Appropriate Standardized Estimates for Moderating Effects in Structural Equation Models

Abstract

The analyses of interactions or moderating effects are very important in psychological,behavioral and management researches.The exogenous(predictors) and endogenous(outcomes) constructs in these studies can be simple manifest variables or latent factors(made up of several observed indicators).The present paper discusses the standardized estimates of the models with moderating effects. Standardized estimates are routinely used to summarize the results of multiple regression models of manifest variables and structural equation models(SEM) of latent variables because they facilitate interpretation and comparison.Although the typical standardized estimates of moderating effects are not appropriate for multiple regression models with moderating terms,straightforward alternatives are well known.Whereas the analogous problem exists for the estimation of latent moderating effects in SEM,the situation is more complicated and apparently the difficulty has not been resolved.Here we propose appropriate standardized parameter estimates which can be easily formulated from the raw and completely standardized estimates routinely available from existing SEM software packages. It can be derived that for a structural equation with an moderating term η=γ1ξ1+γ2ξ2+γ3ξ1ξ2+ζ,the appropriate standardized parameters γ″1,γ″2,γ″3 should be γ″1=γ′1,γ″2=γ′2,γ″3=γ′3112233,where γ′1,γ′2 andγ′3 are the usual completely standardized estimates of γ1,γ2 and γ3,while 11,22 and 33 are the raw estimates of the variances of ξ1,ξ2 and ξ1ξ2.Both γ′1,γ′2,γ′3 and 11,22,33 are available from commercial SEM softwares. Some scale invariant properties of the appropriate standardized estimates are described without proofs,including that the main and moderating effects are scale invariant.That is,irrespective of the metric(e.g.,meter,centimeter) we use for the measurement,the appropriate standardized parameter estimates are identical.These desirable properties of the appropriate standardized estimates are illustrated with a simulation data set using the unconstrained approach to estimate the latent moderating effects.The results support the use of the appropriate standardized estimates in interpreting and comparing SEM estimates in latent moderating effect models. Although the difficult mathematical derivation is a bit beyond most applied SEM users,the ultimate procedures to obtain the appropriate standardized estimates are quite simple and straightforward,easily accomplished in a hand-held calculator or Excel spreadsheets.We demonstrate how applied users can obtain the appropriate standardized parameter estimates for models involving latent moderating terms.