Bayesian Model Selection for Derived Responses

In this article, we adapt a Bayesian model selection criterion for use with derived response variables, motivated by modeling BMI as a function of its components, weight and height. We argue that the best model for the original responses (weight and height) may not be the optimal model for a derived, non-linear quantity like BMI. Existing model selection criteria such as $AIC$ , $DIC$ , and $WAIC$ are limited because they measure the out-of-sample predictive accuracy of the original responses. To address this limitation, we propose to adapt the $WAIC$ . The Derived $WAIC$ ( $DWAIC$ ) is specifically designed to measure the out-of-sample predictive accuracy of the derived quantity of interest. This also allows for direct comparison of models based on their performance in predicting the target variable (BMI), regardless of whether they model the original components (height and weight) or the derived variable directly. We illustrate the utility of $DWAIC$ by comparing a bivariate growth model for height and weight with a spline smoothing model for BMI, a comparison that cannot be done with traditional criteria like $AIC$ or $WAIC$ .