Analysis Of Count Data Using Poisson Regression
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This document describes Poisson regression, a statistical technique for analyzing count data using regression. It compares Poisson regression to ordinary least squares regression, outlines how to perform Poisson regression in the GLIM software package, and provides an example analyzing historical apprentice migration data to Edinburgh. Key aspects include: - Poisson regression is appropriate when the dependent variable is a count, unlike OLS regression which assumes a normal distribution. - It models the logarithm of the mean as a linear combination of predictors rather than the mean directly. - GLIM allows specification of the Poisson error distribution and logarithmic link function required for Poisson regression. - An example apprentice migration data set is analyzed to demonstrate the technique.
Analysis Of Count Data Using Poisson Regression
- 1. Professional zyxwvutsrqpo Geo zyxwvutsrqpon rapher, 41(2), 1989,pp. 190-198 zyxwvut 0Copyright 1 6 9 by Association of American Geographers ANALYSIS OF COUNT DATA USING POISSON REGRESSION* Andrew Lovett and Robin Flowerdew zyx University zyxwvu of Lancaster zyxw In geographical research the data of interest are often in the form of counts. Standard regression analysis is inappropriatefor suck data, but if certain assumptionsare met, a form of regression based on the Poisson distribution can be used. This paper illustrates the use of Poisson regression in the computer package GLIM with an example from historical geography. Apprentice migration zyxwvut to Edinburgh is regressed on a combination zyxw of categorical, count, and continuous explanatory variables. Key Words: Poisson regression, count data, migration, GLIM, Edinburgh. Many geographers receive a basictrain- ing in statistical analysis but lack the op- portunity or desire to keep up to datewith developments in statistical methods that may be useful in contextsreasonably com- mon in geographical research. This paper describes one such development,Poisson regression analysis,in sufficientdetail that interested researchers with moderate knowledge of statistical applications in geography can learn to use it for them- selves. Regression analysis constructs an ex- planatory or predictive model of a de- pendent or response variable (usually de- noted Y) on the basis of one or more independent or explanatory variables (denoted XI, X2, etc.). Poisson regression is appropriate when the dependent vari- able is a count, such as the number of times an event occurs or the number of people in a certain category. It may be particularly useful if some observations have very low values. The method is still not very widely known in geography, or indeed the social or environmental sci- * This work was done under a grant from the Eco- nomic and Social Research Council. The manuscript was prepared while the second author was working with the Institute for Market and Social Analysis, Toronto. We would like to thank Ian Whyte for sup- plying the data, and present and former members of the Centre for Applied Statistics, University of Lan- caster, especially Murray Aitkin, Brian Francis, John Hinde, and Andrew Wood for statisticalhelp. ences generally. It does not yet appear, for example, in any textbook of statistical methods in geography. A growing num- ber of applications exist in the journal lit- erature, however. Dependent variables which have been analyzed using Poisson regression include the number of mi- grants from one city to another (Constan- tine and Gower 1982;Flowerdew and Ait- kin 1982; Flowerdew and Lovett 1988), airline passengers from one city to another (Fotheringham and Williams 1983), origins of US college students (Pickles 1985), interregional industrial move- ments in Britain (Twomey 1986), radia- tion exposure categories (Darby et al. 1985), plant species on islands (Vincent and Haworth 1983),family planning clin- icsin Nigeria (Senior 1987),shoppingtrips (Guy 1987), children immunized against whooping cough (Gatrell1986),heart dis- ease mortality (Lovett et al. 1986), spina bifida (Lovett and Gatrell 1988), looted stores in New York City and national newspapers of the world (Flowerdewand Lovett 1984).Lovett (1984)has produced a detailed guide to the technique as im- plemented in the GLIM statistical pack- age. In the first section of this paper, the characteristics of Poisson regression are introduced by contrasting them with the standard ordinary least squares (OLS)form of regression analysis. An outline of the commands needed to carry out Poisson 190
- 2. VOLUME zyxwvut 41, NUMBER 2, MAY,1989 191 regression in GLIM follows. Finally an illustration of how these can be linked together is provided in the third section through an analysis of a data set from historical geography. Poisson Regression and the Generalized ability. The shape of a Poisson distribu- tion depends on the value of its mean (which is equal to its variance). If the mean is close to zero, then the distribution is strongly skewed; if the mean is larger, the peak occurs further from the vertical axis. If the mean is very large, the Poisson dis- tribution can be approximated by the nor- Linear Model mal. Linear regression analysis assumes that the dependent variable has a linear re- lationship with the independent vari- ables. If there is only one independent variable X, the predicted value of the de- pendent variable Y for case i is given by the equation f i zyxwvutsr = zyxwv PO + PI xi where f i is the predicted value ofzyxwv Y for case i, xi is the value of X for case i, zyxw POis the intercept-the value of f when X is zero, and PI is the slope-the amount by which f changes as X increases by one unit. The observed value yi correspond- ing to xi can be envisaged as a realization of a random variable Yi whose mean is estimated to be Pi. In OLS regression, the Yi variables are assumed to have normal distributions and a constant variance. OLS regression has been used to ana- lyze count dependent variables, but its assumptionsare not valid in this context. The values of the yis are counts and so must be non-negative integers; therefore the random variables generating them cannot have a normal distribution,which is continuous and can take on negative values. If the counts are large, the normal distribution may give a reasonable ap- proximation to the true discrete distri- bution; but when the counts have small values, the use of OLS regression is in- appropriate and the results obtained are unreliable. A type of regression analysis more ap- propriate for small counts is one in which each Yi variable is assumed to have a Pois- son distribution (Frome et al. 1973).The Poisson distribution describes the prob- ability that an event occurs k times in a fixed period given that each occurrence is independent and has a constant prob- In a Poisson regression model the pre- dicted value of the dependent variable for qase i is the maximum likelihoodestimate Xi of the mean of a Poisson-distributed variable Yi. The natural logarithm of this estimate is equal to a linear combination of the corresponding values of the inde- pendent variables; if there is only one in- dependent variable the relationship has the form In(&) = PO + ~ l x i (2) This equation can also be written (3) This is the Poisson regression equivalent of the OLS regression equation (1). Two characteristics of the Poisson regression model are worthy of further discussion. Unlike OLS regression, the Poisson model does not assume that the data are homoscedastic. Indeed, the vari- ance of each case is equal to the corre- sponding predicted value. Consequently the variances associated with the values of Yi cannot be the same and cannot be normally distributed. Second, the ob- served value of Yi is a count of indepen- dent events generated by a Poisson dis- tribution with parameter Xi. This means that the model is not appropriate where the occurrence of one event increases the probability of others. For example, mod- eling the incidence of a contagious dis- ease would not be appropriate. The OLSand Poisson regression models can be considered as variants of what Nelder and Wedderburn (1972) termed generalized linear models,explained more fully in Appendix A of the GLIM User’s Guide (Payne 1985) and by McCullagh and Nelder (1983). Generalized linear models can be used whenever an analyst
- 3. 192 zyxwvut THEPROFESSIONAL GEOGRAPHER wants to estimate the value of a depen- dent variable Y on the basis of a linear predictor (a linear combination of explan- atory variables XI, X2, etc.). An observed value yi is regarded as the sum of a sys- tematic component (pi)and a random (or error) component (ti): yizyxwvu = pi + ti zyxwvu (4) where yi is the observed value of Y for case i, ki is the predicted value of Y for case i, and ti is a randomly distributed error term. Types of generalized linear models are differentiated on the basis of the probability distribution that Yi is as- sumed to have and the link function zyxwv g that associatesthe mean of Yi (pi)with the linear predictor m g(pJ = zyxwv zPjxij (5) j-1 where m is the number of parameters, Pj is the parameter for the jth independent variable, and xi, is the value of the ith observation on the jth independent vari- able. Equation (4) can be rewritten A range of linear models can be con- structed by specifying different types of error term ti and link function g. Options for the link function include identity, log- arithmic, reciprocal, and square root, among others. The error distribution can be any member of the exponentialfamily of probability distributions, including such distributions as the normal, bino- mial, Poisson, and gamma. In OLSregression the error distribution is normal. The predicted value pi is equal to 2Pjxij,the linear predictor, so the link is simply the identity function.In Poisson regression, the error distribution is Pois- son, but it is the naturaj logarithm of the predicted value, not Xi itself, which is equal to the linear predictor, so the link function is logarithmic. Log-linear mod- eling in GLIM (Bowlby and Silk 1982; m j-1 Wrigley 1985)also uses the Poisson prob- ability model and logarithmic link. In- deed, it can be regarded as a special case of Poisson regression when all the inde- pendent variables are categorical. The concept of a generalized linear model is the basis of the computer pack- age GLIMwhich provides a relatively easy means of performing Poisson regression. (It is also available in other packages, in- cluding GENSTATand LIMDEP.)Poisson regression is not available in most of the other major statistical packages, such as SAS, SPSS-X, SYSTAT, or BMDP. This pa- per, therefore, focuses on the implemen- tation of Poisson regression in GLIM. Poisson Regression in GLIM GLIM was developedby the Numerical Algorithms Group in conjunction with the Working Party on Statistical Computing of the Royal Statistical Society (OBrien 1987).In addition to the mainframe ver- sion, a microcomputer version isavailable relatively cheaply from the Numerical Al- gorithms Group in Oxford or in Downers Grove, IL. Payne (1985) provides a de- tailed guide, and several articles have been published describing how to use GLIM for particular purposes. Examples by geographers include Bowlby and Silk (1982)and O’Brien (1983). GLIMallows the user to specifythe error distribution and link function to be used in the regression. Once the data have been entered, any necessary transformations can be done on the variables, and the data can be plotted. The example in the next section illustrates these operations. The main process involved in fitting the mod- el, however, is the selection of indepen- dent variables for inclusion. The results of fitting a model can be evaluated according to the value of scaled deviance, which is a measure of the over- all difference between the observed val- ues of Y and those predicted by the model. Deviance is computed in different ways according to the probability distribution in use, and is based on the log-likelihood ratio. In OLSregression, deviance is equal to the error (or residual) sum of squares.
- 4. VOLUME41, NUMBER 2, MAY,1989 193 In Poisson regression (where it is some- timesdenoted zyxwvut G2), the scaleddeviance(d)is zyxw " d = zyxwv 2 yiln(yi/ii) (7) i=l The deviance is large when the corre- spondence between observed and esti- mated values is poor, and small when it is close. Because the devianceof a Poisson mod- el has a distributionapproximating to chi- square, it is possible to assesswhether the estimated and observed values of Y are significantly different by comparing the calculated deviancewith the critical value of chi-square for the appropriate degrees of freedom and significance level. The number of degrees of freedom is equal to the number of cases minus the number of parameters fitted. If the deviance exceeds the critical value, the null hypothesisthat the model could have generated the data should be rejected, and the model should not be regarded as providing an adequate explanation (at least in statistical terms) of the variation in the values of the de- pendent variable. However, little is known about how good the approxima- tion to chi-square is for small sets of data (Payne 1985,lll), and this test should be regarded only as a general guide in as- sessing goodness of fit. A regression model can be evaluated from several different perspectives. One of these is goodness-of-fit, already men- tioned. Another aspect is the significance of the parameters, which can be assessed using t values, obtained by dividing each parameter estimate by its standard error. A third consideration is interpretability, which can best be evaluated by seeing if the sign and magnitude of parameter es- timates accord with theoretical expecta- tions. Ideally a regression model (Poisson or otherwise)should have a good fit, sig- nificant parameters, and a clear interpre- tation. In practice, however, these are not always attainable simultaneously, and the best compromise must be chosen. There- fore it is sensible to fit models according to a systematic strategy, particularly be- cause the significance of individual pa- rameters is likely to vary between models depending on the other variables includ- ed (especially if they are collinear). One strategy begins by assessing the overall variation in the dependent vari- able by fitting a null model, in which the only parameter isan intercept term (equal to the mean of Y). The deviance of this model can be used as a benchmark to as- sess the effect of adding independent variables. The importance of each vari- able added in turn to the null model can be judged by comparing the reduction in deviance with the critical value of chi- square. Introducing numerical variables involves the loss of only one degree of freedom, and so a decline in deviance of more than zyx 3.84indicates that the variable is significant at the 0.05 level. The same procedure can also be used to test the sig- nificance of factors and interaction terms, though here the degrees of freedom lost will depend on the number of categories of the factor(s) concerned. This procedure can be used to find which variable is the most successful in reducing the deviance. Others are then added one by one in the order defined by the magnitude of their effect on deviance. At each stage the estimates and standard errors are examined to remove any vari- ables whose parameters are insignificant. New variables added to the model can alter the significance of others, so that variables which initially seemed signifi- cant may drop out, and others, not sig- nificant originally, may become so. This procedure is complete when a significant reduction in deviance can no longer be obtained. At this stage, those variables and interaction terms not in the model should be reintroduced to test whether they are now significant. Once this process is com- pleted, the estimates and residuals can be displayed and the overall adequacy of the model evaluated. Application of the Model An analysis of apprentice migration to Edinburgh between 1775and 1799is cho- sen to demonstrate the discussion above with a real and reasonably small data set
- 5. 194 zyxwvut THEPROFESSIONAL GEOGRAPHER TABLEzyxwvuts 1 DATA USED IN THE APPRENTICE MIGRATION EXAMPLE D A POP U R County 21 24 33 33 36 41 41 52 54 56 67 71 78 79 85 86 92 110 110 125 132 156 157 159 174 175 179 234 274 274 283 366 491 225 22 44 3 41 9 2 5 23 zyxwvutsrq 11 9 13 26 0 5 3 1 2 0 1 0 3 0 4 2 0 1 7 4 1 0 0 1 56 000 18000 30 000 7000 94 000 9000 1 1000 5000 147000 31000 34 000 51000 126000 21 000 99 000 55 000 78 000 84 000 29 000 26 000 12000 123000 22 000 36 000 27 000 8000 72 000 74000 55 000 23 000 23 000 29 000 22 000 18.8 3 37.9 2 43.4 3 30.3 1 41.3 1 29.3 3 47.4 1 41.9 3 68.1 2 15.2 3 31.8 3 31.1 2 14.4 1 27.3 2 55.3 1 25.9 3 69.9 2 26.4 2 11.3 3 12.3 1 43.6 2 23.1 1 23.2 3 12.9 1 27.6 1 45.3 1 12.7 2 10.8 1 10.7 1 28.3 1 10.3 1 9.0 1 7.7 1 Midlothian West Lothian East Lothian Kinross Fife Peebles Clackmannan Selkirk Lanark Benvick Roxburgh Stirling Perth Dunbarton Angus Dumfries Renfrew Kirkcud- bright Kincardine Bute Aberdeen Wigtown Banff Moray Nairn Argyll Inverness Ross Caithness Sutherland Orkney Shetland A Yr D = distance, A = number of apprentices, zyxwvut U = degree of urbanization,R = categorical variable (directionof county from Edinburgh).The data set should not include column headings when it is read into GLIM. Source: Lovett 1984. (Lovett et al. 1985; Whyte and Whyte 1986).The dependent variable (A) is the number of apprentices originating from each of the 33 Scottish counties registered in Edinburgh between 1775and 1799.Ap- prentice migration from each county was expected to be positively related to its population (POP), asrecorded in the 1801 census, and negatively related to its dis- tance (D) and its degree of urbanization (U), measured by the percentage of the county population living in urban settle- ments.These relationships might vary re- gionally because of differencesin the ease of travel and the nature of intervening opportunities. A three-level categorical variable (R) was therefore included, de- fined according to whether the county was north, west or south of Edinburgh. The following is an annotated tran- script of a Poisson regression analysis of the data in GLIM 3.77 on a VAX 11 z 1780 running under VMS. Input is indicated by upper case, output by lower case. The following commands read the data from a file called EDIN.DAT(Table 1)on chan- nel (or unit) 10. $UNITS 33 $ $DATA D A POP U R $ $FORMAT zyx (F4.O,F4.0,F7.O,FS.l,F2.0) $ $DINPUT 10 $ File name? EDIN.DAT Population and distance variables are logarithmically transformed, according to usual gravity model practice. The loga- rithmic link function used in Poisson regression makes it unnecessary to trans- form the number of migrants. $CALC LP = %LOG(POP)$ $CALC LD = %LOG(D)$ The regional variable R must be specified as a factor, and A must be identified as the dependent variable. $FACTOR R 3 $ $YVAR A $ $ERROR POISSON $ $LINK LOG $ (Thecommands $ERROR NORMAL $ and $LINK IDENTITY$ would be used at this point if OLS regression was to be done.) Now the null model can be fitted, fol- lowed by each independent variable in turn. $FIT $ scaled deviance = 1350.4 at cycle 5 d.f. = 32 $FIT LP $ scaled deviance = 1217.8 at cycle 5 d.f. = 31
- 6. VOLUME41, NUMBER 2,zyxwv MAY,1989 195 $FIT LDzyxwvut $ scaled deviance = 393.96 at cycle 4 d.f. = 31 $FIT U $ scaled deviance = 1348.9at cycle 6 d.f. = 31 $FIT R $ scaled deviance = 1054.1at cycle 5 d.f. = 30 The cycle number indicates how many iterations were required for the solution. The null model devianceis fairly large, and log distance is the best explanatory variable, followed by region. They can be tried together. $FIT LD + R $ scaled deviance = 329.60 at cycle 5 d.f. = 29 The devianceis reduced by 64 for the loss of two degrees of freedom, so it is worth retaining LD and R together in the model. Next the other variables can be added to the model. In the following commands, the + indicates that the new model con- sists of those variables in the previous model plus the additional one specified. $FIT + LP $ scaled deviance = (change = d.f. = (change = $FIT + zyxwv U $ scaled deviance = (change = d.f. = (change = 97.91 at cycle 4 -231.7) 28 -1) 80.524 at cycle 4 27 -17.390) -1) Both variables produce significant devi- ance reductions, but the overall deviance is stillwell over 40.11,which is the critical chi-square value at 0.05 for 27 degrees of freedom. Before interaction terms are introduced into the model, the parameter estimates and standard errors can be examined. $DISPLAY E $ param- estimate s.e. eter 1 0.7346 0.9885 1 2 -2.055 0.09349 LD 3 -0.1016 0.1726 R(2) 4 0.5044 0.1477 R(3) 5 0.9822 0.07816 LP 6 -0.01705 0.004141 U scale parameter taken as 1.000 The low standard errors of LD, LP and U confirm their significance in accounting for variation in A. Theparameter estimate for LP is near 1,asthe gravity model would suggest. The value for LD is negative as expected, as is that for urbanization. The values shown for R(2) and R(3) are dif- ferences from the intercept; thus region 3 (the south) actually has the highest pa- rameter value, with region l slightly in excess of region 2. Wrigley (1985) pro- vides more detailsabout interpreting this type of GLIM output. Now the interaction of region with the other variables can be investigated.Add- ing the interaction between a factor (R) and a continuous variable (LP) is equiv- alent to estimating a separate parameter for each level of the factor. $FIT f R.LP $ scaled deviance = 76.951 at cycle 4 d.f. = 25 This is not significant, zyx so it can be omitted from the model and the other interaction terms fitted. (change = -3.57) (change = -2) $FIT - R.LP + R.LD $ scaled deviance = 51.836 at cycle 4 (change = -25.11) d.f. = 25 (change = -0) $FIT + R.U $ scaled deviance = 40.919 at cycle 4 (change = -10.92) d.f. = 23 (change = -2)
- 7. 196 THEPROFESSIONAL GEOGRAPHER TABLE 2 PARAMETER ESTIMATES AND STANDARD ERRORS DISPLAYED BY GLIM Estimate SE Parametera zyxwvu ~ 1 -I.623 1.949 zyxwvutsr 2 -1.549 0.2013 3 -3.035 3.837 4 8.657 3.164 5 1.032 0.1386 6 -0.02619 0.008928 7 -2.049 0.8342 8 -1.316 0.2999 9 1.144 0.6820 10 -0.3891 0.2305 11 -0.02834 0.02682 12 0.03025 0.01206 zyxwvuts a Scale parameter taken zyxwvutsrq as 1.000. 1 LD R(2) R(3) LP U LD.R(2) LD.R(3) LP.R(2) LP.R(3) U.R(2) U.R(3) Both terms produce significant reductions in deviance when added to the model, and the overall value is approaching an acceptable fit (the critical value of chi- square at 0.01 is 41.64). Although the interaction term R.LP was not significant earlier, it may be now that the other interaction terms have been added. $FIT + R.LP $ scaled deviance = 32.642at cycle 4 (change = -8.277) d.f. = 21 (change = -2) This term is now significant, and the de- viance is now below the critical value of chi-square at 0.05. This model could therefore have generated the observed data; the null hypothesis that apprentice migration is determined by these vari- ables cannot be rejected on the basis of this data set. The current set of parameter estimates can now be requested with the $DISPLAY E $ command (Table 2). Ap- prentice migration to Edinburgh is posi- tively associated with origin population, and negatively with distance and urban- ization. There are also regional variations; for example, the distance-decay gradient is shallower to the north and steeper to the west. The next step in the analysis might be to display the residuals, and to examine them to see if any observations are out of line with the others. As with other forms of regression analysis, the spatial pattern of residuals can often give the investi- gator ideas for other variables which may be of relevance. In general, the analysis has been sat- isfactory (not always the case!). An ac- ceptable fit has been achieved, and all the parameter values are in accordance with theoretical expectations. The $STOP com- mand concludes the GLIM run. Conclusion This paper has presented the basics of Poisson regression analysis from a prac- tical point of view. The example shows how a researcher may investigate a series of models with speed and flexibility. Not all analyses will produce a model with an adequate fit, especially if data for impor- tant explanatory variables are not avail- able. Even if a good fit is not achieved, however, the results will allow an eval- uation of which variables contribute the most to reducing the deviance. It must be remembered, of course, that any variable or any model, regardless of its goodness of fit, may have no causal link at all to the dependent variable, and may be just a sur- rogate for other variables. The applicability of Poisson regression analysis depends on the assumption that the counts which constitute the depen- dent variable are generated by a Poisson distribution, which is true only if the in- dividual events making up the counts are independent of one another. In our ex- ample, we assume that apprentices acted independently in making their moves to Edinburgh. This assumption is probably not entirely true, as apprenticesmay have been preceded or accompanied by rela- tives or friends from the same origin. In other cases, such as overall interurban mi- gration (Flowerdew and Aitkin 1982), the independence assumption is certainly not true, for families are likely to migrate to- gether. In such a case, some form of gen- eralized or compoundPoisson model may
- 8. VOLUME41, NUMBER 2, MAY, 1989 197 z be appropriate (Flowerdew and Lovett 1989;Hinde 1982);fitting such a model is less straightforward and cannot be done in GLIM without special macro proce- dures. If a Poisson regression of a com- pound or generalized Poisson dependent variable is conducted, the deviance can be expected to be considerably greater than for a Poisson variable. However, Da- vies and Guy (1987)have shown that, for a large class of compound and general- ized Poisson distributions, the parameters derived from Poisson regression are un- biased estimates of the parameters of the compound or generalized process. Poisson regression analysis isa method developed fairly recently and as yet is lit- tle known in geography. Its use is appro- priate in many contexts where the de- pendent variable is a count. The analysis is reasonably easy to apply, given suitable computer software such as GLIM, and geographers should make more use of it in the future. Literature Cited Bowlby, S., and J. Silk. 1982. Analysis of qual- ititative data using GLIM: Two examples based on shopping survey data. zyxwvu The Profes- sional Geographer 34230-90. Constantine, A. G., and J. C. Gower. 1982. Models for the analysis of interregional mi- gration. Environment and Planning zyxwvu A 14477- 97. Darby, S., E. Nakashima, and H. Kato. 1985. A parallel analysis of cancer mortality among atomic bomb survivors and patients with an- kylosing spondylitis given X-ray therapy. lournu1 zyxwvutsrq of the National Cancer Institute 75:l-21. Davies,R. B., and C. M. Guy. 1987.The statistical modeling of flow data when the Poisson as- sumption is violated. Geographical Analysis 19: Flowerdew, R., and M. Aitkin. 1982. A method of fitting the gravity model based on the Poisson distribution. Journal of Regional Sci- ence 22191-202. Flowerdew, R., and A. Lovett. 1984. Poisson regression models for the analysis of geo- graphical data. End-of-grant Report to the Economicand SocialResearch Council, Lon- don. Flowerdew, R., and zyxwvu A. Lovett. 1988. Fitting constrained Poisson regression models to 300-14. inter-urban migration flows. Geographical Analysis 202.97-307. Flowerdew, R.,and A. Lovett. 1989.Compound and generalised Poisson models for inter- urban migration. In Advances in Regional De- mography: Forecasts, Information, Models, ed. P. Congdon and P. Batey, 246-56. London: Belhaven Press. Fotheringham, A. S., and P. A. Williams. 1983. Further discussion on the Poisson interac- tion model. Geographical Analysis 15:343-47. Frome, E. L., M. H. Kutner, and 1.J. Beauchamp. 1973. Regression analysis of Poisson- distribution data. Journalof the American Sta- tistical Association 68:935-40. Gatrell, A. C. 1986. Immunization against whooping cough in Salford: A spatial anal- ysis. Social Science and Medicine 23:1027-32. Guy, C. M. 1987. Recent advances in spatial interaction modelling: An application to the forecasting of shopping travel. Environment and Planning A 19:173-86. Hinde, J. 1982. Compound Poisson regression models. In GLIM 82, ed. R. Gilchrist, 109-21. New York: Springer-Verlag. Lovett, A. 1984. Poisson regression using the GLIM package. Lancaster:Department of Ge- ography, University of Lancaster, Computer Package Guide No. 5. Lovett, A,, C. G. Bentham, and R. Flowerdew. 1986. Analysing geographic variations in mortality using Poisson regression: The ex- ample of ischaemic heart disease in England and Wales 1969-1973. Social Science and Med- icine 23:935-43. Lovett, A,, and Gatrell, A. C. 1988. The geog- raphy of spina bifida in England and Wales. Transactions,Institute of British Geographersn.s. 13:288-302. Lovett, A,, I. D. Whyte, and K. A. Whyte. 1985. Poisson regression analysis and migration fields: The example of the apprenticeship records of Edinburgh in the seventeenthand eighteenth centuries. Transactions,Institute of British Geographers n.s. 10:317-32. McCullagh, P., and J. A. Nelder. 1983. Gener- alized Linear Models. London: Chapman and Hall. Nelder, J. A,, and R. W. M. Wedderburn. 1972. Generalized linearmodels.Journal,Royal Sta- tistical Society A 135:370-84. OBrien, L.zyxw G: 1983. Generalised linear mo- delling using the GLIM system. Area 15:327- 35. OBrien, L. G. 1987. GLIM 3.77 (software re- view). The Professional Geographer 39229-30. Payne, C. D., ed. 1985. The GLIM System Release
- 9. 198 THEPROFESSIONAL zy GEOGRAPHER z 3.77: Manual. Oxford: Numerical Algorithms Group. Pickles, A. 1985. A n Introduction to Likelihood Analysis. Concepts and Techniques in Mod- ern Geography, No. 42. Norwich, UK: Geo Books. Senior, M. L. 1987.The establishment of family planning clinics in south-west Nigeria by 1970: Analyses using logit and Poisson regression. Area 19237-45. Twomey, J. 1986.Establishment migration: an analytical framework. Environment and Plan- ning zyxwvutsr A 18:967-79. Vincent, P. J., and J. M. Haworth. 1983.Poisson regression models of species abundance. Journal of Biogeography 10:153-60. Whyte, I. D., and K. A. Whyte. 1986. Patterns of migration of apprentices into Aberdeen and Inverness during the eighteenth and early nineteenth centuries. Scottish Geograph- ical Magazine 102:81-92. Wrigley, N. 1985. Categorical Data Analysis for Geographers and Environmental Scientists. Lon- don: Longman. ANDREWLOVETT (B.A., University College of Wales, Aberystwyth) is Teaching Fellow, Department of Ge- ography, University of Lancaster, Lancaster LA1 4YB, UK. His research interests are in medical geography and industrial linkage studies. ROBIN FLOWERDEW (Ph.D., Northwestern University) is Lecturer in Ge- ography, University of Lancaster. His research inter- ests are in migration modeling and geographical in- formation systems. zyxw ~~ OPEN FORUM Professional Geo rapher, zyxwvutsrqp 41(2). 1989, pp. 198-203 zyxwvuts 0 Copyright 1889 by Association of American Geographers THE HANFORD ENVIRONMENTAL DOSE RECONSTRUCTION PROJECT Richard Morrill University of Washrngfon A five year, $15 million study is un- derway to estimate radiation doses that people might have received from opera- tions at the Hanford, WA, nuclear reser- vation. The study is funded by the US Department of Energy (DOE),and the re- search is conducted by Battelle’s Pacific Northwest Laboratory, the current main research contractor at Hanford. The work is directed by an independent Technical Steering Panel (TSP)on which I serve as the demographer. Despite the intensely geographic nature of the problem, the possibility of a geographer as such as a panel member was never raised; we have a long way to go. This report provides some background on the issues surround- ing the Hanford controversy, with the hope that readers with expertise in the area or topic will contact me. The Hanford Reservation Hanford is one of the nation’s major nuclear production and research reserv- ations. As many as nine primary produc- tion reactors have existed at various times. Eight water-cooled reactors along the Co- lumbia River have been the source of large