Statistical Tolerance Regions for Flexible Modeling Paradigms

Tolerance intervals in a regression setting allow the user to quantify, with a specified degree of confidence, bounds for a specified proportion of the sampled population when conditioned on a set of covariate values. While methods are available for tolerance intervals in fully-parametric regression settings, the construction of tolerance intervals for semiparametric regression models has been treated in a limited capacity. The first project fills this gap and develops likelihood-based approaches for the construction of pointwise one-sided and two-sided tolerance intervals for semiparametric regression models. A numerical approach is also presented for constructing simultaneous tolerance intervals. An appealing facet of this work is that the resulting methodology is consistent with what is done for fully-parametric regression tolerance intervals. Extensive coverage studies are presented, which demonstrate very good performance of the proposed methods. The proposed tolerance intervals are calculated and interpreted for analyses involving a fertility dataset and a triceps measurement dataset. In the second project, the simultaneous autoregressive (SAR) model is fitted to the data from the CDC’s “500 Cities Project” which uses the Behavioral Risk Factor Surveillance System (BRFSS) to estimate tract-level prevalence of health characteristic and builds up the tolerance regions for each tract. Last but not least, the third project covers the more general development of Bayesian tolerance regions in spatial models. A modified algorithm was developed to built up Bayesian tolerance intervals. The models involved in the simulation study for Bayesian tolerance intervals (regions) are univariate model, simple linear model, linear mixed model and spatial mixed model. As for data analysis, the modified method was applied to the dataset which is Seattle mental health data to build up the tolerance intervals for each tract