A semiparametric Gumbel regression model for analyzing longitudinal data with non-normal tails

Abnormal longitudinal values in biomarkers can be a sign of abnormal status or signal development of a disease. Identifying new biomarkers for early and efficient disease detection is crucial for disease prevention. Compared to the majority of the healthy general population, abnormal values are located within the tails of the biomarker distribution. Thus, parametric regression models that accommodate abnormal values in biomarkers can better detect the association between biomarkers and disease. In this article, we propose semiparametric Gumbel regression models for (1) longitudinal continuous biomarker outcomes, (2) flexibly modeling the time-effect on the outcome, and (3) accounting for the measurement error in biomarker measurements. We adopted the EM algorithm in combination with a two-dimensional grid search to estimate regression parameters and a function of time-effect. We proposed an efficient asymptotic variance estimator for regression parameter estimates. The proposed estimator is asymptotically unbiased in both theory and simulation studies. We applied the proposed model and two other models to investigate associations between fasting blood glucose biomarkers and potential risk factors from a diabetes ancillary study to the Atherosclerosis Risk in Communities (ARIC) study. The real data application was illustrated by fitting the proposed regression model and graphically evaluating the goodness-of-fit value.