Modelling Migration with Poisson Regression

Abstract

Most statistical analysis is based on the assumption that error is normally distributed, but many data sets are based on discrete data (the number of migrants from one place to another must be a whole number). Recent developments in statistics have often involved generalising methods so that they can be properly applied to non-normal data. For example, Nelder and Wedderburn (1972) developed the theory of generalised linear modelling, where the dependent or response variable can take a variety of different probability distributions linked in one of several possible ways to a linear predictor, based on a combination of independent or explanatory variables. Several common statistical techniques are special cases of the generalised linear models, including the usual form of regression analysis, Ordinary Least Squares regression, and binomial logit modelling. Another important special case is Poisson regression, which has a Poisson-distributed dependent variable, linked logarithmically to a linear combination of independent variables. Poisson regression may be an appropriate method when the dependent variable is constrained to be a non-negative integer, usually a count of the number of events in certain categories. It assumes that each event is independent of the others, though the probability of an event may be linked to available explanatory variables. This chapter illustrates how Poisson regression can be carried out using the Stata package, proceeding to discuss various problems and issues which may arise in the use of the method. The number of migrants from area i to area j must be a non-negative integer and is likely to vary according to zone population, distance and economic variables. The availability of high-quality migration data through the WICID facility permits detailed analysis at levels from the region to the output areas. A vast range of possible explanatory variables can also be derived from the 2001 Census data. Model results are discussed in terms of the significant explanatory variables, the overall goodness of fit and the big residuals. Comparisons are drawn with other analytic techniques such as OLS regression. The relationship to Wilson’s entropy maximising methods is described, and variants on the method are explained. These include negative binomial regression and zero-censored and zero-truncated models.