Bootstrapping and the identification of exogenous latent variables within structural equation models

Abstract

In traditional applications of latent variable models, each exogenous latent variable must either have its variance parameter fixed or a loading path to a measured indicator variable fixed (either customarily to 1). Without doing so the measurement model will suffer from underidentification, thereby yielding no unique solution when estimating the parameters of interest. The choice of whether to fix the variance or the loading is somewhat arbitrary, guided primarily by the researcher's need for inference regarding particular parameters within the model. Under conditions of multivariate nonnormal data, the method by which one makes identified the measurement of exogenous latent variables may not be as arbitrary. Specifically, as addressed briefly by Arbuckle (1997), when one is utilizing a bootstrapping approach for generating empirical standard errors for parameters of interest, the researcher must choose to fix an indicator path rather than the latent variable variance in order for the empirical standard errors to be gensrated properly. This article offers an illustrative explanation of why such an approach is necessary. Given the increased attention toward bootstrapping techniques within structural equation modeling, our hope is that a greater awareness and understanding of this unique situation will be facilitated.