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This paper introduces the Normalised Sensitivity Ratio (NSR), a post-hoc diagnostic tool that identifies causal versus spurious features in pre-trained models under a structured-shift regime. By analyzing model sensitivity across environments with varying spurious feature means, NSR achieves exact identification of causal features when applied to a linear structural causal model with three or more environments. Experimental results validate the theoretical framework, demonstrating perfect classification on synthetic data and effective recovery of causal features in real-world datasets, highlighting the method's robustness and practical applicability.
Causal features can be identified post-hoc with 100% accuracy using the Normalised Sensitivity Ratio, even in complex environments.
Given a model that is already trained, which features does it rely on causally versus spuriously? Existing methods require access to the training procedure and cannot answer this post-hoc. We introduce the \textbf{Normalised Sensitivity Ratio~(NSR)}, a post-hoc, model-agnostic diagnostic for this question under a structured-shift regime: environments differ primarily in the mean of spurious features while the causal mechanism and causal marginals remain stable, as in multi-site clinical data or multi-batch genomics. Within this regime, causal features induce constant model sensitivity across environments while spurious features track shift. NSR formalises this as the squared coefficient of variation of per-environment sensitivity. Under a linear structural causal model (SCM) with $K\ge3$ non-degenerate environments, NSR achieves exact identification (Theorem~1). We fully characterise failure: weak shifts ($O(\varepsilon^4)$ collapse), degenerate geometry, and proxy attenuation ($O((1-\alpha)^4)$), giving practitioners quantitative criteria for assessing whether the regime holds. Finite-sample rates are $O_p(n^{-1})$ under the null and $O_p(n^{-1/2})$ under the alternative. Experiments confirm all theoretical predictions on synthetic data (area under the ROC curve [AUROC] $= 1.000$ under conditions satisfying the regime), show consistent rankings across five model families (Kendall $\tau\ge0.529$), and recover six of eight causal features on bike-sharing data (Precision@7 $= 0.75$) without modifying any trained model.
