It has been suggested that children with larger brains tend to perform better on IQ tests or cognitive function tests. Prenatal head growth and head growth in infancy are two crucial periods for subsequent intelligence. Studies have shown that environmental exposure to air pollutants during pregnancy is associated with fetal growth reduction, developmental delay, and reduced IQ. Meanwhile, genetic polymorphisms may modify the effect of environment on head growth. However, studies on gene-environment or gene-gene interactions on growth trajectories have been quite limited partly due to the difficulty to quantitatively measure interactions on growth trajectories. Moreover, it is known that assessing the significance of gene-environment or gene-gene interactions on cross-sectional outcomes empirically using the permutation procedures may bring substantial errors in the tests. We proposed a score that quantitatively measures interactions on growth trajectories and developed an algorithm with a parametric bootstrap procedure to empirically assess the significance of the interactions on growth trajectories under the likelihood framework. We also derived a Wald statistic to test for interactions on growth trajectories and compared it to the proposed parametric bootstrap procedure. Through extensive simulation studies, we demonstrated the feasibility and power of the proposed testing procedures. We applied our method to a real dataset with head circumference measures from birth to age 7 on a cohort currently being conducted by the Columbia Center for Children's Environmental Health (CCCEH) in Krakow, Poland, and identified several significant gene-environment interactions on head circumference growth trajectories.
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