Bias Correction for Relative Importance Measures via Doubly Stochastic Reallocation

2026-07-15

AI Summary

This study addresses systematic biases in existing relative importance metrics: Relative Weights suffers from compression effects, while the Green–Carroll–DeSarbo (GCD) method introduces prior bias due to unequal row sums in its redistribution matrix. The paper is the first to identify the root cause of GCD’s bias and proposes an unbiased correction based on doubly stochastic constraints, transforming the redistribution matrix into a doubly stochastic form via the Sinkhorn–Knopp (SK) algorithm or matrix alternating projections (MAP). Theoretical analysis reveals the excessive shrinkage inherent in Relative Weights, and both simulations and empirical applications demonstrate that the proposed GCD-SK method effectively eliminates row-sum bias. In scenarios dominated by the first principal component, GCD-SK significantly outperforms GCD and often surpasses Relative Weights, offering a more accurate and equitable measure for assessing variable importance.

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.

Problem

• Relative Importance
• Bias Correction
• Doubly Stochastic Matrix
• Reallocation Matrix
• A Priori Bias

Innovation

• bias correction
• doubly stochastic matrix
• relative importance
• Sinkhorn–Knopp algorithm
• reallocation matrix