Since we are dealing with multiple lipid outcomes that are correlated, a natural question may be to investigate whether simultaneously analyzing the phenotypes lead to methods with superior/different performance. In this step we used four phenotypes together to select environmental pollutants by multivariate regression. The micronutrients adjusted for were the union of all phenotype-specific micronutrients selected in Section 1. Specifically, for subject i, the multivariate single-pollutant model is.(4)where is the 4×1 vector of phenotypes, and are 4×1 vectors of intercepts and regression coefficients for one given pollutant, respectively, is the 4× m matrix of regression coefficients for base covariates and micronutrients W, (m×1) and . Similar to the single-phenotype method, we also applied the two-stage analysis. In stage 1, we analyzed the multivariate single-pollutant model for every pollutant using stage 1 samples and calculated the likelihood ratio Chi-squared test statistic with 4 degrees of freedom, namely, χ1 comparing the multivariate single-pollutant model with the base model ( = 0). In stage 2, we repeated the same analysis using stage 2 samples, but only for pollutants with |χ1| > (pre-defined significance
