Generalized Linear Mixed Model and Calibration for Gamma Random Variables: Application to Asbestos Fibers

Statistical methods for analyzing positively skewed data, which commonly arise in environmental monitoring and assessment, are presented. Specifically, I develop relevant methods to estimate the underlying calibration curve and construct the confidence and prediction intervals based on the gamma distribution. Environmental data are subject to measurement errors, and the variance of measurement errors usually depend upon the true concentration level. In addition, when multiple laboratories are involved in analyzing monitoring samples, additional variation at the laboratory level should be incorporated in the analysis. In this dissertation, I propose a mixed-effects gamma regression model to account for the non-constant variability as well as the between-laboratory variability to estimate a calibration curve. I also explore the two-component mixed model to fit environmental data and discuss its applications. The proposed methods borrow strength from all laboratories to estimate the model parameters. I discuss how to estimate unknown true concentrations using the estimated calibration curve when an independent set of samples are obtained. I also derive the global calibration confidence interval that does not require new data from the same set of laboratories, from which background samples were collected. The performances of the estimation procedures, the calibration confidence regions, and their robustness are studied via simulation. I observe that the global calibration confidence intervals based on the gamma mixed model perform robustly in all situations considered. To illustrate the results, a real data set of amosite asbestos fibers is used.