Theory and performance of substitution models for estimating relative causal effects in nutritional epidemiology

Background Estimating relative causal effects (i.e., "substitution effects") is a common aim of nutritional research. In observational data, this is usually attempted using 1 of 2 statistical modeling approaches: the leave-one-out model and the energy partition model. Despite their widespread use, there are concerns that neither approach is well understood in practice. Objectives We aimed to explore and illustrate the theory and performance of the leave-one-out and energy partition models for estimating substitution effects in nutritional epidemiology. Methods Monte Carlo data simulations were used to illustrate the theory and performance of both the leave-one-out model and energy partition model, by considering 3 broad types of causal effect estimands: 1) direct substitutions of the exposure with a single component, 2) inadvertent substitutions of the exposure with several components, and 3) average relative causal effects of the exposure instead of all other dietary sources. Models containing macronutrients, foods measured in calories, and foods measured in grams were all examined. Results The leave-one-out and energy partition models both performed equally well when the target estimand involved substituting a single exposure with a single component, provided all variables were measured in the same units. Bias occurred when the substitution involved >1 substituting component. Leave-one-out models that examined foods in mass while adjusting for total energy intake evaluated obscure estimands. Conclusions Regardless of the approach, substitution models need to be constructed from clearly defined causal effect estimands. Estimands involving a single exposure and a single substituting component are typically estimated more accurately than estimands involving more complex substitutions. The practice of examining foods measured in grams or portions while adjusting for total energy intake is likely to deliver obscure relative effect estimands with unclear interpretations.