Repeated or longitudinal ordinal data occur in many fields such as biology, epidemiology, and finance. These data normally are analyzed using both likelihood and non-likelihood methods. The first part of this dissertation discusses the multivariate ordered probit model which is a likelihood method based on latent variables. We show that this latent variable model belong to a very general class of Copula models. We use the copula representation for the multivariate ordered probit model to obtain maximum likelihood estimates of the parameters. We apply the methodology in the analysis of real life data examples.
Though likelihood methods are preferable, there are computational challenges implementing them. Alternatives are the non-likelihood models. These are partially specified models, that is, in these models only the functional forms of the marginals are known but joint distributions are unknown. In addition, the dependence among the observations is modeled using an appropriate correlation structure. The second part of the dissertation outlines the estimating equations approach for the analysis of longitudinal ordinal data for these non-likelihood models. We study the asymptotic properties of the estimates for both likelihood and non-likelihood methods. Comparisons based on simulations show that the maximum likelihood estimates arising from copula models are more efficient than the estimates obtained from estimating equations.
The third part of this dissertation describes how ordinal data can be viewed as multinomial random vectors and points out the theoretical challenges in finding restrictions on the correlation parameters for dependent multinomial random vectors
