Quantifying variable importance in a multimodel inference framework

The sum of Akaike weights (SW) is often used to quantify relative variable importance (RVI) within the information-theoretic (IT) multimodel inference framework. A recent study (Galipaud etal. 2014, Methods in Ecology and Evolution 5: 983) questioned the validity of the SW approach. Regrettably, this study is flawed because SW was evaluated with an inappropriate benchmark. Irrespective of this study's methodological issues, RVI metrics based on the relative contribution of explanatory variables in explaining the variance in the response variable (partitioned R-2-based) are lacking in multimodel inference. We re-evaluated the validity of SW by repeating Galipaud etal.'s experiment but with an appropriate benchmark. When explanatory variables are uncorrelated, the quantity that SW estimates (i.e. the probability that a variable is included in the actual best IT model) is monotonically related to squared zero-order correlation coefficients (r(2)) between explanatory variables and the response variable. The degree of correspondence between SW and r(2)rankings (not values) of variables in data sets with uncorrelated explanatory variables was therefore used as a benchmark to evaluate the validity of SW as a RVI metric. To address the lack of partitioned R-2-based RVI metrics in multimodel inference, we proposed 2 metrics: (a) I-weighted, the average model probability-weighted partitioned R-2; and (b) I-best, the partitioned R-2 derived from the best IT model. We performed Monte Carlo simulations to evaluate the utility of I-weighted and I-best versus partitioned R-2 derived from the global model (I-global). SW rankings matched r(2) rankings of variables; therefore, SW is a valid measure of RVI. Among partitioned R-2-based metrics, I-weighted and I-global were generally more accurate in estimating the population partitioned R-2. I-weighted performed better when explanatory variables were uncorrelated, whereas I-global was better in smaller data sets with correlated explanatory variables. To improve the utility of I-weighted in such data sets, we proposed approaches to eliminate or reduce the influence of correlated variables. Despite recent criticisms, our results show that SW is a valid RVI metric. To quantify RVI in terms of the R-2 explained by each variable, I-weighted and I-global are the preferred RVI metrics.