Can Collider Bias Explain Paradoxical Associations?

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Limiting the study population to diseased subjects may influence the effect estimates,1–4 because collider bias is introduced. Sperrin et al.5 recently suggested that “the bias is small relative to the causal relationships between the variables.” Furthermore, they stated that collider bias fully explains obesity paradoxes only if “the true causal effect is small, and therefore unlikely to be important; or the effect of the unmeasured confounder on both the mediator and the outcome is very large, therefore unlikely to be missed from the analysis.”
We agree with Sperrin et al.5 that the unknown effects of U must be large to fully explain the paradoxical associations. However, we disagree with the statement that a large, unmeasured effect is implausible. We rather believe that some diseases are characterized by large unknown effects. In this letter, we will consider the effect of obesity in patients with rheumatoid arthritis (RA), a frequently mentioned example of an obesity paradox. We will assess the bias by using a frailty model, and we argue that this model has particularly nice properties to explore selection bias.
Obesity is associated with increased mortality in the general population, and it is also associated with increased risk of developing RA.6 However, in patients with RA, obesity seems to be associated with lower mortality rates. For example, a hazard ratio of 0.93 (95% confidence interval [CI]: 0.89, 0.98) for a unit increase in body mass index (BMI) has been reported in patients with BMI above 20.7 This estimate, however, may be influenced by selection bias due to conditioning on a collider.
In a parallel publication, we suggested methods to evaluate selection bias.8 These methods are based on frailty models, and they are naturally applicable to time-to-event outcomes. We will now apply these methods to assess obesity paradoxes, using published data on RA in a crude example.
We consider the same causal structure as Sperrin et al.5 Let us study subjects at a particular age, say t = 50 years, which is similar to that of Escalante et al.7 Rather than assuming a binary U, we let U (hereby denoted the frailty) be drawn from a probability distribution, in this example the gamma distribution. Without loss of generality, we assume that the distribution has mean equal to 1. Assuming that the probability of developing RA is equal among first-degree siblings, we can now estimate the variance δ from the sibling recurrence risk (familial relative risk, FRR), when the disease is rare9: δ = FRR − 1
For RA, the FRR is reported to be 4.64.10 In contrast to Sperrin et al.,5 our approach therefore allows us to estimate the effect of U on the exposure M using published data. The effect of U on the outcome Y is still unknown, but we let this effect be the same as the effect of U on M Heuristically, we thereby assume that some people with a high risk of developing RA also have a higher mortality rate, for example, due to an aggressive form of RA or general “frailty.” Then, we use the approach described in the appendix of Stensrud et al.8 Consistent with published data,6 we assume that obese people experience an additional hazard rate of RA that is 0.4 times the baseline hazard rate. To get a notion about the size of collider bias, we first assume no causal effect of obesity on mortality, that is, a causal hazard ratio HRc = 1. Then, we would observe a hazard ratio HRo = 0.77 immediately after RA is diagnosed.
Given RA, we may also assume that obesity has a causal, harmful effect on mortality.
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