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  1. There is substantial interest in assessing how exposure to environmental mixtures, such as chemical mixtures, affects child health. Researchers are also interested in identifying critical time windows of susceptibility to these complex mixtures. A recently developed method, called lagged kernel machine regression (LKMR), simultaneously accounts for these research questions by estimating the effects of time‐varying mixture exposures and by identifying their critical exposure windows. However, LKMR inference using Markov chain Monte Carlo (MCMC) methods (MCMC‐LKMR) is computationally burdensome and time intensive for large data sets, limiting its applicability. Therefore, we develop a mean field variational approximation method for Bayesian inference (MFVB) procedure for LKMR (MFVB‐LKMR). The procedure achieves computational efficiency and reasonable accuracy as compared with the corresponding MCMC estimation method. Updating parameters using MFVB may only take minutes, whereas the equivalent MCMC method may take many hours or several days. We apply MFVB‐LKMR to Programming Research in Obesity, Growth, Environment and Social Stressors (PROGRESS), a prospective cohort study in Mexico City. Results from a subset of PROGRESS using MFVB‐LKMR provide evidence of significant and positive association between second trimester cobalt levels andz‐scored birth weight. This positive association is heightened by cesium exposure. MFVB‐LKMR is a promising approach for computationally efficient analysis of environmental health data sets, to identify critical windows of exposure to complex mixtures.

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  2. Exposure to environmental mixtures can exert wide‐ranging effects on child neurodevelopment. However, there is a lack of statistical methods that can accommodate the complex exposure‐response relationship between mixtures and neurodevelopment while simultaneously estimating neurodevelopmental trajectories. We introduce Bayesian varying coefficient kernel machine regression (BVCKMR), a hierarchical model that estimates how mixture exposures at a given time point are associated with health outcome trajectories. The BVCKMR flexibly captures the exposure‐response relationship, incorporates prior knowledge, and accounts for potentially nonlinear and nonadditive effects of individual exposures. This model assesses the directionality and relative importance of a mixture component on health outcome trajectories and predicts health effects for unobserved exposure profiles. Using contour plots and cross‐sectional plots, BVCKMR also provides information about interactions between complex mixture components. The BVCKMR is applied to a subset of data from PROGRESS, a prospective birth cohort study in Mexico city on exposure to metal mixtures and temporal changes in neurodevelopment. The mixture include metals such as manganese, arsenic, cobalt, chromium, cesium, copper, lead, cadmium, and antimony. Results from a subset of Programming Research in Obesity, Growth, Environment and Social Stressors data provide evidence of significant positive associations between second trimester exposure to copper and Bayley Scales of Infant and Toddler Development cognition score at 24 months, and cognitive trajectories across 6‐24 months. We also detect an interaction effect between second trimester copper and lead exposures for cognition at 24 months. In summary, BVCKMR provides a framework for estimating neurodevelopmental trajectories associated with exposure to complex mixtures.

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