Generalized simultaneous confidence regions for regression coefficients and their ratios

This study constructs a simultaneous confidence region for two combinations of coefficients of linear models and their ratios based on the concept of generalized pivotal quantities. Many biological studies, such as those on genetics, assessment of drug effectiveness, and health economics, are interested in a comparison of several dose groups with a placebo group and the group ratios. The Bonferroni correction and the plug-in method based on the multivariate-t distribution have been proposed for the simultaneous region estimation. However, the two methods are asymptotic procedures, and their performance in finite sample sizes has not been thoroughly investigated. Based on the concept of generalized pivotal quantity, we propose a Bonferroni correction procedure and a generalized variable (GV) procedure to construct the simultaneous confidence regions. To address a genetic concern of the dominance ratio, we conduct a simulation study to empirically investigate the probability coverage and expected length of the methods for various combinations of sample sizes and values of the dominance ratio. The simulation results demonstrate that the simultaneous confidence region based on the GV procedure provides sufficient coverage probability and reasonable expected length. Thus, it can be recommended in practice. Numerical examples using published data sets illustrate the proposed methods.