The beta distribution as a latent response model for ordinal data (I): Estimation of location and dispersion parameters

Ordinal data are often modeled using a continuous latent response distribution, which is partially observed through windows of adjacent intervals defined by outpoints. In this paper we propose the beta distribution as a model for the latent response. The beta distribution has several advantages over the other common distributions used, e.g., normal and logistic. In particular, it enables separate modeling of location and dispersion effects which is essential in the Taguchi method of robust design. First, we study the problem of estimating the location and dispersion parameters of a single beta distribution (representing a single treatment) from ordinal data assuming known equispaced cutpoints. Two methods of estimation are compared: the maximum likelihood method and the method of moments. Two methods of treating the data are considered: in raw discrete form and in smoothed continuousized form. A large scale simulation study is carried out to compare the different methods. The mean square errors of the estimates are obtained under a variety of parameter configurations. Comparisons are made based on the ratios of the mean square errors (called the relative efficiencies). No method is universally the best, but the maximum likelihood method using continuousized data is found to perform generally well, especially for estimating the dispersion parameter. This method is also computationally much faster than the other methods and does not experience convergence difficulties in case of sparse or empty cells. Next, the problem of estimating unknown outpoints is addressed. Here the multiple treatments setup is considered since in an actual application, outpoints are common to all treatments, and must be estimated from all the data. A two-step iterative algorithm is proposed for estimating the location and dispersion parameters of the treatments, and the outpoints. The proposed beta model and McCullagh's (1980) proportional odds model are compared by fitting them to two real data sets.