Skip to main content
SpringerLink
Log in

An Efficient MCMC Algorithm to Sample Binary Matrices with Fixed Marginals

  • Theory and Methods
  • Published:
Psychometrika Aims and scope Submit manuscript

Abstract

Uniform sampling of binary matrices with fixed margins is known as a difficult problem. Two classes of algorithms to sample from a distribution not too different from the uniform are studied in the literature: importance sampling and Markov chain Monte Carlo (MCMC). Existing MCMC algorithms converge slowly, require a long burn-in period and yield highly dependent samples. Chen et al. developed an importance sampling algorithm that is highly efficient for relatively small tables. For larger but still moderate sized tables (300×30) Chen et al.’s algorithm is less efficient. This article develops a new MCMC algorithm that converges much faster than the existing ones and that is more efficient than Chen’s algorithm for large problems. Its stationary distribution is uniform. The algorithm is extended to the case of square matrices with fixed diagonal for applications in social network theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Besag, J., & Clifford, P. (1989). Generalized Monte Carlo significance tests. Biometrika, 76, 633–42.

    Article  Google Scholar 

  • Chen, Y. (2006). Simple existence conditions for zero-one matrices with at most one structural zero in each row and column. Discrete Mathematics, 306, 2870–877.

    Article  Google Scholar 

  • Chen, Y., Diaconis, P., Holmes, S., & Liu, J. (2005). Sequential Monte Carlo methods for statistical analysis of tables. Journal of the American Statistical Association, 100, 109–120.

    Article  Google Scholar 

  • Chen, Y., & Small, D. (2005). Exact tests for the Rasch model via sequential importance sampling. Psychometrika, 70, 11–30.

    Article  Google Scholar 

  • Connor, E., & Simberloff, D. (1979). The assembly of species communities: chance or competition. Ecology, 60, 1132–1140.

    Article  Google Scholar 

  • Gale, D. (1957). A theorem on flows in networks. Pacific Journal of Mathematics, 7, 1073–1082.

    Google Scholar 

  • Guttorp, P. (1995). Stochastic modeling of scientific data. London: Chapman and Hall.

    Google Scholar 

  • Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.

    Article  Google Scholar 

  • Kong, A., Liu, J., & Wong, W. (1994). Sequential imputations and Bayesian missing data problems. Journal of the American Statistical Association, 89, 278–288.

    Article  Google Scholar 

  • Marshall, A., & Olkin, I. (1979). Inequalities: theory of majorization and its applications. San Diego: Academic Press.

    Google Scholar 

  • Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., & Teller, E. (1953). Equations of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087–1091.

    Article  Google Scholar 

  • Musalem, A., Bradlow, E., & Raju, J. (2008, in press). Bayesian estimation of random-coefficients models using aggregate data. Journal of Applied Econometrics.

  • Ponocny, I. (2001). Nonparametric goodness-of-fit tests for the Rasch model. Psychometrika, 66, 437–460.

    Article  Google Scholar 

  • Prabhu, N. (1965). Stochastic processes. Basic theory and its applications. New York: Macmillan.

    Google Scholar 

  • Rao, A., Jana, R., & Bandyopadhyay, S. (1996). A Markov chain Monte Carlo method for generating random (0,1)-matrices with given marginals. Sankhya, Series A, 58, 225–242.

    Google Scholar 

  • Roberts, A., & Stone, L. (1990). Island sharing by archipelago species. Oecologia, 83, 560–567.

    Article  Google Scholar 

  • Ryser, H. (1957). Combinatorial properties of matrices with zeros and ones. The Canadian Journal of Mathematics, 9, 371–377.

    Google Scholar 

  • Ryser, H. (1963). Combinatorial mathematics. In Carus mathematical monographs. Washington: The Mathematical Association of America.

    Google Scholar 

  • Snijders, T. (1991). Enumeration and simulation for 0-1 matrices with given marginals. Psychometrika, 56, 397–417.

    Article  Google Scholar 

  • Tanner, M.A. (1996). Tools for statistical inference (Third edn.). New York: Springer.

    Google Scholar 

  • Wasserman, S. (1977). Random directed graph distributions and the triad census in social networks. Journal of Mathematical Sociology, 5, 61–86.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CITO, National Institute for Educational Measurement, P.O. Box 1034, 6801 MG, Arnhem, The Netherlands

    Norman D. Verhelst

Authors
  1. Norman D. Verhelst
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Norman D. Verhelst.

Additional information

I am indebted to my colleague Gunter Maris for his suggestion to add a Metropolis–Hastings step as the finishing touch of the algorithm.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Verhelst, N.D. An Efficient MCMC Algorithm to Sample Binary Matrices with Fixed Marginals. Psychometrika 73, 705–728 (2008). https://doi.org/10.1007/s11336-008-9062-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11336-008-9062-3

Keywords

Search

Navigation