TWO MODELS FOR THE ‘INFERENTIAI. ANALYSIS OF CENTRAL PLACE PATTERNS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY Clifford E. Tiedemann 1916.6 This is to certify that the thesis entitled TWO MODELS FOR THE INFERENTIAL ANALYSIS OF CENTRAL PLACE. PATTERNS presented by Clifford E. Tiedemann has been accepted towards fulfillment of the requirements for kgé/WX/flm Major professor /; ‘ ‘ 3 [ Damcggq 1. HQ 0-169 ABSTRACT TWO MODELS FOR THE INFERENTIAL ANALYSIS OF CENTRAL PLACE PATTERNS by Clifford E. Tiedemann Walter Christaller's central place theory has provided geographers with a logical construct which describes and explains the influences of relative location on the distri- butional patterns of cities. However, the theoretical statements by Christaller are predicated on a set of assump— tions and rules which, it has been stated, are seldom found to be coincident in the observable world. Thus, assessments of real—world spatial patterns of agglomerated settlements in the light of this theory are extremely difficult. Many studies have been made in which analyses of exist- ing settlement patterns are based on selected aspects of central place theory. Although it must be admitted that the spacing of cities is an extremely complex object of study, these efforts are viewed as being partially unsatis— fying, since they all fail to properly account for the dis- tance -- population size relationship. It appears, then, a technique to be used in the evaluation of this association can be of some value in this field of endeavor. Clifford E. Tiedemann Using a stochastic process to define equality among populations of cities, two descriptive models are proposed which can account for this important factor in the spacing of settlements. The two models, one a simple regression and the second a simple correlation, indicate the nature of this bivariate relationship. The independent variable LJ_7l is set -ement population, which serves as an indicator of (T D“ e functional comp exity of each central place. The (—4.1 repenr ,ent variasle is standardized distance, a phenomenon .. which possesses characteristics of particular value in the analvsi of the distribution of central places. U) The parameters of these two models, the y—intercept, the regression coefficient, and the correlation coefficient, each have unique numerical values which are defined by the deterministic character of central place theory. Calculated values for each of these parameters can be compared with the defined figures using standard tests of significance based on the normal and "t" distributions. Relying on the outcome of such tests, statements of confidence can be made concerning the similarity, or lack thereof, between an observed settlement pattern and that theorized by Christaller. In addition, using only slightly modified tests of signifi— cance, it is possible to compare the respective parameters associated with two different settlement patterns. Clifford E. Tiedemann One thousand fifty four agglomerated settlements in Michigan with 1960 populations exceeding 100 are used in the demonstrations of the models. Analyses are reported in which the following comparisons are made: the regres— sion coefficient of the entire state with that defined by central place theory; the regression coefficient of a sub— region with that of the state; the regression coefficient of one subregion with that of another; the y-intercept of a subregion with that defined by central place theory; the y—intercept of one subregion with that of another; and the correlation coefficient of a subregion with that defined by central place theory. In each analysis, an underlying reason for the test is described and an hypothesis is offered, tested, and either accepted or rejected. As a result of the development and demonstrations of the models, several terms are proposed with which the results of the tests of significance may be described. While many of these terms are refinements of previously used ideas, one is particularly interesting —— "differential clustering." This is a situation in which the regression coefficient is found to be significantly different from zero, indicating that one end of the array of populations of places shows a greater deviation from the uniform spatial distribution of central place theory than does the opposite end. Clifford E. Tiedemann Technical discussions are appended in which the stochas— tic process of determining equality among settlement popu— lations and the calculation of metric and standardized distances are reviewed. Approved: Date: TWO MODELS FOR THE INFERENTIAL ANALYSIS OF CENTRAL PLACE PATTERNS by ,4 Clifford E3 Tiedemann A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Geography 1966 Copyright by CLIFFORD EARL TIEDEMANN 1967 ACKNOWLEDGMENTS The preparation of this dissertation has been aided by many individuals, and it is fitting that those who played key roles be cited. In order for proper credit to be given, names of persons are mentioned in groupings according to their activities associated with this work. The first group consists of my academic advisory committee. Chaired by Dr. Donald A. Blome, this group of professors has acted in a most diligent manner in my behalf at all times. Also including Dr. Dieter H. Brunnschweiler, Dr. Harm J. deBlij and Dr. Milton H. Steinmueller of the Department of Resource Development, their quick replies and penetrating comments concerning my efforts are most appre— ciated. Also of assistance in this respect, but no longer at Michigan State University are Professors Allen K. Philbrick, now of Western Ontario University, and Julian Wolpert, now of the University of Pennsylvania. I am particularly indebted to Drs. Blome and Philbrick for their influence on the conceptual framework of my research. These two gentlemen, with their respective interests in central place theory and areal functional organi— zation, have done much to guide me in the selection of a field iii I Illll III-II I. I III. I In! [[[IIK [[[l[l[/|.I: LII of study and a problem. Without their willing and expert assistance, the undertaking of this project would have been at least considerably more difficult. Technical assistance came from two colleagues in the Michigan Interuniversity Community of Mathematical Geog— raphers. Dr. Waldo Tobler of the University of Michigan provided help in the computation of distances between the urban places used in the analyses. Anthony V. Williams, a fellow graduate student in geography at Michigan State University, provided much needed assistance where this writer's computer programming ability proved to be limited or faulty. Finally, in a class by herself, I must recognize my wife, Margaret. Without her faith and constant support, my work as a graduate student would have been doomed from the beginning. Each of the individuals mentioned deserves some credit for whatever good comes from this research. None of them can be held responsible for its shortcomings. My only hope is that someday I will be able to pass on to others the time and effort which they have invested in me. iv TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . LIST OF TABLES O O O O O O O O O O O O O O O 0 LIST OF ILLUSTRATIONS . . . . . . . . . . . . . Chapter I. II. III. IV. INTRODUCTION 0 O C O C O O C C O O I O I Central Place Theory Selected Studies Related to the Analysis Central Place Patterns DEVELOPMENT OF THE MODELS AND TESTS . . The Conceptual Framework The Basic Models descriptive parameters tests of significance influences of error Preliminary Evaluation EMPIRICAL DEMONSTRATIONS OF THE MODELS . The Study Universe Operational Definitions Applications and Tests of the Models the first test of significance the second test of significance the third test of significance CONCLUSION O O O O O O O O O O O O O 0 Evaluation of the Empirical Applications of the Models Evaluation of the Models and Tests of Page iii vii viii 21 44 81 Page APPENDIX A O O O O O O O O O O 0 O O O 0 O O O O O O 91 APPENDIX B O O O O O 0 C O 0 O O O O O O O O O O O O 99 BIBLIOGRAPHY O O O O O O O O O 0 O O O O O O O O O O 104 vi LIST OF TABLES Table Page I—l. Selected Aspects of the Spatial and Functional Hierarchical Patterns of Central Place Theory . . . . . . . . . . . 5 III—l. Number of Settlements with Populations Equal to or Larger than Selected Settlements . . . . . . . . . . . . . . . 51 III—2. Expected Distances Between Selected Settlements . . . . . . . . . . . . . . . 52 vii Figure II—l. 11—2 0 III—l. III—2. III—3. III—4. III-5. III—6 o III—7. III—8o LIST OF ILLUSTRATIONS Expected Distributional Relationships Expected Distributional Relationships Fractile Diagram: Test for Normality (Populations) . . . . . . .‘. . . . Fractile Diagram: Test for Normality (Standardized Distances) . . . . . Michigan: Settlements with More Than 100 Persons . . . . . . . . . . . . The Size—Distance Relationships for Settle— ments with More Than 100 Inhabitants A. For All of Michigan B. For the Western Part of the Upper Peninsula C. For the Lower Peninsula i. inland counties ii. shoreline counties Michigan: Persons in the Western Part of the Upper Peninsula . . . . . . . . . . Michigan: Lower Peninsula . . . . . . . . . . Michigan: Lower Peninsula . . . . . . . . . . Michigan: viii Settlements with More Than 100 Settlements with More Than 100 Persons in the Shoreline Counties of the Settlements with More Than 100 Persons in the Inland Counties of the Settlements with More Than 100 Persons in State Economic Areas 6 and 7 Page 30 33 50 54 56 58 61 64 65 7O LIST OF ILLUSTRATIONS (Continued) Figure Page III—9. The Size—Distance Relationship for Settle- ments with More Than 100 Inhabitants . . . 71 D. For State Economic Area No. 6 E. For State Economic Area No. 8 and No. 9 i. S.E.A.—8 ii. S.E.A.—9 F. For State Economic Area No. 7 III—10. Michigan: Settlements with More Than 100 Persons in State Economic Areas 8 and 9 . 74 ix CHAPTER I INTRODUCTION The distribution of phenomena over the earth is of primary interest to the geographer. One such phenomenon is the city,1 and Walter Christaller’s theoretical formu— lation of the distribution of agglomerated settlements in Southern Germany has done much to direct the interests and efforts of several contemporary geographers and students of other disciplines.2 Since the appearance of this work, many studies have been conducted and papers written on topics closely related to central place theory.3 Central Place Theory In developing the conceptual framework with which Christaller constructs central place theory, he notes that 1The words ”city," "town,” "settlement," and other terms with connotations of human pOpulation agglomeration are used interchangeably for purposes of ease of expres— sion. If a particular definition is required at some point in the text, it is provided at the apprOpriate place. 2Walter Christaller, Central Places in Southern Germany, Carlisle W. Baskin, trans. (Englewood Cliffs: Prentice Hall, Inc., 1966). 3 _ - . Brian J. L. Berry and Alan Pred, Central Place Studies: A Bibliography of Theory and Applications (Philadelphia: Regional Science Research Institute, 1961). 1 2 there exist certain goods and services which are to be acquired only at specific locations. But, the demand for these items is found throughout the pOpulation, regardless of the spatial distribution of individuals. These goods and services he calls "central goods and services," and the locations at which they are available are designated "central places."4 Each central good or service, in order to be made available in any region, must have a demand of sufficient amount to support the necessary marketing activities. A central place, then, is associated with a "complementary region" of a size which contains that number of people capable of generating an aggregate demand such that a pro— fit is made in the sale of the good or service.5 The extent of a complementary region is limited, however, by the cost of the transportation involved in procuring a good or service, and by the relative ease of accessability to another central place offering a similar good or service. Before a particular central good or service will be made available to consumers at a given place, it is axiomatic that the complementary region be of sufficient area to enclose the minimum required demand for the relevant item. 4Christaller, loc. cit., pp. 14—21. 5 3 The function of a central place is "to be [the] center of its rural surroundings and mediator of local commerce 6 The locations of central places, with the outside world." then, are determined to be relative to the region which they serve. Such a locatiOnal definition implicitly elimi— nates from consideration as central places all those set- tlements whose sites are determined to be specifically oriented to some phenomenon. Christaller explicitly lists such nucleated settlements as mining towns, border and ford sites, and towns at other unique locations as monasteries and shrines, and even residential suburbs of large indus— trial urbanlcenters.‘7 It is also pointed out that among those towns which may be considered as central places, "there is a definite connection between the consumption of central goods and the development of those central places. The development of those central places whose inhabitants live by the sale of central goods becomes more pronounced if many central goods are consumed than if few central goods 8 are consumed." Using these terms and relationships, Christaller develOps a theory with which he explains the spatial and functional hierarchical patterns of central place distribution. 6Ibid., p. 16. 7Ibid., pp. 16—7. 81bid., p. 27. 4 He does so using an idealized environmental situation based on the following rules and assumptions: (1) a flat, unbounded plain; (2) a homogeneous resource base; (3) a ubiquitous transportation system; (4) an even distribution of population with uniform, individual purchasing power; (5) the sale prices of central goods of any one type are constant through time and uniform over space; (6) all demands for central goods are satisfied, pro— vided the seller makes a profit; (7) profits for any one seller cannot equal or exceed that amount which can support two such activities, and (8) any central place offering good i must also offer all goods j which require equal or smaller minimum demands. Within this framework, rigid spatial and functional hier— archical patterns of growth and development would occur (see Table I—l). Indeed, Christaller concludes his initial formulations as follows: The result of these theoretical considerations is surprising but clear. First, the central places are distributed over the region according to certain laws. Surrounding a greater place (B— type), there is a wreath of the smallest places (M—type). Furthermore, there is another wreath of the small places (A—type). Towards the periphery there are a second and third wreath .sp .a ..pHnH 000.com.m ooo.ooo.H -ooo.omm ooo.oou ooo.mm ooo.HH oom.m mCOHOTm wo mGOHDoHSQom HoOHQ>H 000.com ooa.mm ooo.oou oom.oH ooo.om oop.m ooo.oH oom.H ooo.e ace ooo.m mmu ooo.H as moUmHm Hmubcov mo dEM .qmv mCOapdajdom coamom Hmufldwe mo mou< .moH a H H.m@ m m .0m 0 @ 5.0m pm we .NH Hm em m.® mwm NQH .v 0mm owe A.Exv ooow mCOHmom >MMD moooam A Houpcou IcoEoHQEOU mo mo omcmm mo muonesz MODESZ m>uoo£8 woman Hmuwcov mo mcuopwom Hmuflboumuoflm HmCOHDOQSE ocm HmHDMQm OLD wo mDOOQm< oowuoHom .HIH oHQoH math mmmHU moouo 6 of the smallest places (M—type), and on the periphery itself we find the middle-sized places of the K—type. The same rules are valid for the develOpment of greater systems. Second, follow— ing the laws of economics, there are necessarily quite definite size—types of central places as well as the complementary regions, and indeed characteristic types, not order classes. Third, the number of central places and their comple— mentary regions which are to be counted for every type form a geometric progression from the highest to the lowest type.l Thus, in his original statement, Christaller has formulated a conceptual model of the patterns of town development which is clearly deterministic in character. That is, under the conditions specified by the rules, assumptions and rela— tionships, certain regular spatial and population—size patterns develop. As can be seen in Table I—l, the populations of central places of different types in the functional hierarchy have "typical" magnitudes. The discreteness of the distribution of these typical populations is associated with the rela— tionship between the functional complexity of a central place and its pOpulation. And, since every central place offers the entire range of central goods and services requiring a demand equal to or less than its highest order item, a step—like hierarchy of central places with a step— like distribution of populations develops under the specified conditions. lOIbid., pp. 66—8. 7 These conditions also lead to the formation of a recog— nizeable spatial pattern—-—the regular hexagonal lattice. In the instance of Table I—l, the lattice contains 486 points, each representing the location of a central place of one type or another. All of the places are centers for the lowest orders (M—type) of goods, and in the example are located approximately eight kilometers apart, or twice the range of that order good. Of the 486 M—type central places, 162 also handle the next higher order of central goods (A—type). These A—type central places are also distributed hexagonally with respect to one another, and since they share locations with some of the M—type places, they are located 13.8 (or 2 - 4 . «8) kilometers apart. The next higher type of central places (K-type) share locations with fifty—four of the A- and M—types of places. The distance between neighboring K—type places is twenty—four (or 2 - 6.9 ° J83 kilometers, since they, like the A and M places, are arranged in a regular hexagonal pattern. Similar rela— tionships are apparent in the spacing of all central places as one moves through the entire functional hierarchy. With— in the context of central place theory, then, it can.be noted that if the distance between two neighboring towns of the same functional order is d, then the distances between neighboring places both of which are m levels above or below the original pair in the hierarchy are d(\/3)m or 8 d(»/§ll/m respectively.11 Thus, a second discrete distribu— tion is produced by Christaller's idealized specifications. Table I—l illustrates that as the "typical" population increases in its step—wise fashion, so does the distance between neighboring central places of the same functional type. It is this distance—population—size relationship which forms the basis of this research. Selected Studies Related to the Analysis of Central Place Patterns One of the earliest discussions of the distribution of cities over space which is based to a great degree on cen— tral place theory is that by L‘osch.l2 L8sch assumes areal conditions quite similar to those of Christaller, and for any system of places offering a particular central good, he generates an hexagonal lattice of central place locations and complementary regions. Although it is not readily apparent in this work, it becomes obvious in a subsequent book that Lbsch does not require that any central place offering good i must also offer all goods j which require smaller minimum demands.l3 Thus, Ldsch's manipulations of llIbid., p. 63. August Lbsch, "The Nature of Economic Regions," Southern Economic Journal V, 2 (July, l938), ppo 71—8. l3August Ldsch, The Economics of Location, W. H. Woglom and W. F. Stolper, trans. (New Haven: Yale Uni— versity Press, 1954), pp. 118—9. 9 the hexagonal trade areas (complementary regions) produces a regular hexagonal lattice with a spacing pattern for cities of similar functional complexity which is different from that of Christaller. Isard criticizes this construct as being inconsistent with Losch's own assumptions, since it produces regions with different concentrations of cities and, therefore, of people.14 An early application of central place theory in America was performed by Brush in his study of settlements in South— western Wisconsin.15 In this effort, Brush recognizes three types of settlements —— hamlets, villages and towns —— each of which is characterized by its degree of functional com- plexity. He then compares the spacing of the various centers with that which would be expected if a regular hexa— gonal pattern were present, and his average distances separ— ating places with equal status in the functional hierarchy are quite close to those which would be generated by central place theory. Criticism of Brush's work is addressed to both his settlement classification scheme and his evaluation of the l4Walter Isard, Location and Space Economy (New York: John Wiley and Sons, Inc., 1956), p. 271. John E. Brush, "The Hierarchy of Central Places in Southwestern Wisconsin," Geographical Review XLIII, 3 (July, 1953), pp. 380—402. lO spatial distribution of the central places. Vining, point— ing out that Brush has not been able to demonstrate a step- like hierarchy of settlement populations, bases his remarks primarily upon the three—fold grouping of settlements. . . .After having specified the criteria by which an enumerator may distinguish among various kinds of activities, the observer has as his basic data the array of communities and the listing of the kinds of activities represented by the establish— ments in each community. There is no evidence that I have seen suggesting that exactly three natural partitions may be observed in this array of numbers of establishments. Like pool, pond and lake, the terms hamlet, village and town are convenient modes of expression, but they do not refer to structurally distinct natural entities. As the number of establishments increases, the number of kinds of activities represented also increased. Clearly it is arbitrary to divide the array into three partitions rather than into a greater or lesser number; and similarly arbi— trary is the determination of where to put the dividing points separating the different classes or types. Having drawn the lines, one may list certain kinds of activities which are typically found within each of the designated classes of center, and . . . [the table in]. . . Mr. Brush's article represents such a listing. It will be noted that not all members of a class will con— tain all the activities listed, and most of the communities within a class will contain activities not listed. Suchaatable is not an independently derived basis for a classification of communities by type. Rather it is itself derived from a previous partitioning of an array which appears as something similar to an arrangement of obser- vations that have been made upon a continuous variable. 16 16 Rutledge Vining, "A Description of Certain Spatial Aspects of an Economic System" Economic DevelOpment and Cultural Change III, 2 (January, 1955), p. 160. 11 Such a criticism, of course, indicates a need for a more rational scheme of city classification in the instance where a relationship between functional complexity and population is to be considered. In a discussion of Vining's paper, Hoover mentions a logical scheme for evaluating the functional and spatial relationships of an observed group of cities within the framework of central place theory.17 This approach takes into account the rank—size of a city and its link with the functional complexity of the city. What is it exactly that makes the series of tri— butary areas of cities follow the rank size rule? It is convenient at this point to think of the series as continuous rather than discrete. In other words, each city is in a class by itself, and performs some function that is performed by pg smaller city, but by all larger cities. Now if we take, say, the 17th biggest area, we can consider it as one of the group of the 17 biggest. With respect to the function exclusively shared by these 17, the whole country is parceled out into 17 equal areas. Similarly the 16th biggest is one of 16 equal areas blanketing the country with respect to the function which only the 16 biggest cities performed. Obviously, the rank—size pro— gression is implicit, then, in the concept of each city's area being determined by equal sharing with all larger cities.1 l7Edgar M. Hoover, "The Concept of a System of Cities: A Comment on Rutledge Vining's Paper," Economic Development and Cultural Change III, 2 (January, 1955), p. 197. 18 For a discussion of several rank—size theories, see Brian J. L. Berry and William L. Garrison, "Alternate Expla— nations of Urban Rank-Size Relationships,” Annals, Association of Americaaneographe s XLVIII, 1 (March, 1958), pp. 83—91. 12 It is apparent in this statement that Hoover is addressing himself at least in part to the spatial aspects of the problem. But, it is to be remembered that according to Christaller's concepts, a direct relationship exists between (1) the size of the area being served by a town and the pOpulation of the area, (2) the population of the area being served and the functional complexity of the central place, and (3) the functional complexity of a central place and its pOpulation. Thus, it is conceivable that a continuous distribution of population is amenable to an analysis based on central place theory, without resorting to the arbitrary groupings criticized by Vining. But this scheme is very restrictive, in that it allows no consideration to be given to places even slightly smaller than a given settlement. In a series of short articles, Berry and Garrison attempt to relate the conditions and constructs of central place theory with observed situations and patterns in the real world.19 Relying on the concepts of the range of a 9 Brian J. L. Berry and William L. Garrison, "The Func- tional Bases of the Central Place Hierarchy," Economic Geography XXXLV, 2 (April, 1958), pp. 145-54. Brian J. L. Berry and William L. Garrison, "A Note on Central Place Theory and the Range of a Good," Economic Geography XXXIV, 4 (October, 1958), pp. 304-11. 13 good and the threshold —- both of which are consistent with Christaller -— these researchers first recognize problems and possible applications of the principles of central place theory. Ultimately many of the more restrictive conditions are relaxed, and, it is reported, a nesting of central places is found to exist. It is also learned that certain central goods require different threshold populations, and a functional hierarchy of cities is formulated based on the numbers and kinds of activities present in cities of various populations. Thus, it is demonstrated that a functional hierarchy of central places will develop under conditions less restrictive than those outlined by Christaller, and that this functional hierarchy is closely related to the distribution of settlement populations. Thomas assumes the presence of an association between city size and spacing, and demonstrates that this relation— ship has not changed significantly during the period 1900- 1950 in Iowa.20 Using inferential statistical methods on a sample of eighty—nine Iowa cities, the author finds that Brian J. L. Berry and William L. Garrison, "Recent Developments of Central Place Theory," Papers and Proceed— ings,Regiona1 Science Association IV (1958), pp. 107-20. OEdwin N. Thomas, "The Stability of Distance—Popula— tion-Size Relationships for Iowa Towns from 1900—1950," Proceedings of the I. G. U. Symposium in Urban Geography, Lund, 1960: Lund Studies in Geography, Series B. Human GeggraphyLyNo. 24 Knut Norborg, ed. (Lund: The Royal University of Lund, Sweden, Department of Geography, 1962), pp. 13-30. 14 a positive correlation exists between the population of a given city and the distance to its nearest neighbor of equal or larger size for the period under study. This, of course, is consistent with central place theory in that increasing population is associated with increasing distance. In this research, Thomas uses a stochastic model to expand upon the definition of "equal size," setting up requirements which he discusses in a subsequent report.21 He states: . . .we do not mean that the population of the sample city and the neighbor city are the same; that is, we do not mean that z ( Si N1. .1) where Si is the population of the ith sample city and Ni is pOpulation of its nearest neighbor. However, we do mean that S. is approximately . 1 equal to Ni; that is S. as N. (2) I i In addition we will say that S. + R. = N. (3) i I I where R. is a variable whose magnitude is due to chance; that is, Ri is a stochastic "error" variable. The phrase "same population size" is defined in terms of (3), and, accordingly, we may say that populations are accepted as having the same size when they differ only by chance. 21Edwin N. Thomas, "Toward an Expanded Central Place Model," Geographical Review LI, 3 (September, 1961), pp. 400—11. 2 2Ibid., p. 403. 15 Thus, a relationship is established between the discrete and typical size features of the distribution of popula- tions according to central place theory and the problem of the continuous distribution of settlement sizes as recog- nized by Vining and Hoover. Using similar definitions of equal population size for towns, King and Blome expand upon the basic analytic model as described by Thomas.23 Both of these works use multivariate statistical techniques in attempts to explain the spatial patterns of settlement distribution. King's effort is aimed at detecting the most important of those variables which he contends influence the spacing of cities at a particular time —- the census year, 1950. Blome, on the other hand, traces the changing relationships between a variety of factors and city spacing over a period of sixty years. Both of these studies, as do those of Thomas, acknowledge that the spacing of settlements is closely related to the dynamic associations between towns and their surroundings, including both contiguous rural and urban developments and similar urban phenomena located some distance away. 23Leslie J. King, "A Multivariate Analysis of the Spacing of Urban Settlements in the United States," Annals, Association_of American_Geographers LI, 2 (June, 1961), pp. 222-33. Donald A. Blome, "An Analysis of the Changing Spatial Relationships of Iowa Towns, 1900—1960,” unpublished Ph.D. dissertation (Iowa City: The State University of Iowa, Department of Geography, 1963). 16 Dacey approaches the spatial aspects of city distribu- tion from a point of view quite different from that of many geographers. His work with central place theory is based primarily upon an interest in the distributional aspects of point patterns through spaces of various dimensions. Much of Dacey's work is based upon nonparametric statistics, with which he describes the spacing of points in terms of their number and the area in which they are distributed. In a discussion of Brush's work in Southwestern Wisconsin, Dacey assumes that the defined classification scheme is valid.24 He then proceeds to demonstrate that the centers of each level of the hierarchy are distributed according to a random pattern. This, of course, does not conform to the regular hexagonal construct of central place theory, and it does not agree with the apparent close com— parison illustrated by Brush.25 In his analysis, Dacey reduces the observed distances between pairs of settlements in Southwestern Wisconsin to a common scale, taking into account the density of the 4Michael F. Dacey, "Analysis of Central Place and Point Patterns by a Nearest Neighbor Method," Proceedings of the I. G. U. Symposium in Urban Geography, Lund, 1960: Lund Studies in Geography, Series B. Human Geography, No. ‘24 Knut Norborg, ed. (Lund: The Royal University of Lund, Sweden, Department of Geography, 1962), pp. 55—76. 2SBrush, loc. cit., p. 393. 17 places being studied. He refers to this reduced value as the "standardized distance," and to the reduction factor as a . . 26 "dimenSIonal constant." The measured map distance from i [a point] to the nearest point (j) is represented by Rij'°° The Ri' measurements reflect the arbitrary map metric. The dimensional constant which elimin— ates effect of scale is dlfl2, where d is the density of points in Q [an area]. Measurements in Q are reduced to standardized distance by the transformation r.. = dl/2R.. ll 11 Thus, the distances between pairs of points —— in this instance, cities -— are rendered dimensionless, and are significant only as to their relative magnitudes. That is, rij has no metric meaning, and will be the same for any one observed distance regardless of whether Rij and Q are meas— ured in terms of miles and square miles, kilometers and square kilometers, or even inches and square inches res— pectively.27 This measure of standardized distance, however, has an additional property which is most useful, and is what formed 26Michael F. Dacey, Imperfections in the Uniform Plane: Discussion Paper No. 4 (Ann Arbor: University of Michigan. Department of Geography for the Michigan Interuniversity Community of Mathematical Geographers, June, 1964). (mimeo.) 27The density of points, d, in an area, Q, is obtained by the following equation: d = n / Q, where n is the number of points in Q. 18 the basis for Dacey's analysis of Brush's work. If in any given area, the places under consideration are arranged in a regular hexagonal manner, all of the rij's will have a common value, say c. Marked deviations from a uniform spatial distribution, however, will produce a mean standard— ized distance, Fij, which will be smaller than c. Tests may be performed in which ng is compared with an expected mean standardized distance, e. which is associated with a random ij’ distribution of points having the same density. The results of these tests allow the researcher to arrive at results to the effect that a given point distribution is "more uniform than random,” "not significantly different from random," and "more clustered than random." All of the models described above fail to account for certain relationships which are of considerable significance in studying the distribution of settlements within the frame— ’ work of central place theory. A few statements will suffice in relating these problems. (1) Brush's work was criticized because of the manner in which he devised his functional classi— fication scheme which he then used to define his hierarchy of central places. (2) Hoover accepts the reality of continuous distribution of settlement populations, but is extremely restrictive in his method of handling i ! a 19 the problem of functional (pOpulation—size) equality. (3) Thomas expands upon Hoover's notions, but in his attempts to relate distance to population, he constructs a model which fails to take into account the spatial relations between cities and the area in which they are distributed. (4) King and Blome, despite their impressive explanatory models, alsodo not account for the nature of point distributions, thus ren— dering their results not comparable with similar studies which might be made elsewhere. (5) Dacey recognizes the spatial dynamics of point distributions, but does not consider anything other than a situation in which all points have a unit value. He also fails to provide a method for the analysis of differ— entials in distribution characteristics either over space or through a hierarchy of point classes. The object of this research is to develop a model or group of models which are capable of providing for these needs. What is proposed and demonstrated in the following chapters is a group of models and tests which are capable of doing the following: 2O (1) describe the relationship between population— sizes of cities and their spacing; (2) test this relationship for: (a) conformity to central place theory over the entire study area, (b) conformity to central place theory over less than the entire study area, (c) a random distribution of cities of all sizes over the entire study area, ‘(d) differential clustering among cities of various population sizes; (3) test for significant differences between the size—distance parameters for different groups of cities; (4) provide a basis for explanatory models which can account for differences in spatial patterns of settlement distribution. Such a group of models can be of great value in analyzing and describing a wide variety of situations and influences relevant to the spacing of agglomerated settlements over the earth's surface. They derive their usefulness from the fact that they can be used both to test deviations from theoretically-based "expected conditions," and to compare observed patterns. CHAPTER II DEVELOPMENT OF THE MODELS AND TESTS Walter Christaller, with his conceptual statements con— cerning the role of economic centrality in the development of settlement patterns, has provided students of such phen— omena with a rigorous theory relating cities to their rural surroundings and to one another. Central place theory takes into account both population—size and distance between simi— lar neighboring towns, and furnishes a logically constructed, "expected" situation with respect to these two variables. Indeed, within the context of the theory, this relationship between size and spacing is deterministic in character, there being matched step functions for both variables and inter— dependent, paired observations for every settlement. Dacey characterizes central place theory as being ". . . a deductive formulation from very restrictive conditions . . . a flat, unbounded, homogeneous area, an evenly distributed 28 population density, and a ubiquitous transportation system." He then observes that these conditions are seldom found to 28Michael F. Dacey, "Analysis of Central Place and Point Patterns. . .," loc.cit., p. 59. 21 22 be coincident in reality. However, it is interesting to note in his criticism of Brush's work that Dacey recognizes five point pattern relationships involving settlement function and spacing and compares the observed respective spatial distributions with those which would be expected to develop if central place theory's idealized conditions prevailed. Thus, it appears that the size—distance associa— tion is one phenomenon which can be used to analyze the distribution of agglomerated settlements —— even if Christaller's specifications are not to be found. The Conceptual Framework In order to achieve consistency with central place theory, it is necessary to take into account certain qualities of the two variables involved in the analysis of the size— distance relationship. Vining's criticism of Brush, as discussed earlier, points out one of two basic problems which one encounters in attempting to analyze a settlement pattern within the context of Christaller's theory. Namely, this problem is centered about the categorization of cities by size and functional complexity, especially in the instance where the distribution of settlement populations is continuous and that of functional complexity appears to be somewhat jumbled, 23 though grossly related to population.29 Hoover's comment on Vining's criticism offers a partial solution, but it is too restrictive to be of great value.30 That is, his qualification, "no smaller city," does not conform to cen— tral place theory, since even Christaller noticed groupings of populations about certain discrete values rather than absolute equality among city sizes within any one functional category. Using a technique known as the "fractile diagram," Thomas expands upon Hoover's contribution, and establishes a link between the discrete and typical size features of central place theory and the problem of the continuous distribution of settlement populations as recognized by Vining and Hoover.31 According to this method, there exist as many "equal size" categories as there are different town pOpulations. And, instead of dividing a region into pro— gressively smaller areal units as we move iteratively through the rank scale, as Hoover suggests, we now divide it into as many units as there are settlements with pOpulations 29Rutledge Vining, loc. cit. 3OEdgar M. Hoover, loc. cit. 31Edwin N. Thomas, "Toward an Expanded. . .," loc. cit. 24 approximately equal to or larger than any given town, regard— less of its rank.32 The fact that Thomas' proposed technique renders a continuous distribution of settlement populations amenable to central place theory, however, forces a second problem into view. That is, since the step—function of city sizes has been generalized, the size-distance relationship is no longer clearly defined in the theory's deterministic terms. A solution to this question is contained in Dacey's criticism of Brush.33 In the work, he points out that the distance between center points of two regular hexagons having one edge and two angles in common is r = 1.075 -\/—H_, where r is the distance between the center points and H is 32The use of fractile diagram for the purpose of establishing equal size categories requires that the popu— lation data be of a normal form. (In addition to providing these categories, the fractile diagram is also an excellent test for normality, see Appendix A.) This requirement, of course, may necessitate some type of transformation of the data in order to render it compatible with the assumptions required by a particular statistical technique. The equal— size categories which are calculated then, are in terms of the transformed data rather than in the mode of the obser- vations. Also, it should be noted that the resultant categories may not be mutually exclusive for places with similar pOpulations. 33 Michael F. Dacey, loc. cit., p. 62. 25 the area of each of two hexagons. In a large region which is completely covered with regular hexagons, the area of each is H = A / n where A is the area of the region and n is the number of regular hexagons it contains. The dimensional constant which removes the map metric from consideration, however, is the square root of the density, d, VEI== (n / A)l/2. Multiplying this conversion factor by R, an observed dis— tance, produces a standardized distance r = (1.075~,/fi) - f5 = 1.075 - (A / n)l/2 - (n / A)l/2 = 1.075. This relationship, r 1.075, holds constant regardless of the number of centers and hexagons present in a region. The distance step function multiplier, J3, is not operative, then, when evaluating the spacing of central places with the use of standardized distances. Instead, the standardized distances associated with any level in an ideal central place hierarchy are equal to the constant value, 1.075. Also, 1.075 is the value of r for any hexagonal lattice, regardless of the number of regular hexagons. Standardized distance, therefore, may be used to evaluate a continuous distribution of the spacing of comparable central 26 places even in areas where a uniform distribution of locations is not to be found.34 This is so, because for any pair of places, if they and all comparable settlements were arranged according to a regular hexagonal lattice, the associated value of r will be 1.075. The Basic Models Descriptive Parameters Assume that the distance separating a given settlement from its nearest neighbor having an equal or larger popula— tion (as discussed earlier) is dependent upon the population) of the given place. This relationship is consistent with central place theory, and is evident in certain of the studies reviewed in the previous chapter. The nature of this relationship may be expressed in the form of a Simple linear regression equation y. = a + bxi, - (l) 34Uniformity in spatial distributions requires that two conditions be satisfied. (1) individual points are evenly spaced from one another, and (2) a maximum number of points must be packed into any small unit area. These specifications in two dimensional space define a regular hexagonal lattice of point locations. 27 where yi is the standardized distance to the nearest neighbor of equal or larger size associated with place 1, xi is the pOpulation of place i, and a and b are parameters which des— cribe the linear relationship between the two variables. Indeed, if both the dependent and independent variables are normally distributed, this model can be easily tested for statistically significant deviations from various expected relationships in several ways. A third descriptive parameter, the correlation coef— ficient, is a measure of how well individual observations conform to the regression model. This index, r, has the value (2) r = b ° sX / sy, where sx is the standard deviation of the independent variable (population) and sy is the standard deviation of the dependent variable (standardized distance). The value of r can also be tested for significant differences from specified values. Tests of Significance The tests of significance which may be utilized in the analysis of these descriptive models are drawn from the realm of inferential statistics. In order to be applied, they require that certain conditions be met. Of primary importance, is the necessity that the distributions of the independent and dependent variables be bivariate normal. 28 In order to conform to the specified distributions, it may be necessary to transform one or both of the observed dis— tributions into a form which more nearly approximates normality. Two other conditions are concerned with the nature of the relationship between the two variables, rather than the characteristics of the distributions themselves. These requirements state that (l) the standard error of the estimate must be constant for all values of the independent variable, and (2) the distribution of the relevant dependent variables is normal for any given value of the independent variable. If all of the conditions stated above are met, then tests of significance may be executed involving the descriptive parameters, a, b and r.35 Based on these tests, statements of confidence may then be made concerning the characteristics of the size—distance relationship for the cities under investigation. The descriptive parameters are products of the des— criptive models (1) and (2). The tests of significance 35Wilfrid J. Dixon and Frank J. Massey, Jr. Introduc— tion to Statistical Analysis (New York: McGraw—Hill Book Co., Inc., 1957), pp. 193—201. 29 involving the descriptive parameters, on the other hand, are analytic devices. The difference, of course, lies in the uses of these two types of formulations. One type, the descriptive model, illustrates one or more characteristics of the relationship between the independent and dependent variables. The second type, the analytic tool, involves the comparing of equivalent descriptive parameters from two relationships, or the comparing of one such parameter from a given relationship with that of an expected situation. In the instance of central place theory, the expected situation is predefined. Figure II—l shows the relation— ship between population size and standardized distance for a group of cities distributed according to a regular hexa- gonal lattice covering an entire study area. In the first case (see Figure II—lA), the idealized central place pattern -— that is, one having step functions for both city size and spacing —— is illustrated by a horizontal regression line. The regression equation for this is . = a + bx. yi i 1.075 + 0.0x. i 1.075. Since there is no deviation from 1.075 for any value of y, the correlation coefficient is r :- 1.0. Fig. 11-1. EXPECTED DISTRIBUTIONAL RELATIONSHIPSB6 A 37 I / F-‘O STANDARDIZED' DISTANCE V-' \. Y_—____- .,— O s-. :3 l’. “““““ at; I ..o ’?o\\\‘.~ o . (:2 I .0 I .0 I..‘\I . 0 ° 0 0 ° ° (4 i 3. I :. I c. 1 u f. . . . . . 0:) I o. o. I 0" . . 3— I .. I .¢// . o I—0 I .0 I // (n | o/l,./ l /,»/ I ,,/” L’ POPULATION POPULATION (A) an idealized central place pattern with the theorized discreet distribution of populations (B) with populations clustered about "typical" magnitudes (C) with a continuous distribution of populations (D) a random dispersal over the prescribed locations 35assuming a regular hexagonal lattice of central place locations 31 In the second and third examples (see Figure II—lB and C), the restrictive step function of populations is replaced by clusters about "typical" size values and by a continuous distribution of populations, although the predefined loca- tional pattern is retained. In both relationships the regression equation has the same parameters as those of the idealized central place patterns. The correlation coeffi- cient, however, is ll 0 '0 0 U) \ U) = 0.0, since the values of y vary about 1.075. Thus, if the slope of the regression line, b, is exactly zero, the correlation coefficient cannot be used as an indicator of conformity of an observed settlement pattern to that specified by central place theory. In the fourth example (see Figure II-lD), the hexagonal pattern of place locations is retained, but the distribution of cities of various sizes among the locations is random. Again, the regression equation is the same as that for the previous three relationships, having a b—value of zero and a y—intercept, a, of 1.075. As in the cases of Figures II—lB and C, the correlation coefficient, r, is zero. The fourth example, of course, is not consistent with central place theory, but resembles the theoretical patterns of settlement 32 location and possesses similar values for the descriptive parameters. The analytic models are designed to test for significant differences between observed and expected size—distances relationships. That is, the observed value for b is com— pared with the expected value (zero), the observed a-value is compared with 1.075, and the observed correlation coef- ficient is compared with unity. Significant differences between the observed values of these descriptive parameters and those dictated by the theory indicate the presence of deviations from central place theory in the distribution of settlements within a study area or group of cities (see Figure II—2). The first test involves the slope of regression line, b. This analytic model is used herein to detect the pres— ence of a bias in the size—distance relationship which is not consistent with central place theory. The test statis— has the value tb=((b—0)°sX-\/N_:—_l)/s in which b is the slope of the regression line in the des- tic, tb, (3) YX’ criptive model yi = a + bxi, where 0 (zero) is the expected slope of the regression line according to central place theory, sX is the standard deviation of the independent variable, N is the number of Fig. 11-2. 0 , O ' . l-IJ __________._'___ s3 ° - a: . . .. o . . o <.— o o O o 252 ° ° . (£3 , . .— o (I) ' . . . STANDARDIZED DISTANCE POPULATION (A) a random spatial pattern 33 EXPECTED DISTRIBUTIONAL RELATIONSHIPS37 POPULATION B increased clustering with increased population 0 decreased clustering with increased population (D an idealized central place pattern covering less than the entire study area 37 populations assuming a continuous distribution of central place 34 observations used in calculating the descriptive parameters, and Syx is the standard error of the estimate, which has the value 2 — bzsx2)) / (N — 2), SyX = \/((N — 1) (By in which sy2 is the variance of the independent variable. The calculated value tb is then compared with a standard statistical table showing percentile values of Student's "t—distribution." If the value of tb is larger or smaller than that range specified in the table for a given number of degrees of freedom (N-2) and a given level of confidence, then the slope of the regression line, b, is assessed to be significantly different from zero. The interpretation of such a result depends in part on the sign of t Figure II—2B illustrates a situation in b' which the slope is negative, showing an inverse relationship between city size and spacing.38 That is, larger cities are more closely spaced than the idealized central place pattern specifies, and the larger a particular city is, the greater is the deviation of its nearest neighbor distance from that which is expected. Figure II—2C, to the contrary, shows a direct relationship between the size of a city and its standardized distance. In this instance, small cities appear 381t must be remembered that the observed pattern is being compared with the ideal central place arrangement. Therefore, real world distances might appear to not conform to such an inverse type of size—distance relationship. 35 to be much more clustered than larger cities.39 In the event that the slope of the regression line is found to be significantly different from zero, further analysis in comparing an observed distribution of settlements with that specified by central place theory is not necessary. If, however, the value of b in the regression equation is found to be not significantly different from zero, fur- ther comparisons with the theoretical distribution may be made. The second analytic model proposed herein involves the y—intercept of the regression equation, a. The test statistic, ta, is calculated a t =( mdkzmomma mfimm mam. mm. mm Om Ow CV ON 0 i NO mOO m.n_m@ 10mm _ WMG — mm _ Om. — Or _ Ofl _ Om — O, N — m 0 W ,0 — 7‘0 I k _ _ _ _ H _ _ _ _ I. _ _ _ _ _ _ I _ _ _ _ _ i _ _ _ _ _ _ _ s \ \ \ \ .\ 63.58311 \ _ x mezqthIE x ....\\ OO_ ZQIP NKOE IHTS WHZmS—mllykmw \o. t\\ kmemmamm :._<<Hq mv Bonmw wommm Homflm pamflwcwsom as Heeom meome emcee mamas Boa momm owmw womb CflmHm Mflmm HBH ombw wmmm 000v moaoxo mom mmmm vvwa ooow OHHH>MOHDDO mmv BNHH mwm OOOH maoocopm vow @mm omv com cpom mmm mmm Bmm 0mm pmHHHE smoe ems see mme mcflezoo DHEHQ umzoq coca mood: umsoq mNHm ummuoq m Mo mmfluommpmu mNHm coepmasaom COHPMHSQOQ >DHU Umpomamm moomHa wo MODESZ Hmsqm mo mpHEHA mpcmEmepmm oopuoHom cosh ummumq no ow Hosqm mCOHDmHSQom Lees mpcoeoauuom wo wonesz .HIHHH mange 52 pHEHH mNHm Hmsqm nozofi och cmcp COHDmHSQom ummumH o mo mmumam who oumcp mm mchOQ >cmE mm OCH>mc OOprmH Hocomoxoc co CH ma comm OCAESmmwa mmsso.e No.04 eloexseeem.s me semawnpsom maeso.e oo.em muoexeemom.e as mass; maeso.e so.mm muoexsmsmw.e see semen mama mmss0.e sm.me mloexmmsmm.m ewe mosmxo wmsso.e we.me muoexmsmeo.m mom meew>uoepso maeso.e es.me muoexsemse.s mms memeqmpm masso.e om.oe muoexemsmo.e sow ream maeso.e No.m muoexmwmes.e 6mm noses: maeso.e .He mm.s Nuoexmomem.e smoe mcflezoo moccamfio mucmpmflo >pflmcmo pHEHQ umzoq corp wpflu Umpomamm pmwflpumccopm Homm mNHm ummumq o mo copummxm powwooxm moomHm wo quESZ mvmpcoEmewmm Umpumamm comzpom mmocmpmfla Umpooaxm .NIHHH mange 53 Tests using the fractile diagram show that the distri- bution of standardized distances, like that of the popula— tions, does not conform to normality. More than a dozen normalizing transformation functions were applied to this data, but even that which provided the best results could arrange only 61.00 percent of the observed values between the specified ninety percent confidence limits centered about a theoretical normal distribution. The transformation function ril/3 in which ri is the observed standardized distance, gives the best visual appearance on the fractile diagram as well as the best calculated comparability to values expected in a normal distribution having the same mean and variance (see Figure III—2, showing ninety—five selected observations). This set of transformed values is then substituted into the regression equation as the dependent variable. Applications and Tests of the Models Arguments have been offered in order to form a basis for a new approach to the analysis of the spatial distribu— tion of cities. Centered about the relationships presented or inferred within the contexts of central place theory and its recent expansions, this group of techniques allows the recognition of patterns of settlement spacing heretofore 54 ) (33,8 NORMAL DEVIATION VALUES, 2.0v 442102 .mw:...<> muzkzwommn. . ow oo oo 0' ON a . No .noo aama _owo — owm — om _ O—a — 0.5 nenomnwfioroll . .omomnndva‘l I . I awm.bon.ov41 AN~._m_0.OVOJI 8530 :I1 Acwwoomn .2 l Ao_n:n.:3l _ 398.0. m ~to_m.oum $02455 03.33245» ”6.00 >._._._<<<~_OZ «On. .53. “2<¢0<_o m.__.pU<~E III—2. Fig. 55 undiscernable with such simple models. The following dis— cussions relate the results of applications of the models, and are based upon the aforementioned arguments and rela— tionships, and upon the Operational definitions of city size and nearest neighbor distance as are specified in the previous section of this paper. The First Test of Significance The first analytic approach involves the regression coefficient of a simple regression relationship, equation (1). 'Based upon the operational definitions of settlement population and standardized nearest neighbor distance, this equation has as a more specific form 1/2 yi = a + b(1ogloxi) , in which xi is the population size of the ith place and yi is the standardized distance from place i to its nearest neighboring place j of equal or larger size. The analytic tool concerned with the regression coefficient is the test statistic in which tb is calculated, equation (3). The first application of this test statistic involves the regression coefficient for the entire study universe (see Figure III—3). In this instance, all of the settle— ments in Michigan with populations of more than 100 are used in calculating a simple regression equation. This equation has the form “7| ,IIIIIillI'lIIII l .l. I .I: .l 56 MICHIGAN SETTLEMENTS WITH MORE THAN I00 PERSONS Scale in Miles 0— EACH DOT REPRESENTS A SETTLEMENT WITH A POPULATION OF MORE THAN IOO INHABITANTS 57 _ _ 1/2 yi — 1.23646 (0.24668 - (logloxi) ) 7 and the regression coefficient, -0.24668, may be used in the analytic model to calculate t (see Figure III-4A). b As part of the analysis of the spatial distribution of agglomerated settlements in Michigan, an hypothesis can be stated concerning the nature of the observed size—distance relationship and its comparison to that which is to be expected according to central place theory. Subjectively, this hypothesis could read: The size-distance relationship for Michigan settle— ments larger than 100, as indicated by the regres— sion coefficient, is not significantly different from that which is dictated by central place theory. Mathematically, the same hypothesis may be stated: b = 0.0. The null hypothesis, that statement which is to be accepted if the hypothesis is found to be not true, is b # 0.0. If the null hypothesis is accepted, it must be concluded that there is a bias in the spacing of Michigan cities which is related to town population and which does not conform with central place theory. The calculated value of t includes the specific value b tb = ((—0.2467 — 0.0) ’ 0.1882 . V1054 — l) / 0.2263 which results in 58 III—4o Fig. zoCthaoa zofidnaaoa nnod hoN.~ 00.... mm... 500.0 0.0 nnod «mud 005.. mm... N000 0.0 p _ .I _ F 0° _ _ _ _ p 00 S S 1 @ $2.0“ G In~_.oV N N O O rmondV rennov a a m a. m [000.0n 1000.03“ 0 0 10.0.00 Io_m.om.u a m umeo._ w. :.I II I Imvo. v N N o m 68.. 3 103.. E 295320.. nn_o.~ bw_~.N 00.x... mm... swmd 00.0 beneficis .332... . moi “3:238 ”.2353 Issov 3356-... 635.. .o N ”muizaoo ozqnz. fl 0 conduv <43m2_zwn_ «£50.. wzh «on. I © 8 0 33nd... .3331. .2 a. $8.0fl 33E; mamzmo .ms .2... «#2 0.20203 M 3:3 .442mz.zwa «mam: MI... .8 ES. zmmpmm; 5.: mo... I G .220 S 3636-... 633.. .o .2 l A_v IIIIIIIIIII T . ed... .39.... .: $3qu 0 23:85 L0 4.2 cc... I @ r..: 3 m._.Z<._._m> thmZmdzpmm m0...— mm_ImZO_.—<._m~_ mUZ<._.m_0..mN_m m1... 59 tb = —6.5654. The calculated value of tb is then compared with a prede— fined entry in a table of the percentiles of Student's t— distribution. If tb does not fall within the range of zero plus or minus t(%ndf)’ then b is found to be significantly different from zero at a prespecified level of confidence.49 A critical value of t from such a table is t(0.5,1052) z lfig6° Since the range 0.0 i 1.96 does not include the calculated value of t it must be concluded that t is significantly b’ b different from zero and, therefore, that b i 0.0 at the .05 level of confidence. Thus, it is reasonable to state that among Michigan cities having more than 100 inhabitants there exists a bias in the size distance relationship. This bias appears to be inconsistent with the pattern which would be generated if the rules and assumptions of central place theory were applied to the study universe. Noting the negative slope of the regression line, it can also be observed that larger cities in Michigan tend to be more clustered with respect 49The subscripts of the value of t from a table of percentiles, 0/o and df, indicate the proportionate size of the rejection region and the number of degrees of freedom allowed. 60 to their nearest neighbors of equal or larger population- size than do smaller settlements in the state. A second test of significance involves the selection of a "sample" region out of the study area and the comparing of its settlement pattern with that of the entire universe. The sample region consists of the counties forming State Economic Area 1, the western portion of Michigan's Upper Peninsula (see Figure III—5). This area was chosen because of the high proportion of the region's employment in mining activities (eighteen percent).50 This characteristic, of course, renders the economic structure different from that of the state as a whole, and gives the area qualities which are incompatible with central place theory. In this test, since the employment structure of the area is quite different from that of the state as a unit, a reasonable hypothesis might contend that the settlement pattern will also differ. It could be stated: The relationship between settlement size and spacing in Michigan's S.E.A.—1, as indicated by the regression coefficient, is significantly different from that of the state as a whole. 50Donald J. Bogue and Calvin L. Beal, "Michigan: Upper Peninsula: Western Areaf'Economic Areas of the United States (New York: The Free Press of Glencoe, Inc., 1961), pp. 756—8. 61 «2:2 5 Boom III—5. fiIII._ _ o o 000 /. IL 0 on / / l . . / xx 1 _ o “a o ./ / / fl _ o o o o o o o /f, d £5?" 0.00 c o o o o o a o. o 000 no...) W I/\ . I\\/\ mezqtqug OO_ 24:» Macs. 1.23 szZMJEIme d m»2wmmmamm Foo 104ml. Fig. mehmw>> mI._. <.5mZ_Zmn. Azqmmam momzmu w 3 ,_ a. I «mud UAZOZOUM wpqhmv «man: 5.: ”.0 :3}. Z_ mZOmmmm oo— Z. i L mezmzmfizm ‘1 «IT IT IT I II I i 62 Or, more mathematically, b # —0.2467. The null hypothesis, by definition, then, must claim that the size—distance relationship, as shown by the regression coefficient, is not significantly different from the state as a unit, and, thus b = —0.2467. The descriptive model for this example has the foam y.l/3 1/2) 1 = 0.1622 +(0.3966 ° (logloxi) (see Figure III-4B), and the significance test has the form tb = ((0.3966 + 0.2467) ° 0.1483 ° V110 — l) / 0.2955, which generates a value of tb = 3.3821. This calculated value of tb is then compared with that range of permissible values as specified by the table of the t— distribution, in which t(.05,108) 1°98° Since the range of 0.0 i 1.98 fails to include the calculated value of tb, it can be concluded that tb is significantly different from zero and, therefore, that b ¢ —0.2467 at the .05 level of confidence. This result leads one to accept the hypothesis that the size—distance relationship for settlements in the western 63 portion of the Upper Peninsula of Michigan, as shown by the regression coefficient, is significantly different from that of the state as a whole. It can be concluded, then, that this sample of towns, based on an arbitrarily delimited subregion, does not reflect the settlement pattern found to be characteristic of the study universe. A small amount of additional information may be acquired by noting that the sign of the regression coefficient is positive, indicating that a greater degree of relative clustering exists among smaller centers than among larger ones. A third illustration of the first test of significance involves the comparison of two "sample" regions. The regions used in this example consist of those counties in MiChigan's Lower Peninsula which border on a shoreline of one of the Great Lakes, and of the remaining or inland counties of the same peninsula (see Figures III—6 and 7). These subregions have been selected because of their obvious association with a fundamental geographic puzzle —— the boundary problem. One area is surrounded on all sides by space which is capable 51A subsequent calculation of t in which the regres— sion coefficient 0.3966 is compared with zero, that which would be expected according to central place theory, pro— duces a value of 2.089. This recomputed tb also lies out— side of the range 0.0 i 1.98, thus giving substance to Christaller's view that mining towns cannot be expected to conform to central place theory. 64 MICHIG SETTLEMENTS WITH MORE THAN I00 PERSONS ‘IN THE S O ”N- COUNTIES OF THE LOWER PENINSU ‘ I . O - EACH DOT REPRESENTS A SETTLEMENT WITH MORE THAN IOO INHABITANTS Fig. III—6. 65 MICHIGN SETTLEMENTS WITH MORE THAN I00 PERSONS IN THE ND COUNTIES OF THE LOWER PENINSULA O — EACH DOT REPRESENTS A SETTLEMENT WITH MORE THAN IOO INHABITANTS Fig. III—7. 66 of being settled in much the same fashion as the subregion itself, whereas the second area posses a boundary beyond which further settlement is impossible. Also, the presence of a boundary to the area of potential development is not consistent with central place theory. In this third example, it is suspected that the pres— ence of a shoreline serves to cause a recognizable disturb— ance of the size—distance relationship of nearby settle— ments. As a result the regression coefficient associated with the settlement pattern of shoreline counties is expected to be significantly different from that of more inland counties. Such an hypothesis may be stated: The relationship between city population and distance to the nearest neighbor of equal or larger size for settlements in the shoreline counties, as indicated by the regression co— efficient, is significantly different from that of inland county settlements. Or, in the terms of an inequality, bS ¢ bi’ where bS is the regression coefficient for the settlements of the shoreline counties, and bi is the regression coefficient for the inland places. The null hypothesis, must lead to the conclusion that the size—distance relationship 67 of the shoreline counties, as indicated by the relevant regression coefficients, is not significantly different from that of the inland counties (see Figure III—4C). Calculating the required parameters for this test of significance produces two descriptive models. For the settlements in the inland counties, the simple regression equation has the form 1/3 Y1 /2 l = 1.2778 — (0.2741 (logloxi) ). For those in the shoreline counties, the descriptive model has the form 1/3 Y1 1/2 = 1.3875 — (0.3440 . (loglei) ). The two regression coefficients are then used in the test of significance. The analytic approach, in this instance, has the form tb = ((bS — bi) - sXS ~ VTE:ETI) / Syxs’ in which the subscript s indicates values associated with the shoreline counties, and the subscript i designates the regression coefficient of the inland counties. These vari— ables are used, because the object is to find if bS is significantly different from bi' If the object of the test were to find out if bi is significantly different from bs’ the relevant subscripts would be reversed. In calculating t the test of significance incorpor- b7 ates the specific values: tb = ((—0.3440 - 0.2741) - 0.2091 - 363 — l) / 0.2227 68 which generate the value tb = -l.2456 This calculated value of tb is then compared with a critical figure in a table of percentiles of the t-distributions, t(.05,361) = 1°97' Since the calculated value of tb, —l.2456, falls within the range of 0.0 i 1.97. tb is found to be not significantly different from zero and, therefore, bs is not significantly different from bi at the .05 level of significance. Thus, the size—distance relationship for the settle- ments in the shoreline counties of Michigan's Lower Penin— sula is not significantly different from that of the inland counties. A reasonable conclusion, then, is to accept the null hypothesis which proposes that the rates of change in the dependent variables for a unit change in the inde— pendent variables are found to be not significantly different from one another. In this instance, both regres- sion coefficients indicate that relative clustering is greater among larger cities than among smaller ones. The Second Test of Significance The second analytic approach is a statistical test of significance involving the y—intercept of the same descrip- tive model, the simple regression equation, .1/3 1/2). yl = a + Kb -.(loglOXi) 69 The test statistic for this descriptive parameter is equa- tion (3), in which Y'is either the arithmetic mean of the dependent variable or some previously specified figure —- an expected value according to a set of predefined condi- tions. The calculated value ta is compared with a table of percentiles of Student's t—distribution from which is selected a suitable relevant figure, in order to measure the comparability of Y and a. The testing of the y-intercept should be performed only after it has been ascertained that the slope (or slopes) of the regression 1ine(S) is (or are) not signi- ficantly different from zero, since as b nears zero, the y-intercept, a, becomes more representative of the average spacing characteristics of the settlement pattern being analyzed. Thus, differences between a group of data and a hypothetical situation, or between two groups of data may be discerned, even if the slopes of the relevant regression lines are not significantly different from zero. The second test of significance compares the spacing characteristics of the settlements in Michigan's State Economic Area number six with that which would be expected if the rules, assumptions and relationships of central place theory applied to the entire state (see Figure III—8). The descriptive, simple regression equation upon which the analytic model is based contains the following specific parameters (see Figure III—9D): 7O Z. - m...z<._._mame:State Economic Areas eight and nine, both of which are found in the southern portion of the Lower Peninsula (see Figure III—10). In both of 74 w 532'. A E E E. U) I Z 3 o 0 t 2 '32 z I“ n «f. a. 2 mi 0 < §§ zg ° K 2 k‘ <§3 855 I Id 3. =2 m: 1 U c o— 2 2 1 ' " 2 <23 ® ' ..'°' 3 :0 o . . EB . o ’0 o . :2 “.1 - 3.’ 3' z < m :7, ° .- 5 2 o E “I i: .— I” U) 75 these subregions, the size—distance relationship shows no significant bias with respect to differential clustering from one end of the array of pOpulation sizes to the other. State Economic Area nine is described by Bogue and Beal as lying "near the northern border of the Corn Belt."53 Agriculture accounts for a large portion of this area's economic activity, although a variety of manufacturing establishmentsare found in the towns included in S.E.A.—9. The other subregion, State Economic Area eight is des— cribed as being "submetropolitan," both socially and economically.54 Flanking the Detroit urban complex, the economics of this subregion are closely linked with those of the urban area, both from the point of view of there being a large number of manufacturing firms located within S.E.A.—8, and from the standpoint that the nearby Detroit area serves as an employment source for many residents of this subregion. Thus, State Economic Area nine appears to possess certain of the qualities conducive to the development of a settlement pattern approaching that which was theorized 53Donald J. Bogue and Calvin L. Beal, "Michigan: Southern Michigan: Eastern Area," and ". . . Western Area," loc. cit., pp. 769—71. Donald J. Bogue and Calvin L. Beal, "Michigan: Southeast Michigan Area," loc. cit., p. 767—9. 76 by Christaller, whereas the S.E.A.—8 is a suburban area near a large metropolitan complex —— one type of situation which central place theory fails to consider. State Economic Area nine, then, might be expected to conform more closely with central place theory than S.E.A.—8. In this test, the spacing characteristics of the agricultural area are accepted as the expected situa— tion, and those of the suburban areas are suspected of being significantly different from the theoretical pattern. Therefore, a reasonable hypothesis to be tested would state: The spacing of settlements in State Economic Area eight is significantly smaller, as indi— cated by the y—intercepts, than that of the settlements in State Economic Area nine. That is, since the regression coefficient for both sub— regions are not significantly different from zero, it is to be expected that a8.<:a9, where the subscripts 8 and 9 indicate the respective economic areas. Such a result would be compatible with Christaller's logic, whereas the null hypothesis, a8;>.a9 would not. Subjectively, the null hypothesis must claim that the spacing of the suburban settlement pattern found 77 in S.E.A.-8 is not significantly more clustered than that of S.E.A.—9. The descriptive models for the two subregions are as follows (see Figure III—9E): for S.E.A.-8 yil/3 = 0.7718- (0.0125 ° (logloxi)i/2), and for S.E.A.-9 1/3 _ . 1/2? 55 yi — 0.9174 - (0.0534 (logloxi) ). The analytic approach with which a8 is tested for a signifi— cant difference from a9 is -! ta + ((0.7719 - 0.9174) 757) / 0.1741. The calculated value of ta is —6.3l36. This figure does not fall within the rejection region, —l.67 to plus infinity. Therefore, it is reasonable to accept the hypothesis that a8 be successful. It was finally decided that the one which came closest (61.06 percent inside the ninety percent confidence limits) was to be adopted (see Figure III-2). This transformation had the form X. = 3/ standardized distancei. APPENDIX B FORMULAE USED IN THE COMPUTATION OF DISTANCE BETWEEN URBAN PLACES Because of the number of observations being con— sidered in this research, approximations of road distances between each place and its order neighbors of equal or larger size are used. In order to do this, two situa— tions must be considered in making the calculations. First, it is possible that the points might be connected by a fairly straight road passing through both of them. Second, since much of Michigan is served by section line roads, it was considered quite possible that two such places might be connected by a route running east—west and north—south and possessing a right angle intersection. Therefore, two different distances were calculated for each pair of points, one straight—line and one right— angle distance. In addition, the fact that Michigan is made up of two peninsulas created another problem. In order to account for the fact that the only connection for land travel between the two is by way of the Mackinac Bridge, straight line distances between points on Opposite peninsulas were 99 100 computed in two segments —— one from place i to the bridge, and the second from the bridge to place j. (It iS' shown that onlyr two cities had their nearest neighbor on the Opposite peninsula: Saint Ignace was the nearest neighbor for Mackinac City, and Cheboygan was Saint Ignace's nearest neighbor.) The approximated distances were calculated for both major categories, straight—line and right—angle, and were ranked in order of ascending size. Then, the smaller of these order neighbor distances for each settlement were visually compared with the map situation to see which best depicted the road situation. The one was considered to be most representative was selected as the value to be used in the statistical analyses of the size—distance relationships of Michigan settlements. The formulae used were standard half—angle trigonome— tric functions. In the instance of straight distances between places i and j on the same peninsula, the formula was as follows: sin(theta / 2) = ((cos(lati) - cos(latj) - sin2('loni — lonjl / 2)) + sinZHlati — latjl / 2))1/2 in which theta = the distance between places i and j in radians, lati = the latitude of place i, r‘r___a..=.: M- 7 , , , lOl latj = the latitude of place j, loni = the longitude of place i, and lonj = the longitude of place j. In order to find the distance between i and j in terms of miles, it is necessary to multiply theta by the radius of the earth at an apprOpriate latitude. In the case of right—angle distance between places on the same peninsula, a modification of the previous formulae were used, sin(dlon / 2) = (cos2((lati + latj) / 2) . 1/2 and sin2(lloni — lonj' / 2)) sin(dlat / 2) = sin(llati — latj' / 2) , in which dlon = the relevant east—west distance in radians, and dlat = the relevant north—south distance in radians. Because of the convergence of meridians toward the poles, the east—west distance was calculated along a parallel midway between those of the two cities. The straight—line distance by way of Macknica Bridge was calculated by using the straight-line formula twice. They had the following appearance: sin(thetai / 2) = ((cos(lati) ' cos(latb) ° 102 sin2( loni - lonb| / 2)) + sin2(|lati — latb} / 2))1/2 and sin(thetaj / 2) = ((cos(latb) ° cos(latj) ° sin2(|lon — lonj / 2)) + sin2(|lat — latjl / 2))1/2, b b in which thetai = the straight—line distance between place i and the bridge in radians, thetaj = the straight—line distance between the bridge and place j in radians, latb = the latitude of the Mackinac Bridge, and lonb = the longitude of the Mackinac Bridge. The right—angle distance between places on Opposite peninsulas involved four formulae, each a modification of the basic straight—line function. They appeared as below: sin(dloni / 2) = (cosZ((lati + latb) / 2) - sin2(|loni - lonbl / 2))1/2, sin(dlati / 2) = sin(llati — latbl / 2), sin(dlonj / 2) = (cosZ((latb + latj) / 2) - sin2(|lonb — lonj| / 2))1/2, and sin(dlatj / 2) = sin(llatb — latjl / 2), in which dloni = the relevant east—west distance between place i and the bridge, dlati = the relevant north—south distance between place i and the bridge, 103 dlonj = the relevant east—west distance between the bridge and place j, dlatj = the relevant north—south distance between the bridge and place j. These computed distances are then standardized by multiplying them by the square root of the density associated with place i. This ith density is calculated by dividing the number of settlements equal to or larger than the ith city by the area of the state. 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XLVI, NO. 6 (May, 1941), pp. 853—64. Vining, Rutledge. "A Description of Certain Aspects of an Economic System," Economic Development and Cul— tural Change. III, NO. 2 (January, 1955), pp. 147—95. Page 30, Page 58, Page 71, Figure II—l. Figure III—4. Figure III—9. ERRATA "discreet" should read "discrete" "SHORLINE" should read "SHORELINE" POPULATIONS are transformed by the factor \/LoglO population STANDARDIZED DISTANCES are trans— formed by the factor Q/standardized distance POPULATIONS are transformed by the factor \/LoglO population STANDARDIZED DISTANCES are trans- formed by the factor 3 \//standardized distance M'Tll'lllllLTIEMIMllllll'lflllwlllllms