Statistical Methods in Medical Research 4ed

356 Modelling continuous data
If there are interactions then the Type III and IV SSq for main effects
correspond to averages of heterogeneous effects. The use of Type II SSq corresponds
to a more commendable strategy of testing interactions in the presence of
main effects, and either testing main effects without interactions (if the latter can
safely be ignored) or not testing main effects at all (if interactions are regarded as
being present), since it would rarely be useful to correct a main effect for an
interaction. Type I SSq are also useful since they allow an effect to be corrected
for any other effects as required in the context of the nature of the variables and
the purpose of the analysis; in some cases the extraction of all the required Type I
SSq may require several analyses, with the variables introduced in different
orders. To summarize, in our view, appropriately chosen Type I SSq and Type
II SSq are useful, but Types III and IV are unnecessary.
11.9 Checking the model
In this section we consider methods that can be used to check that a fitted
regression model is valid in a statistical sense, that is, that the values of the
regression coefficients, their standard errors, and inferences made from test
statistics may be accepted.
Selecting the best regression
First we consider how the `best' regression model may be identified. For a fuller
discussion of this topic see Berry and Simpson (1998), Draper and Smith (1998,
Chapter 15) or Kleinbaum et al. (1998, Chapter 16).
The general form of the multiple regression model (11.38), where the
variables are labelled x1, x2, ..., xp, includes all the explanatory variables on
an apparently equal basis. In practice it would rarely, if ever, be the case that all
variables had equal status. One or more of the explanatory variables may be of
particular importance because they relate to the main objective of the analysis.
Such a variable or variables should always be retained in the regression model,
since estimation of their regression coefficients is of interest irrespective of
whether or not they are statistically significant. Other variables may be included
only because they may influence the response variable and, if so, it is important
to correct for their effects but otherwise they may be excluded. Other variables
might be created to represent an interactive effect between two other explanatory
variablesÐfor example, the wi variables in (11.64)Ðand these variables must be
assessed before the main effects of the variables contributing to the interaction
may be considered. All of these considerations imply that the best regression is
not an unambiguous concept that could be obtained by a competent analyst
without consideration of what the variables represent. Rather, determination of
the best regression requires considerable input based on the purpose of the