Advertisement

Modeling Heterogeneous Peer Assortment Effects Using Finite Mixture Exponential Random Graph Models

  • Teague R. HenryEmail author
  • Kathleen M. Gates
  • Mitchell J. Prinstein
  • Douglas Steinley
Original Research

Abstract

This article develops a class of models called sender/receiver finite mixture exponential random graph models (SRFM-ERGMs). This class of models extends the existing exponential random graph modeling framework to allow analysts to model unobserved heterogeneity in the effects of nodal covariates and network features without a block structure. An empirical example regarding substance use among adolescents is presented. Simulations across a variety of conditions are used to evaluate the performance of this technique. We conclude that unobserved heterogeneity in effects of nodal covariates can be a major cause of misfit in network models, and the SRFM-ERGM approach can alleviate this misfit. Implications for the analysis of social networks in psychological science are discussed.

Keywords

p* exponential random graphs finite mixture modeling individual differences modeling 

Notes

Acknowledgements

Funding was provided by National Science Foundation (US) (DGE-1650116) and National Institute on Alcohol Abuse and Alcoholism (US) (Grant No. 1R21AA022074).

Supplementary material

11336_2019_9685_MOESM1_ESM.pdf (89 kb)
Supplementary material 1 (pdf 88 KB)
11336_2019_9685_MOESM2_ESM.zip (284 kb)
Supplementary material 2 (zip 283 KB)

References

  1. Achenbach, T. M. (1991). Manual for the Youth self report and 1991 profile. Burlington: VT University of Vermont.Google Scholar
  2. Baudry, J.-P. (2015). Estimation and model selection for model-based clustering with the conditional classification likelihood. Electronic Journal of Statistics, 9, 1041–1077.CrossRefGoogle Scholar
  3. Bauer, D. J. (2011). Evaluating individual differences in psychological processes. Current Directions in Psychological Science, 20(2), 115–118.CrossRefGoogle Scholar
  4. Bauer, D. J., & Cai, L. (2008). Consequences of unmodeled nonlinear effects in multilevel models. Journal of Educational and Behavioral Statistics, 34(1), 97–114.CrossRefGoogle Scholar
  5. Bauer, D. J., & Curran, P. J. (2003). Distributional assumptions of growth mixture models: implications for overextraction of latent trajectory classes. Psychological methods, 8(3), 338–363.CrossRefPubMedGoogle Scholar
  6. Biernacki, C., Celeux, G., & Govaert, G. (1999). An improvement of the NEC criterion for assessing the number of clusters in a mixture model. Pattern Recognition Letters, 20(3), 267–272.CrossRefGoogle Scholar
  7. Brechwald, W. A., & Prinstein, M. J. (2011). Beyond homophily: A decade of advances in understanding peer influence processes. Journal of Research on Adolescence, 21(1), 166–179.CrossRefPubMedPubMedCentralGoogle Scholar
  8. Bryant, P. G. (1991). Large-sample results for optimization-based clustering methods. Journal of Classification, 8(1), 31–44.CrossRefGoogle Scholar
  9. Celeux, G., & Govaert, G. (1993). Comparison of the mixture and the classification maximum likelihood in cluster analysis. Journal of Statistical Computation and Simulation, 47(3–4), 127–146.CrossRefGoogle Scholar
  10. Celeux, G., & Soromenho, G. (1996). An entropy criterion for assessing the numbers of clusters in a mixture model. Journal of Classification, 13, 195–212.CrossRefGoogle Scholar
  11. Chatterjee, S., & Diaconis, P. (2013). Estimating and understanding exponential random graph models. Annals of Statistics, 41, 2428–2461.CrossRefGoogle Scholar
  12. Comets, F., & Janžura, M. (1998). A central limit theorem for conditionally centred random fields with an application to Markov fields. Journal of Applied Probability, 35(3), 608–621.CrossRefGoogle Scholar
  13. Daudin, J.-J., Picard, F., & Robin, S. (2008). A mixture model for random graphs. Statistics and Computing, 18(2), 173–183.CrossRefGoogle Scholar
  14. Dempster, Laird, & Rubin, N. M. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. B, 39(1), 1–38.Google Scholar
  15. DeSarbo, W. S., & Cron, W. L. (1988). A maximum likelihood methodology for clusterwise linear regression. Journal of Classification, 5(2), 249–282.CrossRefGoogle Scholar
  16. Efron, B. (1978). The geometry of exponential families. The Annals of Statistics, 6(2), 362–376.CrossRefGoogle Scholar
  17. Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81(395), 832–842.CrossRefGoogle Scholar
  18. Gilman, S. R., Iossifov, I., Levy, D., Ronemus, M., Wigler, M., & Vitkup, D. (2011). Rare De Novo variants associated with autism implicate a large functional network of genes involved in formation and function of synapses. Neuron, 70(5), 898–907.CrossRefPubMedPubMedCentralGoogle Scholar
  19. Govaert, G., & Nadif, M. (1996). Comparison of the mixture and the classification maximum likelihood in cluster analysis with binary data. Computational Statistics and Data Analysis, 23(1), 65–81.CrossRefGoogle Scholar
  20. Handcock, M. S. (2003). Assessing degeneracy in statistical models of social networks. Technical Report 39, University of Washington.Google Scholar
  21. Handcock, M. S., Hunter, D. R., Butts, C. T., Goodreau, S. M., & Morris, M. (2003). statnet: Software tools for the statistical modeling of network data (Version 2). Seattle, WA.Google Scholar
  22. Handcock, M. S., Raftery, A. E., & Tantrum, J. M. (2007). Model-based clustering for social networks. Journal of the Royal Statistical Society, Series B, 170, 1–22.CrossRefGoogle Scholar
  23. Hanneke, S., Fu, W., & Xing, E. P. (2010). Discrete temporal models of social networks. Electronic Journal of Statistics, 4, 585–605.CrossRefGoogle Scholar
  24. Hipp, J. R., & Bauer, D. J. (2006). Local solutions in the estimation of growth mixture models. Psychological Methods, 11(1), 36–53.CrossRefPubMedGoogle Scholar
  25. Hoff, P. D., Raftery, A. E., & Handcock, M. S. (2002). Latent space approaches to social network analysis. Journal of the American Statistical Association, 97(460), 1090–1098.CrossRefGoogle Scholar
  26. Holland, P., & Leinhardt, S. (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Associationerican Statistical, 76(373), 33–50.CrossRefGoogle Scholar
  27. Hubert, L., & Arabie, P. (1985). Comparing partitions. Journal of Classification, 2(1), 193–218.CrossRefGoogle Scholar
  28. Hunter, D. R., Goodreau, S. M., & Handcock, M. S. (2008). Goodness of fit of social network models. Journal of the American Statistical Association, 103(481), 248–258.CrossRefGoogle Scholar
  29. Hunter, D. R., & Handcock, M. S. (2006). Inference in curved exponential family models for networks. Journal of Computational and Graphical Statistics, 15(3), 565–583.CrossRefGoogle Scholar
  30. Jaccard, J., Turrisi, R., & Wan, C. K. (1990). Interaction effects in multiple regression. In Sage University Paper Series on Quantitative Applications in the Social Sciences (07-072).Google Scholar
  31. Kearns, M., Mansour, Y., & Ng, A. Y. (1997). An information-theoretic analysis of hard and soft assignment methods for clustering. In Proceedings of conference on uncertainty in artificial intelligence (pp. 282–293).Google Scholar
  32. Kiuru, N., Burk, W. J., Laursen, B., Salmela-Aro, K., & Nurmi, J. E. (2010). Pressure to drink but not to smoke: Disentangling selection and socialization in adolescent peer networks and peer groups. Journal of Adolescence, 33(6), 801–812.CrossRefPubMedGoogle Scholar
  33. Koskinen, J. H. (2009). Using latent variables to account for heterogeneity in exponential family random graph models. In Proceedings of the 6th St. Petersburg workshop on simulation (Vol. II, pp. 845–849).Google Scholar
  34. Krivitsky, P. N., & Handcock, M. S. (2014). A separable model for dynamic networks. Journal of the Royal Statistical Society. Series B: Statistical Methodology, 76(1), 29–46.CrossRefPubMedGoogle Scholar
  35. Kullback, S. (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22(1), 79–86.CrossRefGoogle Scholar
  36. Lubbers, M. J., & Snijders, T. A. B. (2007). A comparison of various approaches to the exponential random graph model: A reanalysis of 102 student networks in school classes. Social Networks, 29(4), 489–507.CrossRefGoogle Scholar
  37. MacQueen, J. B. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of symposium on mathematical statistics and probability (pp. 281–297).Google Scholar
  38. McLachlan, G., & Peel, D. (2005). ML fitting of mixture models. Finite mixture models (pp. 40–80)., Wiley series in probability and statistics Hoboken: Wiley.CrossRefGoogle Scholar
  39. Molenaar, P. C. M. (2004). A manifesto on psychology as idiographic science: Bringing the person back into scientific psychology, this time forever. Measurement: Interdisciplinary Research & Perspective, 2(4), 201–218.Google Scholar
  40. Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society. Series A (General), 135(3), 370.CrossRefGoogle Scholar
  41. Nowicki, K., & Snijders, T. A. B. (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96, 1077–1087.CrossRefGoogle Scholar
  42. Rubin, D. B. (1987). Multiple imputation for nonresponse in surveys (pp. 15–19). New York: Wiley.CrossRefGoogle Scholar
  43. Schweinberger, M., & Handcock, M. S. (2015). Local dependence in random graph models: Characterization, properties and statistical inference. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(3), 647–676.CrossRefGoogle Scholar
  44. Snijders, T. A., van de Bunt, G. G., & Steglich, C. E. (2010). Introduction to stochastic actor-based models for network dynamics. Social Networks, 32(1), 44–60.CrossRefGoogle Scholar
  45. Steinley, D. (2004). Properties of the Hubert-Arabie adjusted rand index. Psychological Methods, 9(3), 386–396.CrossRefPubMedGoogle Scholar
  46. Steinley, D., & Brusco, M. J. (2011a). Evaluating mixture modeling for clustering: Recommendations and cautions. Psychological Methods, 16(1), 63–79.CrossRefPubMedGoogle Scholar
  47. Steinley, D., & Brusco, M. J. (2011b). K-means clustering and mixture model clustering: Reply to McLachlan (2011) and Vermunt (2011). Psychological Methods, 16(1), 89–92.CrossRefGoogle Scholar
  48. Steinley, D., Brusco, M. J., & Wasserman, S. (2011). Clusterwise p* models for social network analysis. Statistical Analysis and Data Mining, 4(5), 487–496.CrossRefGoogle Scholar
  49. Strauss, D., & Ikeda, M. (1990). Pseudolikelihood estimation for social networks. Journal of the American Statistical Association, 85(409), 204–212.CrossRefGoogle Scholar
  50. Sweet, T. M. (2015). Incorporating covariates into stochastic blockmodels. Journal of Educational and Behavioral Statistics, 40(6), 635–664.CrossRefGoogle Scholar
  51. Symons, M. J. (1981). Clustering criteria and multivariate normal mixtures. Biometrics, 37(1), 35.CrossRefGoogle Scholar
  52. Tallberg, C. (2004). A Bayesian approach to modeling stochastic blockstructures with covariates. The Journal of Mathematical Sociology, 29(1), 1–23.CrossRefGoogle Scholar
  53. Thiemichen, S., Friel, N., Caimo, A., & Kauermann, G. (2016). Bayesian exponential random graph models with nodal random effects. Social Networks, 46, 11–28.CrossRefGoogle Scholar
  54. van Duijn, M. A. J., Gile, K. J., & Handcock, M. S. (2009). A framework for the comparison of maximum pseudo likelihood and maximum likelihood estimation of exponential family random graph models. Social Networks, 31(1), 52–62.CrossRefPubMedPubMedCentralGoogle Scholar
  55. Van Duijn, M. A. J., Snijders, T. A. B., & Zijlstra, B. J. H. (2004). p2: A random effects model with covariates for directed graphs. Statistica Neerlandica, 58(2), 234–254.CrossRefGoogle Scholar
  56. Wager, T. D., Kang, J., Johnson, T. D., Nichols, T. E., Satpute, A. B., & Barrett, L. F. (2015). A Bayesian model of category-specific emotional brain responses. PLOS Computational Biology, 11(4), e1004066.CrossRefPubMedPubMedCentralGoogle Scholar
  57. Wasserman, S., & Pattison, P. (1996). Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and \(p^*\). Psychometrika, 61(3), 401–425.CrossRefGoogle Scholar
  58. Wedel, M., & DeSarbo, W. S. (1995). A mixture likelihood approach for generalized linear models. Journal of Classification, 12(1), 21–55.CrossRefGoogle Scholar
  59. Zijlstra, B. J. H., Duijin, Ma J V, & Snijders, Ta B. (2006). The multilevel p 2 model social networks. Methodology, 2(1), 42–47.CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2019

Authors and Affiliations

  1. 1.University of North Carolina at Chapel HillChapel HillUSA
  2. 2.University of MissouriColumbiaUSA

Personalised recommendations