Collider bias is only a Partial Explanation for the Obesity Paradox

Background: Obesity paradox refers to the commonly reported association between obesity and reduced mortality where obesity is a risk factor for selection into the population under investigation, for example a case-cohort within type 2 diabetes. A common explanation is collider stratification bias, which proposes that the obesity paradox is driven by unmeasured confounding induced by selection bias. Methods: Using a directed acyclic graph and regression models to characterise relationships between variables, we derive equations that give the extent of bias caused by collider stratification in a selected population using counterfactual causal analysis. We illustrate the bias for a range of parameter settings, describing different associations between the exposure (obesity), the outcome (mortality), the mediator (diabetes status) and an unmeasured confounder. Results: Collider stratification leads to biased estimation of the causal effect of the exposure on the outcome. However, the bias induced is small relative to the causal relationships between the variables. In particular, if there were a causal effect between the exposure and the outcome, the confounding would need to be at least three times stronger than this effect to observe an association between the exposure and outcome in the opposite direction. Conclusions: While collider bias can be a partial explanation of the obesity paradox, it is unlikely to explain an observed association between obesity and mortality in the reverse direction to a true causal relationship. Alternative explanations of the obesity paradox should continue to be explored.