Confounding in air pollution epidemiology: when does two-stage regression identify the problem?

A two-stage approach has recently been proposed to assess confounding by copollutants or other variables in time-series epidemiology studies for airborne particulate matter (PM), using independent series from different cities. In the first stage of the proposed method, two regression models are fitted for each city in the analysis. The first relates the health effect to the putative causal variable such as PM without including any copollutant or confounder. The other first-stage model relates a putative confounding variable to PM. In the second stage of the analysis, the estimated city-specific regression slopes for the health-effect-versus-PM model are regressed against the estimated city-specific regression slopes for the confounder-versus-PM model. Under the proposed method, a nonzero intercept estimate in the second-stage regression would be interpreted as indicating a direct pathway from PM to the health effect, and a nonzero slope estimate would be interpreted as indicating at least partial confounding of PM with the putative confounder. A simple counterexample using an additional copollutant variable shows that inferences based on this method could be misleading.