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Goodness of fit for log-linear network models: dynamic Markov bases using hypergraphs

Author

Listed:
  • Elizabeth Gross

    (San José State University)

  • Sonja Petrović

    (Illinois Institute of Technology)

  • Despina Stasi

    (Illinois Institute of Technology)

Abstract

Social networks and other sparse data sets pose significant challenges for statistical inference, since many standard statistical methods for testing model/data fit are not applicable in such settings. Algebraic statistics offers a theoretically justified approach to goodness-of-fit testing that relies on the theory of Markov bases. Most current practices require the computation of the entire basis, which is infeasible in many practical settings. We present a dynamic approach to explore the fiber of a model, which bypasses this issue, and is based on the combinatorics of hypergraphs arising from the toric algebra structure of log-linear models. We demonstrate the approach on the Holland–Leinhardt $$p_1$$ p 1 model for random directed graphs that allows for reciprocation effects.

Suggested Citation

  • Elizabeth Gross & Sonja Petrović & Despina Stasi, 2017. "Goodness of fit for log-linear network models: dynamic Markov bases using hypergraphs," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(3), pages 673-704, June.
    • Handle: RePEc:spr:aistmt:v:69:y:2017:i:3:d:10.1007_s10463-016-0560-2
    DOI: 10.1007/s10463-016-0560-2
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    References listed on IDEAS

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    1. Satoshi Aoki & Akimichi Takemura, 2003. "Minimal Basis for a Connected Markov Chain over 3 × 3 ×K Contingency Tables with Fixed Two‐Dimensional Marginals," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 45(2), pages 229-249, June.
    2. Hara, Hisayuki & Takemura, Akimichi & Yoshida, Ruriko, 2009. "A Markov basis for conditional test of common diagonal effect in quasi-independence model for square contingency tables," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1006-1014, February.
    3. Hunter, David R. & Goodreau, Steven M. & Handcock, Mark S., 2008. "Goodness of Fit of Social Network Models," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 248-258, March.
    4. Mitsunori Ogawa & Hisayuki Hara & Akimichi Takemura, 2013. "Graver basis for an undirected graph and its application to testing the beta model of random graphs," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 191-212, February.
    5. Aleksandra Slavković & Xiaotian Zhu & Sonja Petrović, 2015. "Fibers of multi-way contingency tables given conditionals: relation to marginals, cell bounds and Markov bases," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(4), pages 621-648, August.
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