Log-Contrast Regression with Functional Compositional Predictors: Linking Preterm Infant's Gut Microbiome Trajectories in Early Postnatal Period to Neurobehavioral Outcome

When compositional data serve as predictors in regression, the log-contrast model is commonly applied. A prominent feature of the model is that it complies with the simplex geometry and enables the regression analysis to have various desirable invariance properties. Motivated by the needs in understanding how the trajectories of gut microbiome compositions during early postnatal stage impact later neurobehavioral outcomes among preterm infants, we develop a sparse log-contrast regression with functional compositional predictors. The functional simplex structure is preserved by a set of zero-sum constraints on the parameters, and the compositional predictors are allowed to have sparse, smoothly varying, and accumulating effects on the outcome through time. Through basis expansion, the problem boils down to a linearly constrained group lasso regression, for which we develop an efficient augmented Lagrangian algorithm and obtain theoretical performance guarantees. The proposed approach yields interesting results in the preterm infant study. The identified microbiome markers and the estimated time dynamics of their impact on the neurobehavioral outcome shed lights on the functional linkage between stress accumulation in early postnatal stage and neurodevelpomental process of infants.