  Mixed effects (ME) models inform a vast array of problems in the physical and
social sciences, and are pervasive in meta-analysis. We consider ME models
where the random effects component is linear. We then develop an efficient
approach for a broad problem class that allows nonlinear measurements, priors,
and constraints, and finds robust estimates in all of these cases using
trimming in the associated marginal likelihood.
  The software accompanying this paper is disseminated as an open-source Python
package called LimeTr. LimeTr is able to recover results more accurately in the
presence of outliers compared to available packages for both standard
longitudinal analysis and meta-analysis, and is also more computationally
efficient than competing robust alternatives. Supplementary materials that
reproduce the simulations, as well as run LimeTr and third party code are
available online. We also present analyses of global health data, where we use
advanced functionality of LimeTr, including constraints to impose monotonicity
and concavity for dose-response relationships. Nonlinear observation models
allow new analyses in place of classic approximations, such as log-linear
models. Robust extensions in all analyses ensure that spurious data points do
not drive our understanding of either mean relationships or between-study
heterogeneity.
