Selected Mathematical and Statistical Results
R. Darrell Bock, Robert D. Gibbons · 2021
This chapter presents, mostly without proof, various definitions, results, and algorithms needed in the sequel. The statistical methods applied in the sequel to item response data are for the most part random-effects extensions of standard fixed-effects procedures for the analysis of binomial and multinomial data. In particular, they are generalizations of well-known methods of probit and logit analysis. The chapter introduces the notation and concepts of item response theory. It considers the distributions of linear and nonlinear functions of random variables. When the functions are expressed in terms of matrices, the required results can be obtained more easily by vector and matrix differentiation than by the more familiar methods of scalar calculus. Classical test theory develops certain implications of a simple measurement model with additive errors. The object of classical test theory is to account for the statistical properties of test scores under sampling.