Fitting data by a certain generalized linear model means choosing appropriate forms for the predictor, the link function, and the exponential family or variance function. In the previous chapters Pearsons’s X 2, the deviance and, in the multinomial case, the power-divergence family were introduced as general goodness-of-fit statistics. This chapter considers more specific tools to select and check models. Section 4.1 deals with variable selection, i.e., which variables should be included in the linear predictor. Diagnostic methods based on the hat matrix and on residuals are described in Section 4.2, and Section 4.3 covers general misspecification tests, such as Hausman-type tests and tests for nonnested models. We do not treat tests for specific directions, such as testing the correct form of the link function by embedding it in a broader parametric class of link functions. A survey of tests of this type is contained in Chapter 11.4 of McCullagh & Nelder (1989). In addition to the methods of this chapter, nonparametric approaches, as in Chapter 5, may also be used to check the adequacy of certain parametric forms.