ABSTRACT
To evaluate how effective a model is in describing the outcome variable, we need to assess the quality of its fit. The lack-of-fit of a regression model is investigated by testing the hypothesis that a function has a prescribed parametric form. The function of interest is typically the mean of the response, but it might also be its variance, or the correlation between different outcomes. In other cases, it might be a complete density function of which we want to investigate the goodness-of-fit. The parametric testing methods of Chapter 7 are designed to detect very specific types of departures from the hypothesized model. For example, likelihood ratio, Wald or score tests are employed to contrast a constant and a linear dose-response curve. While very powerful for this particular class of alternative models, these tests quickly lose power when the truth is more complicated. The omnibus nonparametric methods of this chapter are appealing in that they are consistent against virtually any departure from the hypothesized parametric model. Section 9.1 describes an adaptation of the Hosmer-Lemeshow (1989) ap-
proach, for application to clustered binary data. In the remaining sections of this chapter, we discuss order selection tests based on orthogonal series estimators. In particular, Section 9.2 defines the different testing strategies for simple regression, while Section 9.3 gives extensions to multiple regression and to somewhat more specific alternative models. Section 9.4 applies these ideas to test whether density functions belong to a specific class.
