The R Book

Misspecified Model

The model may have the wrong terms in it, or the terms may be included in the model in the wrong way. We deal with the selection of terms for inclusion in the minimal adequate model in Chapter 9. Here we simply note that transformation of the explanatory variables often produces improvements in model performance. The most frequently used transformations are logs, powers and reciprocals.

When both the error distribution and functional form of the relationship are unknown, there is no single specific rationale for choosing any given transformation in preference to another. The aim is pragmatic, namely to find a transformation that gives:

• constant error variance;
• approximately normal errors;
• additivity;
• a linear relationship between the response variables and the explanatory variables;
• straightforward scientific interpretation.

The choice is bound to be a compromise and, as such, is best resolved by quantitative comparison of the deviance produced under different model forms. Again, in testing for non-linearity in the relationship between y and x we might add a term in x² to the model; a significant parameter in the x² term indicates curvilinearity in the relationship between y and x.

A further element of misspecification can occur because of structural non-linearity. Suppose, for example, that we were fitting a model of the form

but the underlying process was ...