Assessment of Model Adequacy
Model-based inferences depend completely on the fitted statistical model. For these inferences to be “valid” in any sense of the word, the fitted model must provide an adequate summary of the data upon which it is based. Hence a complete and thorough examination of model adequacy is just as important as careful model development.
The goal of statistical model development is to obtain the model that best describes the “middle” of the data. The specific definition of “middle” depends on the particular type of statistical model, but the idea is basically the same for all statistical models. In the normal errors linear regression model setting, we can describe the relationship between the observed outcome variable and one of the covariates with a scatterplot. This plot of points for two or more covariates is often described as the “cloud” of data. In model development, we find the regression surface (i.e., line, plane, or hyperplane) that best fits/splits the cloud. The notion of “best” in this setting means that we have equal distances from observed points to fitted points above and below the regression surface. A “generic” main effects model with some nominal covariates, which treats continuous covariates as linear, may not have enough tilts, bends, or turns to fit/split the cloud. Each step in the model development process is designed to tailor the regression surface to the observed cloud of data.
In most, if not all, applied ...