Papers of the Regional Science Association

, Volume 52, Issue 1, pp 187–205 | Cite as

Log-linear modelling of spatial interaction

  • Frans Willekens
Developments in Theory and Methodology


The research reported in this paper is part of a larger research effort to develop a methodology for estimating spatial interaction (migration) flows. The first section of the paper summarises the equivalences between the log-linear model and conventional spatial interaction models. It is shown under what conditions the values of the balancing factors of the gravity model coincide with the parameter values of the loglinear model. The second section focuses on theanalysis of (known) spatial interaction flows. The effects associated with the region of origin, region of destination and spatial separation are identified and quantified. A ‘distance effect’ is derived from the flow matrix. A measure of accessibility is developed and compared with accessibility measures derived from different forms of distance functions. The third section deals with theestimation of spatial interaction flows. It is shown how the quality of the estimates (and hence the model performance) can be improved by efficiently adding information. This is particularly relevant in the estimation of disaggregated spatial interaction flows.


Model Performance Research Effort Distance Function Interaction Model Gravity Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Batten, D. 1982. The location-production-interaction model revisited. Paper presented at the XXII European Congress of the Regional Science Association, Groningen.Google Scholar
  2. Baxter, M. J. and Ewing, G. O. 1979. Calibration of production-constrained trip distribution models and the effect of intervening opportunities.Journal of Regional Science 19: 319–30.Google Scholar
  3. Birch, M. 1963. Maximum likelihood in three-way tables.Journal of the Royal Statistical Society, Series B 25: 220–33.Google Scholar
  4. Bishop, Y. M., Fienberg, S. E. and Holland, P. W. 1975.Discrete multivariate analysis: theory and practice. Cambridge, Massachusetts: M.I.T. Press.Google Scholar
  5. Caussinus, H. 1965. Contribution a l'analyse statistique des tableaux de corrélation.Annales de la Faculté des Sciences de l'Université de Toulouse 29: 77–182.Google Scholar
  6. Charnes, A., Haynes, K. A., Phillips, F. and White, G. 1977. Dual extended geometric programming problems and the gravity model.Journal of Regional Science 17: 71–76.Google Scholar
  7. Charnes, A., Haynes, K. A. and Phillips F. 1976. A ‘generalized distance’ estimation procedure for intraurban interaction.Geographical Analysis 8: 289–94.Google Scholar
  8. Charnes, A., Raike, W. and Bettinger, C. 1972. An extremal and information-theoretic characterization of some interzonal transfer models.Socio-Economic Planning 6: 531–37.CrossRefGoogle Scholar
  9. Darroch, J. N. and Ratcliff, O. 1972. Generalized iterative scaling for loglinear models.Annals of Mathematical Statistics 43: 1470–80.Google Scholar
  10. Davis, J. A. 1978. Hierarchical models for significance tests in multivariate contingency tables: an exegesis of Goodman's recent papers. In L. Goodman,Analyzing qualitative/categorical data, ed. J. Magidson, pp. 233–75. Cambridge, Massachusetts: ABT Books.Google Scholar
  11. Deming, W. and Stephan, F. 1940. On a least squares adjustment of a sampled frequency table when expected marginal totals are known.Annals of Mathematical Statistics 11: 427–44.Google Scholar
  12. Dinkel, J., Kochenberger, G. and Wong, S. 1977. Entropy maximization and geometric programming,Environment and Planning 9: 419–27.Google Scholar
  13. Evans, S. P. 1973. A relationship between the gravity model for trip distribution and the transportation problem in linear programming.Transportation Research 7: 39–61.CrossRefGoogle Scholar
  14. Goodman, L. A. 1968. The analysis of cross-classified data: independence, quasi-independence and interactions in contingency tables with or without missing entries.Journal of the American Statistical Association 324: 1091–131.Google Scholar
  15. Haberman, S. J. 1979.Analysis of qualitative data 2 Vols. New York: Academic Press.Google Scholar
  16. Hauser, R. M. 1979. Some exploratory methods for modelling mobility tables and other cross-classified data. InSociological Methodology 1980, ed. K. F. Schuessler, pp. 413–58. San Francisco: Jossey-Bass Publishers.Google Scholar
  17. Hua, C. and Porell, F. 1979. A critical review of the development of the gravity model.International Regional Science Review 4: 97–126.Google Scholar
  18. MacFadden, D. 1978. Modelling the choice of residential location. InSpatial interaction theory and planning models, eds. A. Karlqvist, L. Lundqvist, F. Snickars and J. Weibull, pp. 75–96. Amsterdam: North-Holland Publ. Co.Google Scholar
  19. Macgill, S. M. 1975.Balancing factor methods in urban and regional analysis. Leeds: Department of Geography, University of Leeds, Working Paper 124.Google Scholar
  20. Macgill, S. M. 1977. Theoretical properties of biproportional matrix adjustments.Environment and Planning A 9: 687–701.Google Scholar
  21. March, L. 1971. Urban systems: a generalized distribution function. InUrban and regional planning (London Papers in Regional Science, Vol. 2), ed. A. G. Wilson. London: Pion Ltd.Google Scholar
  22. Nijkamp, P. 1975. Reflections on gravity and entropy models.Regional Science and Urban Economics 5: 203–25.CrossRefGoogle Scholar
  23. Openshaw, S. 1979. Alternative methods of estimating spatial interaction models and their performance in short-term forecasting. InExploratory and explanatory statistical analysis of spatial data, eds. C. P. A. Bartels and R. H. Ketellapper, pp. 201–25. Boston: Martinus Nijhoff Publishing.Google Scholar
  24. Payne, C. 1977. The log-linear model of contingency tables. InAnalysis of survey data. Vol. 2: model fitting, eds. C. O'Muircheartaigh and C. Payne, pp. 105–44. New York: Wiley.Google Scholar
  25. Scholte, H. 1982. Prescribing and forecasting migration within a housing market area. The loglinear model used as a spatial interaction model.Geografische en Planologische Notities no. 20. Amsterdam: Institute for Geographical Studies and Urban and Regional Planning, Free University.Google Scholar
  26. Sikdar, P. K. and Hutchinson, B. G. 1981. Empirical studies of work trip distribution models.Transportation Research Journal: A 15A, 3: 233–43.Google Scholar
  27. Snickars, F. and Weibull, J. W. 1977. A minimum information principle. Theory and practice.Regional Science and Urban Economics 7: 137–68.CrossRefGoogle Scholar
  28. Whitney, J. B. and Boots, B. N. 1979. An examination of residential mobility through the use of log-linear model. Part II: empirical results.Regional Science and Urban Economics 9: 393–409.CrossRefPubMedGoogle Scholar
  29. Willekens, F. 1980. Entropy, multiproportional adjustment and analysis of contingency tables.Systemi Urbani 2: 171–201.Google Scholar
  30. Willekens, F. 1982. Multidimensional population analysis with incomplete data. InMultidimensional mathematical demography, eds. K. Land and A. Rogers, pp. 43–111. New York: Academic Press.Google Scholar
  31. Willekens, F. and Baydar, N. 1983. Hybrid log-linear models. Voorburg: NIDI Working Paper No. 41.Google Scholar
  32. Willekens, F., Por, A. and Raquillet, R. 1979.Entropy, multiproportional and quadratic techniques for inferring detailed migration patterns from aggregate data. Mathematical theories, algorithms, applications and computer programs. Laxenburg, Austria: I.I.A.S.A., Working Paper WP-79-88.Google Scholar
  33. Willekens, F. and Tan, E. 1980. Entropy maximization, geometric programming and biproportional adjustment. Unpublished research note. Available from author.Google Scholar
  34. Wilson, A. G. 1970.Entropy in urban and regional modelling. London: Pion Ltd.Google Scholar

Copyright information

© The Regional Science Association 1983

Authors and Affiliations

  • Frans Willekens
    • 1
  1. 1.Netherlands Interuniversity Demographic InstituteVoorburgThe Netherlands

Personalised recommendations