Epidemiologists frequently aim to quantify geospatial heterogeneity in disease occurrence to identify relevant hidden health disparities. With the growing prevalence of individual-level point-referenced data, generalized additive models (GAMs) are becoming increasingly popular to map geospatial disease risk patterns while adjusting for confounding effects when the study is a cross-sectional one with an exponential family response. In the meanwhile, local regression smoothers are frequently adopted for spatial effects estimation in GAM framework by researchers partially due to their intuitive ideas and adaptation to changing population density.
However, studies with records over a (potentially long) period of time, including those with repeated measurements on subjects, commonly come into play nowadays. For these studies, traditional GAMs could be problematic. Firstly, since data could be recorded over a period of time while spatial risk patterns should not be assumed to be invariant in many cases, statistical tools to access time-varying spatial effects are required. On the other hand, if the study is longitudinally designed, traditional GAMs could lead to incorrect inference due to their incapability of accomodating within-individual correlation.
This dissertation work sought to develop statistical methodologies to address these problems under the GAM framework with kernel smoothers, using local regression smoothers in particular. In Chapter 3, we proposed GAMs with stratified kernel smoothers that could be applied for time-specific spatial effects modeling. Based on the new class of GAMs, we further designed a hypothesis testing procedure to formally detect temporal heterogeneity of spatial effects. In Chapter 4 and 5, we incorporated random effects, as well as kernel smoothers, into GAM, resulting in a class of generalized additive mixed models (GAMMs) with kernel smoothers. We further elaborated the novel fitting and inference procedures for the proposed models.
Relevant empirical results showed the utility and advantages in model fitting under some fairly designed scenarios, with comparison to classic models. We further applied our proposed methods in a study on birth defects in Massachusetts in Chapter 3 and a study on residents' serum PFOA concentration in Lubeck, WV, and Little Hocking, OH region.