MODELING SPARSE INTERACTION MATRICES - INTERWARD MIGRATION IN HEREFORD AND WORCESTER, AND THE UNDERDISPERSION PROBLEM

Recently, it has been argued that migration models in which ordinary l east squares regression is used are inappropriate, because the depende nt variable (number of migrants) is a count. Instead, a Poisson regres sion approach can be adopted. Goodness of fit can be evaluated by usin g a deviance (log-likelihood) or X2 statistic, whose significance can be compared with a chi2 distribution with appropriate degrees of freed om. In this paper, such an approach is used to model ward-level migrat ion flows within the county of Hereford and Worcester. However, it is shown that for this exceedingly sparse data set the deviance figures a ttained are very low, suggesting that there may be a problem of underd ispersion. This is in contrast to the overdispersion which has been id entified as a common problem in Poisson models. The low deviance figur es arise from the large number of zeros and small flows in the data ma trix, which invalidate the usual chi2 goodness-of-fit test. A simulati on approach to the assessment of model goodness of fit is suggested, a nd the results from applying it to the Hereford and Worcester data set are described.