PROJECT SUMMARY Although it is well known that humans are exposed to a complex mixture of different chemicals, having constit- uents that change dynamically as an individual ages, very little is known about how these exposures interact to impact health outcomes. The overarching focus in the toxicology and epidemiology literatures has been on ex- amining the health effects of chemicals one at a time. One reason for the lack of consideration of more holistic approaches for simultaneously assessing the health effect of multiple chemicals is the lack of appropriate sta- tistical methods that are interpretable and reliable at disentangling the impact of each chemical in the mixture. When attempts are made to include different chemicals simultaneously in statistical models, most of the focus has been on generic multivariate statistical methods that often fail to have adequate performance. For exam- ple, simply including different exposures in nonparametric regression models can lead to unstable estimates due to the so-called curse of dimensionality, particularly if the different exposures are moderately to highly cor- related. The overarching goal of this proposal is to develop novel statistical approaches, which are specifically tailored for mixture exposure problems, incorporating mechanistic constraints and supplemental data on chem- ical structure and toxicological responses to improve performance. An initial focus is on developing restricted nonparametric regression methods, which constrain the response surface to be monotone with possible down- turns at low and high doses, consistent with prior data and mechanistic knowledge. Such constraints substan- tially improve stability and performance in estimating dose response, while facilitating interpretation. Another key advance is the development of mechanistic interaction models, which reduce dimensionality and enable disentangling of main effects and chemical-chemical interactions, allowing no interaction, synergy or antago- nism. A further thread designs a novel class of mechanistic response surface models, which directly incorpo- rate supplemental data on chemical structure and borrow information from one-chemical-at-a-time toxicological studies. These models enable de novo prediction of dose response and interactions for new chemicals, which have known structure but have not been studied in toxicology and epidemiology studies. These predictions in- clude an accurate characterization of uncertainty, highlighting cases in which more data are needed. To be ap- propriate for a rich variety of epidemiological study designs, the methods are generalized to account for covari- ate adjustments, longitudinal and nested data structures, censoring, and other complications. A key focus of the project is on producing user-friendly software that non-statistician scientists can use to analyze and visual- ize the health effects of mixture exposures, provided on the project's GitHub site and beta tested. Methods will be tested in a multi-tiered fashion through theoretical studies, comprehensive simulation experiments including comparisons to a rich variety of existing approaches under challenging scenarios, and applications to multiple epidemiology studies. These studies include the MSSM Children's Cohort, NHANES, and CHAMACOS.