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

Published online by Cambridge University Press:  01 January 2025

Teague R. Henry*
Affiliation:
University of North Carolina at Chapel Hill
Kathleen M. Gates
Affiliation:
University of North Carolina at Chapel Hill
Mitchell J. Prinstein
Affiliation:
University of North Carolina at Chapel Hill
Douglas Steinley
Affiliation:
University of Missouri
*
Correspondence should be made to Teague R. Henry, University of North Carolina at Chapel Hill, Chapel Hill, USA. Email: trhenry@email.unc.edu

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.

Type
Original Research
Copyright
Copyright © 2019 The Psychometric Society

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Footnotes

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11336-019-09685-2) contains supplementary material, which is available to authorized users.

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References

Achenbach, T.M., (1991). Manual for the Youth self report and 1991 profile Burlington:VT University of Vermont.Google Scholar
Baudry, J.-P., (2015). Estimation and model selection for model-based clustering with the conditional classification likelihood. Electronic Journal of Statistics, 9, 1041107710.1214/15-EJS1026CrossRefGoogle Scholar
Bauer, D.J., (2011). Evaluating individual differences in psychological processes. Current Directions in Psychological Science, 20(2), 115118.CrossRefGoogle Scholar
Bauer, D.J.,&Cai, L., (2008). Consequences of unmodeled nonlinear effects in multilevel models. Journal of Educational and Behavioral Statistics, 34(1), 97114.CrossRefGoogle Scholar
Bauer, D.J.,&Curran, P.J., (2003). Distributional assumptions of growth mixture models: implications for overextraction of latent trajectory classes. Psychological methods, 8(3), 338363.CrossRefGoogle ScholarPubMed
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), 267272.CrossRefGoogle Scholar
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), 166179.CrossRefGoogle ScholarPubMed
Bryant, P.G., (1991). Large-sample results for optimization-based clustering methods. Journal of Classification, 8(1), 3144.CrossRefGoogle Scholar
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), 127146CrossRefGoogle Scholar
Celeux, G.,&Soromenho, G., (1996). An entropy criterion for assessing the numbers of clusters in a mixture model. Journal of Classification, 13 , 195212CrossRefGoogle Scholar
Chatterjee, S.,&Diaconis, P., (2013). Estimating and understanding exponential random graph models. Annals of Statistics, 41 , 24282461CrossRefGoogle Scholar
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), 608621.CrossRefGoogle Scholar
Daudin, J-J.,Picard, F.,&Robin, S., (2008). A mixture model for random graphs. Statistics and Computing, 18(2), 173183.CrossRefGoogle Scholar
Dempster, Laird,&Rubin, N.M., (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. B, 39(1), 138.CrossRefGoogle Scholar
DeSarbo, W.S.,&Cron, W.L., (1988). A maximum likelihood methodology for clusterwise linear regression. Journal of Classification, 5(2), 249282.CrossRefGoogle Scholar
Efron, B., (1978). The geometry of exponential families. The Annals of Statistics, 6(2), 362376.CrossRefGoogle Scholar
Frank, O.,&Strauss, D., (1986). Markov graphs. Journal of the American Statistical Association, 81(395), 832842.CrossRefGoogle Scholar
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), 898907.CrossRefGoogle ScholarPubMed
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), 6581.CrossRefGoogle Scholar
Handcock, M. S.(2003). Assessing degeneracy in statistical models of social networks. Technical Report 39, University of Washington.Google Scholar
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
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 , 122Google Scholar
Hanneke, S.,Fu, W.,&Xing, E.P.,2010). Discrete temporal models of social networks. Electronic Journal of Statistics, 4 , 585605CrossRefGoogle Scholar
Hipp, J.R.,&Bauer, D.J., (2006). Local solutions in the estimation of growth mixture models. Psychological Methods, 11(1), 3653.CrossRefGoogle ScholarPubMed
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), 10901098.CrossRefGoogle Scholar
Holland, P.,&Leinhardt, S.,(1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Associationerican Statistical, 76(373), 3350.CrossRefGoogle Scholar
Hubert, L.,&Arabie, P.,(1985). Comparing partitions. Journal of Classification, 2(1), 193218.CrossRefGoogle Scholar
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), 248258.CrossRefGoogle Scholar
Hunter, D.R.,&Handcock, M.S.,(2006). Inference in curved exponential family models for networks. Journal of Computational and Graphical Statistics, 15(3), 565583.CrossRefGoogle Scholar
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
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
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), 801812.CrossRefGoogle Scholar
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
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), 2946.CrossRefGoogle ScholarPubMed
Kullback, S.,(1951). On information and sufficiency. The Annals of Mathematical Statistics, 22(1), 7986.CrossRefGoogle Scholar
Lubbers, M.J.,&Snijders, TAB,(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), 489507.CrossRefGoogle Scholar
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
McLachlan, G.,&Peel, D.,(2005). ML fitting of mixture models.Finite mixture models, HobokenWiley 4080Google Scholar
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), 201218.Google Scholar
Nelder, J.A.,&Wedderburn, RWM,(1972). Generalized linear models. Journal of the Royal Statistical Society. Series A (General), 135 3370CrossRefGoogle Scholar
Nowicki, K.,&Snijders, TAB,(2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96 , 10771087CrossRefGoogle Scholar
Rubin, D.B., Multiple imputation for nonresponse in surveys 1987 New YorkWiley 1519CrossRefGoogle Scholar
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), 647676.CrossRefGoogle ScholarPubMed
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), 4460.CrossRefGoogle Scholar
Steinley, D.,(2004). Properties of the Hubert-Arabie adjusted rand index. Psychological Methods, 9(3), 386396.CrossRefGoogle ScholarPubMed
Steinley, D.,&Brusco, M.J.,(2011). Evaluating mixture modeling for clustering: Recommendations and cautions. Psychological Methods, 16(1), 6379.CrossRefGoogle ScholarPubMed
Steinley, D.,&Brusco, M.J.,(2011). K-means clustering and mixture model clustering: Reply to McLachlan (2011) and Vermunt (2011). Psychological Methods, 16(1), 8992.CrossRefGoogle Scholar
Steinley, D.,Brusco, M.J.,Wasserman, S.,(2011). Clusterwise p* models for social network analysis. Statistical Analysis and Data Mining, 4(5), 487496.CrossRefGoogle Scholar
Strauss, D.,&Ikeda, M.,(1990). Pseudolikelihood estimation for social networks. Journal of the American Statistical Association, 85(409), 204212.CrossRefGoogle Scholar
Sweet, T.M.,(2015). Incorporating covariates into stochastic blockmodels. Journal of Educational and Behavioral Statistics, 40(6), 635664.CrossRefGoogle Scholar
Symons, M.J.,(1981). Clustering criteria and multivariate normal mixtures. Biometrics, 37 135CrossRefGoogle Scholar
Tallberg, C.,(2004). A Bayesian approach to modeling stochastic blockstructures with covariates. The Journal of Mathematical Sociology, 29(1), 123.CrossRefGoogle Scholar
Thiemichen, S.,Friel, N.,Caimo, A.,&Kauermann, G.,(2016). Bayesian exponential random graph models with nodal random effects. Social Networks, 46 , 1128CrossRefGoogle Scholar
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), 5262.CrossRefGoogle ScholarPubMed
Van Duijn, MAJ,Snijders, TAB,&Zijlstra, B.J.H,(2004). p2: A random effects model with covariates for directed graphs. Statistica Neerlandica, 58(2), 234254.CrossRefGoogle Scholar
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 4e1004066CrossRefGoogle ScholarPubMed
Wasserman, S.,&Pattison, P.,(1996). Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^*$$\end{document}. Psychometrika, 61(3), 401425.CrossRefGoogle Scholar
Wedel, M.,&DeSarbo, W.S.,(1995). A mixture likelihood approach for generalized linear models. Journal of Classification, 12(1), 2155.CrossRefGoogle Scholar
Zijlstra, B.J.H,Duijin, MaJV,&Snijders, Ta.B,(2006). The multilevel p 2 model social networks. Methodology, 2(1), 4247.CrossRefGoogle Scholar
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