An evaluation of likelihood-based approaches in nonlinear structural equation modelling

Abstract

Research projects in various fi elds often incorporate observable and latent variables. Observable variables are quantities which can be directly measured, such as temperature, disposable income and heart rate. Latent variables are measurements which cannot be directly quanti fied, and thus need to be measured indirectly through other manifest (observable) variables. These variables occur in a multitude of fi elds, ranging from the social sciences to the behavioural, psychological, financial and biological sciences. In biology, for instance, obesity which is a latent variable is measured through several manifest variables including height and weight. The tools to model and analyse relationships between observed and latent variables and also between latent variables themselves are provided by Structural Equation Modelling (SEM). Structural Equation Modelling is used to create statistical models which describe the relationship between observable and latent variables, and between latent variables themselves. The simplest form of SEM assumes linearity in the relationship between the latent factors and the variables used to measure them. This assumption results in restrictions when complex relationships, which may be of a nonlinear nature, are considered. Consequently, in such instances, a class of more general models needs to be considered and hence the necessity arises for nonlinear structural equation models. Allowing relationships between observable and latent variables to be nonlinear, provides a wider fi eld of work where more elaborate models may be considered. The theory of nonlinear structural equation modelling allows for several approaches which may be utilised to estimate model parameters. In this dissertation, three likelihood-based approaches will be considered, namely: the Latent Moderated Structural (LMS) equation approach; the Quasi-Maximum Likelihood (QML) approach; and the Nonlinear Structural Equation Mixture Modelling (NSEMM) approach. The behaviour of these approaches will be studied under different conditions such as: the effect of sample size, and the departure from normality assumption. The study will not only take a theoretical stance, but also a more practical approach through a Monte-Carlo simulation study where the behaviour of the model estimates, achieved using the three different approaches considered, will be analysed. An application of nonlinear structural equation modelling using an empirical dataset is also presented.