Linear projection inference for Individual Variable Importance

Quantifying how the predictive value of biomarkers varies across patient characteristics is central to precision medicine. While Individual Variable Importance (IVI) offers a rigorous framework for this task, existing non-parametric estimators often prove unstable and inefficient in clinical studies with limited sample sizes ($N\lt 500$). To address this, we propose a Linear Projection IVI framework that prioritizes inferential stability over functional flexibility. We formulate the estimator as a first-order approximation to high-dimensional importance surfaces using linear interaction structures. Theoretical results establish that, even under model misspecification, the estimator converges to a well-defined pseudo-true parameter corresponding to the best linear projection of the underlying IVI function. We derive closed-form expressions for local and global importance, prove $\sqrt{N}$-consistency, and introduce a principled "Gatekeeper" hypothesis test. This procedure utilizes a robust joint Wald statistic to control Type I error for heterogeneity detection prior to subgroup stratification, effectively preventing spurious discovery. Simulation studies demonstrate superior finite-sample efficiency relative to kernel-based alternatives and robustness to binary outcome misspecification. Applications to NHANES and Pima Indian diabetes datasets illustrate the framework's utility in distinguishing reproducible biological signals from noise in data-constrained settings.