Hostname: page-component-5f745c7db-szhh2 Total loading time: 0 Render date: 2025-01-06T06:29:52.137Z Has data issue: true hasContentIssue false
  • English
  • Français

Logit Models and Logistic Regressions for Social Networks: III. Valued Relations

Published online by Cambridge University Press:  01 January 2025

Garry Robins ,
Philippa Pattison  and
Stanley Wasserman
Garry Robins*
Affiliation:
Deakin University
Philippa Pattison
Affiliation:
University of Melbourne
Stanley Wasserman
Affiliation:
University of Illinois
*
Requests for reprints should be sent to Garry Robins, Faculty of Health and Behavioural Sciences, School of Psychology, Deakin University, Geelong Victoria 3217, AUSTRALIA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper generalizes the p* model for dichotomous social network data (Wasserman & Pattison, 1996) to the polytomous case. The generalization is achieved by transforming valued social networks into three-way binary arrays. This data transformation requires a modification of the Hammersley-Clifford theorem that underpins the p* class of models. We demonstrate that, provided that certain (non-observed) data patterns are excluded from consideration, a suitable version of the theorem can be developed. We also show that the approach amounts to a model for multiple logits derived from a pseudo-likelihood function. Estimation within this model is analogous to the separate fitting of multinomial baseline logits, except that the Hammersley-Clifford theorem requires the equating of certain parameters across logits. The paper describes how to convert a valued network into a data array suitable for fitting the model and provides some illustrative empirical examples.

Keywords

social networksp* modelsautologistic modelspseudo-likelihood estimation
Type
Original Paper
Information
Psychometrika , Volume 64 , Issue 3 , September 1999 , pp. 371 - 394
Copyright
Copyright © 1999 The Psychometric Society

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.)

Footnotes

This research was supported by grants from the Australian Research Council, the National Science Foundation (#SBR96-30754), and the National Institute of Health (#PHS-1RO1-39829-01).

References

Agresti, A. (1990). Categorical data analysis, New York: John Wiley and Sons.Google Scholar
Anderson, C.J., & Wasserman, S. (1995). Log multiplicative models for valued social relations. Sociological Methods & Research, 24, 96–127.CrossRefGoogle Scholar
Anderson, C.J., & Wasserman, S., & Crouch, B. (in press). Ap* primer: logit models for social networks. Social Networks.Google Scholar
Bearman, P. (1997). Generalized exchange. American Journal of Sociology, 102, 1383–1415.CrossRefGoogle Scholar
Begg, C.B., & Gray, R. (1984). Calculation of polychotomous logistic regression parameters using individualized regressions. Biometrika, 71, 11–18.CrossRefGoogle Scholar
Besag, J.E. (1972). Nearest neighbour systems and the auto-logistic model for binary data. Journal of the Royal Statistical Society, Series B, 34, 75–83.CrossRefGoogle Scholar
Besag, J.E. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). Journal of the Royal Statistical Society, Series B, 36, 192–236.CrossRefGoogle Scholar
Besag, J.E. (1975). Statistical analysis of nonlattice data. The Statistician, 24, 179–195.CrossRefGoogle Scholar
Besag, J.E. (1977). Some methods of statistical analysis for spatial data. Bulletin of the International Statistical Association, 47, 77–92.Google Scholar
Besag, J.E. (1977). Efficiency of pseudo-likelihood estimation for simple Gaussian random fields. Biometrika, 64, 616–618.CrossRefGoogle Scholar
Besag, J.E., & Clifford, P. (1989). Generalized Monte Carlo significance tests. Biometrika, 76, 633–642.CrossRefGoogle Scholar
Crouch, B., & Wasserman, S. (1998). Fitting p*: Monte Carlo maximum likelihood estimation. Paper presented at International Conference on Social Networks, Barcelona, Spain.Google Scholar
Faust, K., & Wasserman, S. (1993). Association and correlational models for studying measurements on ordinal relations. In Marsden, P.V. (Eds.), Sociological methodology 1993 (pp. 177–215). Cambridge, MA: Basil Blackwell.Google Scholar
Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81, 832–842.CrossRefGoogle Scholar
Geyer, C. J., & Thompson, E.A. (1992). Constrained Monte Carlo maximum likelihood for dependent data. Journal of the Royal Statistical Society, Series B, 54, 657–699.CrossRefGoogle Scholar
Hammersley, J. M., & Clifford, P. (1971). Markov fields on finite graphs and lattices. Unpublished manuscript.Google Scholar
Hosmer, D.W., & Lemeshow, S. (1989). Applied logistic regression, New York: John Wiley & Sons.Google Scholar
Ising, E. (1925). Beitrag zur theorie des ferromagnetismus. Zeitschrift fur Physik, 31, 253–258 [Contribution to the theory of ferromagnetism]CrossRefGoogle Scholar
Johnsen, E.C. (1986). Structure and process: agreement models for friendship formation. Social Networks, 8, 257–306.CrossRefGoogle Scholar
Lauritzen, S.L. (1996). Graphical models, Oxford: Clarendon Press.CrossRefGoogle Scholar
Lazega, E., & Pattison, P. (1998). Social capital, multiplex generalized exchange and cooperation in organizations: A case study. Submitted to Social Networks.Google Scholar
Norusis, M.J. (1990). SPSS advanced statistics user's guide, Chicago: SPSS.Google Scholar
Pattison, P., & Wasserman, S. (in press). Logit models and logistic regressions for social networks: II. Multivariate relations. British Journal of Mathematical and Statistical Psychology.Google Scholar
Preisler, H. (1993). Modeling spatial patterns of trees attacked by bark-beetles. Applied Statistics, 42, 501–514.CrossRefGoogle Scholar
Rennolls, K. (1995). p 1/2. In Everett, M.G., & Rennolls, K. (Eds.), Proceedings of the 1995 International Conference on Social Networks, Vol. 1. (pp. 151–160). London: Greenwich University Press.Google Scholar
Robins, G.L. (1997, February). p* models of social influence. Paper presented at the International Sunbelt Social Network Conference, San Diego, CA.Google Scholar
Robins, G.L. (1998). Personal attributes in inter-personal contexts: Statistical models for individual characteristics and social relationships, Australia: University of Melbourne.Google Scholar
Robins, G.L., Pattison, P., & Langan-Fox, J. (1995, July). Group effectiveness: A comparative analysis of interactional structure and group performance in organizational workgroups. Paper presented at International Social Networks Conference, London.Google Scholar
Strauss, D. (1992). The many faces of logistic regression. The American Statistician, 46, 321–327.CrossRefGoogle Scholar
Strauss, D., & Ikeda, M. (1990). Pseudolikelihood estimation for social networks. Journal of the American Statistical Association, 85, 204–212.CrossRefGoogle Scholar
Vickers, M. (1981). Relational analysis: An applied evaluation. Unpublished Master of Science thesis, Department of Psychology, University of Melbourne.Google Scholar
Vickers, M., & Chan, S. (1981). Representing classroom social structure, Melbourne: Victoria Institute of Secondary Education.Google Scholar
Wasserman, S. (1987). Conformity of two sociometric relations. Psychometrika, 52, 3–18.CrossRefGoogle Scholar
Wasserman, S., & Faust, K. (1989). Canonical analysis of composition and structure of social networks. In Clogg, C.C. (Eds.), Sociological methodology 1989 (pp. 1–42). Cambridge, MA: Basil Blackwell.Google Scholar
Wasserman, S., & Faust, K. (1994). Social network analysis: Methods and applications, New York: Cambridge University Press.CrossRefGoogle Scholar
Wasserman, S., Faust, K., & Galaskiewicz, J. (1990). Correspondence and canonical analysis of relational data. Journal of Mathematical Sociology, 15, 11–62.CrossRefGoogle Scholar
Wasserman, S., & Iacobucci, D. (1986). Statistical analysis of discrete relational data. British Journal of Mathematical and Statistical Psychology, 39, 41–64.CrossRefGoogle Scholar
Wasserman, S., & Pattison, P. (1996). Logit models and logistic regressions for social networks. I: An introduction to Markov graphs and p*. Psychometrika, 60, 401–425.CrossRefGoogle Scholar
Wasserman, S., & Pattison, P. (1999). Multivariate random graph distributions, New York: Springer-Verlag.Google Scholar
Wong, G.Y., & Wang, Y.J. (1995). Exponential models for polytomous stochastic networks. Unpublished manuscript.Google Scholar