To the Editor:
I read with great interest the article by Naimi et al,1 assessing the implications of the use of inverse-probability-weighted Cox models in the context of the healthy-worker survivor effect. The authors conclude that, under certain scenarios, this method may produce less biased results than those produced by crude and adjusted Cox models. Their conceptual framework and methods raise several questions.
First, the authors use the term “nonpositivity bias” without having explicitly defined it, and so it remains unclear what exactly they take this bias to be. Specifically, what is the rationale for presenting nonpositivity bias as conceptually distinct and separate from confounding bias?
Second, the authors quantify the magnitude of bias of the hazard ratio (HR) estimates produced by the 3 methods against the “true” HR estimated by a Cox model with inverse-probability weighting for the unobserved confounder U. This suggests that some of the numeric differences between the HR estimates produced by the weighted versus adjusted Cox models could have been due to the fact that the latter models estimate the conditional HR (with respect to work status), whereas the former estimate the marginal HR. For example, according to the scenario presented in row 5 in Table 1, eβ is equal to 1, which means that baseline exposure X(0) has no effect on work status at follow-up W(1). Thus, conditioning on W(1) in the adjusted Cox model would not induce any correlation between U and X(0), there would be no collider-stratification bias, and the estimated HR for the association between cumulative exposure x and outcome T would be valid. Yet, the authors take the results of the simulations for this scenario to indicate that the HR estimates from the adjusted Cox model are biased (p.720).
Finally, given that among those who have left the workplace (ie, W(1) = 0), the inverse-probability weights for X(1) = 1 are undefined, the idea of applying the weights only to those with X(1) = 0 appears rather questionable, as this fails to ensure a lack of correlation between X(1) and W(1) in the population upon the weighting. Thus, the mechanism whereby inverse probability weighting could possibly reduce the magnitude of the confounding bias in the context of nonpositivity is left without explication.
Department of Social and Preventive Medicine
University of Montreal
Montreal, Quebec, Canada
The Research Institute of the University of Montreal Hospital Centre
Montreal, Quebec, Canada
1. Naimi AI, Cole SR, Westreich DJ, Richardson DB. A comparison of methods to estimate the hazard ratio under conditions of time-varying confounding and nonpositivity. Epidemiology. 2011;22:718–723.