AREAL DATA ANALYSIS. . More importantly, one can represent values that have no particular point locations at all, such as the population of R
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1 AEAL DATA ANALYSIS 1. Overvew of Areal Data Analyss The key dfference between areal data and contnuous data s bascally n terms of the form of the data tself. Whle contnuous data nvolves pont samples from a contnuous spatal dstrbuton (such as temperature readngs at varous pont locatons), areal data nvolves aggregated quanttes for each areal unt wthn some relevant spatal partton of a gven regon (such as census tracts wthn a cty, or countes wthn a state). Such dfferences are llustrated n Fgures 1.1 and 1.2 below. s 4 s 1 s 2 s c 1 c 3 c 2 c Fgure 1.1. Pont Samples Fgure 1.2. Areal Unts Here Fgure 1.1 shows four sample ponts, s, n regon,, (whch s qualtatvely the same as Fgure 1.1 of Part II ), and Fgure 1.2 represents a partton of regon,, nto four areal unts { 1, 2, 3, 4}. Such areal unts,, are often represented by approprate central locatons, c, such as major ctes, or geometrc centrods (to be defned below). 1 But the data values assocated wth these ponts represent summary measures for the areal unt as a whole. For example, rather than measurng the temperature at locaton, c, one could assgn (an estmate of) the average temperature over all ponts n areal unt. More mportantly, one can represent values that have no partcular pont locatons at all, such as the populaton of or the average ncome of all household unts n. The practcal sgnfcance of areal data for purposes of analyss s that most socoeconomc data comes n ths form. For example, whle ndvdual ncome data s generally regarded as propretary n nature, such data s often made publcally avalable n terms of averages (such as per capta ncome at the state or county level). More generally, most publcally avalable data (such as US Census data) s only of ths type. 2 1 Ths type of representaton n terms of pont locatons has led to the alternatve descrpton of areal data as lattce data, as for example n Cresse (1993, Secton 6.1). 2 There are exceptons however, such as the Center for Economc Studes (CES) run by the Census Bureau, whch allows restrcted access to ndvdual mcro data by qualfed researchers. ESE 502 III.1-1 Tony E. Smth
2 NOTEBOOK FO SPATIAL DATA ANALYSIS As n Parts I and II above, t s approprate to llustrate some of the key features of areal data n terms of specfc examples (agan drawn from [BG, Part D]). 1.1 Extensve versus Intensve Data epresentatons Areal data s most easly represented vsually n terms of choropleth maps, such as the chld mortalty data for each of the 167 Census Dstrcts n the cty of Auckland, New Zealand over the nne year perod from 1977 to 1985 (taken from [BG, pp.249, ]). Here we focus only on the populatons at rsk,.e., chldren under the age of 5 n each dstrct. Two possble representatons of ths data are shown n Fgures 1.3 and 1.4 below. CBD CBD 0 10 mles 0 10 mles Fgure 1.3. aw Populaton Data Fgure 1.4. Populaton Densty Data The representaton n Fgure 1.3, whch shows the actual number of chldren under 5 n each dstrct, appears to suggest that the most substantal concentraton of these chldren les n dstrcts to the southeast of the Central Busness Dstrct (CBD). But t s mportant to note that census dstrcts are specfcally desgned to nclude roughly the same populaton totals n each dstrct. So the smaller dstrcts around the CBD ndcate that populaton denstes are much hgher n ths area. An alternatve representaton of ths populaton n gven n Fgure 1.4, whch dsplays the densty of such chldren n each dstrct,.e., the number of chldren per square mle (approx.) Here t s clear that the most dense concentratons are precsely n the smallest dstrcts, ncludng the CBD. So ths representaton suggests (not surprsngly) that chldren under fve are n fact qute evenly spread throughout the populaton as a whole. ESE 502 III.1-2 Tony E. Smth
3 NOTEBOOK FO SPATIAL DATA ANALYSIS Ths example serves to underscore the fact that the dstrbuton of areal data s usually more accurately represented n terms of densty values. More generally, representatons n terms of actual data totals (such a populaton counts) are desgnated as extensve representatons of areal data, and representatons n terms of denstes (such as populaton denstes) are desgnated as ntensve representatons of areal data. The key dfference s that ntensve representatons allow more drect comparsons between values n each areal unt. For example, populaton per square mle has the same meanng everywhere, and s ndependent of the sze of each areal unt Spatal Pattern Analyss The above example also demonstrates that when ntensve data representatons are used, choropleth maps can serve to reveal meanngful patterns n areal data. In ths case, chldren under fve (and ndeed all people) are more concentrated around the CBD than n outlyng areas. But there are many more nterestng pattern examples that ths. One example of comparatve pattern analyss s provded the Chnese soco-economc data n [BG, p ]. Fgures 1.5 and 1.6 below show the shft n per capta gross domestc product (pcgdp) n the provnces of Chna from 1984 to (Note agan that ths data s n ntensve form, where per capta ndcates that the relevant unts of comparson here are ndvduals rather than square mles.) mles mles Fgure Per Capta GDP Fgure Per Capta GDP Here t s clear at a glance that coastal regon of Chna has been the hgh growth area. Statstcal analyss can of course be appled to confrm ths. But the key pont here s that vsual pattern analyss s a powerful heurstc tool for dscernng relatons that may not be mmedately evdent n the data tself. 3 For a more detaled dscusson of ntensve versus extensve data representatons, see the classc paper by Goodchld and Lam (1980). 4 To allow drect comparson, data on both maps has been normalzed to have unt maxmum values. ESE 502 III.1-3 Tony E. Smth
4 NOTEBOOK FO SPATIAL DATA ANALYSIS A second example s provded by the Irsh blood group data from [BG, p.253] for the 26 countes of Ere. From an hstorcal perspectve, there s strong reason to beleve that the Anglo-Norman colonzaton of Ireland n the 12 th Century had a lastng effect on the populaton composton. Fgure 1.7 below shows the estmated proporton of adults n each county wth blood group A n 1958 (where values ncrease from blue to red). Fgure 1.8 shows the orgnal colonzed area of Ere, known as the Pale. Snce blood group A s much more common among ndvduals wth Anglo-Norman hertage, a vsual comparson of these two fgures strongly suggests a contnued pattern of Anglo-Norman nfluence n the regon around the Pale. We shall later confrm these fndngs wth spatal regresson mles 0 50 mles Fgure 1.7. Blood Group A Percentages Fgure 1.8. Countes n the Pale 1.3 Spatal egresson Analyss A fnal example of areal data s provded by the Englsh Mortalty data from [BG, pp ]. Here the areal unts are the 190 Health Authorty Dstrcts throughout England, and the data used nvolve deaths from myocardal nfarctons among males (35-64), as shown n Fgure 1.9 below. Ths data s n standardzed rates, defned here to be the number of deaths n the perod dvded by the expected number of deaths durng that perod based on natonal averages. Such standardzed rates are qute typcal for medcal data, and help to dentfy those areas where death rates are much hgher than expected relatve to natonal averages. In partcular, the darkest areas on the map ndcate rates well above average. Whle such hgher rates may be nfluenced by many factors, the present analyss focuses on aspects of socal deprvaton as summarzed by the Jarman underprvleged areas score, or Jarman score (whch s a weghted average of factors ncludng levels of unemployment and overcrowdng). Ths measure for each Health ESE 502 III.1-4 Tony E. Smth
5 NOTEBOOK FO SPATIAL DATA ANALYSIS Authorty Dstrct s shown n Fgure 1.10, where darker areas here show hgher levels of socal deprvaton km km Fgure 1.9. Myocardal Infarctons Fgure Jarman Scores A vsual comparson of these two maps suggests that there may ndeed be some postve correlaton between these two patterns, especally n Northern England where the hghest levels of both death rates and socal deprvaton seem to occur. Ths relaton can be readly confrmed by a smple regresson of log Myocardal Infarcton (lnmi) rates on log Jarman scores (lnjaman). The results n Fgure 1.11 below show that there s ndeed a very strong relaton between the two. Parameter Estmates Term Estmate Std Error t ato Prob> t Intercept <.0001* lnjaman <.0001* Parameter Estmates Term Estmate Std Error t ato Prob> t Intercept es_nn <.0001* Fgure egresson esults Fgure NN-esdual Analyss ESE 502 III.1-5 Tony E. Smth
6 NOTEBOOK FO SPATIAL DATA ANALYSIS However, t s also clear from Fgures 1.9 and 1.10 above that there s a strong correlaton between both MI rates and Jarman scores n neghborng dstrcts. Moreover, snce t s hghly unlkely that the correlatons among Jarman scores could completely account for those among MI rates, one can expect there to be a strong spatal autocorrelaton among the regresson resduals. Ths s confrmed by the smple nearest-neghbor analyss of these regresson resduals shown n Fgure 1.12 (where nearest neghbors are here defned wth respect to centrod dstances between dstrcts). In fact the correlaton among these resduals s even stronger than that between lnmi and lnjaman (as can be seen by comparng the t-ratos of es_nn versus lnjamon). Whle much of ths resdual correlaton could n prncple be removed by ncludng a range of other relevant explanatory varables, t s qute apparent from Fgure 1.9 that sgnfcant autocorrelaton wll reman. Wth these observatons, our ultmate objectve s to extend ths smple nearest-neghbor analyss to a broader and more rgorous framework for spatal autocorrelaton analyses of areal data. But to do so, we must frst address the dffcult ssue of defnng approprate measures of dstance between areal unts. ESE 502 III.1-6 Tony E. Smth
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