Group Testing for Longitudinal Data

We consider how to test for group differences of shapes given longitudinal data. In particular, we are interested in differences of longitudinal models of each group's subjects. We introduce a generalization of principal geodesic analysis to the tangent bundle of a shape space. This allows the estimation of the variance and principal directions of the distribution of trajectories that summarize shape variations within the longitudinal data. Each trajectory is parameterized as a point in the tangent bundle. To study statistical differences in two distributions of trajectories, we generalize the Bhattacharyya distance in Euclidean space to the tangent bundle. This not only allows to take second-order statistics into account, but also serves as our test-statistic during permutation testing. Our method is validated on both synthetic and real data, and the experimental results indicate improved statistical power in identifying group differences. In fact, our study sheds new light on group differences in longitudinal corpus callosum shapes of subjects with dementia versus normal controls.