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Variable importance (VI) tools describe how much covariates contribute to aprediction model's accuracy. However, important variables for onewell-performing model (for example, a linear model$f(\mathbf{x})=\mathbf{x}^{T}\beta$ with a fixed coefficient vector $\beta$)may be unimportant for another model. In this paper, we propose model classreliance (MCR) as the range of VI values across all well-performing model in aprespecified class. Thus, MCR gives a more comprehensive description ofimportance by accounting for the fact that many prediction models, possibly ofdifferent parametric forms, may fit the data well. In the process of derivingMCR, we show several informative results for permutation-based VI estimates,based on the VI measures used in Random Forests. Specifically, we deriveconnections between permutation importance estimates for a single predictionmodel, U-statistics, conditional variable importance, conditional causaleffects, and linear model coefficients. We then give probabilistic bounds forMCR, using a novel, generalizable technique. We apply MCR to a public data setof Broward County criminal records to study the reliance of recidivismprediction models on sex and race. In this application, MCR can be used to helpinform VI for unknown, proprietary models.

All Models are Wrong, but Many are Useful: Learning a Variable's Importance by Studying an Entire Class of Prediction Models Simultaneously