Bias Correction for Relative Importance Measures via Doubly Stochastic Reallocation

Abstract

Relative importance (RI) analysis quantifies each predictor's contribution to the explained variance of a linear model. General Dominance (GD), a widely used benchmark, requires evaluating $2^p-1$ sub-models and becomes computationally intensive as the number of predictors $p$ grows. Orthogonalization-Reallocation Measures (ORMs), including Relative Weights (RW) and the Green--Carroll--DeSarbo measure (GCD), provide efficient alternatives by assigning importance to orthogonalized predictors and reallocating it to the original predictors. Each, however, has a structural limitation: RW exhibits a leveling problem that compresses differences among predictor importance values, whereas GCD exhibits an a priori bias that systematically favors certain predictors before a response is observed. We show that this bias is governed by the row-sums of the reallocation matrix. A closed-form analysis under compound symmetry relates the reallocations underlying GCD and RW to a GD-based benchmark, showing that homogeneous multicollinearity alone does not induce an a priori bias and formalizing RW's leveling problem as excess shrinkage relative to the benchmark. We correct GCD's bias by mapping its reallocation matrix to a doubly stochastic matrix using the Method of Alternating Projections (MAP) and the Sinkhorn--Knopp (SK) algorithm, yielding GCD-MAP and GCD-SK. Comprehensive simulations show that GCD-SK removes the structural row-sum bias, substantially improves upon GCD, and often outperforms RW when the first principal component is dominant. We conclude with empirical guidelines for selecting among the measures.