Fungible weights in logistic regression

Abstract

In this paper we discuss and develop methods for assessing parameter sensitivity in logistic regression models. To set the stage for this work, we review Waller's (2008) equations for computing standardized fungible weights in multiple linear regression (MLR) and we then extend this work by showing how to compute unstandardized fungible weights. Next, we describe what factors affect fungible weight variability for a given model. To complete this section, we describe a likelihood based method for computing fungible weights that may be appropriate when standard MLR distributional assumptions are satisfied. Having reviewed fungible weights in linear regression, we next describe two methods for computing fungible weights in logistic regression. To demonstrate the utility of these methods, we computed fungible logistic weights for two well known datasets with binary outcome variables. To better understand what factors affect logistic fungible weights, we used Monte Carlo simulations to evaluate coefficient sensitivity in samples drawn from 64 populations with a wide range of data characteristics. Interestingly, our results showed that for many data-parameter configurations---e.g., latent coefficient of determination (R_{\boldsymbol{\beta}}^{2}), structure of the predictor correlation matrix (\boldsymbol{R_{xx}}), orientation of the vector of regression coefficients with respect to the eigenvectors of \boldsymbol{R_{xx}}, and the number of predictors, p---that both methods are affected by these design factors in a similar manner. Specifically, R_{\boldsymbol{\beta}}^{2} determined the size of the fungible weights and both \boldsymbol{R_{xx}} and the orientation of \boldsymbol{\beta} determined the shape of the fungible weight contours in \mathbb{R}^{p}. The number of predictors did not have an appreciable affect other than increasing the computation time of the two algorithms. Finally, we discuss future directions for research on fungible weights in other models.