. This approach leaves the practitioner to choose the PIs to be included in the model, introducing arbitrariness into the modeling. In contrast to previous Monte Carlo studies, we evaluated the PI approach by three post-hoc analyses applied to a real-world case adopted from a research effort in social psychology. The measurement design applied three and four indicators for the two latent first-order variables, leaving the researcher with a choice among more than 4000 possible PI configurations. Sixty so-called matched-pair configurations that have been recommended in previous literature are of special interest. In the first post-hoc analysis we estimated the interaction effect for all PI configurations, keeping the real-world sample fixed. The estimated interaction effect was substantially affected by the choice of PIs, also across matched-pair configurations. Subsequently, a post-hoc Monte Carlo study was conducted, with varying sample sizes and data distributions. Convergence, bias, type I error and power of the interaction test were investigated for each matched-pair configuration and the all-pairs configuration. Variation in estimates across matched-pair configurations for a typical sample was substantial. The choice of specific configuration significantly affected convergence and the interaction test’s outcome. The all-pairs configuration performed overall better than the matched-pair configurations. A further advantage of the all-pairs over the matched-pairs approach is its unambiguity. The final study evaluates the all-pairs configuration for small sample sizes, and compares it to the non-PI approach of LMS.
Foldnes, Njål and Knut Arne Hagtvet. 2014. “The choice of product indicators in latent variable interaction models: post hoc analyses.” Psychological Methods, 19(3): 444-457