Mixed effects envelope models

Yuyang Shi, Linquan Ma, Lan Liu

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article numbere313
Issue number1
StatePublished - 2020

Bibliographical note

Funding Information:
This research was supported by NSF DMS 1916013, NIH U24‐DK‐060990 and NIH R01HL155417‐01. We would like to thank the Editor, Associate Editor, and two referees for their thorough reviews of the manuscript.

Publisher Copyright:
© 2020 John Wiley & Sons, Ltd.


  • efficiency gain
  • envelope method
  • mixed effects model
  • sufficient dimension reduction


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