# Uses and Limitation of Continuous-Time Models to Examine Dyadic Interactions

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## Abstract

In this chapter we present an application of the exact discrete model, first proposed by Bergstrom, to model daily interactions among romantic couples. The theoretical model is based on work by Felmlee and Greenberg (J Math Soc 23(3):155–180, 1999), which specifies that change in affect results from the combination of a weighted difference between long-term expectations and daily ratings as well as daily ratings between partners in the dyad. To verify the correct specification, we used simulated models using the LSDE SAS/IML package developed by Singer.

## References

- Baltes, P. B., Reese, H. W., & Nesselroade, J. R. (1988).
*Life-span developmental psychology: Introduction to research methods*(reprint of 1977 edition). Mahwah: Erlbaum.Google Scholar - Bergstrom, A. R. (1988). The history of continuous-time econometric models.
*Econometric Theory, 4*, 365–383. https://doi.org/10.1017/S0266466600013359 MathSciNetCrossRefGoogle Scholar - Boker, S. M. (2001). Differential structural equation modeling of intraindividual variability. In L. M. Collins & A. G. Sayer (Eds.),
*New methods for the analysis of change*(pp. 5–27). Washington: American Psychological Association. https://doi.org/10.1037/10409-001.CrossRefGoogle Scholar - Boker, S. M., Neale, M. C., Maes, H. H., Wilde, M. J., Spiegel, M., Brick, T. R., …Driver, C. (2015). OpenMx 2.3.1 user guide [Computer software manual].Google Scholar
- Box, G. E. P. (1950). Problems in the analysis of growth and wear curves.
*Biometrics, 6*(4), 362–389. https://doi.org/10.2307/3001781.CrossRefGoogle Scholar - Box, G. E. P., & Pierce, D. A. (1970). Distribution of residual correlations in autoregressive-integrated moving average time series models.
*Journal of the American Statistical Association*,*65*, 1509–1526.MathSciNetCrossRefGoogle Scholar - Canova, F. (2007).
*Methods for applied macroeconomic research*(vol. 13). Princeton: Princeton University Press.zbMATHGoogle Scholar - Chow, S.-M., Ho, M. H. R., Hamaker, E. J., & Dolan, C. V. (2010). Equivalences and differences between structural equation and state-space modeling frameworks.
*Structural Equation Modeling*,*17*, 303–332. https://doi.org/10.1080/10705511003661553 MathSciNetCrossRefGoogle Scholar - Coleman, J. S. (1964).
*Introduction to mathematical sociology*. New York: Free Press.Google Scholar - Coleman, J. S. (1968). The mathematical study of change. In H. M. Blalock & A. Blalock (Eds.),
*Methodology in social research*(pp. 428–478). New York: McGraw-Hill.Google Scholar - Cranford, J. A., Shrout, P. E., Iida, M., Rafaeli, E., Yip, T., & Bolger, N. (2006). A procedure for evaluating sensitivity to within-person change: Can mood measures in diary studies detect change reliably?
*Personality and Social Psychology Bulletin*,*32*(7), 917–929.CrossRefGoogle Scholar - Cronbach, L. J., & Furby, L. (1970). How we should measure change—or should we?
*Psychological Bulletin*,*74*, 68–80.CrossRefGoogle Scholar - Davidson, M. L. (1972). Univariate versus multivariate tests in repeated measures experiments.
*Psychological Bulletin, 77*(6), 446–452. https://doi.org/10.1037/h0032674.CrossRefGoogle Scholar - Driver, C. C., Oud, J. H. L., & Voelkle, M. C. (2017). Continuous time structural equation modelling with R package ctsem.
*Journal of Statistical Software*,*77*(5), 1–35. https://doi.org/10.18637/jss.v077.i05.CrossRefGoogle Scholar - Felmlee, D. H. (2006). Application of dynamic systems analysis to dyadic interactions. In A. Ong & M. V. Dulmen (Eds.),
*Oxford handbook of methods in positive psychology*(pp. 409–422). Oxford: Oxford University Press.Google Scholar - Felmlee, D. H., & Greenberg, D. F. (1999). A dynamic systems model of dyadic interaction.
*The Journal of Mathematical Sociology*,*23*(3), 155–180. https://doi.org/10.1080/0022250X.1999.9990218.CrossRefGoogle Scholar - Ferrer, E., & Steele, J. S. (2011). Dynamic systems analysis of affective processes in dyadic interactions using differential equations. In G. R. Hancock & J. R. Harring (Eds.),
*Advances in longitudinal methods in the social and behavioral sciences*(pp. 111–134). Charlotte: Information Age Publishing.Google Scholar - Ferrer, E., & Steele, J. S. (2014). Differential equations for evaluating theoretical models of dyadic interactions. In
*Handbook of developmental systems theory and methodology*(pp. 345–368). New York: Guilford Press.Google Scholar - Gilbert, P. D. (2006). Brief user’s guide: Dynamic systems estimation [Computer software manual]. Retrieved from http://cran.r-project.org/web/packages/dse/vignettes/Guide.pdf.Google Scholar
- Hosking, J. R. M. (1980). The multivariate portmanteau statistic.
*Journal of the American Statistical Association, 75*, 602–608. https://doi.org/10.1080/01621459.1980.10477520 MathSciNetCrossRefGoogle Scholar - Jazwinski, A. H. (1970).
*Stochastic processes and filtering theory*. New York: Academic Press.zbMATHGoogle Scholar - Juhl, R. (2015). Ctsmr: Ctsm for R [Computer software manual]. R package version 0.6.8-5.Google Scholar
- Kalman, R. E. (1960). A new approach to linear filtering and prediction problems.
*Journal of Basic Engineering*,*82*(1), 35–45.CrossRefGoogle Scholar - Ljung, G. M., & Box, G. E. P. (1978). On a measure of lack of fit in time series models.
*Biometrika*,*65*, 297–303. https://doi.org/10.1093/biomet/65.2.297.CrossRefGoogle Scholar - MacCallum, R., & Ashby, F. G. (1986). Relationships between linear systems theory and covariance structure modeling.
*Journal of Mathematical Psychology*,*30*(1), 1–27.MathSciNetCrossRefGoogle Scholar - McDonald, R. P., & Swaminathan, H. (1973). A simple matrix calculus with application to multivariate analysis.
*General Systems, XVIII*, 37–54.Google Scholar - Meredith, W., & Tisak, J. (1990). Latent curve analysis.
*Psychometrika*,*55*(1), 107–122. https://doi.org/10.1007/BF02294746.CrossRefGoogle Scholar - Miller, M. L., & Ferrer, E. (2017). The effect of sampling-time variation on latent growth curve model fit.
*Structural Equation Modeling*,*24*, 831–854. https://doi.org/10.1080/10705511.2017.1346476.MathSciNetCrossRefGoogle Scholar - Neale, M. C., Hunter, M. D., Pritikin, J. N., Zahery, M., Brick, T. R., Kickpatrick, R. M., …Boker, S. M. (2016). OpenMx 2.0: Extended structural equation and statistical modeling.
*Psychometrika, (81)*, 535–549. https://doi.org/10.1007/s11336-014-9435-8.MathSciNetCrossRefGoogle Scholar - Nielsen, F., & Rosenfeld, R. A. (1981). Substantive interpretations of differential equation models.
*American Sociological Review, 46*(2), 159–174. https://doi.org/10.2307/2094976.CrossRefGoogle Scholar - O’Brien, R. G., & Kaiser, M. K. (1985). MANOVA method for analyzing repeated measures designs: An extensive primer.
*Psychological Bulletin, 97*(2), 316. https://doi.org/10.1037/0033-2909.97.2.316.CrossRefGoogle Scholar - Oud, J. H. L. (2004). SEM state space modeling of panel data in discrete and continuous time and its relationship to traditional state space modeling. In K. van Montfort, J. H. L. Oud, & A. Satorra (Eds.),
*Recent developments on structural equation models: Theory and applications*(pp. 13–40). Dordrecht: Kluwer.CrossRefGoogle Scholar - Oud, J. H. L. (2007). Comparison of four procedures to estimate the damped linear differential oscillator for panel data. In K. van Montfort, J. H. L. Oud, & A. Satorra (Eds.),
*Longitudinal models in the behavioral and related sciences*(pp. 19–40). Mahwah: Lawrence Erlbaum.Google Scholar - Oud, J. H. L., & Jansen, R. A. R. G. (2000). Continuous time state space modeling of panel data by means of SEM.
*Psychometrika, 65*(2), 199–215. https://doi.org/10.1007/BF02294374.MathSciNetCrossRefGoogle Scholar - Oud, J. H. L., & Singer, H. (2008). Continuous time modeling of panel data: SEM versus filter techniques.
*Statistica Neerlandica, 62*, 4–28.MathSciNetCrossRefGoogle Scholar - SAS Institute Inc. (2002–2008). SAS 9.2 Help and documentation [Computer software manual]. Cary, NC: SAS Institute Inc.Google Scholar
- Singer, H. (1991a). Continuous-time dynamical systems with sampled data, errors of measurement and unobserved components.
*Journal of Time Series Analysis*,*14*, 527–545.MathSciNetCrossRefGoogle Scholar - Singer, H. (1991b).
*LSDE- A program package for the simulation, graphical display, optimal filtering and maximum likelihood estimation of Linear Stochastic Differential Equations*. Meersburg: Author.Google Scholar - Steele, J. S., & Ferrer, E. (2011). Latent differential equation modeling of self-regulatory and coregulatory affective processes.
*Multivariate Behavioral Research*,*46*(6), 956–984.CrossRefGoogle Scholar - Steele, J. S., Ferrer, E., & Nesselroade, J. R. (2014). An idiographic approach to estimating models of dyadic interactions with differential equations.
*Psychometrika*,*79*(4), 675–700. https://doi.org/10.1007/s11336-013-9366-9.MathSciNetCrossRefGoogle Scholar - Tucker, L. R. (1958). Determination of parameters of a functional relation by factor analysis.
*Psychometrika*,*23*(1), 19–23. https://doi.org/10.1007/BF02288975.CrossRefGoogle Scholar - Voelkle, M. C. (2017). A new perspective on three old methodological issues: The role of time, missing values, and cohorts in longitudinal models of youth development. In A. C. Petersen, S. H. Koller, F. Motti-Stefanidi, & S. Verma (Eds.),
*Positive youth development in global contexts of social and economic change*(pp. 110–136). New York: Routledge.Google Scholar

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