Bayesian Variable Selection for Multivariate Spatially Varying Coefficient Regression

Summary Physical activity has many well‐documented health benefits for cardiovascular fitness and weight control. For pregnant women, the American College of Obstetricians and Gynecologists currently recommends 30 minutes of moderate exercise on most, if not all, days; however, very few pregnant women achieve this level of activity. Traditionally, studies have focused on examining individual or interpersonal factors to identify predictors of physical activity. There is a renewed interest in whether characteristics of the physical environment in which we live and work may also influence physical activity levels. We consider one of the first studies of pregnant women that examines the impact of characteristics of the built environment on physical activity levels. Using a socioecologic framework, we study the associations between physical activity and several factors including personal characteristics, meteorological/air quality variables, and neighborhood characteristics for pregnant women in four counties of North Carolina. We simultaneously analyze six types of physical activity and investigate cross‐dependencies between these activity types. Exploratory analysis suggests that the associations are different in different regions. Therefore, we use a multivariate regression model with spatially varying regression coefficients. This model includes a regression parameter for each covariate at each spatial location. For our data with many predictors, some form of dimension reduction is clearly needed. We introduce a Bayesian variable selection procedure to identify subsets of important variables. Our stochastic search algorithm determines the probabilities that each covariate's effect is null, non‐null but constant across space, and spatially varying. We found that individual‐level covariates had a greater influence on women's activity levels than neighborhood environmental characteristics, and some individual‐level covariates had spatially varying associations with the activity levels of pregnant women.