From Modeling Long-Term Growth to Short-Term Fluctuations: Differential Equation Modeling Is the Language of Change

Many applied statistical problems address how the change in one variable is related to change in another variable. While the change of one variable with respect to another is the very definition of a derivative, the language of derivatives is not commonplace among social scientists. In this chapter we present derivatives as a language framework that is ideal for describing changes in variables, particularly changes with respect to time. This language can be used to understand many common models as relationships between derivatives rather than as disparate entities. Derivatives also can be used to provide statisticians and substantive researchers a common language that can be used to create better matches between models and theory. Moreover, this chapter will present derivatives as a language that has the potential to change the kinds of questions asked from variables measured repeatedly over time. Three differential equation models will be introduced for addressing questions related to the shortterm fluctuation often described as intraindividual variability.

[1]  R. MacCallum,et al.  Studying Multivariate Change Using Multilevel Models and Latent Curve Models. , 1997, Multivariate behavioral research.

[2]  Steven M Boker,et al.  A Method for Modeling the Intrinsic Dynamics of Intraindividual Variability: Recovering the Parameters of Simulated Oscillators in Multi-Wave Panel Data , 2002, Multivariate behavioral research.

[3]  Steven M Boker,et al.  Social support as a predictor of variability: an examination of the adjustment trajectories of recent widows. , 2006, Psychology and aging.

[4]  H. F. Gollob,et al.  Taking account of time lags in causal models. , 1987, Child development.

[5]  Daniel J. Bauer.  Estimating Multilevel Linear Models as Structural Equation Models , 2003 .

[6]  J. Oud,et al.  Continuous time state space modeling of panel data by means of sem , 2000 .

[7]  Johan H. L. Oud,et al.  Recent developments on structural equations models: theory and applications , 2004 .

[8]  Steven M Boker,et al.  Maternal depressive symptomatology and child behavior: transactional relationship with simultaneous bidirectional coupling. , 2011, Developmental psychology.

[9]  C S Bergeman,et al.  The reservoir model: a differential equation model of psychological regulation. , 2013, Psychological methods.

[10]  C S Bergeman,et al.  Resilience-as-process: negative affect, stress, and coupled dynamical systems. , 2010, Psychology and aging.

[11]  Jean-Philippe Laurenceau,et al.  Dynamical Systems Modeling: An Application to the Regulation of Intimacy and Disclosure in Marriage. , 2006 .

[12]  Steven M. Boker,et al.  Latent Differential Equation Modeling with Multivariate Multi-Occasion Indicators , 2004 .

[13]  Robert Cudeck,et al.  A Version of Quadratic Regression with Interpretable Parameters , 2002, Multivariate behavioral research.

[14]  M. Mehl,et al.  Handbook of research methods for studying daily life , 2012 .

[15]  Pascal R. Deboeck,et al.  Estimating Dynamical Systems: Derivative Estimation Hints From Sir Ronald A. Fisher , 2010, Multivariate behavioral research.

[16]  J. Oud,et al.  An SEM approach to continuous time modeling of panel data: relating authoritarianism and anomia. , 2012, Psychological methods.

[17]  Theodore A. Walls,et al.  Models for intensive longitudinal data , 2006 .

[18]  E. Leibenluft,et al.  Mood Oscillations and Coupling Between Mood and Weather in Patients with Rapid Cycling Bipolar Disorder. , 2008, International journal of child health and human development : IJCHD.

[19]  S. Maxwell,et al.  Testing mediational models with longitudinal data: questions and tips in the use of structural equation modeling. , 2003, Journal of abnormal psychology.

[20]  D. Watson,et al.  Development and validation of brief measures of positive and negative affect: the PANAS scales. , 1988, Journal of personality and social psychology.