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
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Damcggq 1. HQ
0169
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 yintercept 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 IIIII 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.
III5.
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
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of the smallest places (M—type), and on the
periphery itself we find the middlesized 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 (Ktype) 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 RankSize 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. 14554.
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. 30411.
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. 10720.
OEdwin N. Thomas, "The Stability of Distance—Popula—
tionSize 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. 1330.
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. 22233.
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
theoreticallybased "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 sizedistance 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. 111. 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 IIlD), 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 avalue
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. 112.
0 , O ' .
lIJ __________._'___
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 (N2) 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 III4A).
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 sizedistance 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.
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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
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o m
68.. 3 103..
E 295320..
nn_o.~ bw_~.N 00.x... mm... swmd 00.0
beneﬁcis .332... . moi
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3356... 635.. .o N
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conduv
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.8 ES. zmmpmm; 5.: mo... I G .220
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3636... 633.. .o .2 l
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0
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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 III4B), 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 tdistributions,
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 yintercept 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
yintercept, 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 III2). 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 straightline 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.
In order to compare the observed sizedistance rela—
tionship with that which might be expected according to
central place theory, it is necessary to calculate
expected standardized distances. This is done using the
following formula:
in which
Bi = the expected real distance between place i
and its nearest neighbor, place j,
A = the area of the study region (58,216 in the
case of Michigan), and
ni = the number of places equal to or larger than
place i.
This expected distance is standardized in the same manner
as are the observed distances. That is,
' _ . 1/2
standardized Bi. — (1.075\/A 7 ni) (ni / A) ,
or J
standardized Bi = 1.075.
J
BIBLIOGRAPHY
Anderson, Theodore R. "Potential Models and Spatial Dis—
tribution of Population," Papers and Proceedings,
Regional Science Association. II (1956), pp. 175—82.
Arkin, Herbert and Raymond R. Colgon. Tables for Statisti—
cians: College Outline Series. New York: Barnes and
Noble, Inc., 1963.
Artle, Roland. "On Some Methods and Problems In the Study
of Metropolitan Economics," Papers and Proceedings,
Regional Science Association. VIII (1962), pp. 71—88.
Bachi, Roberto. "Standard Distance Measures and Related
Methods for Spatial Analysis," Papers and Proceedings,
Regional Science Association. X (1963), pp. 83—132.
Beckman, Martin J. "City Hierarchies and the Distribution
of City Size," Economic Development and Cultural
Change. VI, No. 3 (April, 1958), pp. 243—8.
Berry, Brian J. L. "The Impact of Expanding Metropolitan
Communities Upon the Central Place Hierarchy," Annals,
Association of American Geographers. L, No. 2
(June, 1960), pp. 112—6.
and H. Gardiner Barnum. "Aggregate Relations and
Elemental Components Of Central Place Systems,"
Journal of Regional Science. IV, NO. 1 (Summer,
1962), pp. 35—68.
, and Robert J. Tennent. "Retail Loca
tion and Consumer Behavior," Papers and Proceedings,
Regional Science Association. IX (1962), pp. 65—105.
and William L. Garrison. "Recent Developments of
Central Place Theory," Papers and Proceedings,
Regional Science Association. IV (1958), pp. 107—20.
and . "A Note on Central Place Theory and
the Range of a Good," Economic Geography, XXXIV, No.
(October, 1958), pp. 304—11.
104
4
105
and . "The Functional Bases of the
Central Place Hierarchy," Economic Geography. XXXIV,
No. 2 (April, 1958), pp. 145—54.
and . "Alternate Explanations of Urban
Rank—Size Relationships," Annals, Association of
American Geographers. XLVIII, NO. 1 (March, 1958),
pp. 83—90.
and Allen Pred. Central Place Studies: A
Bibliography of Theory and Applications. Philadelphia:
Regional Science Research Institute. 1961.
Blome, Donald A. "An Analysis of the Changing Spatial
Relationships of Iowa Towns, 1900—1960," (unpub—
lished Ph.D. dissertation) Iowa City: State Uni—
versity Of Iowa, 1963.
Bogue, Donald J. and Calvin L. Beal. Economic Areas of
the United States. New York: The Free Press of
Glencoe, Inc., 1961.
Brush, John E. ”The Hierarchy of Central Places in
Southwestern Wisconsin," Geographical Review.
XLIII, No. 3 (July, 1953), pp. 380—402.
Bunge, William. Theoretical Geography: Lund Studies in
Geography. Series C. General and Mathematical
Geography, NO. l. Lund: Department of Geography,
The Royal University of Lund, 1962.
Bureau of the Census. "Table 8., Populations of All Incor
porated Places and Unincorporated Places of 1000 or
More, 1960," United States Census of Population, 1960:
Michigan: Number of Inhabitants PC(1)24A. Washing—
ton: United States Department of Commerce, 1961,
pp. 23—50
Carol, Hans. "Hierarchy of Central Functions Within the
City," Annals, Association of American Geographers.
L, NO. 4 (December, 1960), pp. 419—38.
Christaller, Walter. Central Places in Southern Germany.
(Carlislevm Baskin, trans.) Englewood Cliffs:
Prentice Hall, Inc., 1966.
Clark, Philip J. and Francis C. Evans, "Distance to Nearest
Neighbor as a Measure of Spatial Relationships in
POpulations," Ecology. XXXV, NO. 4 (October, 1954),
pp. 445—530
106
Dacey, Michael F. "A Note on Some Number Properties of
a Hexagonal Hierarchical Plane Lattice," Journal of
Regional Science. V, No. 2 (Winter, 1964), pp. 63—9.
. "Two Dimensional Point Patterns: A Review
and Interpretation," Papers, Regional Science Assoc—
iation. XIII (1964), pp. 41—58.
"Imperfections in the Uniform Plane," Discus—
sion Paper No. 4 of the Michigan Interuniversity
Community of Mathematical Geographers. Ann Arbor:
Department of Geography, University of Michigan for
the Michigan Interuniversity Community of Mathe—
matical Geographers, June, 1964. (mimeo.)
. "Analysis of Central Place and Point Patterns
by a Nearest Neighbor Method," Proceedings of the
I.G.U. Symposium on Urban Geography, Lund, 1960:
Lund Studies in Geography, Seriesy Human Geography,
NO. 24. (Knut Norborg, ed.). Lund: Department of
Geography, Royal University of Lund, 1962, pp. 55—76.
Dixon, Wilfrid J. and Frank J. Massey, Jr. Introduction
to Statistical Analysis. New York: McGraw—Hill
Book Co., Inc., 1957.
Geological Survey. "Michigan," National Topographic Map
Seriesl l:250,000. Washington: United States Depart—
ment of the Interior, 1954—64.
Getis, Arthur. "Temporal Land—Use Pattern Analysis With
the Use of Nearest Neighbor and Quadral Methods,"
Annals, Association of American Geographers. LIV,
No. 3 (September, 1964), pp. 391—99.
Gibbs, Jack P. "The Evolution of Population Concentra—
tion," Economic Geography. XXXIX, NO. 2 (April,
1963), pp. 11929.
Hald, Anders. "The Normal Distribution," Statistical
Theory with Engineering Applications: Wiley Publi—
cations in Statistics. (G. Seidelin, transt
New York: John Wiley and Sons, Inc., 1952, pp. 119—43.
Hoover, Edgar M. "The Concept of a System of Cities: A
Comment on Rutledge Vining's Paper," Economic Devel—
opment and Cultural Change. III, No. 2 (January,
1955), pp. 196—8.
107
. The Location of Economic Activipy. New York:
McGraw—Hill Book Co., Inc., 1948.
Isard, Walter. Location and Space Economy. New York:
John Wiley and Sons, Inc., 1958.
King, Leslie J. "A Multivariate Analysis of the Spacing
of Urban Settlements in the United States," Annals,
Association of American Geographers. LI, No. 2
(June, 1961), pp. 222—33.
LOsch, August. The Economics of Location. (W. H. Woglom
and W. F. Stolper, trans.) New Haven: Yale Uni—
versity Press, 1954.
. "The Nature of Economic Regions," Southern
Economic Journal. V, No. 1 (July, 1938), pp. 71—8.
"Michigan: Index to Cities, Towns, Counties, Transporta—
tion Lines, Airports, Banks, and Post Offices,"
Commercial Atlas and Marketinnguide: 1962. Chicago:
Rand McNally and Co., 1962, pp. 233—7.
Michigan: Official Highway Map: 1961. Lansing:
Michigan State Highway Department, 1961.
Morrill, Richard L. "The Development of Spatial Distri—
bution of Towns in Sweden: An Historical—Predictive
Approach," Annals, Association of American Geographers.
LIII, NO. 1 (March, 1963), pp. 1—14.
"Simulation of Central Place Patterns Over Time."
Proceedings of the I.G.U. Symposium on Urban Geog—
raphy, Lund, 1960: Lund Studies in Gepgraphyi Series
B. Human Geographynyo. 24. (Knut Norborg, ed.),
Lund: Department of Geography, Royal University of
Lund, 1962, pp. 109—20.
Nelson, Howard J. "Some Characteristics of the Population
of Cities in Similar Service Classifications,"
Economic Geogrgphy. XXXIII, No. 1 (January, 1957),
pp. 95—108.
"A Service Classification of American Cities,"
Economic Geggraphy. XXXI, No. 3 (July, 1955), pp.
189—210.
108
Philbrick, Allen K. "Principles of Areal Functional
Organization in Regional Human Geography," Economic
Gepgraphy. XXXIII, NO. 4 (October, 1957), pp. 299—336.
"Areal Functional Organization in Regional
Geography," Papers and Proceedings, Regional Science
Association. III (1957), pp. 87—98.
Schnor, Leo F. "The Functions of MetrOpOlitan Suburbs,"
Papers and Proceedings, Regional Science Association.
V (1959), pp. 272—9.
"Satellites and Suburbs," Social Forces. XXXVI,
No. 4 (December, 1957), pp. 121—7.
Stewart, Charles T., Jr. "The Size and Spacing of Cities,"
Geographical Review. XLVIII, No. 2 (April, 1958),
pp. 222—45.
Thomas, Edwin N. "The Stability of DistancePopulation—
Size Relationships for Iowa Towns from 1900—1950,"
Proceedings of the I.G.U. Symposium in Urban Geog—
raphy, Lund, 1960: Lund Studies in Geography.
Series B. Human Geography, NO. 24. (Knut Norborg,
ed.), Lund: Department of Geography, Royal University
Of Lund, 1962, pp. 3—12.
. "Toward an Expanded Central Place Model,"
Geographical Review. LI, NO. 3 (September, 1961),
pp. 400—11.
Ullman, Edward L. "A Theory of Location for Cities,"
American Journal of Sociology. 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
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