Modelling sparse interaction matrices: interward migration in Hereford and Worcester, and the underdispersion problem

Recently, it has been argued that migration models in which ordinary least squares regression is used are inappropriate, because the dependent variable (number of migrants) is a count. Instead, a Poisson regression approach can be adopted. Goodness of fit can be evaluated by using a deviance (log-likelihood) or <I>X</i><SUP>2</SUP>statistic, whose significance can be compared with a<FONT SIZE="3" FACE="Brush Script MT"> X</FONT><SUP>2 </SUP>distribution with appropriate degrees of freedom. In this paper, such an approach is used to model ward-level migration flows within the county of Hereford and Worcester. However, it is shown that for this exceedingly sparse data set the deviance figures attained are very low, suggesting that there may be a problem of underdispersion. This is in contrast to the overdispersion which has been identified as a common problem in Poisson models. The low deviance figures arise from the large number of zeros and small flows in the data matrix, which invalidate the usual<FONT SIZE="3" FACE="Brush Script MT"> X</FONT><SUP>2 </SUP>goodness-of-fit test. A simulation approach to the assessment of model goodness of fit is suggested, and the results from applying it to the Hereford and Worcester data set are described.