To the Editor:
Recently, Sperrin et al1 claim that collider bias is unlikely to be the main explanation for the “obesity paradox.” However, the authors neglected the fact that
in their causal model and their conclusions are questionable. In particular, the bias they report for
sensibly underestimates its actual value, and collider bias can fully explain the obesity paradox under the simple generative model they consider.
Sperrin et al1 consider the causal system depicted in Figure 1, where A, M, Y, and U correspond to obesity, diabetes status, early death, and some typically unobserved binary confounder, respectively. See Section 1 in the eAppendix (http://links.lww.com/EDE/B216) for details. They first introduce the association measure:
Several epidemiologic studies reported a negative estimate for
, suggesting that obesity was associated with a decreased mortality among diabetic patients, hence the obesity paradox. However,
has to be related to a causal effect to support this paradox. Because the focus is on diabetic patients, i.e., patients with M = m, Sperrin et al1 consider the causal effect of A on Y, conditioned on M being at level m,
The authors incorrectly establish that
after claiming, on the left column of page 526, that
But equation (2) is only guaranteed if
which, on turn, is only guaranteed if
. However, this conditional independence does generally not hold under the model depicted in Figure 1: M being on a causal path between A and Y, the set (M, U) does not satisfy the back-door criterion; see 2, 3 as well as Fine point 7.2 in 4 where the authors use the single world intervention graph approach5 in a related causal model and Section 4.2 in the eAppendix (http://links.lww.com/EDE/B216). As a result,
is generally different from
under the causal model of Figure 1. The reason for this difference is precisely collider bias, which makes
different. See Section 4 in the eAppendix (http://links.lww.com/EDE/B216) for more insights on the difference between
can not be expressed in terms of the distribution of the variables (A, M, U, Y) without further assumptions on the causal model. By specifying the structural functions and the distributions of the disturbances in the generative model considered by Sperrin et al,1 an analytic formula for
can be derived and the bias
can be evaluated (Section 3.2 in the eAppendix; http://links.lww.com/EDE/B216). As shown in Figure 2, this bias can be severe under this model, which invalidates the conclusions by Sperrin et al.1
Moreover, the premise that we are interested in
is actually questionable, and the controlled direct effect of A on Y at M = m,
, may be better suited6,7 (Sections 2 and 5 in the eAppendix; http://links.lww.com/EDE/B216). Under the model of Figure 1, it can be shown that
Comparing equations (2) and (3), the difference between
is only due to the discrepancy between P (U = u) and P (U = u|M = m, A = a). In the absence of the confounder U,
are equal. The bias between
can then be interpreted as a confounding bias and is generally limited under the generative model considered by Sperrin et al1 (Figure 2). Note, however, that we can still have a negative association measure
and a positive controlled direct effect
, as illustrated in eFigure 3 (http://links.lww.com/EDE/B216) after considering additional interaction terms in this model.
To recap, contrary to what Sperrin et al1 claim, collider bias is likely to make the bias between
substantial and can fully explain the obesity paradox. The bias between
is only due to confounding and is likely to be weaker.
Univ Lyon, Université Lyon 1
F-69373 Lyon, France
1. Sperrin M, Candlish J, Badrick E, Renehan A, Buchan ICollider bias is only a partial explanation for the obesity paradox. Epidemiology. 2016;27:525–530.
2. Pearl JCausality: Models, Reasoning, and Inference. 2000.Cambridge, United Kingdom; New York: Cambridge University Press;
3. Pearl JAn introduction to causal inference. Int J Biostat. 2010;6:Article 7.
4. Hernán MA, Robins JMCausal Inference. 2017.Boca Raton, Florida: Chapman & Hall/CRC,
5. Richardson T, Robins JMSingle world intervention graphs (SWIGs): a unification of the counterfactual and graphical approaches to causality. (2013) University of Washington, Seattle. Center for the Statistics and the Social Sciences, Working Paper 128.
6. Imai K, Keele L, Tingley DA general approach to causal mediation analysis. Psychol Methods. 2010;15:309–334.
7. Vanderweele TJ, Vansteelandt SOdds ratios for mediation analysis for a dichotomous outcome. Am J Epidemiol. 2010;172:1339–1348.