Multilevel models for the analysis of ordinal data

The scope for application of multilevel models is very wide. The term multilevel refers to a hierarchical relationship among units in a system. In an education system, for example, multilevel data is obtained from samples of randomly drawn students (level 1) from randomly drawn classes (level 2) from randomly drawn schools (level 3). Multilevel analysis allows characteristics of each group (for example the students of a specific class of a specific school) to be incorporated into models of individual behaviour. General multilevel theory is discussed. The fixed parameter linear regression model is extended to a random parameter linear regression model. Marginal maximum likelihood and the E-M algorithm are given combined as a means for estimating the unknown model parameters. A general expression for the two-level model is obtained. Maximum likelihood estimation and iterative generalized least squares are discussed as estimation procedures. The multilevel logit model is emphasized as a form of the general two-level model, and illustrated with an example. The two-level model is then extended to the general three-level model. Ordinal variables are often treated as qualitative, being analysed using methods for nominal variables. But, in many aspects ordinal variables more closely resemble interval variables. Often in analysis numerical scores are assigned to ordinal categories. This approach though is subjective. In a new approach, three models are described. These models are the logit model, the cumulative logit model and McCullagh's proportional odds model. To estimate the unknown model parameters, generalized least squares estimation is applied. The three models used for analysing data with an ordinal dependent variable is described in the context of multilevel theory. Iterative generalized least squares is discussed in this new framework. In particular Cholesky decomposition is used to obtain a positive definite matrix estimate of the covariance matrix of the explanatory variables whose coefficients are random at level 2. Examples of the logit, cumulative logit and McCullagh 's proportional odds models are used to illustrate the effect of the multilevel approach.Dissertation (MSc (Mathematical Statistics))--University of Pretoria, 1994.StatisticsMSc (Mathematical Statistics)Unrestricte