Abstract
We explore a model for social networks that may be viewed either as an extension of logistic regression or as a Gibbs distribution on a complete graph. The model was developed for data from a mental health service system which includes a neighborhood structure on the clients in the system. This neighborhood structure is used to develop a Markov chain Monte Carlo goodness-of-fit test for the fitted model, with pleasing results.
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References
Besag, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion)Journal of Royal Statistical Society Series B, 36, 192–236.
Besag, J. E. (1975). Statistical analysis of non-lattice dataThe Statistician, 24, 179–195.
Cométs, F. (1992). On consistency of a class of estimators for exponential families of Markov random fields on the latticeAnnals of Statistics, 20, 455–468.
Cressie, N. (1993).Statistics for Spatial DataNew York: John Wiley&Sons.
Doreian, P. (1980). Linear models with spatially distributed data: spatial disturbances or spatial effectsSociological Methods Research, 9, 29–61.
Doreian, P. (1982). Maximum likelihood methods for linear models: spatial effect and spatial disturbance termsSociological Methods e.4 Research, 10, 243–269.
Doreian, P. (1989). Network autocorrelation models: problems and prospects. Paper presented at 1989 Symposium“Spatial Statistics: Past Present Future”Department of Geography, Syracuse University.
Galaskiewicz, J. and Wasserman, S. (1993). Social network analysis: Concepts, methodology, and directions for the 1990sSociological Methods Research, 22, 3–22.
Gibbs, J. W. (1902).Elementary Principles of Statistical MechanicsYale University Press.
Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and Bayesian restoration of imagesIEEE Transactions on Pattern Analysis and Machine Intelligence,6, 721–741.
Georgii, H. O. (1988).Gibbs Measures and Phase TransitionsBerlin: Walter de Gruyter.
Geyer, C. J. and Thompson, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data (with discussion)Journal of Royal Statistical Society Series B, 54, 657–699.
Gould, R. (1991). Multiple networks and mobilization in the Paris Commune, 1871American Sociological Review, 56, 716–29.
Ji, C. and Seymour, L. (1996). A consistent model selection procedure for Markov random fields based on penalized pseudolikelihoodAnnals of Applied Probability,6, 423–443.
Lehman, A., Postrado, L., Roth, D., McNary, S., and Goldman, H. (1994). An evaluation of continuity of care, case management, and client outcomes in the Robert Wood Johnson Program on chronic mental illnessThe Milbank Quarterly, 72, 105–122.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953). Equations of state calculations by fast computing machinesJournal of Chemical Physics, 21, 1087–1092.
Morrissey, J. P., Calloway, M., Bartko, W. T., Ridgley, S., Goldman, H., and Paulson, R. I. (1994). Local mental health authorities and service system change: Evidence from the Robert Wood Johnson Foundation Program on Chronic Mental IllnessThe Milbank Quarterly, 72, 49–80.
Nash, J. C. (1990).Compact Numerical Methods for Computers - Linear Algebra and Function Minimisation(2nd edition), Bristol: Adam Hilger.
Possolo, A. (1991). Subsampling a random field, InSpatial Statistics and Imaging(Ed., A. Possolo), Vol. 20, pp. 286–294, IMS Lecture Notes - Monograph Series.
Seymour, L. (2000). A note on the variance of the maximum pseudo-likelihood estimator, Submitted to Proceedings of the Symposium on Stochastic ProcessesAthens, Georgia.
Seymour, L. and Ji, C.(1996). Approximate Bayes model selection criteria for Gibbs-Markov random fieldsJournal of Statistical Planning and Inference, 51, 75–97.
Seymour, L., Smith, R., Calloway, M., and Morrissey, J. P. (2000).Lattice models for social networks with binary data, Technical Report 2000–24Department of Statistics, University of Georgia.
Strauss, D. and Ikeda, M. (1990). Pseudolikelihood estimation for social networks.Journal of the American Statistical Association, 85, 204–212.
Wasserman, S. and Pattison, P. (1996). Logit models and logistic regressions for social networks: L An introduction to Markov graphs andp* Psychometrika, 61, 401–425.
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Seymour, L. (2002). Gibbs Regression and a Test for Goodness-of-Fit. In: Huber-Carol, C., Balakrishnan, N., Nikulin, M.S., Mesbah, M. (eds) Goodness-of-Fit Tests and Model Validity. Statistics for Industry and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0103-8_12
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DOI: https://doi.org/10.1007/978-1-4612-0103-8_12
Publisher Name: Birkhäuser, Boston, MA
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