On detecting the effect of exposure mixture

To study the effect of exposure mixture on the continuous health outcomes, one can use the linear model with a weighted sum of multiple standardized exposure variables as an index predictor and its coefficient for the overall effect. The unknown weights typically range between zero and one, indicating contributions of individual exposures to the overall effect. Because the weight parameters present only when the parameter for overall effect is non-zero, testing hypotheses on the overall effect can be challenging, especially when the number of exposure variables is above two. This paper presents a working model based approach to estimate the parameter for overall effect and to test specific hypotheses, including two tests for detecting the overall effect and one test for detecting unequal weights when the overall effect is evident. The statistics are computationally easy and one can apply existing statistical software to perform the analysis. A simulation study shows that the proposed estimators for the parameters of interest may have better finite sample performance than some other estimators.