Can collider bias fully explain the obesity paradox?

The "obesity paradox" has been reported in several observational studies, where obesity was shown to be associated to a decreased mortality in individuals suffering from a chronic disease, such as diabetes or heart failure. Causal arguments have recently been given to explain this apparently paradoxical fact: because the chronic disease is caused by obesity, the observed "protective effect" of obesity among patients with, say, diabetes, actually has no causal value. Recently, Sperrin et al. (2016) relaunched the debate and claimed that the resulting bias, the so-called collider bias, was unlikely to be the main explanation for the obesity paradox. However, a number of issues in their work make their conclusions questionable. In this article, we first study the bias between (i) the association between obesity and early death among patients suffering from the chronic disease $\Delta_{AS}$ and (ii) the causal effect considered by Sperrin et al. Under the usual framework of structural causal models, we explain why this bias can be much higher than what these authors reported. We further consider alternative causal effects of potential interest and study their difference with $\Delta_{AS}$. Numerical examples are presented to illustrate the magnitude of these differences under realistic scenarios. We show that it is possible to have a negative $\Delta_{AS}$, while the causal effects we considered are all positive. Therefore, even under the very simple generative model we considered, collider bias can be the sole cause of the obesity paradox.