The Annals of Applied Statistics

A mixed effects model for longitudinal relational and network data, with applications to international trade and conflict

Anton H. Westveld and Peter D. Hoff

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The focus of this paper is an approach to the modeling of longitudinal social network or relational data. Such data arise from measurements on pairs of objects or actors made at regular temporal intervals, resulting in a social network for each point in time. In this article we represent the network and temporal dependencies with a random effects model, resulting in a stochastic process defined by a set of stationary covariance matrices. Our approach builds upon the social relations models of Warner, Kenny and Stoto [Journal of Personality and Social Psychology 37 (1979) 1742–1757] and Gill and Swartz [Canad. J. Statist. 29 (2001) 321–331] and allows for an intra- and inter-temporal representation of network structures. We apply the methodology to two longitudinal data sets: international trade (continuous response) and militarized interstate disputes (binary response).

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Ann. Appl. Stat., Volume 5, Number 2A (2011), 843-872.

First available in Project Euclid: 13 July 2011

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Bayesian inference international trade longitudinal data militarized interstate disputes network data relational data


Westveld, Anton H.; Hoff, Peter D. A mixed effects model for longitudinal relational and network data, with applications to international trade and conflict. Ann. Appl. Stat. 5 (2011), no. 2A, 843--872. doi:10.1214/10-AOAS403.

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  • Airoldi, E., Blei, D. Xing, E. and Fienberg, S. (2005). A latent mixed membership model for relational data. In Proceedings of the 3rd International Workshop on Link Discovery 82–89. ACM, New York.
  • Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. J. Amer. Statist. Assoc. 88 669–679.
  • Anderson, J. E. (1979). A theoretical foundation for the gravity equation. American Economic Review 69 106–116.
  • Barabasi, A.-L. and Oltvar, Z. N. (2004). Network biology: Understanding the cell’s functional organization. Nat. Rev. Genet. 5 101–113.
  • Chib, S. and Greenberg, E. (1998). Analysis of multivariate probit models. Biometrika 2 347–361.
  • Erosheva, E., Fienberg, S. and Lafferty, J. (2004). Mixed-membership models of scientific publications. In Proceedings of the National Academy of Sciences of the United States of America 101 5220–5227.
  • Frank, O. and Strauss, D. (1986). Markov graphs. J. Amer. Statist. Assoc. 81 832–842.
  • Gill, P. S. and Swartz, T. B. (2001). Statistical analyses for round robin interaction data. Canad. J. Statist. 29 321–331.
  • Givens, G. H. and Hoeting, J. A. (2005). Computational Statistics. Wiley, Hoboken, NJ.
  • Handcock, M. S., Raftery, A. E. and Tantrum, J. (2007). Model-based clustering for social networks. J. Roy. Statist. Soc. Ser. A 170 301–354.
  • Hanneke, S., Fu, W. and Xing, E. P. (2010). Discrete temporal models for social networks. Electronic Journal of Statistics 4 585–605.
  • Hoff, P. D. (2003). Random effects models for network data. In Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers (R. Breiger, K. Carley and P. Pattison, eds.) 303–312. National Academies Press, Washington, DC.
  • Hoff, P. D. (2005). Bilinear mixed-effects models for dyadic data. J. Amer. Statist. Assoc. 100 286–295.
  • Hoff, P. D. (2007). Model averaging and dimension selection for the singular value decomposition. J. Amer. Statist. Assoc. 102 674–685.
  • Hoff, P. D., Raftery, A. E. and Handcock, M. S. (2002). Latent space approaches to social network analysis. J. Amer. Statist. Assoc. 97 1090–1098.
  • Hoff, P. D. and Ward, M. D. (2003). Modeling dependencies in international networks. Technical report. Center for Statistics and the Social Sciences, Univ. Washington, Seattle, WA.
  • Huisman, M. and Snijders, T. A. B. (2003). Statistical analysis of longitudinal network data with changing composition. Sociol. Methods Res. 32 253–287.
  • Hunter, D. R. and Handcock, M. S. (2006). Inference in curved exponential family models for networks. J. Comput. Graph. Statist. 15 565–583.
  • Jones, D. M., Bremer, S. A. and Singer, J. D. (1996). Militarized interstate disputes, 1816–1992: Rationale, coding rules, and emirical patterns. Conflict Manegment and Peace Science 15 163–213.
  • Li, H. (2002). Modeling through group invariance: An interesting example with potential applications. Ann. Statist. 30 1069–1080.
  • Li, H. and Loken, E. (2002). A unified theory of statistical analysis and inference for variance component models for dyadic data. Statist. Sinica 12 519–535.
  • McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models. Chapman & Hall/CRC, Washington, DC.
  • Radcliffe-Brown, A. R. (1940). On social structure. The Journal of the Royal Anthropological Institute of Great Britain and Ireland 70 1–12.
  • Reinsel, G. C. (1997). Elements of Multivariate Time Series Analysis. Springer, New York.
  • Snijders, T. A. B, Steglich, C. E. G. and Schweinberger, M. (2007). Modeling the co-evolution of networks and behavior. In Longitudinal Models in the Behavioral and Related Sciences (K. van Montfort, J. Oud and A. Satorra, eds.) 41–71. Routledge Academic, London.
  • Snijders, T. A. B., van de Bunt, G. G. and Steglich, C. E. G. (2010). Introduction to stochastic actor-based models for network dynamics. Social Networks 32 44–60.
  • Snijders, T. A. B., Koskinen, J. and Schweinberger, M. (2010). Maximum likelihood estimation for social dynamics. Ann. Appl. Statist. 4 567–588.
  • Tinbergen, J. (1962). Shaping the World Economy-Suggestions for an International Economic Policy. The Twentieth Century Fund, New York.
  • Ward, M. D. and Hoff, P. D. (2007). Persistent patterns of international commerce. Journal of Peace Research 44 157–175.
  • Ward, M. D., Siverson, R. M. and Cao, X. (2007). Disputes, democracies, and dependencies: A reexamination of the Kantian Peace. American J. Political Sci. 51 583–601.
  • Warner, R. M., Kenny, D. A. and Stoto, M. (1979). A new round robin analysis of variance for social interaction data. Journal of Personality and Social Psychology 37 1742–1757.
  • Wasserman, S. and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge Univ. Press, Cambridge.
  • Westveld, A. H. (2007). Statistical methodology for longitudinal social network data. Ph.D. thesis, Dept. Statistics, Univ. Washington, Seattle, WA.
  • Westveld, A. H. and Hoff, P. D. (2010). Supplement to “A mixed effects model for longitudinal relational and network data, with applications to international trade and conflict.” DOI: 10.1214/10-AOAS403SUPP.
  • Wong, G. Y. (1982). Round robin analysis of variance via maximum likelihood. J. Amer. Statist. Assoc. 77 714–724.
  • Xing, E. P., Fu, W. and Song, L. (2010). A state-space mixed membership blockmodel for dynamic network tomography. Ann. Appl. Statist. 4 535–566.

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