  Mixed effect modeling for longitudinal data is challenging when the observed
data are random objects, which are complex data taking values in a general
metric space without linear structure. In such settings the classical additive
error model and distributional assumptions are unattainable. Due to the rapid
advancement of technology, longitudinal data containing complex random objects,
such as covariance matrices, data on Riemannian manifolds, and probability
distributions are becoming more common. Addressing this challenge, we develop a
mixed-effects regression for data in geodesic spaces, where the underlying mean
response trajectories are geodesics in the metric space and the deviations of
the observations from the model are quantified by perturbation maps or
transports. A key finding is that the geodesic trajectories assumption for the
case of random objects is a natural extension of the linearity assumption in
the standard Euclidean scenario. Further, geodesics can be recovered from noisy
observations by exploiting a connection between the geodesic path and the path
obtained by global Fr\'echet regression for random objects. The effect of
baseline Euclidean covariates on the geodesic paths is modeled by another
Fr\'echet regression step. We study the asymptotic convergence of the proposed
estimates and provide illustrations through simulations and real-data
applications.
