Advertisement

Continuous-Time Modeling of Panel Data with Network Structure

  • Nynke M. D. Niezink
  • Tom A. B. Snijders
Chapter

Abstract

In some panel data studies, respondents are nested in social contexts, like classrooms or organizations. The social relations, such as friendship or collaboration, between the individuals in such contexts can be represented by social networks. Like individual outcomes (e.g., behavior or performance), the social relations between individuals are not static. Social networks and individual outcomes change over time and can mutually affect each other. In this chapter, we present a statistical model to study the interdependent dynamics (or coevolution) of network structure and individual outcomes. We assume panel observations of networks and individual attributes to be the discrete-time realizations of an underlying continuous-time process, which is modeled by a stochastic differential equation for the attribute dynamics and a Markov chain model for the network dynamics. We illustrate the proposed method by a study of the coevolution of friendship ties and mathematics grades among 1160 students in 39 classrooms.

References

  1. Agnew, R. (1991). The interactive effects of peer variables on delinquency. Criminology, 29, 47–72. https://doi.org/10.1111/j.1745-9125.1991.tb01058.x CrossRefGoogle Scholar
  2. Bergstrom, A. R. (1984). Continuous time stochastic models and issues of aggregation over time. In Z. Griliches & M. D. Intriligator (Eds.), Handbook of econometrics, volume II (pp. 1146–1212). Amsterdam: Elsevier Science Publishers BV.Google Scholar
  3. Block, P. (2015). Reciprocity, transitivity, and the mysterious three-cycle. Social Networks, 40, 163–173. https://doi.org/10.1016/j.socnet.2014.10.005.CrossRefGoogle Scholar
  4. Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to meta-analysis. Chichester: Wiley. https://doi.org/10.1002/9780470743386.
  5. Caravita, S. C. S., Sijtsema, J. J., Rambaran, A. J., & Gini, G. (2014). Peer influences on moral disengagement in late childhood and early adolescence. Journal of Youth and Adolescence, 43, 193–207. https://doi.org/10.1007/s10964-013-9953-1.CrossRefGoogle Scholar
  6. Cochran, W. G. (1954). The combination of estimates from different experiments. Biometrics, 10(1), 101–129. https://doi.org/10.2307/3001666.CrossRefGoogle Scholar
  7. Gandolfo, G. (1993). Continuous-time econometrics has come of age. In G. Gandolfo (Ed.), Continuous-time econometrics (pp. 1–11). London: Chapman & Hall.CrossRefGoogle Scholar
  8. Greenan, C. C. (2015). Diffusion of innovations in dynamic networks. Journal of the Royal Statistical Society, Series A, 178, 147–166.  https://doi.org/10.1111/rssa.12054.MathSciNetCrossRefGoogle Scholar
  9. Hamerle, A., Nagl, W., & Singer, H. (1991). Problems with the estimation of stochastic differential equations using structural equation models. Journal of Mathematical Sociology, 16(3), 201–220. https://doi.org/10.1080/0022250X.1991.9990088.CrossRefGoogle Scholar
  10. Holland, P. W., & Leinhardt, S. (1977). A dynamic model for social networks. Journal of Mathematical Sociology, 5(1), 5–20. https://doi.org/10.1080/0022250X.1977.9989862.MathSciNetCrossRefGoogle Scholar
  11. Huisman, M., & Steglich, C. (2008). Treatment of non-response in longitudinal network studies. Social Networks, 30, 297–308. https://doi.org/10.1016/j.socnet.2008.04.004.CrossRefGoogle Scholar
  12. Hyde, J. S., Fennema, E., & Lamon, S. J. (1990). Gender differences in mathematics performance: A meta-analysis. Psychological Bulletin, 107(2), 139–155. https://doi.org/10.1037/0033-2909.107.2.139.CrossRefGoogle Scholar
  13. Kadushin, C. (2012). Understanding social networks: Theories, concepts, and findings. New York: Oxford University Press.Google Scholar
  14. Kalbfleisch, J. D., & Lawless, J. F. (1985). The analysis of panel data under a Markov assumption. Journal of the American Statistical Association, 80, 863–871. https://doi.org/10.1080/01621459.1985.10478195.MathSciNetCrossRefGoogle Scholar
  15. Kandel, D. B. (1978). Similarity in real-life adolescent friendship pairs. Journal of Personality & Social Psychology, 36(3), 306–312. https://doi.org/10.1037/0022-3514.36.3.306.CrossRefGoogle Scholar
  16. Knecht, A. (2004). Network and actor attributes in early adolescence. DANS.  https://doi.org/10.17026/dans-z9b-h2bp
  17. Knecht, A., Burk, W. J., Weesie, J., & Steglich, C. E. G. (2011). Friendship and alcohol use in early adolescence: A multilevel social network approach. Journal of Research on Adolescence, 21, 475–487. https://doi.org/10.1111/j.1532-7795.2010.00685.x.CrossRefGoogle Scholar
  18. Knecht, A., Snijders, T. A. B., Baerveldt, C., Steglich, C. E. G., & Raub, W. (2010). Friendship and delinquency: Selection and influence processes in early adolescence. Social Development, 19(3), 494–514. https://doi.org/10.1111/j.1467-9507.2009.00564.x.CrossRefGoogle Scholar
  19. Koskinen, J. H., & Snijders, T. A. B. (2007). Bayesian inference for dynamic social network data. Journal of Statistical Planning and Inference, 137, 3930–3938. https://doi.org/10.1016/j.jspi.2007.04.011.MathSciNetCrossRefGoogle Scholar
  20. Lehmann, E. L. (1999). Elements of large sample theory. New York: Springer. https://doi.org/10.1007/b98855.zbMATHGoogle Scholar
  21. Lindberg, S. M., Hyde, J. S., & Petersen, J. L. (2010). New trends in gender and mathematics performance: A meta-analysis. Psychological Bulletin, 136(6), 1123–1135. https://doi.org/10.1037/a0021276.CrossRefGoogle Scholar
  22. Manger, M. S., & Pickup, M. A. (2016). The coevolution of trade agreement networks and democracy. Journal of Conflict Resolution, 60(1), 164–191. https://doi.org/10.1177/0022002714535431.CrossRefGoogle Scholar
  23. Marsden, P. V. (2005). Recent developments in network measurement. In P. Carrington, J. Scott, & S. Wasserman (Eds.), Models and methods in social network analysis (pp. 8–30). New York: Cambridge University Press.  https://doi.org/10.1017/CBO9780511811395.002.CrossRefGoogle Scholar
  24. McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior. In P. Zarembka (Ed.), Frontiers in economics (pp. 105–142). New York: Academic Press.Google Scholar
  25. Niezink, N. M. D., & Snijders, T. A. B. (2017). Co-evolution of social networks and continuous actor attributes. The Annals of Applied Statistics, 11(4), 1948–1973. https://doi.org/10.1214/17-AOAS1037.MathSciNetCrossRefGoogle Scholar
  26. Norris, J. R. (1997). Markov chains. New York: Cambridge University Press.  https://doi.org/10.1017/CBO9780511810633.
  27. Oud, J. H. L. (2007). Continuous time modeling of reciprocal effects in the cross-lagged panel design. In S. M. Boker & M. J. Wenger (Eds.), Data analytic techniques for dynamical systems (pp. 87–129). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  28. 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
  29. Polyak, B. T. (1990). New method of stochastic approximation type. Automation and Remote Control, 51, 937–946.MathSciNetzbMATHGoogle Scholar
  30. Ripley, R. M., Snijders, T. A. B., Boda, Z., Vörös, A., & Preciado, P. (2018). Manual for RSiena (version May 2018) [Computer software manual]. Oxford: University of Oxford, Department of Statistics, Nuffield College.Google Scholar
  31. Robbins, H., & Monro, S. (1951). A stochastic approximation method. Annals of Mathematical Statistics, 22, 400–407.  https://doi.org/10.1214/aoms/1177729586.MathSciNetCrossRefGoogle Scholar
  32. Robins, G. (2015). Doing social network research: Network-based research design for social scientists. London: Sage.Google Scholar
  33. Ruppert, D. (1988). Efficient estimation from a slowly convergent Robbins-Monro process (Tech. Rep.). Cornell University, School of Operations Research and Industrial Engineering.Google Scholar
  34. Schweinberger, M., & Snijders, T. A. B. (2007). Markov models for digraph panel data: Monte Carlo-based derivative estimation. Computational Statistics & Data Analysis, 51, 4465–4483. https://doi.org/10.1016/j.csda.2006.07.014.MathSciNetCrossRefGoogle Scholar
  35. Scott, J., & Carrington, P. J. (Eds.). (2011). The SAGE handbook of social network analysis. London: SAGE Publications Ltd.Google Scholar
  36. Snijders, T. A. B. (2001). The statistical evaluation of social network dynamics. In M. Sobel & M. Becker (Eds.), Sociological methodology (pp. 361–395). Boston: Basil Blackwell.Google Scholar
  37. Snijders, T. A. B. (1996). Stochastic actor-oriented models for network change. Journal of Mathematical Sociology, 21, 149–172. https://doi.org/10.1080/0022250X.1996.9990178.CrossRefGoogle Scholar
  38. Snijders, T. A. B. (2005). Models for longitudinal network data. In P. J. Carrington, J. Scott, & S. S. Wasserman (Eds.), Models and methods in social network analysis. New York: Cambridge University Press.Google Scholar
  39. Snijders, T. A. B., & Baerveldt, C. (2003). A multilevel network study of the effects of delinquent behavior on friendship evolution. Journal of Mathematical Sociology, 27, 123–151. https://doi.org/10.1080/00222500305892.CrossRefGoogle Scholar
  40. Snijders, T. A. B., Koskinen, J., & Schweinberger, M. (2010a). Maximum likelihood estimation for social network dynamics. The Annals of Applied Statistics, 4(2), 567–588. https://doi.org/10.1214/09-AOAS313.MathSciNetCrossRefGoogle Scholar
  41. Snijders, T. A. B., & Pickup, M. (2016). Stochastic actor-oriented models for network dynamics. In J. N. Victor, A. M. Montgomery, & M. Lubell (Eds.), The Oxford handbook of political networks. Oxford: Oxford University Press.Google Scholar
  42. Snijders, T. A. B., Van de Bunt, G. G., & Steglich, C. E. G. (2010b). Introduction to actor-based models for network dynamics. Social Networks, 32, 44–60. https://doi.org/10.1016/j.socnet.2009.02.004.CrossRefGoogle Scholar
  43. Snijders, T. A. B., Steglich, C. E. G., & Schweinberger, M. (2007). Modeling the co-evolution of networks and behavior. In K. van Montfort, J. H. L. Oud, & A. Satorra (Eds.), Longitudinal models in the behavioral and related sciences (pp. 41–71). New York: Cambridge University Press.Google Scholar
  44. Stadtfeld, C. (2012). Events in social networks: A stochastic actor-oriented framework for dynamic event processes in social networks. Phd dissertation, Karlsruher Institut für Technologie.Google Scholar
  45. Stadtfeld, C., & Geyer-Schulz, A. (2011). Analyzing event stream dynamics in two-mode networks: An exploratory analysis of private communication in a question and answer community. Social Networks, 33, 258–272. https://doi.org/10.1016/j.socnet.2011.07.004.CrossRefGoogle Scholar
  46. Stadtfeld, C., Hollway, J., & Block, P. (2017). Dynamic network actor models: Investigating coordination ties through time. Sociological Methodology, 47, 1–40. https://doi.org/10.1177/0081175017709295.CrossRefGoogle Scholar
  47. Steglich, C. E. G., Snijders, T. A. B., & Pearson, M. (2010). Dynamic networks and behavior: Separating selection from influence. Sociological Methodology, 40, 329–392. https://doi.org/10.1111/j.1467-9531.2010.01225.x CrossRefGoogle Scholar
  48. Voelkle, M. C., 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. https://doi.org/10.1037/a0027543.CrossRefGoogle Scholar
  49. Wasserman, S. S. (1980). A stochastic model for directed graphs with transition rates determined by reciprocity. Sociological Methodology, 11, 392–412. https://doi.org/10.2307/270870.CrossRefGoogle Scholar
  50. Wasserman, S. S., & Faust, K. (1994). Social network analysis: Methods and applications. Cambridge: Cambridge University Press.  https://doi.org/10.1017/CBO9780511815478.
  51. Weerman, F. (2011). Delinquent peers in context: A longitudinal network analysis of selection and influence effects. Criminology, 49, 253–286. https:10.1111/j.1745-9125.2010.00223.x CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistics & Data ScienceCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of SociologyUniversity of GroningenGroningenThe Netherlands

Personalised recommendations