Empirical Bayes Derivative Estimates

Abstract

A dynamic system is a set of interacting elements characterized by changes occurring over time. The estimation of derivatives is a mainstay for exploring dynamics of constructs, particularly when the dynamics are complicated or unknown. The presence of measurement error in many social science constructs frequently results in poor estimates of derivatives, as even modest proportions of measurement error can compound when estimating derivatives. Given the overlap in the specification of latent differential equation models and latent growth curve models, and the equivalence of latent growth curve models and mixed models under some conditions, derivatives could be estimated from estimates of random effects. This article proposes a new method for estimating derivatives based on calculating the Empirical Bayes estimates of derivatives from a mixed model. Two simulations compare four derivative estimation methods: Generalized Local Linear Approximation, Generalized Orthogonal Derivative Estimates, Functional Data Analysis, and the proposed Empirical Bayes Derivative Estimates. The simulations consider two data collection scenarios: short time series (≤10 observations) from many individuals or occasions, and long individual time series (25–500 observations). A substantive example visualizing the dynamics of intraindividual positive affect time series is also presented.

Keywords:

• Differences
• Derivatives
• Generalized Local Linear Approximation
• Generalized Orthogonal Local Derivative Estimates
• Functional Data Analysis
• Empirical Bayes

Author note

Updated copies of the code provided in Appendix B will be maintained by the author. The author would like to acknowledge Dr C. S. Bergeman and The Notre Dame Study on Wellbeing (Wallace et al., 2002) for providing the data that were used in the substantive example. The authors would also like to acknowledge Jonathan Butner’s input on the R function provided for implementing the proposed method.

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.

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: Updated copies of the code provided in Appendix B will be maintained by the author. The author would like to acknowledge Dr. C. S. Bergeman and The Notre Dame Study on Wellbeing (Wallace et al., 2002) for providing the data that were used in the substantive example. The author would also like to acknowledge Jonathan Butner's input on the R function provided for implementing the proposed method. 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 There is the possibility of producing second differences with a higher reliability than the original scale. One case where this can occur is when the correlation between adjacent observations is positive and the correlation of observations separated by an observation is negative. Consequently, when observations are positively correlated for more than one interval, the reduction in reliability or higher orders of differences is substantial.

2 SAS, HLM, and lme4 in R all provide empirical Bayes estimates of the level-1 random effects. Note that these estimates are conditional on the estimated model parameters, which can differ based on factors such as the estimator used.

3 The endpoints themselves are relatively arbitrary, but a consistent interval of time was utilized so that the distributional properties of the derivatives were the same while varying sampling interval.

4 For time series of 25 observations, 6 basis functions were used, the minimum allowed when estimating cubic splines.