On variable importance in linear regression

The paper examines in detail one particular measure of variable importance for linear regression that was theoretically justified by Pratt (1987), but which has since been criticized by Bring (1996) for producing "counterintu itive" results in certain situations, and by other authors for failing to g uarantee that importance be non-negative. In the article, the "counterintui tive" result is explored and shown to be a defensible characteristic of an importance measure. It is also shown that negative importance of large magn itude can only occur in the presence of multicollinearity of the explanator y variables, and methods for applying Pratt's measure in such cases are des cribed. The objective of the article is to explain and to clarify the chara cteristics of Pratt's measure, and thus to assist practitioners who have to choose from among the many methods available for assessing variable import ance in linear regression.