Functional marginality and response-surface fitting

Well-formed polynomials contain the marginal terms of all terms; for example, they contain both x1 and x2 if x1x2 is present. Such models have a goodness of fit that is invariant to linear transformations of the x variables. Recently, selection procedures have been proposed which may not give well-formed polynomials. Analysis of two data sets for which non-well-formed polynomials have been selected shows that conversion to well-formed polynomials is beneficial in terms of goodness of fit, as well as giving fits invariant to linear transformation of the x variables. It is concluded that selection procedures should search among well-formed polynomials only.