  When multiple measures are collected repeatedly over time, redundancy
typically exists among responses. The envelope method was recently proposed to
reduce the dimension of responses without loss of information in regression
with multivariate responses. It can gain substantial efficiency over the
standard least squares estimator. In this paper, we generalize the envelope
method to mixed effects models for longitudinal data with possibly unbalanced
design and time-varying predictors. We show that our model provides more
efficient estimators than the standard estimators in mixed effects models.
Improved accuracy and efficiency of the proposed method over the standard mixed
effects model estimator are observed in both the simulations and the Action to
Control Cardiovascular Risk in Diabetes (ACCORD) study.
