we conducted joint analysis to combine z1 and z2 and get a new statistic that allows for between-stage heterogeneity [36],(2)where was the proportion of samples in stage 1 (0.7 in our case). zjoint was compared with a significance threshold Cjoint. Thresholds C1 and Cjoint were selected to control for the false positive rate. Details for the calculation can be found in Skol et al. [36]. Pollutants with |z 1|>C 1 and |zjoint|>Cjoint were selected for ERS and in our study, we chose C 1 and Cjoint to be 2.58 and 3.57, respectively (corresponding to a significance level of 0.01 for the Wald test in both stage 1 and stage 2 analyses). The choice of these thresholds can be optimized for enhanced power at a given false positive rate; however, we wanted to be liberal in the choice of these thresholds. Our primary goal was to identify pollutants to be included in the construction of the ERS that can be used for prediction of health risks, not just identification of individual pollutants, thus, we are less concerned about the false positive rate of the discovery process at this step. We denote the set of pollutants selected in this step as Es.
