Bayesian Data Analysis with the Bivariate Hierarchical Ornstein-Uhlenbeck Process Model

ABSTRACT

In this paper, we propose a multilevel process modeling approach to describing individual differences in within-person changes over time. To characterize changes within an individual, repeated measures over time are modeled in terms of three person-specific parameters: a baseline level, intraindividual variation around the baseline, and regulatory mechanisms adjusting toward baseline. Variation due to measurement error is separated from meaningful intraindividual variation. The proposed model allows for the simultaneous analysis of longitudinal measurements of two linked variables (bivariate longitudinal modeling) and captures their relationship via two person-specific parameters. Relationships between explanatory variables and model parameters can be studied in a one-stage analysis, meaning that model parameters and regression coefficients are estimated simultaneously. Mathematical details of the approach, including a description of the core process model—the Ornstein-Uhlenbeck model—are provided. We also describe a user friendly, freely accessible software program that provides a straightforward graphical interface to carry out parameter estimation and inference. The proposed approach is illustrated by analyzing data collected via self-reports on affective states.

Keywords:

• Intensive longitudinal data analysis
• dynamical modeling
• Ornstein-Uhlenbeck
• Bayesian modeling
• individual differences

Article information

Conflict of Interest Disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.

Ethical Principles: The authors affirm 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: The research reported in this paper was sponsored in part by Belgian Federal Science Policy within the framework of the Interuniversity Attraction Poles program IAP/P7/06 (FT), in part by the grant GOA/15/003 from the University of Leuven (FT), in part by the grant G.0806.13 from the Research Foundation—Flanders (FT), in part by the grant #48192 from The John Templeton Foundation (ZO and JV) and in part by the grant #1230118 from the National Science Foundation's Methods, Measurements, and Statistics panel (JV).

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.

Acknowledgements: The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institutions or The John Templeton Foundation is not intended and should not be inferred. The authors are grateful to Chelsea Muth, Marlies Houben, Madeline Pe and Peter Kuppens for their beta-testing efforts of BHOUM as well as for their helpful comments on the manuscript.

Notes

1 As a result of the mean-reverting specification (as shown in Equation [1(1) ]) the OU process does not have an ever-expanding variance expectation as in basic random walk processes.

2 Formally, , where stands for the vector of all parameters in the model. The normalization constant, , where stands for the data, does not depend on the parameter and is therefore not considered here.

3 The stand-alone BHOUM version with the accompanying free MATLAB Compiler Runtime (MCR) has been tested for Windows 32bit and 64bit. If users do not want to install MCR because they have a MATLAB license already, that MATLAB should be run in 32bit mode.

4 Given the simultaneous inclusion of the predictors, the estimated regression weights of each predictor here are conditional on the other predictors in the model (i.e., they indicate effects over and above those of the other predictors.)

5 Note that, while the self-esteem measure was collected at every measurement occasion, the person-specific variance in self-esteem is time-invariant.