Simulation studies of structural equation models with covariates in a redundancy analysis framework

A recent approach to structural equation modelling based on so-called extended redundancy analysis (ERA) has been proposed in the literature, enhanced with the added characteristic of generalizing redundancy analysis (GRA) models for more than two blocks. In this approach, the relationships between the observed exogenous variables and the observed endogenous variables are moderated by the presence of unobservable composites, estimated as linear combinations of exogenous variables. However, in presence of direct effects linking exogenous and endogenous variables, the composite scores are estimated by ignoring the presence of the specified direct effects. In this paper, we generalize the ERA, extending this new class of models to allow for external covariate effects. In particular, covariates are allowed to affect endogenous indicators indirectly through the composites and/or directly. The method proposed herein is called GRA, which allows us to specify and fit a variety of relationships among composites and endogenous variables. In the paper we propose two simulation studies aimed to illustrate the advantages of GRA over ERA in terms of recovery of the original underlying structure of the data in small samples. Moreover, other than the proposal of this new methodology, a second aspect of originality of this paper is that, to our knowledge, no existing empirical research addresses the behaviour of ERA with external covariate effect in simulation studies.