| Ordinal data are often modeled using a continuous latent response distribution. In this thesis I propose the beta distribution as a model for the latent response. The beta distribution enables separate modeling of location and dispersion effects which is essential in the Taguchi method of robust design.; First, I study the problem of estimating the location and dispersion parameters of a single beta distribution from ordinal data assuming known equispaced cutpoints. Two methods of treating the data are considered: in raw discrete form and in smoothed continuousized form. Studies from a large scale simulation show that no method is universally the best, but the maximum likelihood method using continuousized data (MLE-C) is found to perform generally well, with the maximum likelihood method using discrete data (MLE-D) being the close second.; The problem of estimating unknown cutpoints is also addressed. A two-step iterative algorithm is proposed. In the first step, either the MLE-C or the MLE-D method can be used for estimating the location and dispersion parameters. In the second step only the MLE-D method is found to be appropriate for estimating the cutpoints.; Next, the test procedures for the location and dispersion effects in single factor experiments are developed. Simulation studies show that the beta model method and Nair's method exhibit similar power for detecting significant location and dispersion effects. However, Nair's method often falsely detects dispersion effect at a higher rate.; Finally, the beta model method is extended to analyzing two-level factorial experiments with ordinal responses. Simulation studies show that the beta model method provides good type I error rate control as well as good power for identifying location and dispersion effects in multifactor experiments.; In conclusion, the beta model method is both an estimation and testing method that can be used to detect significant location and dispersion effects, set the optimal and robust factor levels, and predict future responses. |
