Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-30T05:17:13.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Likelihoods for fixed rank nomination networks

Published online by Cambridge University Press:  20 December 2013

PETER HOFF ,
BAILEY FOSDICK ,
ALEX VOLFOVSKY  and
KATHERINE STOVEL
PETER HOFF
Affiliation:
Departments of Statistics and Biostatistics, University of Washington, Seattle, WA 98195, USA (e-mail: pdhoff@uw.edu)
BAILEY FOSDICK
Affiliation:
Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, NC 27709, USA (e-mail: bfosdick@samsi.info)
ALEX VOLFOVSKY
Affiliation:
Department of Statistics, Harvard University, Cambridge, MA 02138, USA (e-mail: volfovsky@fas.harvard.edu)
KATHERINE STOVEL
Affiliation:
Department of Sociology, University of Washington, Seattle, WA 98195, USA (e-mail: stovel@u.washington.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Many studies that gather social network data use survey methods that lead to censored, missing, or otherwise incomplete information. For example, the popular fixed rank nomination (FRN) scheme, often used in studies of schools and businesses, asks study participants to nominate and rank at most a small number of contacts or friends, leaving the existence of other relations uncertain. However, most statistical models are formulated in terms of completely observed binary networks. Statistical analyses of FRN data with such models ignore the censored and ranked nature of the data and could potentially result in misleading statistical inference. To investigate this possibility, we compare Bayesian parameter estimates obtained from a likelihood for complete binary networks with those obtained from likelihoods that are derived from the FRN scheme, and therefore accommodate the ranked and censored nature of the data. We show analytically and via simulation that the binary likelihood can provide misleading inference, particularly for certain model parameters that relate network ties to characteristics of individuals and pairs of individuals. We also compare these different likelihoods in a data analysis of several adolescent social networks. For some of these networks, the parameter estimates from the binary and FRN likelihoods lead to different conclusions, indicating the importance of analyzing FRN data with a method that accounts for the FRN survey design.

Keywords

censoringlatent variablemarginal likelihoodmissing dataordinal dataranked datanetworksocial relations model
Type
Research Article
Information
Network Science , Volume 1 , Issue 3 , December 2013 , pp. 253 - 277
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Airoldi, E. M., Blei, D. M., Fienberg, S. E., & Xing, E. P. (2008). Mixed membership stochastic blockmodels. Journal of Machine Learning Research, 9, 1981–2014.Google ScholarPubMed
Breiger, R. L., Boorman, S. A., & Arabie, P. (1975). An algorithm for clustering relational data with applications to social network analysis and comparison with multidimensional scaling. Journal of Mathematical Psychology, 12 (3), 328–383.Google Scholar
Currarini, S., Jackson, M. O., & Pin, P. (2010). Identifying the roles of race-based choice and chance in high school friendship network formation. Proceedings of the National Academy of Sciences, 107 (11), 4857–4861.CrossRefGoogle ScholarPubMed
Fletcher, J. M., & Ross, S. L. (2012). Estimating the effects of friendship networks on health behaviors of adolescents. Technical report, National Bureau of Economic Research, Cambridge, MA.Google Scholar
Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81 (395), 832–842. ISSN .CrossRefGoogle Scholar
Geweke, J. (1991). Efficient simulation from the multivariate normal and student-t distributions subject to linear constraints and the evaluation of constraint probabilities. In Computing Science and Statistics: Proceedings of the 23rd symposium on the interface, pp. 571–578.Google Scholar
Gill, P. S., & Swartz, T. B. (2001). Statistical analyses for round robin interaction data. Canadian Journal of Statistics, 29 (2), 321–331. ISSN .Google Scholar
Goodreau, S. M., Kitts, J. A., & Morris, M. (2009). Birds of a feather, or friend of a friend? using exponential random graph models to investigate adolescent social networks.* Demography, 46 (1), 103–125.CrossRefGoogle ScholarPubMed
Handcock, M. S., & Gile, K. J. (2010). Modeling social networks from sampled data. Annals of Applied Statistics, 4 (1), 5–25. ISSN . 10.1214/08-AOAS221 Retrieved from http://dx.doi.org/10.1214/08-AOAS221.Google Scholar
Harris, K. M., Halpern, C. T., Whitsel, E., Hussey, J., Tabor, J., Entzel, P., & Udry, J. R. (2009). The national longitudinal study of adolescent health: Research design. Retrieved from http://www.cpc.unc.edu/projects/addhealth/design (December 15, 2012).Google Scholar
Hoff, P. D. (2005). Bilinear mixed-effects models for dyadic data. Journal of the American Statistical Association, 100 (469), 286–295. ISSN .Google Scholar
Hoff, P. D. (2007). Extending the rank likelihood for semiparametric copula estimation. Annals of Applied Statistics, 1 (1), 265–283. ISSN .Google Scholar
Hoff, P. D. (2009a). Multiplicative latent factor models for description and prediction of social networks. Computational and Mathematical Organization Theory, 15 (4), 261–272.Google Scholar
Hoff, P. D. (2009b). A first course in bayesian statistical methods. Springer Texts in Statistics. New York, NY: Springer. ix, 270 p. EUR 64.15; SFR 99.50.CrossRefGoogle Scholar
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. ISSN .Google Scholar
Holland, P. W., & Leinhardt, S. (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, 76 (373), 33–50.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), 801–812.Google Scholar
Krivitsky, P. N., & Butts, C. T. (2012). Exponential-family random graph models for rank-order relational data. Retrieved from http://arxiv.org/abs/1210.0493 (December 15, 2012).Google Scholar
Li, H., & Loken, E. (2002). A unified theory of statistical analysis and inference for variance component models for dyadic data. Statistics Sinica, 12 (2), 519–535. ISSN .Google Scholar
Macdonald-Wallis, K., Jago, R., Page, A. S., Brockman, R., & Thompson, J. L. (2011). School-based friendship networks and children's physical activity: A spatial analytical approach. Social Science & Medicine, 73 (1), 6–12.CrossRefGoogle ScholarPubMed
Moody, J., Brynildsen, W. D., Osgood, D. W., Feinberg, M. E., & Gest, S. (2011). Popularity trajectories and substance use in early adolescence. Social Networks, 33 (2), 101–112.Google Scholar
Moreno, J. L. (1953). Who shall survive? Foundations of sociometry, group psychotherapy and socio-drama. Mustang, OK: Beacon House.Google Scholar
Moreno, J. L. (1960). The sociometry reader. New York, NY: Free Press.Google Scholar
Nowicki, K., & Snijders, T. A. B. (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96 (455), 1077–1087. ISSN .Google Scholar
Pettitt, A. N. (1982). Inference for the linear model using a likelihood based on ranks. Journal of the Royal Statistical Society B, 44 (2), 234–243. ISSN .Google Scholar
Rodriguez-Yam, G., Davis, R. A., & Scharf, L. L. (2004). Efficient Gibbs sampling of truncated multivariate normal with application to constrained linear regression. Unpublished manuscript.Google Scholar
Sampson, S. F. (1969). Crisis in a cloister. Unpublished doctoral dissertation, Cornell University, Ithaca, NY.Google Scholar
Severini, T. A. (1991). On the relationship between Bayesian and non-Bayesian interval estimates. Journal of the Royal Statistical Society B, 53 (3), 611–618. ISSN .Google Scholar
Sherman, L. (2002). Sociometry in the classroom: How to do it. Retrieved from http://www.users.muohio.edu/shermalw/sociometryfiles/socio_introduction.htmlx (December 15, 2012).Google Scholar
Snijders, T. A. B., Pattison, P. E., Robins, G. L., & Handcock, M. S. (2006). New specifications for exponential random graph models. Sociological Methodology, 36 (1), 99–153.Google Scholar
Thomas, S. L. (2000). Ties that bind: A social network approach to understanding student integration and persistence. Journal of Higher Education, 71, 591–615.Google Scholar
van Duijn, Marijtje A. J., Snijders, Tom A. B., & Zijlstra, Bonne J. H. (2004). p 2: A random effects model with covariates for directed graphs. Statistica Neerlandica, 58 (2), 234–254. ISSN .CrossRefGoogle Scholar
Warner, R., Kenny, D. A., & Stoto, M. (1979). A new round robin analysis of variance for social interaction data. Journal of Personality and Social Psychology, 37, 1742–1757.Google Scholar
Wasserman, S., & Pattison, P. (1996). Logit models and logistic regressions for social networks: I. an introduction to markov graphs andp. Psychometrika, 61 (3), 401–425.Google Scholar
Weerman, F. M. (2011). Delinquent peers in context: A longitudinal network analysis of selection and influence effects. Criminology, 49 (1), 253–286.Google Scholar
Weerman, F. M., & Smeenk, W. H. (2005). Peer similarity in delinquency for different types of friends: A comparison using two measurement methods. Criminology, 43 (2), 499–524.Google Scholar
Wong, G. Y. (1982). Round robin analysis of variance via maximum likelihood. Journal of the American Statistical Association, 77 (380), 714–724. ISSN .Google Scholar