Relative Importance by Averaging over Orderings

Many ways have been suggested for explicating the ambiguous concept of relative importance for independent variables in a multiple regression setting. There are drawbacks to all the explications, but a relatively acceptable one is available when the independent variables have a relevant, known ordering: consider the proportion of variance of the dependent variable linearly accounted for by the first independent variable; then consider the proportion of remaining variance linearly accounted for by the second independent variable; and so on. When, however, the independent variables do not have a relevant ordering, that approach fails. The primary suggestion of this article is to rescue the idea by averaging relative importance over all orderings of the independent variables. Variations and extensions of the idea are described.