Dynamic Process for Ordinal Data Simulation and Application

Ordinal data are widely used in psychology research. There are statistical models to deal with ordinal data, such as the probit model and the logit model. However, for data with complex structures, for example longitudinal data, researchers often treat ordinal variables as continuous. The goal of this work is to identify methods for modeling longitudinal ordinal data. Several longitudinal models are reviewed and this dissertation is focused on differential equation modeling. The threshold probit model is combined with the latent differential equation model as one solution to analyze longitudinal ordinal data. In addition, two novel methods named “Mirror Model”, which reduces bias by creating bias in the opposite direction, and “Hopper Model”, which reduces bias by averaging overestimated frequency and underestimated frequency, are proposed as alternative solutions. The bias caused by fitting differential equation models to ordinal data is evaluated. Simulation results suggest that the Naive Model which blindly treats ordinal data as continuous data leads to bias, especially when the ordinal data have very few levels and the levels are divided by unequal intervals. Simulation also suggests that the Mirror Model is an unbiased and efficient estimator under ordinal data conditions, whereas the Threshold Model and the Hopper Model are unbiased and efficient under binary data condition. For power and type I error, the Threshold Model has smallest power but least type I error. Other three models suffer from α inflation. Based on the simulation results, the Mirror Model was applied to an empirical data set related with substance-use, and the Hopper Model was applied to a data set related with thought-suppression as robust estimators. The results of real data analysis confirmed the simulation findings. The Naive Model and the Mirror Model produced similar conclusion in the first application in which the ordinal data has four levels, but the Native Model and the Hopper Model produced contradictory conclusion in the second application in which the ordinal data are binary.