Letter to the Editor: Marginal structural models and the healthy worker survivor effect
Ashley Isaac Naimi, University of North Carolina at Chapel Hill
7 November 2011
Ashley I. Naimi*, Alexander Keil
* Correspondence: Epidemiology, CB7435, University of North Carolina, Chapel Hill, NC 27599
We read with interest the article by Thygesen et. al. (1) on the quantification of the healthy worker effect in a nationwide study of electricians in Denmark. The authors sought to assess the magnitude of the healthy worker survivor effect in this cohort by comparing the mortality rate in electricians who left work to those who remained at work. They further sought to minimize the bias induced by the healthy worker survivor effect using several methods, including marginal structural models. However, we were concerned about two aspects of their analysis using marginal structural models: First, the model used to estimate the inverse probability (IP) of exposure weights may not have accounted for some important time-varying confounders. Second, even with an appropriate set of measured confounders, marginal structural models fit using IP weights could not have resolved the bias induced by the healthy worker survivor effect due to violations of the positivity assumption.
Regarding the first point, we agree that the healthy worker survivor effect is an example of time-varying confounding affected by prior exposure. The authors adjusted for this time-varying confounding using a marginal structural model fit using inverse probability (IP) of exposure weights. Those weights were defined as the inverse of the probability of being an electrician in year 2 (the exposure), conditional on a measure of comorbidity, social events, and an indicator of work status in the prior year, and stabilized using age and calendar year. However, employment as an electrician in year 1, and employment status in year 2 are both time-varying covariates that confound the association between the exposure and mortality. Not including these variables in the model for the IP weights may have resulted in a bias from residual time-varying confounding.
More importantly, the healthy worker survivor effect is also characterized by what is known as a violation of the positivity assumption (or nonpositivity) (2-4): individuals who have left work cannot, by definition, be exposed. Thus, there is a zero probability of exposure for individuals who are not at work. Consequently, the inverse of a zero probability of being exposed is undefined, and marginal structural models are biased in the presence of nonpositivity (see ref 5, Appendix 2). Thygesen et al seemed to have circumnavigated this problem by modeling the probability of exposure that is not conditional on employment as an electrician in year 1, and employment status in year 2; in effect, using a model for the weights that may not account for all of the time-varying confounding of the exposure-outcome association.
We have recently explored the characteristics of the bias under conditions of nonpositivity in a simulation study (4). As expected, our findings suggest that marginal structural models do not fully account for the bias induced by the healthy worker survivor effect. Alternative methods which may be better suited to handle time-varying confounding with nonpositivity include the parametric g-formula (6) and g-estimation of structural nested models (7). Researchers interested in mitigating the bias due to the healthy worker survivor effect should consider such methods as more appropriate alternatives to marginal structural models.
1. Thygesen L, Hvidtfeldt U, Mikkelsen S, Bronnum-Hansen H. Quantification of the healthy worker effect: A nationwide cohort study among electricians in Denmark. BMC Public Health 2011;11:571. 2. Westreich D, Cole SR. Invited Commentary: Positivity in Practice. American Journal of Epidemiology 2010;171:674-677. 3. Cole SR, Hernán MA. Constructing inverse probability weights for marginal structural models. American Journal of Epidemiology 2008;168:656-64. 4. Naimi AI, Cole SR, Westreich D, Richardson D. A Comparison of Methods to Estimate the Hazard Ratio under Conditions of Time-Varying Confounding with Nonpositivity. Epidemiology 2011;22:718-23. 5. Robins JM, Hernán MA, Brumback B. Marginal Structural Models and Causal Inference in Epidemiology. Epidemiology 2000;11:550-560. 6. Taubman SL, Robins JM, Mittleman MA, Hernán MA. Intervening on risk factors for coronary heart disease: an application of the parametric g-formula. Int J Epidemiol 2009;38:1599-611. 7. Hernán MA, Cole SR, Margolick J, Cohen M, Robins JM. Structural accelerated failure time models for survival analysis in studies with time-varying treatments. Pharmacoepidemiol Drug Saf 2005;14:477-91.
Competing interests
None of the authors have personal or financial conflicts of interest
Letter to the Editor: Marginal structural models and the healthy worker survivor effect
7 November 2011
Ashley I. Naimi*, Alexander Keil
* Correspondence: Epidemiology, CB7435, University of North Carolina, Chapel Hill, NC 27599
We read with interest the article by Thygesen et. al. (1) on the quantification of the healthy worker effect in a nationwide study of electricians in Denmark. The authors sought to assess the magnitude of the healthy worker survivor effect in this cohort by comparing the mortality rate in electricians who left work to those who remained at work. They further sought to minimize the bias induced by the healthy worker survivor effect using several methods, including marginal structural models. However, we were concerned about two aspects of their analysis using marginal structural models: First, the model used to estimate the inverse probability (IP) of exposure weights may not have accounted for some important time-varying confounders. Second, even with an appropriate set of measured confounders, marginal structural models fit using IP weights could not have resolved the bias induced by the healthy worker survivor effect due to violations of the positivity assumption.
Regarding the first point, we agree that the healthy worker survivor effect is an example of time-varying confounding affected by prior exposure. The authors adjusted for this time-varying confounding using a marginal structural model fit using inverse probability (IP) of exposure weights. Those weights were defined as the inverse of the probability of being an electrician in year 2 (the exposure), conditional on a measure of comorbidity, social events, and an indicator of work status in the prior year, and stabilized using age and calendar year. However, employment as an electrician in year 1, and employment status in year 2 are both time-varying covariates that confound the association between the exposure and mortality. Not including these variables in the model for the IP weights may have resulted in a bias from residual time-varying confounding.
More importantly, the healthy worker survivor effect is also characterized by what is known as a violation of the positivity assumption (or nonpositivity) (2-4): individuals who have left work cannot, by definition, be exposed. Thus, there is a zero probability of exposure for individuals who are not at work. Consequently, the inverse of a zero probability of being exposed is undefined, and marginal structural models are biased in the presence of nonpositivity (see ref 5, Appendix 2). Thygesen et al seemed to have circumnavigated this problem by modeling the probability of exposure that is not conditional on employment as an electrician in year 1, and employment status in year 2; in effect, using a model for the weights that may not account for all of the time-varying confounding of the exposure-outcome association.
We have recently explored the characteristics of the bias under conditions of nonpositivity in a simulation study (4). As expected, our findings suggest that marginal structural models do not fully account for the bias induced by the healthy worker survivor effect. Alternative methods which may be better suited to handle time-varying confounding with nonpositivity include the parametric g-formula (6) and g-estimation of structural nested models (7). Researchers interested in mitigating the bias due to the healthy worker survivor effect should consider such methods as more appropriate alternatives to marginal structural models.
1. Thygesen L, Hvidtfeldt U, Mikkelsen S, Bronnum-Hansen H. Quantification of the healthy worker effect: A nationwide cohort study among electricians in Denmark. BMC Public Health 2011;11:571.
2. Westreich D, Cole SR. Invited Commentary: Positivity in Practice. American Journal of Epidemiology 2010;171:674-677.
3. Cole SR, Hernán MA. Constructing inverse probability weights for marginal structural models. American Journal of Epidemiology 2008;168:656-64.
4. Naimi AI, Cole SR, Westreich D, Richardson D. A Comparison of Methods to Estimate the Hazard Ratio under Conditions of Time-Varying Confounding with Nonpositivity. Epidemiology 2011;22:718-23.
5. Robins JM, Hernán MA, Brumback B. Marginal Structural Models and Causal Inference in Epidemiology. Epidemiology 2000;11:550-560.
6. Taubman SL, Robins JM, Mittleman MA, Hernán MA. Intervening on risk factors for coronary heart disease: an application of the parametric g-formula. Int J Epidemiol 2009;38:1599-611.
7. Hernán MA, Cole SR, Margolick J, Cohen M, Robins JM. Structural accelerated failure time models for survival analysis in studies with time-varying treatments. Pharmacoepidemiol Drug Saf 2005;14:477-91.
Competing interests
None of the authors have personal or financial conflicts of interest