King 1961 a Multivariate Analysis of the Spacing of Urban Settlements in the United States

8/3/2019 King 1961 a Multivariate Analysis of the Spacing of Urban Settlements in the United States

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A Multivariate Analysis of the Spacing of Urban Settlements in the United StatesAuthor(s): Leslie J. KingReviewed work(s):Source: Annals of the Association of American Geographers, Vol. 51, No. 2 (Jun., 1961), pp.222-233Published by: Taylor & Francis, Ltd. on behalf of the Association of American GeographersStable URL: http://www.jstor.org/stable/2561348 .

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A MULTIVARIATE ANALYSIS OF THE SPACING OF URBANSETTLEMENTS IN THE UNITED STATESLESLIE J. KING

University of Canterbury, New ZealandA ubiquitous feature of the American settle-ment pattern is the urban agglomeration,be it a hamlet, village, town, or city. Whilethese population aggregations display a widerange of internal characteristics, from a func-tional point of view they frequently have thesame raison d'etre, that is they provide someform of service, however simple it may be, tothe dispersed population residing in the sur-rounding areas.Insofar as these settlement clusters areunevenly distributed over the surface of theearth, it behooves the geographer to attemptsome explanation of their distribution pattern.The most outstanding contribution yet madetowards the development of a general theoryconcerning the distribution, size, and numberof urban places within a large area, is that ofChristaller. As outlined in his work, Diezentralen Orte in Sfiddeutschland, publishedin 1933,1 Christaller's model, generally knownas the "central place model," is derived fromcertain basic premises within a framework ofseveral limiting assumptions. These premisesand assumptions have already been discussedby several authors,2 and for this reason noattempt is made to treat them in this context.However, many of the concepts presented inChristaller's work, for example, the notions ofa central place, centrality, complementary re-gion, central goods and services, and the rangeof a good, are continually stressed throughoutthis study.That Christaller's model is inadequate asan approximation of reality is a point whichhas been emphasized by several writers.3

1 Walter Christaller, Die zentralen Orte in Siid-deutschland (Jena, 1933). Trans. by C. W. Baskinin, A Critique and Translationof Walter Christaller'sDie zentralen Orte in Siiddeutschland (Unpub. Ph.D.dissertation, Dept. of Economics, University of Vir-ginia, 1957).2 See for example: B. J. L. Berry and W. L. Gar-rison, "A Note on Central Place Theory and theRange of a Good," Economic Geography, Vol. 34(1958), pp. 304-312.3 For a pointed criticism along these lines see Egon

Bergel, Urban Sociology (New York: McGraw-Hill,1955), p. 64.

These inadequacies stem not only from thevery limiting nature of the assumptions, butalso from Christaller's use of discrete popula-tion-size groups with which there are associ-ated "typical" minimum distances betweenurban places. Thomas has recently drawn at-tention to a number of research findings whichsuggest that such discrete groupings do notappear to be necessary for any explanationof reality.4 While the studies mentioned byThomas have ignored any consideration of thespacing of settlements, their findings, by con-trast, are extremely relevant to the latter prob-lem. For if discrete population-size groups areunnecessary, then a different basis must befound on which the distances separating settle-ments can be derived. If the values for popu-lation-size appear to be unimodally distributedwhen they are formed into an array, then itmay well be that the distances between urbanplaces are similarly distributed. However, inthe quarter of a century or so following thepublication of Christaller's work, a majorityof studies concerned with the spacing of urbansettlements have continued to emphasize theexistence of discrete population-size groupsand average distances.5 In addition, little ef-fort has been made towards incorporating ad-ditional variables other than the size of popu-lation into the analyses.The present study was undertaken with theexpressed need in mind for discovering thenature of the relationships between the spac-ing of towns on the one hand, and variousphysical, social, and economic factors on theother. The conclusions drawn from this anal-ysis should contribute towards a firmer foun-

4Edwin N. Thomas, "Towardsan Expanded Cen-tral Place Model" (Abstract of paper read at theannual meeting of the Association of American Geo-graphers,Dallas, April 17-21, 1960).5 See for example: John E. Brush, "The Hierarchyof Central Places in Southwestern Wisconsin," Geo-graphical Review, Vol. 43(1953), pp. 380-402. Au-gust Lbsch, Die rdumliche Ordnung der Wirtschaft(Jena, 1944). Trans. by W. H. Woglom and W. F.Stolper as, The Economics of Location (New Haven,1954).

222


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1961 SPACING OF URBAN SETTLEMENTS IN THE UNITED STATES 223

LOCATION OF SAMPLE TOWNS

FIGURE 1

dation for the construction of a general theoryconcerning the distribution pattern of townsover the face of the earth.Specifically, with reference to a randomsample of two hundred urban settlements, thisstudy will attempt to explain the areal varia-tion throughout the United States in the dis-tances separating the towns from their nearestneighbors of the same population-size. Theuse of simple and multiple regression analyseswill allow for consideration of the effects,either independently or simultaneously, of se-lected physical, social, and economic variablesupon this set of distances.THE SAMPLE TOWNS AND THE

DEPENDENT VARIABLESA sample of two hundred towns was drawn

randomly from the list of Incorporated Placesand Unincorporated Places of over 1,000 in-habitants, contained in the 1950 United States

Census of Population.6 The towns were chosenwith the aid of a random-numbers table afterthe members of the entire universe had beennumbered consecutively. The settlementswhich were thus selected (Fig. 1), rangedin population-size from 5 (Slaughter Beach,Delaware) to over 467,000 (Seattle, Wash-ington).If, as an initial assumption, it is acceptedthat each of these sample towns is a centralplace in the sense that it exists to serve theinhabitants of some surrounding areas, thenin the light of "central place theory" the rela-tive location of each of these towns can beviewed as a function of the extent to which ithas been able to exist in proximity to the near-est neighboring town of the same population-6 U. S. Bureau of the Census, Seventeenth Decen-nial Census of the United States: Census of popula-tion, 1950, Vol. 1, "Number of Inhabitants" (Wash-ington: Govt. Printing Office, 1952), Table 7.


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224 LESLIE J.KING June

FRACTILEIAGRAMSF THEINDEPENDENTARIABLESLOGPOPULATION SIZE LOG.LOG VER. SIZE OF FARM LOG:DENSITYPER SQ. MILERURAL FARM POPULATION5-0 g g | ' X 60 W | , , | , 18 |4-0- -48- - 1463.0- -*36 - 1-42-0- -24- - 1-2-0 , - 12 .0 l

0 l I ~ I I ? 0 -'I I I f I I I I I I I1 3 1020 40 60 80 95 995 1 3 1020 40 60 80 95 995 3 1020 4060 80 95 99.5PERCENTILE PERCENTILE PERCENTILE

ARCSIN:PER CENT OF EMPLOYED LOG.LOG;ENSITY POPULATION LOG:VALUE LAND ANDIN MANUFACTURING BUILDINGS PER ACRE50- r7 .40 - -58- - 26 -30 - -46- - 22 -20 - - 34 - 18-10 - -*22- - 14 -

I 3 1020 4060 80 95 995 I 3 1020 4060 80 95 995 1 3 1020 4060 80 95 995PERCENTILE PERCENTILE PERCENTILE

FiGuRE

size and functional complexity as itself. Im-plicit in the latter part of this statement isalso the assumption of a positive relationshipbetween the population-size of a town and thenumber and complexity of goods and servicesoffered in it. That these initial assumptionsmay be unrealistic need cause no great con-cern since the discrepancies between the modeland reality are expected to show up as largeresidual values in the subsequent regressionand correlation analysis.Areal variations in the distances separatingtowns of the same population-size, should re-flect not only the population-size and func-tional complexity of the towns concerned, butalso the influence of any factors which affectdemand and the range of the goods and ser-vices offered by the towns.

The problem of providing an operationaldefinition for the variable distance betweentwo towns of the samepopulation-size,has al-

ready been considered in detail by Thomas.7With respect to the i th city, the nearest neigh-bor of the same population-size is definedas "that place which is located spatially near-est to the sample city and has a populationdiffering from the population of the samplecity only by chance." A more formal defini-tion in terms of fiducial limits is given asSt- xEi C Ni C Si + xEi, where Si is the popu-lation of the sample town, Nj the populationof the nearest neighbor, Et is a random errorvalue, and x is the standard abscissa of thenormal curve associated with a desired con-fidence level. The definition is not withoutits weaknesses. As Thomas points out, thefact that some normalizing transformation ofthe population-size data is frequently required,is one such weakness. Furthermore, the modelis essentially a static one. Two urban placesmay be of the same population-size in 1950,

Thomas, op. cit.


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1961 SPACING OF URBAN SETTLEMENTS IN THE UNITED STATES 225but when population changes through timeare taken into account, then it is likely thatthis relationship between the two towns willvary considerably. These weaknesses notwith-standing, Thomas appears to have provideda reasonably precise and logically acceptabledefinition of the notion of same population-size. The use of this stochastic model, how-ever, is conditional upon the fulfillment of thebasic assumptions of a random sample drawnfrom a normally distributed population withrespect to population-size. It has already beennoted that the sample of towns was randomlychosen. The use of a fractile diagram, as atest for normality,8 revealed that the distribu-tion of the population-size data was consider-ably skewed in character and a logarithmictransformation was necessary to ensure nor-mality at the ninety-five percent confidencelevel (Fig. 2).Having established the population-size in-tervals, and subsequently identified the near-est neighbor of the same population-size foreach sample town, then the set of distanceswhich is the dependent variable could be de-rived. As was the case with population-size,the distribution of these distances was mark-edly skewed to the right. Since the level ofconfidence associated with the subsequent sta-tistical analyses will be verified by tests ofsignificance which presuppose samples fromnormally distributed populations, it is desir-able that all the variables used in this study betransformed when necessary to ensure a nor-mal distribution.9 In the case of the depen-dent variable, a logarithmic transformationappeared adequate (Fig. 3).

THE FORMULATION AND TESTING OF ANINITIAL SET OF HYPOTHESES

In formulating the following hypothesesconcerning the spatial association of the de-8A. Hald, Statistical Theory with EngineeringApplications(New York: John Wiley and Sons, 1952),pp. 119-158.9The various transformationsused in this studyare shown in Figure 2. For discussions of data trans-formation see: J. Aitchison and J. A. C. Brown,The LognormalDistribution (Cambridge: UniversityPress, 1957). Also: Edwin N. Thomas, An Analysisof the Areal AssociationsBetween Population Growth

and Selected Factors Within Outlying Cities of theChicago Urbanized Area. (Unpub. Ph.D. dissertation,Department of Geography, Northwestern University,1958).

pendent variable and selected independentvariables, it must be emphasized that the vari-ables chosen are the ones which appear to belogically the most relevant. The degree towhich they are spatially associated with thedependent variable and thereby provide a sat-isfactory explanation of the areal variation inthe magnitude of the distances separatingtowns of the same population-size, will be re-vealed by the subsequent statistical analyses.The amount of unexplained variation in the de-pendent variable, will be indicative not only ofthe aptness of the operational definitions usedin this study, but also of the relative impor-tance of the variables chosen and those whichhave been neglected. In addition, each hy-pothesis is formulated within a probabilityframework, although this may not always bemade explicit. However, the acceptance orrejection of any hypothesis is with respect tosome predetermined level of probability orconfidence.First, the distance between two towns of thesame population-size should reflect the actualpopulation-size of the two towns in question.10That is to say, the distance separating a largecity from its nearest neighbor of the same sizewill be much greater than the correspondingdistance between two small hamlets, for thelarger urban settlement offers more specializedservices which have a far greater range thando the more basic services contained in thesmaller centers. The greater the range of theservices offered by a town, the farther apartit will be located from the nearest town offer-ing similar services.Second, in the areas of more intensive farm-ing, where there are relatively higher inputsper unit area of the factors of production, thedemand for central services should be greaterthan in the areas where these inputs are on amuch lower level. Consequently, if the avail-ability of transportation facilities is consideredrelatively constant throughout the country, itis expected that towns should be located closertogether in the areas of intensive farming, andconversely, farther apart in the areas of more

10 Population data obtained from U. S. Bureau ofthe Census, op. cit. It should be noted at this pointthat the distances mentioned in this study are the air-line distances between the approximate geographiccenters of the towns, as measuredon the U. S. Bureauof Census Maps (Washington: Govt. Printing Office,1940).


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226 LESLIE J.KING June

FRACTILEDIAGRAMOF THE DEPENDENTVARIABLELOG:DISTANCE O NEARESTNEIGHBOROF SAME SIZE3.3

2.8 /

2.3-

0.8 I1 3 10 20 30 40 50 60 70 80 90 95 98 99.5CUMULATIVE ER CENT

FIGURE 3

extensive farming. For the purposes of thisinvestigation the scale of farming operationsis defined in terms of the average size of farmin 1950 for the county in which the sampletown is located.11 A positive relationship be-tween farm size and distance is hypothesized.At this point attention should be drawn to thefact that county averages and density figuresare used throughout this study as dimension-less terms indicative of the character of thetrade area of the town concerned. It has al-ready been suggested that the distance be-tween two towns of the same population-sizeshould in part reflect the character of the in-tervening area which they serve.Third, as a corollary of the preceding hy-pothesis it is expected that the density of ruralfarm population will be inversely associated

11 Data obtained from U. S. Bureau of the Census,1954 Census of Agriculture (Washington, D. C.:Govt. Printing Office, 1956).

with distance. Christaller maintained that agreater rural farm population density wouldencourage a higher consumption of goods, agreater degree of labor specialization, and agreater use of capital necessary for centralgoods.12 Therefore, where rural farm popula-tion density is high, the system of service cen-ters should be more strongly organized andthe distance between towns shorter than inareas characterized by a low density of ruralfarm population.'3Fourth, in the discussion so far, towns havebeen considered purely as service centers.However, there are many, perhaps even a ma-jority of urban settlements which serve a dif-ferent function. For those towns which exist

12 Christaller, op. cit., pp. 148-151.13 U. S. Bureau of the Census, Seventeenth Decen-nial Census of the United States: Census of Popula-tion, 1950, Vol. 2 (Washington, D. C.: Govt. PrintingOffice, 1952).


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1961 SPACING OF URBAN SETTLEMENTS IN THE UNITED STATES 227primarily as manufacturing centers, an addi-tional set of assumptions must now be intro-duced. These involve factors known in generallocation theory as agglomeration economies,14and the effects of these might well be evi-denced in a closer spacing of larger manufac-turing centers. Since published data on manu-facturing is unavailable for towns of less than2,500 population, which constitute two-thirdsof the present sample, reliance was placedupon county data,15 and the importance ofthe agglomerative factors was defined in termsof the percentage of the employed populationengaged in manufacturing in the county for1950. The specific hypothesis is that wherethe percentage is high, the towns will be lo-cated closer together.Fifth, of the two hundred towns selectedfor this study, thirty-four are located in Stand-ard Metropolitan Areas, but are not in them-selves central cities in terms of the Censusdefinitions.16 The distances separating thesetowns from their nearest neighbors of thesame size are often very small, as is the casefor example with McCook (Chicago), CapitolHeights (Washington, D. C.), and WestBrookfield (Worcester, Massachusetts). Foreach of these towns the distance measurementlies at least two standard deviation units be-low the sample mean. In addition, many ofthese towns are quite large in population-sizeand it appears likely that the positive relation-ship between population-size and distancewhich has already been hypothesized, may notexist in these highly urbanized areas. It ispertinent, therefore, to consider a possible re-lationship between total population densityand the spacing of towns of the same popula-tion-size.17

Sixth, in areas of higher agricultural pro-ductivity there is presumably a higher levelof rural purchasing power which should bereflected in a greater consumption and de-mand for central goods and services. Chris-taller insisted that if such is the case thenthe system of central places should be morestrongly developed and as a result towns14Walter Isard, Location and Space Economy (NewYork: John Wiley and Sons, 1956), pp. 172-188.15 U. S. Bureau of the Census, op. cit.16 That is to say they are not in terms of the Censusdefinitions, Principal central cities of 50,000 inhab-itants or greater, or Central cities of 25,000 or more.17 U. S. Bureau of the Census, op. cit.

TABLE 1Coefficient "b" ValuesScirple of for multipleVariable correlation determi- regressioncoefficient nation analysis

Population of town .15* .02 .13*Average size of farm .33* .10 -.02Density ruralfarm pop. -.24* .05 -.14Percent in manufac-turing -.34* .11 -.05Density total popu-lation -.41* .17 -.12*Value of land andbldgs. per acre -.30* .09 -.10* Significant at the ninety-five percent confidence level.

should be more closely spaced.18 The valueof land and buildings per acre by county hasbeen suggested as one index of agriculturalproductivitys and, notwithstanding the weak-nesses associated with this index, it is hy-pothesized that in areas characterized by highvalues of land and buildings per acre townswill be spaced closer together than in areasof lower agricultural productivity.20A simple regression analysis in which theregression of the dependent variable on eachindependent variable is considered withoutreference to the effects of the other independ-ent variables, substantiated each of the abovehypotheses. That is to say, there was a statis-tically significant relationship between eachindependent variable and the dependent vari-able (Table 1).However, the coefficient of determination(r2) is in each case comparatively small. Whilethe variable, density of total population,ac-counts for seventeen percent of the variationin the dependent variable, only in the casesof three of the remaining variables is there asmuch as ten percent explained variation. Theseresults would seem to support the tentativeconclusion that the spacing of towns is a verycomplex phenomenon, and that an explanationof it will involve consideration of a greaternumber of variables than has so far been thecase. The multiple regression analysis addsgreater weight to this conclusion, for when

18 Christaller,op. cit., p. 150.19 N. E. Salisbury, "Agricultural Productivity and

the Physical Resource Base of Iowa," Iowa BusinessDigest, Vol. 31(1960), pp. 27-31.20 U. S. Bureau of the Census, 1954 Census of Agri-culture, op. cit., Chap. B., Table 1.


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228 LESLIEJ. KING JuneTABLE 2.-MATRIX OF INTERCORRELATIONS FOR

THE SrMPLE REGRESSION ANALYSIS

Xi X2 X3 X4 X5 Xo

Xi Population oftown ... -.13 -.03 .15 .21 .16X2 Average sizeof farm ... -.53 -.49 -.78 -.60X3 Density ruralfarm population .25 .45 .24X4 Percent in

manufacturing ... .65 .30X5 Density of total

population ... .71X6 Value of land andbuildings per acre

all six independent variables are consideredsimultaneously, only twenty-five percent ofthe variation in the dependent variable couldbe accounted for. While this represents some-what of an improvement over the results ofthe simple regression analysis it is neverthe-less a disappointing result. Furthermore, inthe multiple regression analysis only two ofthe variables appear to contribute significantlyin explaining the variation in the dependentvariable (Table 1, Column 4). That the re-maining four variables are not statisticallysignificant, appears to reflect in part the com-paratively high intercorrelations between eachof these variables and the density of total pop-ulation (Table 2).The analysis henceforth, can proceed ineither of two directions. On the one hand,an expected value of the dependent variablecould be derived for each sample city by theuse of the regression equation, Y= 28.6 + .13X1-.02X2-.14X3-.05X4-.12X5- .10X6. The nu-merical difference between this expected valueand the observed value is termed the residual,and it provides for each city a measure of howclosely the regression estimate fits the ob-served value. On the basis of a map of theseresiduals and knowledge pertaining to someof the deviant cases, new hypotheses couldbe formulated and additional variables as ameans of testing these hypotheses could thenbe incorporated into the analysis. Such anapproach, with the addition and deletion ofvariables, the formulation and testing of newhypotheses, would be pursued until an accept-able level of explanation was achieved. Alter-natively at this point an attempt might bemade to assess, by means of covariance and

tests of significance, the importance of certainregionalizations and classificatory groupingsin explaining the variation in the dependentvariable. These regional divisions may bemade with reference to some factor which isqualitative rather than quantitative in nature,or they might represent multivariate regions inwhich numerous subtle forces are known to beoperating. It is along the lines of this secondapproach that the remainder of this study isoriented. The following hypotheses are out-lined as the framework within which this sub-sequent analysis is set:First, there are many towns included withinthe sample which are obviously not true cen-tral places in the sense of existing primarily asservice centers for surrounding rural areas(Fig. 4). Christaller designated such townsas "pointly-bound" places.21 It is anticipatedthat a grouping into central and non-centralplaces along these lines will be significant inhelping to explain the variation in the depend-ent variable.Second, as yet no explicit recognition hasbeen accorded to physical factors such as re-lief, elevation, or roughness. Intuition alonesuggests that these factors may be critical inthe location of any urban settlement. How-ever, the problems of isolating and then meas-uring the relevant variables have precludedtheir consideration. A regional classificationon the basis of slope may reveal some interest-ing conclusions. For the purposes of this studyHammond's classification of the terrain typesof North America was used.22 The system wasgeneralized in that Hammond's eight maintypes were combined into two. The first in-cludes those areas which have at least fiftypercent of their area in "near level land,"which is defined as land characterized byslopes of less than eight percent, while thesecond group includes those areas with lowerpercentages of "near level land."Third, although variables relating to someaspects of agriculture have already been in-cluded within the analysis, it is hypothesizedthat a classification of the towns on the basisof the type of farming area in which they arelocated, may reveal some interesting facets of

21 Christaller,op cit., p. 116.22 Edwin H. Hammond, "Small-Scale ContinentalLandform Maps," Annals, Association of AmericanGeographers,Vol. 44(1954), pp. 33-42.


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1961 SPACING OF URBAN SETTLEMENTS IN THE UNITED STATES 229

NON-CENTRALLACES

0 ~ ~ ~ ~ ~ ~~00~~~~~ O(5

0 ~~~~~~~00 00~~~~~~~~~~

0~~~~~~0~~~~~> o~~~~~~~~o

* TOWNS LOCATEDWITHINA STANDARDMETROPOLITANAREA0 OTHER NON-CENTRALPLACES

FIGURE 4

the problem. Throughout the United States,farming has developed in response to numer-ous physical, biological, social, and economicforces which are explicitly recognized in theUnited States Department of Agriculture'sclassification of generalized farming types.23However, since some of the farming types in-cluded within this scheme contained only oneor two if any, of the sample towns, a simplifica-tion was made by combining several of thetypes according to the general scale on whichthe type of farming is practiced. Thus GroupI includes the extensive farming types, "Wheatand Small Grains" and "Range Livestock."Group II on the other hand, is made up ofthe more specialized farming types including"Fruit, Truck, and Mixed Farming," "Cotton,""Tobacco and General Farming," and "Special

23U. S. Dept. of Agriculture, "Generalized Typesof Farming in the United States," Agric. InformationBull. No. 3 (Washington, D. C.: Govt. Printing Of-fice, 1953).

Crops and General Farming." Group III is the"General Farming" type, while Group IV rep-resents the "Feed Grains and Livestock" com-plex. Finally, the "Dairy"areas are consideredas Group V.THE COVARIANCE MODEL AND THE

GROUPINGS EXAMINEDAssuming that the sample units have beengrouped or regionalized on some logical basis,then it is of interest to know whether or notthe introduction of this subdivision accountsfor a significant amount of the variation in thedependent variable, apart from the regressionof this variable on the independent variables.The importance of regionalizations or clas-sificatory groupings in this respect can beestablished by an analysis of covariance, pro-viding the assumptions underlying this model

are fulfilled. The assumptions are firstly thatthe variance of the dependent variable doesnot differ significantly from group to group,


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230 LESLIEJ. KING JuneTABLE 3

Coefficient ofcorrelation' Central places Non-central placesY.123456 .51* .65*Y.X1 .29* .08Y.X2 .27* .36*Y.X3 -.17* -.39*Y.X4 -.32* -.35*Y.X5 -.32* -.60*Y.X6 -.26* -.33*

1 In this table the dependent variable is designated by Y,while the independent variables are listed in the same orderas in Table 1. (*Significant at the ninety-five percent con-fidence level.)

and secondly that the subgroup regressionlines are parallel. When these assumptionsappear unwarranted, then tests of the statis-tical significance of the difference betweencoefficients of correlation may be employedto determine the relevance of the subdivision.24

Central and Non-central PlacesAn analysis of covariance was precluded inthis case by unequal variances. However, theresults of the regression analysis for eachgroup were significant. In the simple regres-sion analysis, for example, five of the inde-

pendent variables showed up as being signifi-cantly related to the dependent variable in bothgroups. The exception was the variable, pop-ulation-size of the sample town, which did notappear significant in the group of non-centralplaces (Table 3). In the group of centralplaces the most closely related variable wasthe percent of the employed engaged in manu-facturing, and, as was the case for the wholesample, the relationship with distance was aninverse one. However, the amount of ex-plained variation accounted for by this manu-facturing variable was only ten percent. Inthe group of non-central places, on the otherhand, the variable density of total populationexplained almost thirty-seven percent of thevariation in the distances separating towns of

24For discussions of the regional problem and theuse of covariance and tests of statistical significance inthis regard, see: Donald J. Bogue and Dorothy L.Harris, "Comparative Population and Urban Researchvia Multiple Regression and Covariance Analysis,"Studies in Population Distribution: No. 8 (Oxford,Ohio: Scripps Foundation, 1954). Also Leslie J.King, The Spacing of Urban Places in the UnitedStates (Unpub. Ph.D. dissertation, Department ofGeography, State University of Iowa, 1960).

TABLE 4.-BETA VALUES FOR THE INDEPENDENTVARIABLES SIGNIFICANT IN THE MULTIPLE RE-

GRESSION ANALYSES FOR THE SUBGROUPS

c -O - _ e

~~~~~~~~~~~~~~~~~~~~~~~~~X1 .32 - .50 - - .52 -

X2 - - .71 - - - .66X3 - - - - .26 - -X4 .24 - .60 .41 - - -

Xi 2- .6 .76 - - - .83X6 - - -

the same population-size. The direction of thisrelationship was in agreement with that forthe sample as a whole, but in this case thelevel of explained variation was much higherthan had previously been the case.The coefficients of multiple correlation forthe two subgroups did not differ significantlyeither with respect to the multiple correlationcoefficient for the complete sample or withrespect to one another, and therefore thegrouping appeared to have little significance.However, in the case of the non-central placesthe amount of explained variation was almostforty-two percent which represents a consider-able improvement on the level achieved forthe sample as a whole. At the same time itshould be noted that only one variable, densityof total population, showed up as being sig-nificant in the multiple regression analysis forthis subgroup (Table 4), and it will be re-membered that this same variable alone ac-counted for as much as thirty-seven percentof the variation in the dependent variable inthe simple regression analysis. For the groupof central places two variables were significantin the multiple analysis, namely, the popula-tion-size of the sample town, and the percentof the employed engaged in manufacturing,and the first of the two appeared to be themore important in explaining the variation inthe dependent variable (Table 4).

Physical Slope RegionsIn contrast to the preceding subdivision, theregionalization on the basis of slope could

be evaluated by means of an analysis of co-variance since the assumptions of equal vari-ance and parallel regression planes were ful-


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1961 SPACING OF URBAN SETTLEMENTS IN THE UNITED STATES 231TABLE 5.-ANALYSIS OF COVARIANCE

Source of Variance of Y Errors of estimatevariation df Jy2 Mean Sq. df Sum of Sq. Mean Sq.Total 199 2794 .25 193 2092.02Subgrp.means 1 41 41Withinsubgrps. 198 2753 14 .32 192 1871.34 9.74For test of the significanceofthe adjustedsubgroupmeans 1 220.68 220.68

Test for Parallel Regressions: F = 41/14 = 2.9. Not sig-nificant.Test for Adjusted Means: F = 220.68/9.74 = 22.63. Sig-nificant.ryy =.57

filled.25 Formal evidence as to the statisticalsignificance of the regional grouping is pre-sented in Table 5.The regionalization appeared significant andthe amount of explained variation in the de-pendent variable was increased from twenty-five to thirty-two percent.26Types of Farming

While the assumption of equal varianceswas fulfilled in this case, the fact that the re-gression planes were not parallel precludedany meaningful analysis of covariance. How-ever, some significant information was forth-coming from an examination of the regressionanalyses for the subgroups.The simple regression analysis emphasizedthe relative importance of different variablesin the different subgroups (Table 6). Forexample, in the areas of extensive farming(Group 1), over forty percent of the variationin the dependent variable could be attributedto the comparatively close relationship be-tween distance and the population-size of thesample town. In addition, in the same areasthe density of rural farm population accountedfor almost twenty percent of the variation inthe dependent variable. In the areas of spe-cialized farming (Group II), only one variablethe percent of the employed engaged in man-ufacturing appeared significantly related to dis-tance and even then the amount of explained

25 The tests of statistical significance used in thisanalysis may be found in: George W. Snedecor, Sta-tistical Methods (Ames: Iowa State College Press,1956), pp. 316-319.26 See Bogue and Harris, op. cit., pp. 69-71.

TABLE 6Coefficient of Groupcorrelation I II III IV VY.123456 .81* .44 .81* .58* .60*YX1 .65* .11 -.27 .47* .21YX2 -.01 .24 .49* .17 .01YX3 -.44* -.15 .16 -.24 .01YX4 .39 -.30* -.50* -.11 -.36*YX5 -.18 -.22 -.62* -.22 -.30*YX6 .00 -.22 -.74* -.16 -.19

* Significant at the ninety-five percent level.

variation accounted for by this variable wasvery low (r2 -.09). By contrast, as many asfour variables appeared significant in the gen-eral farming region (Group III), and of thesethe value of land and buildings per acre ac-counted for as much as fifty-five percent ofthe variation in the dependent variable forthis subgroup. In the areas of feed grain andlivestock economy the distance separating twotowns of the same population-size again ap-peared to be some function of the population-size of the sample town, this being the onlyindependent variable which in this groupshowed up as being significantly related to thedependent variable. Finally, in the dairy re-gion both the density of total population andthe percent of the employed engaged in manu-facturing were significantly related to distancein an inverse direction, but the level of ex-plained variation in either case was not veryhigh (r2 = .09 and .13 respectively).This pattern in which certain relationshipsappear to be significant in one region but notin another was emphasized by the results ofthe multiple regression analysis. In only oneof the groups, that of specialized farming, wasthe coefficient of multiple correlation insig-nificant. In the remaining groups, by con-trast, the multiple relationship between dis-tance and the six independent variables wasoften very close, and in the cases of Groups Iand III the coefficient of multiple correlationdiffered significantly from that of the sampleas a whole, thereby suggesting that this clas-sification based upon farming types is of somesignificance in helping to explain the arealvariation in the distances separating towns ofthe same population-size. For Groups I andIII the level of explained variation was in eachcase higher than sixty-five percent. However,it was again apparent from Table 4, that not


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232 LESLIEJ.KING Juneall of the variables contributed significantly tothe multiple regression within each subgroup.Also, it is important to note that in some casesa variable which was not significant in thesimple regression analysis proved to be sig-nificant in the multiple analysis. Such was thecase, for example, with the variable, density ofrural farm population, in the general farminggroup. This emphasizes the fact that manygeographic problems are extremely complexin nature and the holding of certain factorsconstant in the multiple regression analysismay reveal important relationships which asimple regression analysis may in fact conceal.

CONCLUSIONWhen an overall appraisal is made, the pre-

ceding analysis appears most inadequate as anexplanation of the areal variation in the dis-tances separating towns from their nearestneighbors of the same population-size. Thisis emphasized by the fact that when all sixindependent variables were considered simul-taneously, only two proved to be significantin contributing to an explanation of the varia-tion in the dependent variable, and even thenthese two could account for no, more than onequarter of the total variation. Nor did the in-troduction of certain regionalizations and clas-sificatory groupings raise the overall level ofexplanation much higher, except on a levelwithin subgroups.At this point it is relevant to consider thepossible sources of these inadequacies. Theobvious need for the inclusion of additionalvariables into the analysis has already beenstressed. In addition, some of the operationaldefinitions used in this study may be inade-quate. For example, it was assumed that thepopulation-size of a town was a good index ofthe town's functional complexity, but such aclose positive relationship between these twofactors may be non-existent with respect to asample of towns drawn from the whole of theUnited States. The index of rural purchasingpower, namely, the value of land and build-ings per acre, also appears to have serious lim-itations and preferably some direct incomestatistic might be sought.The use of the county area as an approxi-mation of the trade area of a town is alsoburdened with difficulties. Not only do countyareas vary considerably in size-a fact thatcan be partially compensated for by the use

of dimensionless statistics-but in addition itis often the case that more than one sampletown of varying population-size are locatedin the same county area and are thereforeassigned the same data for many of the vari-ables.27The results of the regression analysis mustalso be supplemented by some acknowledg-ment of the chance factor in the location oftowns. The system of towns with which thisstudy deals cannot logically be conceived ofas a completely deterministic system in whichevery variable is in fact a function of theothers, but rather it represents a system inwhich chance elements are likely to be pres-ent. The relative importance of these sto-chastic elements, however, has yet to be de-termined. Nevertheless, such considerationsappear to be very pertinent when dealing withsuch a diverse area as the United States inwhich "pointly-bound"places may well be therule rather than the exception. In this sensethe distance separating two towns of the samepopulation-size in 1950 might be simply arandom phenomenon. On the other hand, ifallowance is made for the notion of a func-tional interdependence between a town andthe surrounding rural area, then it is apparentthat more detailed information is requiredconcerning such topics as the actual extent ofa town's trade area and the nature of con-sumer trip behavior. Also, as far as the rela-tive location of any town is concerned, thereis a need for greater insight into the numberand size of the towns which were already inexistence within an area at the time that thetown in question came into being. Further-more, the fact that the operational definitionof same population-size which is employed inthis study is essentially a static one emphasizesthe need for projecting the study back intothe past.These are but a few of the questions whichmight be raised concerning the problem of

27 Such was the case for example, with the townsof Ossian and Spillville in Winneshiek County, Iowa.The former with a population of 804 would neces-sarily have a larger trade area in accordance with theassumptions underlying this study, than would Spill-ville with a population of only 363. However, in eachcase Winneshiek County alone was accepted as anapproximation to the character of their respective tradeareas.


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1961 SPACING OF URBAN SETTLEMENTS IN THE UNITED STATES 233explaining the spatial distribution of urbanplaces. Despite its many limitations the pres-ent study has contributed some understandingof this problem, in that it has demonstratedthe existence of meaningful relationships be-

tween distance and certain independent vari-ables within the universe of the United States,while at the same time it has focussed atten-tion upon the meagerness of present under-standing in this very same area.

Documents

King 1961 a Multivariate Analysis of the Spacing of Urban Settlements in the United States