The Partial Derivative Framework for Substantive Regression Effects

Regression models are ubiquitous in the psychological sciences. The standard practice in reporting and interpreting regression models are to present and interpret coefficient estimates and the associated standard errors, confidence intervals and p-values. However, coefficient estimates have limited inferential utility if the outcome is modeled nonlinearly with respect to the substantively interpreted predictors. This is problematic in common modeling strategies, such as nonlinear predictor designs and/or generalized linear models. In the former, coefficients may correspond to product, power, log, and/or exponentially transformed units. In the latter, the relationship between the predictors and outcome are modeled via a function of the outcome, rather than the outcome in its original units. In both cases, the interpretation of the coefficients alone do not provide straightforward summaries of the data, and in fact may be misleading. We address these issues by developing a framework of regression effects by integrating two critical features. First, we explicitly model substantive variables in the units that provide the desired interpretation. Second, we use partial derivatives to summarize the relations between the substantive predictors and outcome variables to account for nonlinearities arising from modeling strategies. We show how to derive estimates and standard errors for quantities of interest in the interpretive units, as well as techniques to present the relationships between variables in meaningful ways. Finally, we provide demonstrations in both simulated and real data over a wide variety of models and estimation procedures. Translational Abstract Regression models are used in many aspects of the psychological sciences. Typically, researchers report and interpret coefficient estimates, standard errors, confidence intervals, and/or p-values after a statistical data analysis. However, these quantities may not provide straightforward interpretations of the results if the regression model is nonlinear. We address this issue by using partial derivatives to describe how the outcome relates to a specific predictor of interest. The partial derivative can be thought as a measure of sensitivity or a rate of change with respect to a given variable. In this article, we show to use partial derivatives to obtain useful quantities of interest, as well as their standard errors in a wide variety of situations. Examples in both simulated and real data are provided.