Mathematical Theory of Collinearity Effects on Machine Learning Variable Importance Measures

In many machine learning problems, understanding variable importance is a central concern. Two common approaches are Permute-and-Predict (PaP), which randomly permutes a feature in a validation set, and Leave-One-Covariate-Out (LOCO), which retrains models after permuting a training feature. Both methods deem a variable important if predictions with the original data substantially outperform those with permutations. In linear regression, empirical studies have linked PaP to regression coefficients and LOCO to $t$-statistics, but a formal theory has been lacking. We derive closed-form expressions for both measures, expressed using square-root transformations. PaP is shown to be proportional to the coefficient and predictor variability: $\text{PaP}_i = \beta_i \sqrt{2\operatorname{Var}(\mathbf{x}^v_i)}$, while LOCO is proportional to the coefficient but dampened by collinearity (captured by $\Delta$): $\text{LOCO}_i = \beta_i (1 -\Delta)\sqrt{1 + c}$. These derivations explain why PaP is largely unaffected by multicollinearity, whereas LOCO is highly sensitive to it. Monte Carlo simulations confirm these findings across varying levels of collinearity. Although derived for linear regression, we also show that these results provide reasonable approximations for models like Random Forests. Overall, this work establishes a theoretical basis for two widely used importance measures, helping analysts understand how they are affected by the true coefficients, dimension, and covariance structure. This work bridges empirical evidence and theory, enhancing the interpretability and application of variable importance measures.

Model agnostic local variable importance for locally dependent relationships

Global variable importance measures are commonly used to interpret machine learning model results. Local variable importance techniques assess how variables contribute to individual observations rather than the entire dataset. Current methods typically fail to accurately reflect locally dependent relationships between variables and instead focus on marginal importance values. Additionally, they are not natively adapted for multi-class classification problems. We propose a new model-agnostic method for calculating local variable importance, CLIQUE, that captures locally dependent relationships, contains improvements over permutation-based methods, and can be directly applied to multi-class classification problems. Simulated and real-world examples show that CLIQUE emphasizes locally dependent information and properly reduces bias in regions where variables do not affect the response.