Chapter 5 Flexible approaches for complex settings

In the previous sections we have discussed the challenges that arise when evaluating environmental mixtures and the several available techniques based on regression modeling that can be used to to address different research questions in this context. The final section of section 3 discussed the two major limitations shared by all regression techniques, namely the difficulties in estimating overall mixture effects and to include additional model complexities such as non-linearities and (possibly high-order) interactions. In the previous section we have discussed WQS as a useful tool to address the first limitation. Note than, interestingly, this technique can be actually seen as yet another regression extension, as it is based on integrating a summary score into a generalized linear model.

To tackle the second challenge, let’s first note that any regression would allow integrating interactions of any order (this is done by simply including product terms between any pair, or higher combination, of exposures) as well as non-linear associations. Splines modeling is probably the best way of accounting for non-linear effects in regression modeling, or one can also consider using generalized additive models (GAM), which have been successfully applied in the context of environmental mixtures (Zheng et al. (2020)). Nevertheless, both the inclusion of product terms and spline transformations will rapidly increase the number of parameters that have to be estimated, and we might be in need of alternative techniques that can more flexibly tackle these issues. We are going to describe two approaches: first, Bayesian Kernel Machine Regression (BKMR), a method specifically developed for evaluating environmental mixtures that is increasing in popularity because of its several advantages and flexibility (Bobb et al. (2015)),(Bobb et al. (2018)). Second, the use of machine learning techniques, and specifically tree-based modeling such as boosted regression trees (Lampa et al. (2014)),(Bellavia et al. (2021)). Additional techniques that can be considered when the specific focus is on detecting interactions will not be discussed here, and the reader can refer to these publications summarizing and discussing methodologies in this context: Barrera-Gómez et al. (2017), Sun et al. (2013).


Barrera-Gómez, Jose, Lydiane Agier, Lützen Portengen, Marc Chadeau-Hyam, Lise Giorgis-Allemand, Valérie Siroux, Oliver Robinson, et al. 2017. “A Systematic Comparison of Statistical Methods to Detect Interactions in Exposome-Health Associations.” Environmental Health 16 (1): 1–13.
Bellavia, Andrea, Aisha S Dickerson, Ran S Rotem, Johnni Hansen, Ole Gredal, and Marc G Weisskopf. 2021. “Joint and Interactive Effects Between Health Comorbidities and Environmental Exposures in Predicting Amyotrophic Lateral Sclerosis.” International Journal of Hygiene and Environmental Health 231: 113655.
Bobb, Jennifer F, Birgit Claus Henn, Linda Valeri, and Brent A Coull. 2018. “Statistical Software for Analyzing the Health Effects of Multiple Concurrent Exposures via Bayesian Kernel Machine Regression.” Environmental Health 17 (1): 1–10.
Bobb, Jennifer F, Linda Valeri, Birgit Claus Henn, David C Christiani, Robert O Wright, Maitreyi Mazumdar, John J Godleski, and Brent A Coull. 2015. “Bayesian Kernel Machine Regression for Estimating the Health Effects of Multi-Pollutant Mixtures.” Biostatistics 16 (3): 493–508.
Lampa, Erik, Lars Lind, P Monica Lind, and Anna Bornefalk-Hermansson. 2014. “The Identification of Complex Interactions in Epidemiology and Toxicology: A Simulation Study of Boosted Regression Trees.” Environmental Health 13 (1): 1–17.
Sun, Zhichao, Yebin Tao, Shi Li, Kelly K Ferguson, John D Meeker, Sung Kyun Park, Stuart A Batterman, and Bhramar Mukherjee. 2013. “Statistical Strategies for Constructing Health Risk Models with Multiple Pollutants and Their Interactions: Possible Choices and Comparisons.” Environmental Health 12 (1): 1–19.
Zheng, Yinnan, Cuilin Zhang, Marc G Weisskopf, Paige L Williams, Birgit Claus Henn, Patrick J Parsons, Christopher D Palmer, Germaine M Buck Louis, and Tamarra James-Todd. 2020. “Evaluating Associations Between Early Pregnancy Trace Elements Mixture and 2nd Trimester Gestational Glucose Levels: A Comparison of Three Statistical Approaches.” International Journal of Hygiene and Environmental Health 224: 113446.