To the Editor:
We appreciate that Goldberg et al.1 read our article2 with interest. We would like to clarify that the objective of this article was to address confounding adjustment and exposure prediction simultaneously. We focused on the simplified context of a cross-sectional study where the health outcome and the exposure prediction models are linear regression models. This setting allowed us to derive in closed form the bias of the health effect estimate under several scenarios regarding which covariates are included in the health model for confounding adjustment and which covariates are included in the exposure model for better prediction. Using the mathematical definition and properties of ordinary least squares, we demonstrated that even under the true health outcome model (i.e., fully adjusting for confounding) using a predicted exposure can lead to a biased health effect estimate.
We will address several of the points raised by Goldberg et al.1 First, by confounding bias, we mean the bias of the health effect estimate if we fail to control for any of the available covariates. This definition of confounding bias corresponds to the bias associated with an unadjusted analysis and provides a point of reference for all of the results of our article.
Second, we never claim that our results hold in the context of any epidemiologic design. In the final two paragraphs of the discussion, we wrote that the theoretical results of this article were derived in the context of cross-sectional analyses using linear regression models. The calculation of the bias in the health effect estimate under different epidemiologic designs, more complex health outcome models, and more sophisticated approaches for confounding adjustment are important topics of future work.
Third, personal risk factors should be considered as covariates in the health outcome model, and our results hold in this setting. Finally, Goldberg et al.1 state “In our experience, there are very few area-level variables included in exposure models that are true causal variables for health outcomes.” This is inconsequential. Our results suggest that if one or more of the area-level variables included in the exposure model are correlated with at least one of the confounders, then the estimated health effect is likely biased. This highlights the need for simultaneous consideration of the covariates used for confounding adjustment and the covariates used for exposure prediction.
See the results of the article for more discussion.
Santa Monica, CA
Harvard T.H. Chan School of Public Health
1. Goldberg M, Villeneuve P, Crouse D. Re: Does exposure prediction bias health-effect estimation?: the relationship between confounding adjustment and exposure prediction. Epidemiology. 2016; 28:e2–e3.
2. Cefalu M, Dominici F. Does exposure prediction bias health-effect estimation?: the relationship between confounding adjustment and exposure prediction. Epidemiology. 2014;25:583–590.