A Mixed-Effects Model in Which the Parameters of the Autocorrelated Error Structure Can Differ between Individuals

Abstract

Research in psychology has seen a rapid increase in the usage of experience sampling methods and daily diary methods. The data that result from using these methods are typically analyzed with a mixed-effects or a multilevel model because it allows testing hypotheses about the time course of the longitudinally assessed variable or the influence of time-varying predictors in a simple way. Here, we describe an extension of this model that does not only allow to include random effects for the mean structure but also for the residual variance, for the parameter of an autoregressive process of order 1 and/or the parameter of a moving average process of order 1. After we have introduced this extension, we show how to estimate the parameters with maximum likelihood. Because the likelihood function contains complex integrals, we suggest using adaptive Gauss-Hermite quadrature and Quasi-Monte Carlo integration to approximate it. We illustrate the models using a real data example and also report the results of a small simulation study in which the two integral approximation methods are compared.

Keywords:

• Mixed-effects models
• longitudinal data
• autocorrelation
• moving average
• ARMA

Article Information

Conflict of Interest Disclosures: The author signed a form for disclosure of potential conflicts of interest. The author did not report any financial or other conflicts of interest in relation to the work described.

Ethical Principles: The author affirms having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was not supported by a grant.

Role of the Funders/Sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The author would like to thank the reviewers for their comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the author alone, and endorsement by the author’s institution is not intended and should not be inferred.

Notes

1 In a pre-simulation, we examined the performance of QMC with 500 and 1,000 points and found that the bias was very large and the coverage rates very far away from the nominal value for a larger set of the parameters. Therefore, we report only the results for QMC with 2,000 points.