A MULTIVARIATE ANALYSIS OF THE SPACING
OF CENTRAL PLACES IN COUNTY TIPPERARY
PATRICK N. O'FARRELL
The Queen's University of Belfast
INTRODUCTION
Geography is concerned with the variation of phenomena over space
and therefore it is of interest to the geographer to attempt to explain the
distributional pattern of central places. Walter Christaller developed a
general deductive theory to explain the size, number and distribution of
towns which was derived from a series of basic assumptions.1 The nature
of this classic model, together with its assumptions, has already been
discussed by several authors and no detailed account of it will be included
here.2 In Christaller's theoretical landscape the central places, supported
exclusively by tertiary activity, are regularly distributed over an isotropic
surface serving evenly distributed, identical, consumers. On the basis of
his threshold assumption—that groups of key functions have similar
threshold values—Christaller postulated a functional and demographic
hierarchical class system of central places with each class possessing
specific groups of central functions and being characterised by a discrete
population level of its centres. An optimum pattern, without excess
profits, can only be achieved for functions whose market areas are equal
to that of the market of the initial highest-order function or equal to onethird, one-ninth, one-twenty-seventh (etc.) that size.3 The classes are
arranged in a hierarchy such that central places of high-order classes
perform all the functions of lower-order centres plus a group of functions
distinguishing them from the lower-order centres. Consequently all
distances between neighbouring places on the same hierarchical level are
constant, as they have equal-sized market areas for their highest-order
goods with the same number of consumers located in those market areas.
The straight-line distance, in miles, from a specific centre to its nearest
neighbour of the same or larger centrality was used in this study as a
measure of the location of a central place relative to other places4 (Table i).
As Thomas and others have done, it is reasonable to use the distance to the
nearest neighbour of the same or larger size, whenever a centre of higher
centrality is located closer to another centre than its nearest neighbour of
the same size.5 This is also consistent with one of Christaller's principal
428
postulates already outlined, namely that each higher-order central place
performs all the functions of lower-order central places. Assuming
that consumers are highly rational and are interested in minimising travel
costs, the distance between a given centre and its nearest neighbour of
equal or greater centrality is a meaningful statistic.
Since the publication of Christaller's work, the majority of studies
concerned with the spacing of urban centres have continually emphasised
the existence of discrete population-size groups and average distances.6
Thomas, however, has shown that empirical evidence exists to demonstrate
that population-sizes of central places are in general unimodally distributed.7 Adequate testing of Christaller's formulations calls for an
explanation of spatial patterns ; it also requires that spacing be regarded
as a continuous variable. In Christaller's theoretical model, with homogeneous purchasing power and rational consumers, population size or
centrality accounts for the total variation in spacing but when tested
against reality other factors must be incorporated into an explanatory
model. Also, if distance is expressed as a continuous function, the generality imposed on an observation by adopting a taxonomic system is avoided
and the precision of each individual measurement is retained. King and
Vuicich considered spacing as a complex function and used multiple
correlation and regression methods for their analysis of the relationships
between the spacing of places in the United States and selected physical,
social and economic factors.8
The present work was conducted within the framework of a general
central-place study and has the object of explaining the nature of the
relationship between the spacing of central places and three independent
variables : the population of the centres, centrality of the centres and
population density.9 Statistical evidence will be presented which will
tend to support the deductive formulations from the theoretical model.
The use of simple and multiple regression analysis will permit consideration
of the influences, either independently or simultaneously, of the independent variables. Most investigations concerned with the spacing of central
places have used the population-size of centres as one of the variables.
Olsson and Persson in their study of the spacing of places in Sweden
quantified the size of a centre, not as its population, but as a functional
measure, an index of centrality,10 derived by Persson.11 Both of these
approaches are employed in this study. As Christaller's model is applicable
only to the tertiary sector the inclusion of a centrality measure, based on
tertiary functions, represents a more valid test of the theory than population-size and should raise the level of explanation. Lösch and Isard
empirically modified the classic model and brought it closer to reality by
incorporating variations in population density and thereby analysing
429
-markets of different areal extent but containing equal numbers of consumers.12 Therefore, central places of the same order are not separated
by constant distances but by equal numbers of consumers. However,
since Christaller 's isotropic plain is populated by evenly distributed
consumers, the ideas of Lösch and Isard represent simply the removal of
the assumption of even population distribution. Olsson and Persson
analysed the spacing of central places initially with regard to and secondly
without incorporating the number of inhabitants in the service areas.13
The major limitation of such an approach is that of defining precisely the
service area of a.centre; trade areas differ for every function and in
addition are seldom discrete but rather represent continuous density
surfaces of consumer movement with associated probabilities decreasing
with increasing distance from the centre. Variations in population
•density were accounted for in the present study by incorporating population density as one of the independent variables in the multivariate model.
DATA COLLECTION AND MEASUREMENT OF CENTRALITY
In the study area, Co. Tipperary, the total number of central functions
of all places performing more than one function was recorded in the field.
Data relating to the number of establishments, number and type of
functional units and other components of central-place systems were also
collected.14 Centrality values for each function were calculated from a
simple formula, which is analogous to that derived by Davies :15
Ca = K(ioT a )- 1
Where Ca = centrality of a function (a) ; Ta = number of functional
units of (a) in the study area ; K is the total number of functional units of
all kinds in the area (i.e., a constant representing total demand) and 10
is a constant.18 A centrality value may therefore be calculated for a
central function (a), and a centre providing only functions (a) possesses
the corresponding degree of centrality Most places have several central
functions located in varying numbers of units and as a measure of the
relative importance of the settlement a Functional Index (centrality
measure for a central place) may be calculated so that :
k
(F.I.)j = äl ("1 A + n2J C2 + n3J C3 + "¡I Ci + ""i Ck) ' ' ' • M
where (FJ.)j = Functional Index of the j t h place of the series (FJ.j,
F.I2, F.I3 .. F.Ij ..) ; Cj = a centrality value of a central function of the
series (C1( C2> C3 .. Cj ..) ; and niS is the number of functional units of the
Cj01 function in the j " 1 place. This data was organised into a matrix in
which the columns were central places, ranked according to centrality
430
(Functional Index), and the rows were the centrality values of the functions
present, weighted, as shown above, according to number of functional
units. Spearman rank correlation coefficients of association were calculated.
between pairs of central places in the matrix and a grouping test applied
to the results.17 Four grades of centres were defined and classified as five
Towns, seven Minor Towns, forty Villages and ninety-three Hamlets.
As none of the hamlets, with less than io central functions and a maximum
Fig. 1. The distribution of central places in the study area.
431
of 13 functional units, have any trade areas with a radious of greater than
half a mile18 they were eliminated from this study and attention focused
•on the spacing of the fifty-two higher-order places, ranging in size from
Clonmel (11,457) to Kilteely (55) (Fig. 1).
FORMULATION AND TESTING OF HYPOTHESES
In formulating the following hypotheses relating to the degree of
association between the dependent variable, distance, and certain independent variables, it should be emphasised that two of the variables
tested, centrality19 and population-size, are derived from the theoretical
model. The other independent variable, population density, is one that
appears intuitively to be relevant. The theoretical rationale supporting
•the first two hypotheses has been outlined in the introduction to this
paper. First, as a consequence of the hierarchical class structure of
•centres, with each class being characterised by a discrete population level
of its centres, the distance between two towns of the same population
•size should reflect the population of the two towns. It may therefore be
hypothesised that there is a positive relationship between the distance
separating a given centre from its nearest neighbour of the same or greater
population and the population of the given centre. Christaller's concept of
•discrete population levels of centres is not consistent with his basic
assumption of even population distribution, so the population-size
variable is only being tested to permit comparison with other empirical
"findings and will not be incorporated into the multivariate model.
Secondly, as a consequence of the hierarchical class structure of centres,
"with specific tiers of the functional hierarchy being associated with
•characteristic groups of central functions, it follows that centres of equal
•centrality are the same distance apart. It may therefore be hypothesised
that there is a positive relationship between the distance separating a
given centre from its nearest neighbour of equal or greater centrality and
"the centrality of the given centre (Table 1).
Thirdly, where population densities are high, the distance between two
centres of equal rank should be less than the distance separating two
centres of equivalent rank in an area of low population density.20
Therefore one may hypothesise an inverse relationship between density of
population and distance.21
TESTING OF HYPOTHESES
The three hypotheses were tested initially by means of simple correlation
and regression analysis ; the regression of distance, being the dependent
432
TABLE 1
POPULATION, CENTRALITY AND DISTANCE DATA*
Popula- Centraltion
ity
Distance
Clonmel
11,457
3,496
26
Ballyporeen
276
Thurles
6,949
1,948
23
Moneygall
296
57
4
Tipperary
4,507
1,760
21
Portroe
126
55
6.5
Nenagh
4,609
1,674
21
Bansha
223
53
4.3
Carrick-on-Suir
4,874
1,103
12
Silvermines
224
51
5
Roscrea
3,511
1,051
18
Piltown
418
46
3.5
Cashel
2,682
778
11.5
Shinrone
359
49
6
Caher
1,740
642
9
Newcastle
90
40
4
Templemore
2,031
577
8
Kilcommon
60
37
7.5
Borrisokane
751
340
8
New Inn
154
37
5
Fethard
997
320
8
Drangan
98
33
3.5
7
Golden
172
31
4
4
Central
Place
Killenaule
583
272
Central
Place
Popula- Centraltion
ity
60
Distance
4.5
Clogheen
572
202
7.5
Rear Cross
72
30
Newport
584
184
7
Hollyford
86
30
4
Doon
354
158
4
Oola
319
27
5
Borrisoleigh
470
156
6
Templetuohy
162
25
5
Cloughjordan
449
138
5
Littleton
338
24
4
Cappàwhite
311
129
7.3
Gortnahoe
83
24
3
Mullinahone
274
95
5
Ballymacarbery
113
23
4.4
Ardfinnan
529
94
5
Rathcabban
92
22
5
Ballmgarry
193
88
5.5
Templederry
75
19
5.5
GalbJly
246
86
3
Pallas Grean
94
19
3
Borris-in-Ossory
230
83
6
Lorrha
81
19
4
Toomyvara
246
80
6.5
Kilteely
55
18
3
93
74
5.3
Kilsheelin
157
15
5
161
63
6.5
Lisvernane
63
17
3
Dundrum
Emly
*Distances are straight-line distances in miles from specific centres to their nearest neighbours of equal or larger centrality.
433
variable, was considered on each independent without reference to the
effect of other independent variables. A linear product correlation model
was employed to test the degree of association between distance and
population size and yielded a correlation coefficient of 0.924. However,
two important assumptions of the regression model which require to be
satisfied are : (i) that the errors in the 'y' data be normally distributed
with the means of the distribution located along a line and (ii) that the
variances of the errors be constant. Transformations of the variables were
tested and the most normal and homoscedastic transformation functions
were common logarithms of both variables. After double-log transformation the correlation was 0.861 with the regression of distance on population
size having the form (where Y = distance and P — population size) i22
Log Y = .1174 + 0.3317 Log P
(2)
The coefficient of determination was 0.742 and thus the first hypothesis,
that the distance between a given centre and its nearest neighbour of equal
or greater population is a function of the population size of the given
centre, is supported.
In testing the relationship between distance and centrality logarithms
were again used as input data for the regression model to ensure greater
homoscedasticity and normality of errors, notwithstanding a higher
correlation coefficient when both variables were expressed as a linear
function.23 The coefficient of correlation was 0.886, thus verifying the
second hypothesis ; the relationship took the form (where Y = distance
and Xj = centrality)24 :
Log Y = 0.1174 + 0.3317 Log Xi
(3)
This revealed that there is no great difference between the correlations
obtained for population-size and centrality when tested against distance
to nearest neighbour (r values of 0.861 and 0.886 respectively). It is
logical therefore that there should be a significant correlation between
the population-size of centres and their centrality and testing this relationship yielded a correlation coefficient of 0.945 and a regression equation of :
Log P = 0.8332 + 0.8433 Log Xj
(4)
where P = population size and Xx = centrality.
The third hypothesis, that there is an inverse relationship between the
distance to nearest neighbour of greater centrality and population density,
is not substantiated by the zero order correlation coefficient of 0.2273.
However, a more valid statement of thé degree of explanation due to
population density is obtained when the partial correlation coefficient
between distance and population density is examined, holding centrality
of centres constant. This yielded a partial correlation coefficient, where
X2 = population density, of r yx .X =-0.1717 (Tables). A degree of
434
multicollinearity between the independent variables, r x X . Y = 0.3096,
was sufficient to convert a negative partial correlation coefficient into a
positive zero-order one when the centrality variable was not held constant.
This serves to emphasise the importance of experimenting within a
multivariate framework. The relationship between distance and population
density was of the form (where Y = distance and X2 = population
density) :
Log Y = 0.0632 + 0.4287 Log X2
(5)
Table 2 confirms that the centrality measure is clearly the most important variable—explaining approximately 78% of the variation in
the dependent variable—and thus supporting one of Christaller's principal
formulations. The multiple regression analysis adds weight to this
conclusion for when all the variables are considered simultaneously
an R = .889 value is obtained and thus it is possible to account for 79%
of the variation in the dependent variable (Table 2).25 This represents
TABLE 2
THE CORRELATION OF Y ON Xx AND X 2
Variables (Log of all variables are used)
Dependent Variable :
Y
Independent Variables :
Xx
X2
Distance of a centre, a, from its nearest neighbour,
b, of equal or larger centrality.
Centrality of a given centre—calculated from
Equation (1).
Population density between a centre, a, and its
nearest neighbour, b, of equal or larger centrality.
Zero-order Correlation Matrix
X
Y
I
.886
Xx
Coefficients of Determination Matrix
"V*
Ä
2
"V
l
Ä
.227
Y
.341
Xi
R
=
.8891
Rj
=
.7905
.7841
=
.883,
r8
=
.7791
=
—0.1717,
r2
=
.0295
=
.3096,
r2
=
.0959
435
.0517
.1166
Partial Correlation Coefficients
r
"V"
¿H
only a marginal improvement over the partial correlation analysis
(r2YX .X = : -7791) an< i demonstrates the predominance of the centrality
1
2
variable in explaining the spatial pattern. The partial correlation coefficients show that the other independent variable, population density,
accounts for only 3 % of the variation in the dependent variable ; these
coefficients also evidence a small degree of multicollinearity between the
two independent variables: r 2 x x . Y = - ° 9 5 9 (Table 2). The partial
correlation coefficients confirm that the strongest relationship is between
RESIDUALS
from
Logy—3S7J+-3H6 logX,+2992 LogX2
A.
•
> +az
+J-1 -
A
0
+0-2
—+0-1
O
0
•
-01
-
•
-0-2
-IM
0-2
>-0-3
0-3
'
B0UN04RV OF STUDY AREA
MUES
Fig. 2. Residuals from the multiple-regression equation.
436
distance and centrality—accounting for almost four-fifths of the variation.
The analysis was carried a stage further by mapping the residuals from
the multiple regression equation
Log Y = -.3978 + .3426 Log Xx + .2992 Log X2
(6)
The distribution of errors on the resulting map was random with no
tendency for positive or negative residuals to cluster in distinct areas
(Fig. 2). It may therefore be concluded that there is probably no localised
factor accounting for a proportion of the unexplained variation.
CONCLUSION
This paper has demonstrated that those parts of the Christaller model
concerned with the spacing of central places can be applied to the study
area, without modification. The independent variable, centrality, derived
directly from the theoretical model, accounted for four-fifths of the
variation in the dependent variable. Incorporation of population density
into the multivariate model produced a marginal improvement in the
proportion of explained variation. However, the fact that 21% of the
variation remains unexplained by centrality and population density
shows that the spacing of central places, like other geographic phenomena,
is the result of complex inter-relationships between many variables.
The findings come from too small an area (2,500 square miles) to have
general application for the country as a whole. Work by the author,
currently in progress, is concerned with explaining the spacing of centres
with more than 1,500 inhabitants over the Republic of Ireland and
preliminary results indicate that spacing becomes more complex when
this larger area is considered. Other variables are being tested and added
to a multivariate model in this attempt to explain the spacing of urban
places throughout the country. The finding of greatest heuristic value
from the present study is the fact that a measure of the functional content
of a centre has proved to be predominant in contributing to an explanation
of the variation in the spacing of central places. This is confirmed by the
findings of Thomas in Iowa and Olsson and Persson in Sweden, although
when an area as large as the United States is considered spacing becomes a
more complex function.26 The results of the regression analysis must be
supplemented by an acknowledgement of stochastic and historical factors
in the location of central places and a realisation that they may probably
account for a proportion of the unexplained variation. More empirical
evidence is needed to illuminate regional differences but the few results
so far seem to indicate that some measure of urban-size function is the
most important independent variable—thus confirming Christaller's
ideas. Further empirical evidence may establish this as a universal factor
437
affecting the spacing of central places and assist in the development of
general laws. This objective must be regarded as being more important to
geography than the identification of regional differences and it is hoped
that this paper has made a small contribution towards the attainment of
such a goal.
REFERENCES AND NOTES
1
W. Christaller, Central Places in Southern Germany, translated by C. W. Baskin,
New Jersey, 1966, from Die Zentralen Orte in Süddeutschland, Jena, 1933.
2
See for example : B. J. L. Berry and A. Pred, Central Place Studies : A Bibliography of Theory and Applications, Philadelphia Regional Science Research Institute,
1961 ; E. von Böventer, " Towards a United Theory of Spatial Economic Structure,"
Regional Science Association, Paper X, Zurich Congress, 1962, 163-191 ; and B. J. L.
Berry, Geography of Market Centres and Retail Distribution, New Jersey, 1967.
3
E. von Böventer, " Towards a United Theory of Spatial Economic Structure,"
Regional Science Association, Paper X, Zurich Congress, 1962, 169.
4
It was shown, for a sample of towns, that there was a high correlation between
straight-line distance and road distance. Also nearest neighbours outside the study
area were included where appropriate.
5
E. N. Thomas, " The Stability of Distance Population-Size-Relationships for
Iowa from 1900-1950," Proceedings of the I.G.U. Symposium in Urban Geography,
Lund 1960, Royal University of Lund, 1962, p. 21.
6
See for example : J. E. Brush, " The Hierarchy of Central Places in Southwestern
Wisconsin," Geographical Review, Vol. 43, 1953, 380-420 ; A. Lösch, Die räumliche
Ordnung der Wirtschaft, Jena, 1944, translated by W. H. Woglom and W. F. Stolper
as, The Economics of Location, New Haven, 1954 ; and J. E. Brush and H. E. Bracey,
" R u r a l Service Centres in Southwestern Wisconsin and Southern England," Geographical Review, Vol. 45, 1955, 559-569.
7
E. N. Thomas, " Toward an Expanded Central-Place Model," Geographical
Review, Vol. 51, 1961, 400-411.
8
L. J. King, A Multivariate Analysis of the Spacing of Urban Settlements in
the United States," Annals of the Association of American Geographers, Vol. 51, No.
2, June 1961, 222-233 ; and G. Vuicich, An Analysis of the Spacing of Small Towns
in Iowa, unpublished Ph.D. thesis, Dept. of Geography, State University of Iowa,
1960.
9
P. N. O'Farrell, Central Place Analysis in County Tipperary, Unpublished Ph.D.
thesis, Dept. of Geography, Trinity College Dublin, 1967.
10
G. Olsson and A. Persson, " The Spacing of Central Places in Sweden," Regional
Science Association : Papers, XII, Lund Congress, 1963, 87-93. The index of centrality used did not take all central functions into consideration and was obtained from
the formula
1.000.P
—
. Ic
100
where C= centrality ; S p = sales of durables in the place ; P/100 = percentage of
per capita income available for durable goods; and Ic = per capita income in province
in which a central place is located.
11
A. Persson, Handelsorternas Hierarki, Fil. lic. thesis, Department of Geography,
University of Uppsala, 1963.
12
A. Lösch, The Economics of Location, op.cit. ; and Walter Isard, Location and
Space-Economy, New York, 1956.
13
G. Olsson and A. Persson, " The Spacing of Central Places in Sweden," op.cit.
(note 10), p. 90.
14
The following definitions are used in this study :
Central function : Any type of tertiary activity, such as chemist, public house
or newsagent, regardless of size.
Central place : A location at which one or more central functions are performed ;
438
a location from which tertiary goods and services are provided for the surrounding area.
Establishment : The premises in which an activity is performed, e.g., the surgery
of a doctor or the building in which an insurance office is located.
Functional unit : Each individual occurrence of a central function constitutes
one functional unit, which may also be defined as that part of a building performing a single central function, in the event that more than one function is performed in that building.
15
W. K. D. Davies, " Some Considerations of Scale in Central Place Analysis, "
Tijdschrift Voor Econ. En. Soc. Geografie, Nov./Dec., 1965, p. 223. The centrality
measure devised by Davies was :
t
C
=
— .100
T
where C = location coefficient of function t, t = one outlet of function t, T = t o t a l
n u m b e r of outlets of function t in t h e whole system.
16
P . N . O'Farrell, " A Proposed Methodological Basis for t h e D e t e r m i n a t i o n of t h e
Centrality a n d R a n k of Central Places," Administration, Vol. 16, N o . 1, 1968, 17-32.
17
Ibid., p . 28
18
T h e conclusion t h a t t h e hamlets h a d t r a d e areas w i t h a radius of less t h a n half
a mile w a s reached after a n investigation of consumer travel b e h a v i o u r based u p o n
t h e results of 2,000 interviews representing a 1 0 % systematic r a n d o m sample of rural
households t h r o u g h o u t t h e s t u d y area. T h e respondents were asked questions relating
t o place a n d frequency of shopping for hardware, clothing, footwear, furniture a n d
for obtaining banking, chemist, doctor, dental, veterinary a n d solicitor services. Of
interest also were centres normally frequented on visits t o t h e cinema a n d t h e places
used t o obtain m o s t of t h e goods a n d services. See P . N . O'Farrell, Central Place
Analysis in County Tipperary, op.cit. (note 9), p p . 167-206.
19
T h e t e r m ' centrality ' hereinafter refers t o t h e centrality of a central place,
calculated from (1).
20
Localised population-density increase adjacent t o central places, is a phenomenon
t h a t occurs within one quarter t o one half a mile of t h e largest centres only (greater
t h a n 1,500 population). These centres a r e far enough a p a r t (12-26 miles) for t h i s
feature t o b e regarded as having a negligible affect on overall population density.
I n addition t h e limits of centres were defined as t h e outer limit of continuous built-up
area a n d t h u s included t h e ' environs ' of urban districts.
21
D e n s i t y of population w a s quantified a s t h e m e a n density of all t h e District
Electoral Divisions lying along a straight line between a given centre a n d a centre
of equal or greater centrality.
22
This m a y b e compared w i t h a correlation coefficient of 0.78 obtained from I o w a
d a t a , 1950. See, E . N . Thomas, " T h e Stability of Distance-Population-Size Relationships for Iowa from 1900-1950," op. cit. (note 5), p . 27.
23
F o r t h e t w o variables expressed as a linear function t h e correlation coefficient
was 0.947. I n t h e case of all correlation a n d regression analyses carried o u t a n u m b e r
of transformations h a v e been tested for homoscedasticity a n d normality of error
distribution a n d t h e transformation w i t h t h e greatest degree of error-normality a n d
t h e m o s t homogeneous variance h a s been used i n all cases.
24
This correlation of 0.886 between distance a n d centrality m a y b e compared w i t h
one of 0.31. obtained for Swedish centres. See, G. Olsson a n d A . Persson, " T h e
Spacing of Central Places in Sweden," op.cit. (note 10), p . 88.
25
This may be compared with an explained percentage of 25 % for a random sample
of two hundred towns in the United States. See, L. J. King, " A Multivariate Analysis
of the Spacing of Urban Settlements in the United States," op.cit. (note 8), p. 228.
26
E. N. Thomas, "The Stability of Distance-Population-Size Relationships for
Iowa from 1900-1950," op.cit. (note 5), and G. Olsson and A. Persson, " The Spacing
of Central Places in Sweden," op.cit. (note 10).
ACKNOWLEDGMENTS
The author wishes to express his thanks to Miss J. Orr for her help and to Dr.
' F. W. Boal and Mr. M. A. Poole, both of the Department of Geography, Queen's
University Belfast, for their advice and suggestions.
439