Sequential confidence regions of generalized linear models with adaptive designs

Abstract

The methods of adaptive designs are some statistical methods which allow us to choose new design variables according to the information obtained from the previous design variables and their corresponding responses. They have been implemented into many practical applications such as in engineering-control problems, educational/psychological testing and biological assays. For example, in the modern educational testing theory, they apply the idea of adaptive designs to the tailored testing or the computerized adaptive testing, in which a new test item is chosen for each individual test-taker according to the estimate of his/her ability, which is based on his/her performance to the previous test items assigned to him/her. In this paper, we propose a sequential procedure for constructing fixed size confidence regions with a prescribed precision and a given coverage probability for the regression parameters of generalized linear models with adaptive designs. The procedure is based on the maximum quasi-likelihood estimate and without assuming canonical link functions for generalized linear models. Some large sample properties and simulation results of the proposed sequential procedures are obtained. The parallel results for generalized linear models with fixed or i.i.d. design variables can be obtained as corollaries.