Bilevel factor analysis models

Abstract

The theory of ordinary factor analysis and its application by means of software packages do not make provision for data sampled from populations with hierarchical structures. Since data are often obtained from such populations - educational data for example ¬the lack of procedures to analyse data of this kind needs to be addressed. A review of the ordinary factor analysis model and maximum likelihood estimation of the parameters in exploratory and confirmatory models is provided, together with practical applications. Subsequently, the concept of hierarchically structured populations and their models, called multilevel models, are introduced. A general framework for the estimation of the unknown parameters in these models is presented. It contains two estimation procedures. The first is the marginal maximum likelihood method in which an iterative expected maximisation approach is used to obtain the maximum likelihood estimates. The second is the Fisher scoring method which also provides estimated standard errors for the maximum likelihood parameter estimates. For both methods, the theory is presented for unconstrained as well as for constrained estimation. A two-stage procedure - combining the mentioned procedures - is proposed for parameter estimation in practice. Multilevel factor analysis models are introduced next, and subsequently a particular two-level factor analysis model is presented. The general estimation theory that was presented earlier is applied to this model - in exploratory and confirmatory analysis. First, the marginal maximum likelihood method is used to obtain the equations for determining the parameter estimates. It is then shown how an iterative expected max¬imisation algorithm is used to obtain these estimates in unconstrained and constrained optimisation. This method is applied to real life data using a FORTRAN program. Secondly, equations are derived by means of the Fisher scoring method to obtain the maximum likelihood estimates of the parameters in the two-level factor analysis model for exploratory and confirmatory analysis. A FORTRAN program was written to apply this method in practice. Real life data are used to illustrate the method. Finally, flowing from this research, some areas for possible further research are pro¬posed.