Bayesian Model Checking in Multivariate Discrete Regression Problems

by Dong, Fanglong

Ordinal data are common in the academic area such as a student grade, A, B, C, D, or F, also ordinal data are common in other area such as customer satisfaction survey. It is straightforward to fit a regression model to reflect the relationship between the response and the predictors. Since the response in an ordinal data set is a vector, it is not clear how the traditional statistics define residuals and detect outliers because of the dimension of response. Since the introduction of latent variable, we can model the data using the latent variable and we have a new type of residual called latent residual. With the help of introduction of latent variable into the model, it is easy to define residuals and detect outliers. In practice there are usually more than one predictor in the data set and we need to decide to choose variable that should be included in the model. We look at from a frequentist's perspective and a Bayesian perspective. Also when we fit a model to a data set, we care about how well this model fit the data set, and we look from both a frequentist's perspective and a Bayesian perspective. Usually methods from a frequentist's perspective rely on the asymptotic distribution to draw a conclusion and sometime this will become a problem especially when the sample size is small, on the contrary, methods from a Bayesian perspective use simulation and thus it removes the reliance on the asymptotic distribution. Chapter 3 talks about methods for outlier detection problems and Chapter 4 talks about goodness-of-fit and model selection problems, in Chapter 5 we apply the methods from Chapter 3 and Chapter 4 to the BGSU student data set. Chapter 6 summarized the whole dissertation and possible future research interest and applications.