On Covariate Importance for Regression Models with Multivariate Response

Abstract

We address the question of identifying the relative importance of covariates for model response, a form of sensitivity analysis. Relative importance is typically implemented as part of the model building procedure, e.g., forward variable selection or backward elimination. Here, we take a different perspective. We assume a model with multiple covariates and multivariate response has been selected and formulate criteria to assess covariate importance. Hence, with regard to covariates, our approach is joint, post model fitting, rather than conditional or sequential model creation. The noteworthy challenge we accommodate is the handling of multivariate response where individual regressions may give differing, perhaps conflicting, relative importances. In addition, we recognize that, according to the model specification, importance/sensitivity to covariates may be a global or a local issue. For models with multivariate response, we provide a criterion that (i) produces one sensitivity coefficient for each covariate, (ii) takes into account the model specification of uncertainty, and (iii) is based only on the model parameters but does not require a distribution on the covariates. However, with a prior on the covariates, in special cases, we show that comparison of covariates using this criterion gives the same results as comparison of marginal variances of the inverse predictive distributions of the covariates. We illustrate with an application examining sensitivity of tree abundance to climate.