A Continuous-Time Approach to Intensive Longitudinal Data: What, Why, and How?

  • Oisín RyanEmail author
  • Rebecca M. Kuiper
  • Ellen L. Hamaker


The aim of this chapter is to (a) provide a broad didactical treatment of the first-order stochastic differential equation model—also known as the continuous-time (CT) first-order vector autoregressive (VAR(1)) model—and (b) argue for and illustrate the potential of this model for the study of psychological processes using intensive longitudinal data. We begin by describing what the CT-VAR(1) model is and how it relates to the more commonly used discrete-time VAR(1) model. Assuming no prior knowledge on the part of the reader, we introduce important concepts for the analysis of dynamic systems, such as stability and fixed points. In addition we examine why applied researchers should take a continuous-time approach to psychological phenomena, focusing on both the practical and conceptual benefits of this approach. Finally, we elucidate how researchers can interpret CT models, describing the direct interpretation of CT model parameters as well as tools such as impulse response functions, vector fields, and lagged parameter plots. To illustrate this methodology, we reanalyze a single-subject experience-sampling dataset with the R package ctsem; for didactical purposes, R code for this analysis is included, and the dataset itself is publicly available.



We thank an editor and anonymous reviewer for helpful comments that led to improvements in this chapter. The work of the authors was supported by grants from the Netherlands Organization for Scientific Research (NWO Onderzoekstalent 406-15-128) to Oisín Ryan and Ellen Hamaker, and (NWO VENI 451-16-019) to Rebecca Kuiper.

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  1. Aalen, O., Gran, J., Røysland, K., Stensrud, M., & Strohmaier, S. (2017). Feedback and mediation in causal inference illustrated by stochastic process models. Scandinavian Journal of Statistics, 45, 62–86. MathSciNetCrossRefGoogle Scholar
  2. Aalen, O., Røysland, K., Gran, J., Kouyos, R., & Lange, T. (2016). Can we believe the DAGs? A comment on the relationship between causal DAGs and mechanisms. Statistical Methods in Medical Research, 25(5), 2294–2314. MathSciNetCrossRefGoogle Scholar
  3. Aalen, O., Røysland, K., Gran, J., & Ledergerber, B. (2012). Causality, mediation and time: A dynamic viewpoint. Journal of the Royal Statistical Society: Series A (Statistics in Society), 175(4), 831–861.MathSciNetCrossRefGoogle Scholar
  4. Bisconti, T., Bergeman, C. S., & Boker, S. M. (2004). Emotional well-being in recently bereaved widows: A dynamical system approach. Journal of Gerontology, Series B: Psychological Sciences and Social Sciences, 59, 158–167. CrossRefGoogle Scholar
  5. Boker, S. M. (2002). Consequences of continuity: The hunt for intrinsic properties within parameters of dynamics in psychological processes. Multivariate Behavioral Research, 37(3), 405–422. CrossRefGoogle Scholar
  6. Boker, S. M., Deboeck, P., Edler, C., & Keel, P. (2010a). Generalized local linear approximation of derivatives from time series. In S. Chow & E. Ferrar (Eds.), Statistical methods for modeling human dynamics: An interdisciplinary dialogue (pp. 179–212). Boca Raton, FL: Taylor & Francis.Google Scholar
  7. Boker, S. M., & McArdle, J. J. (1995). Statistical vector field analysis applied to mixed cross-sectional and longitudinal data. Experimental Aging Research, 21, 77–93. CrossRefGoogle Scholar
  8. Boker, S. M., Montpetit, M. A., Hunter, M. D., & Bergeman, C. S. (2010b). Modeling resilience with differential equations. In P. Molenaar & K. Newell (Eds.), Learning and development: Individual pathways of change (pp. 183–206). Washington, DC: American Psychological Association. CrossRefGoogle Scholar
  9. Boker, S. M., Neale, M., & Rausch, J. (2004). Latent differential equation modeling with multivariate multi-occasion indicators. In K. van Montfort, J. H. L. Oud, & A. Satorra (Eds.), Recent developments on structural equation models (pp. 151–174). Dordrecht: Kluwer.CrossRefGoogle Scholar
  10. Boker, S. M., & Nesselroade, J. R. (2002). A method for modeling the intrinsic dynamics of intraindividual variability: Recovering parameters of simulated oscillators in multi-wave panel data. Multivariate Behavioral Research, 37, 127–160.CrossRefGoogle Scholar
  11. Boker, S. M., Staples, A. D., & Hu, Y. (2016). Dynamics of change and change in dynamics. Journal for Person-Oriented Research, 2(1–2), 34. CrossRefGoogle Scholar
  12. Bolger, N., & Laurenceau, J.-P. (2013). Intensive longitudinal methods: An introduction to diary and experience sampling research. New York, NY: The Guilford Press.Google Scholar
  13. Bringmann, L., Lemmens, L., Huibers, M., Borsboom, D., & Tuerlinckx, F. (2015). Revealing the dynamic network structure of the beck depression inventory-ii. Psychological Medicine, 45(4), 747–757. CrossRefGoogle Scholar
  14. Bringmann, L., Pe, M., Vissers, N., Ceulemans, E., Borsboom, D., Vanpaemel, W., …Kuppens, P. (2016). Assessing temporal emotion dynamics using networks. Assessment, 23(4), 425–435.
  15. Bringmann, L., Vissers, N., Wichers, M., Geschwind, N., Kuppens, P., Peeters, …Tuerlinckx, F. (2013). A network approach to psychopathology: New insights into clinical longitudinal data. PLoS ONE, 8, e60188.
  16. Browne, M. W., & Nesselroade, J. R. (2005). Representing psychological processes with dynamic factor models: Some promising uses and extensions of ARMA time series models. In A. Maydue-Olivares & J. J. McArdle (Eds.), Psychometrics: A festschrift to Roderick P. McDonald (pp. 415–452). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  17. Chow, S., Ferrer, E., & Hsieh, F. (2011). Statistical methods for modeling human dynamics: An interdisciplinary dialogue. New York, NY: Routledge.CrossRefGoogle Scholar
  18. Chow, S., Ferrer, E., & Nesselroade, J. R. (2007). An unscented Kalman filter approach to the estimation of nonlinear dynamical systems models. Multivariate Behavioral Research, 42(2), 283–321. CrossRefGoogle Scholar
  19. Chow, S., Ram, N., Boker, S., Fujita, F., Clore, G., & Nesselroade, J. (2005). Capturing weekly fluctuation in emotion using a latent differential structural approach. Emotion, 5(2), 208–225.CrossRefGoogle Scholar
  20. De Haan-Rietdijk, S., Voelkle, M. C., Keijsers, L., & Hamaker, E. (2017). Discrete- versus continuous-time modeling of unequally spaced ESM data. Frontiers in Psychology, 8, 1849. CrossRefGoogle Scholar
  21. Deboeck, P. R., & Preacher, K. J. (2016). No need to be discrete: A method for continuous time mediation analysis. Structural Equation Modeling: A Multidisciplinary Journal, 23(1), 61–75.MathSciNetCrossRefGoogle Scholar
  22. Denollet, J., & De Vries, J. (2006). Positive and negative affect within the realm of depression, stress and fatigue: The two-factor distress model of the Global Mood Scale (GMS). Journal of Affective Disorders, 91(2), 171–180. CrossRefGoogle Scholar
  23. Dormann, C., & Griffin, M. A. (2015). Optimal time lags in panel studies. Psychological Methods, 20(4), 489. CrossRefGoogle Scholar
  24. Driver, C., Oud, J. H. L., & Voelkle, M. (2017). Continuous time structural equation modelling with r package ctsem. Journal of Statistical Software, 77, 1–35. CrossRefGoogle Scholar
  25. Driver, C. C., & Voelkle, M. C. (2018). Hierarchical Bayesian continuous time dynamic modeling. Psychological Methods. Advance online publication. Google Scholar
  26. Fisher, M. (2001). Modeling negative autoregression in continuous time.
  27. Gault-Sherman, M. (2012). It’s a two-way street: The bidirectional relationship between parenting and delinquency. Journal of Youth and Adolescence, 41, 121–145. CrossRefGoogle Scholar
  28. Gollob, H. F., & Reichardt, C. S. (1987). Taking account of time lags in causal models. Child Development, 58, 80–92. CrossRefGoogle Scholar
  29. Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica, 37, 424–438. CrossRefGoogle Scholar
  30. Hamaker, E. L., Dolan, C. V., & Molenaar, P. C. M. (2005). Statistical modeling of the individual: Rationale and application of multivariate time series analysis. Multivariate Behavioral Research, 40(2), 207–233. CrossRefGoogle Scholar
  31. Hamaker, E. L., & Grasman, R. P. P. P. (2015). To center or not to center? Investigating inertia with a multilevel autoregressive model. Frontiers in Psychology, 5, 1492. CrossRefGoogle Scholar
  32. Hamaker, E. L., Kuiper, R., & Grasman, R. P. P. P. (2015). A critique of the cross-lagged panel model. Psychological Methods, 20(1), 102–116. CrossRefGoogle Scholar
  33. Hamerle, A., Nagl, W., & Singer, H. (1991). Problems with the estimation of stochastic differential equations using structural equations models. Journal of Mathematical Sociology, 16(3), 201–220. CrossRefGoogle Scholar
  34. Hamilton, J. D. (1994). Time series analysis. Princeton, NJ: Princeton University Press.zbMATHGoogle Scholar
  35. Horn, E. E., Strachan, E., & Turkheimer, E. (2015). Psychological distress and recurrent herpetic disease: A dynamic study of lesion recurrence and viral shedding episodes in adults. Multivariate Behavioral Research, 50(1), 134–135. CrossRefGoogle Scholar
  36. Ichii, K. (1991). Measuring mutual causation: Effects of suicide news on suicides in Japan. Social Science Research, 20, 188–195. CrossRefGoogle Scholar
  37. Johnston, J., & DiNardo, J. (1997). Econometric methods (4th ed.). New York, NY: McGraw-Hill.Google Scholar
  38. Kim, C.-J., & Nelson, C. R. (1999). State-space models with regime switching: Classical and Gibbs-sampling approaches with applications. Cambridge, MA: The MIT Press.
  39. Kossakowski, J., Groot, P., Haslbeck, J., Borsboom, D., & Wichers, M. (2017). Data from critical slowing down as a personalized early warning signal for depression. Journal of Open Psychology Data, 5(1), 1.CrossRefGoogle Scholar
  40. Koval, P., Kuppens, P., Allen, N. B., & Sheeber, L. (2012). Getting stuck in depression: The roles of rumination and emotional inertia. Cognition and Emotion, 26, 1412–1427.CrossRefGoogle Scholar
  41. Kuiper, R. M., & Ryan, O. (2018). Drawing conclusions from cross-lagged relationships: Re-considering the role of the time-interval. Structural Equation Modeling: A Multidisciplinary Journal.
  42. Kuppens, P., Allen, N. B., & Sheeber, L. B. (2010). Emotional inertia and psychological maladjustment. Psychological Science, 21(7), 984–991. CrossRefGoogle Scholar
  43. Kuppens, P., Sheeber, L. B., Yap, M. B. H., Whittle, S., Simmons, J., & Allen, N. B. (2012). Emotional inertia prospectively predicts the onset of depression in adolescence. Emotion, 12, 283–289. CrossRefGoogle Scholar
  44. Meier, B. P., & Robinson, M. D. (2004). Why the sunny side is up: Associations between affect and vertical position. Psychological Science, 15(4), 243–247. CrossRefGoogle Scholar
  45. Moberly, N. J., & Watkins, E. R. (2008). Ruminative self-focus and negative affect: An experience sampling study. Journal of Abnormal Psychology, 117, 314–323. CrossRefGoogle Scholar
  46. Moler, C., & Van Loan, C. (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review, 45(1), 3–49.MathSciNetCrossRefGoogle Scholar
  47. Oravecz, Z., & Tuerlinckx, F. (2011). The linear mixed model and the hierarchical Ornstein–Uhlenbeck model: Some equivalences and differences. British Journal of Mathematical and Statistical Psychology, 64(1), 134–160. CrossRefGoogle Scholar
  48. Oravecz, Z., Tuerlinckx, F., & Vandekerckhove, J. (2011). A hierarchical latent stochastic difference equation model for affective dynamics. Psychological Methods, 16, 468–490. CrossRefGoogle Scholar
  49. Oravecz, Z., Tuerlinckx, F., & Vandekerckhove, J. (2016). Bayesian data analysis with the bivariate hierarchical Ornstein-Uhlenbeck process model. Multivariate Behavioral Research, 51(1), 106–119. CrossRefGoogle Scholar
  50. Oud, J. H. L. (2007). Continuous time modeling of reciprocal relationships in the cross-lagged panel design. In S. M. Boker & M. J. Wenger (Eds.), Data analytic techniques for dynamic systems in the social and behavioral sciences (pp. 87–129). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  51. Oud, J. H. L., & Delsing, M. J. M. H. (2010). Continuous time modeling of panel data by means of SEM. In K. van Montfort, J. H. L. Oud, & A. Satorra (Eds.), Longitudinal research with latent variables (pp. 201–244). New York, NY: Springer. CrossRefGoogle Scholar
  52. Oud, J. H. L., & Jansen, R. A. (2000). Continuous time state space modeling of panel data by means of SEM. Psychometrika, 65(2), 199–215. MathSciNetCrossRefGoogle Scholar
  53. Oud, J. H. L., van Leeuwe, J., & Jansen, R. (1993). Kalman filtering in discrete and continuous time based on longitudinal lisrel models. In Advances in longitudinal and multivariate analysis in the behavioral sciences (pp. 3–26). Nijmegen: ITS.Google Scholar
  54. Reichardt, C. S. (2011). Commentary: Are three waves of data sufficient for assessing mediation? Multivariate Behavioral Research, 46(5), 842–851.CrossRefGoogle Scholar
  55. Rovine, M. J., & Walls, T. A. (2006). Multilevel autoregressive modeling of interindividual differences in the stability of a process. In T. A. Walls & J. L. Schafer (Eds.), Models for intensive longitudinal data (pp. 124–147). New York, NY: Oxford University Press. CrossRefGoogle Scholar
  56. Schuurman, N. K., Ferrer, E., de Boer-Sonnenschein, M., & Hamaker, E. L. (2016). How to compare cross-lagged associations in a multilevel autoregressive model. Psychological methods, 21(2), 206–221. CrossRefGoogle Scholar
  57. Steele, J. S., & Ferrer, E. (2011). Latent differential equation modeling of selfregulatory and coregulatory affective processes. Multivariate Behavioral Research, 46(6), 956–984. CrossRefGoogle Scholar
  58. Strogatz, S. H. (2014). Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering. Boulder, CO: Westview press.zbMATHGoogle Scholar
  59. Voelkle, M., & Oud, J. H. L. (2013). Continuous time modelling with individually varying time intervals for oscillating and non-oscillating processes. British Journal of Mathematical and Statistical Psychology, 66(1), 103–126. MathSciNetCrossRefGoogle Scholar
  60. Voelkle, M., Oud, J. H. L., Davidov, E., & Schmidt, P. (2012). An SEM approach to continuous time modeling of panel data: Relating authoritarianism and anomia. Psychological Methods, 17, 176–192. CrossRefGoogle Scholar
  61. Watkins, M. W., Lei, P.-W., & Canivez, G. L. (2007). Psychometric intelligence and achievement: A cross-lagged panel analysis. Intelligence, 35, 59–68. CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Oisín Ryan
    • 1
    Email author
  • Rebecca M. Kuiper
    • 1
  • Ellen L. Hamaker
    • 1
  1. 1.Department of Methodology and Statistics, Faculty of Social and Behavioural SciencesUtrecht UniversityUtrechtThe Netherlands

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