Quantifying relative importance: Computing standardized effects in models with binary outcomes

Abstract

Scientists commonly ask questions about the relative importances of processes, and then turn to statistical models for answers. Standardized coefficients are typically used in such situations, with the goal being to compare effects on a common scale. Traditional approaches to obtaining standardized coefficients were developed with idealized Gaussian variables in mind. When responses are binary, complications arise that impact standardization methods. In this paper, we review, evaluate, and propose new methods for standardizing coefficients from models that contain binary outcomes. We first consider the interpretability of unstandardized coefficients and then examine two main approaches to standardization. One approach, which we refer to as the Latent-Theoretical or LT method, assumes that underlying binary observations there exists a latent, continuous propensity linearly related to the coefficients. A second approach, which we refer to as the Observed-Empirical or OE method, assumes responses are purely discrete and estimates error variance empirically via reference to a classical R2 estimator. We also evaluate the standard formula for calculating standardized coefficients based on standard deviations. Criticisms of this practice have been persistent, leading us to propose an alternative formula that is based on user-defined “relevant ranges”. Finally, we implement all of the above in an open-source package for the statistical software R. Results from simulation studies show that both the LT and OE methods of standardization support a similarly-broad range of coefficient comparisons. The LT method estimates effects that reflect underlying latent-linear propensities, while the OE method computes a linear approximation for the effects of predictors on binary responses. The contrast between assumptions for the two methods is reflected in persistently weaker standardized effects associated with OE standardization. Reliance on standard deviations for standardization (the traditional approach) is critically examined and shown to introduce substantial biases when predictors are non-Gaussian. The use of relevant ranges in place of standard deviations has the capacity to place LT and OE standardized coefficients on a more comparable scale. As ecologists address increasingly complex hypotheses, especially those that involve comparing the influences of different controlling factors (e.g., top-down versus bottom-up or biotic versus abiotic controls), comparable coefficients become a necessary component for evaluations.