Modeling the health effects of time‐varying complex environmental mixtures: Mean field variational Bayes for lagged kernel machine regression

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 and z‐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.