Item response theory (IRT) models such as the Rasch model, one, two, or three parameter logistic models are measurement models that are used to estimate the latent trait of individuals. Traditional formulations of IRT models do not allow explaining individual differences. However, it is possible to use these models as statistical models when they are formulated under the generalized linear and nonlinear mixed models (GLMM and NLMM) framework. Including person properties to explain the differences among person abilities and/or item properties to explain, the differences among item difficulties into traditional IRT models that are formulated as descriptive item response models under the generalized linear and nonlinear mixed models framework, explanatory item response models (EIRM) are obtained. In this study, the application of four basic descriptive and explanatory item response models - Rasch model, latent regression Rasch model, linear logistic test model (LLTM), and latent regression LLTM - was illustrated using TIMSS 2007 mathematics data for eight grade Turkish students.