Download spatial ordering in social area typology

SPATIAL ORDERING
IN SOCIAL AREA TYPOLOGY
Larry B. Bubacz
Mr. Bubacz has an M.A . in Geography from the University of Wyoming. He has taught Geography in the
public schools of Cloquet , Minnesota and is now a
planner with a consulting firm in Minneapolis.
The primary purpose of this paper
is to provide tentative evaluation of
the general applicability of two statistics, the weighted mean areal center
and standard distance, to the question
of spatial ordering in social area analysis. More specifically, does the use of
these statistics identify whether social
phenomena tends to conform to either
a concentric (do ughnut) ring or a sector (p ie-shaped) pattern? The weighted
mean areal center is defined as the
" balancing point" or " center of gravity" for a population distributed in twodimensions (x-y coordinates) which is
comparable to the arithmetic mean of
the co nventional linear frequency distribution .' The descriptive spatia l statistic, that is, sta ndard distance, which
is comparable to the standard deviation in one-dimension is designed to
measure the average dispersion in distances of all points from the mean areal
center.2
Since Burgess first posted the concentric growth of cities, and cited Chicago as an example substantiating such
a hypothesis, numerous studies have
focused on the spatial arrangement or
ordering of cultural phenomena in
cities.3 In 1939, Homer Hoyt studied
nineteen selected cities and concluded
that similar rental values in cities were
located along axial patterns of growth.
As a result, Hoyt formulated the hypothesis thatsocio-economic status followed a sector pattern. 4 For a number
47
of years these two studies provided the
basis for characterizing patterns of urban growth in expanding modern, industrialized cities. More than a decade
passed before these hypotheses were
subjected to rigorous examination.
In 1949, Shevky and Williams introduced the concept of social area analysis, a methodology for classifying
demographic data in census tracts. s
The concept was later extended in
1955 when Shevky and Bell formulated
the theoretic as well as computational
procedures for the analysis of social
area typology.6 In this study, Shevky
and Bell developed three indexes: (a)
social rank which is the result of analyzing thechangingdistribution of skills
and degree of employment specialization as well as education; (b) urban
rank which is based on fertility rates
along with measures of house types and
women in the work force ; and (c) segregation which is composed of changes
in the ethnic composition of a population . These indexes were derived from
census data and serve as the basis for
the differentiation and stratification of
the urban-cultural fabric in modern,
industrialized society. However, the
three indexes were not universally
accepted by American sociologists.
Faced with criticism, Bell in 1955 advanced alternative designations for the
three basic constructs. These were : (a)
economic status for social rank ; (b)
family status for urban rank ; and (c)
ethnic status for segregation .7
With the advent of computer technology, it has been possible to extend
the number of socio-economic variables examined within the framework
of social area typology. An example of
this technique is found in urban factorial ecology. Factorial ecology developed from the application of factor
analysis to an extended set of socioeconomic variables including those
48
originally isolated by Shevky and Bell.
Major studies employing factorial
ecology have been conducted by Berry
and Rees on Calcutta, and by Murdie
on Toronto.s
Spatial Components of Social
Area Analysis
The principal areal statistical unit
utilized in the Shevky-Bell typology is
the census tract. By aggregating census
tracts having similar index scores, or
typologies, a city can be analyzed for
patterns of regional homogeneity. By
mapping the social area indexes, spatial ordering of concentric, sectorial,
or multiple nucleation arrangements
can be established amongst the various
census tracts. Though much effort has
been given over to emphasizing social
areas in various Standard Metropolitan
Statistical Areas within social area
analysis, the spatial component has not
received similar consideration. One of
the few studies discussing and testing
the spatial model of urban growth and
structure was authored by Anderson
and Egeland in 1961.9 These two authors studied the residential areas of
four cities (Akron and Dayton, Ohio;
Indianapolis, Indiana; and Syracuse,
New York) . The cities were selected on
the basis of : (a) having populations between 200,000 and 500,000 in 1950; (b)
outlying territory that was tracted; and
(c) circular shape and topographic uniformity.'o Anderson and Egeland calculated the indexes of social and urban
rank for census tracts in each of four
arbitrarily delineated sectors. An analysis of variance suggested that social
rank, or economic status, organizes
itself within a sector pattern while family status, or urban rank, was spatially
ordered in a concentric manner.
Another study dealing with the spatial components of social area analysis
was by McElrath on Rome." He first
applied social area analysis to 1951
census data in co njunction with an
analysis of variance upon the Shevky
indexes. McElrath concludes his study
by noting that economic status is patterned sectorially and family status is
patterned concentrically.
In 1969, Murdie published a factorial ecology on metropolitan Toronto.12 As in previous studies, Murdie
concluded that economic status displayed a dominantly sectorial pattern
and family status was distributed spatially by concentric zones.
Statements pertaining to the various
spatial aspects of the Shevky-Bell typology have also been discussed by
Berry and Rees.H Berry and Rees, quoting from Berry's " Internal Structure of
the City" (1965) note that : " It becomes
increasingly evident that each of these
dimensions (Shevky's) captures the essential features of one of the classic
spatial models (socio-economic status
- Hoyt, family status-B urgess ... ).14
These studies have provided the incentive for the following analysis using
spatial statistics to assess the existence
of ordering within metropolitan centers.
Experimental Design and Data
The analysis discussed herein focuses
upon testing two hypotheses concerning spatial ordering in socia l area typologies. First, concentric zonation wi ll
be indicated when: (a) the weighted
mean areal centers of each subpopulation tend to concide; and (b) the standard distances on the other hand will
show significant increments as a test
population is further from the center.
Second, sector arrangements will be indicated when: (a) in an idealized sector the weighted mean areal center
(WMSC) will be found toward the midpoint of the sector, rather than all
WMAC being clustered together at the
center of the entire population; and
(b) the standard distances will have approximately the same values.
In order to assess the above questions of spatial ordering, a test statistic
with known characteristics of behavior
under a variety of previously specified
THEORETICAL MODEL
CONCENTRIC
SECTOR
St. Dev.
0 - 2
o
m
2-3
IM1 3-5
Figure 1
49
conditions must be available. Therefore, an idealized model was first developed to test for the characteristic
behavior of the weighted mean areal
center and standard distance. The results from the theoretical model can
be used comparatively with output
from the actual data.
The first theoretical model consisted
of three idealized concentric circles,
whereas the second model was based
centers and standard distances were
calculated via an XDS Sigma 7 computer in Fortran IV language (Tab le
1 ).15
Examination of Table 1 reveals several properties which characterize behavior of the weighted mean areal centers and standard distances for the two
idealized models. As anticipated the
weighted mean areal centers for the
concentric model are locationally sim-
Table 1
Weighted Mean Areal Centers and Standard
Distances for Theoretical Models
Model
No . of Tracts
Weighted MAC
(x-y coordinates)
Standard Distance
Concentric
Circle 1
Circle 2
Circle 3
20
49
86
2.88,
2.88,
3.03,
3.27
3.28
3.32
.704
1.614
2.570
Sector
Sector 1
Sector 2
Sector 3
50
50
55
3.94,
3.73,
1.48,
4.76
1.82
3.40
1.326
1.344
1.440
on three similarly idealized sectors
(Figure 1). A grid overlay was then superimposed over each of the two
models. The tracts within the concentric model and sector model were assigned random values ranging from 0
to 500. Circle # 1 and sector # 1 were
assigned values of 0-200; circle # 2 and
sector #2 received values of 200-300 ;
and ci rcle # 3 and sector # 3 were
loaded with values 300-500. The nature of the three above categories was
established so that it approximately
corresponds to the field data tested.
After the index values had been randomly assigned to the two hypotheti cal models, the weighted mean areal
50
ilar, that is, their x-y coordinate values
tend to coincide. It is quite clear that
in an idealized concentric model each
weighted mean areal center would be
located similarly. It can also be noted
in Table 1 that there are progressive
increments in the standard distances
for concentric organization.
In the case of the sectors, however,
just the opposite condition exists for
the weighted mean areal centers and
the standard distances. In the idealized
sector model, the weighted mean areal
centers are found at different locations
while the standard distances are quite
similar.
It should, therefore, be possible to
COLORADO
SPRINGS
CENSUS TRACTS
_ _ _ _ _ NO.
T'hl'MP1'PIU'\.oft.hl",por\.~
pC "!!!Il. um and ",",11M·
VIMl "'port PlClp . n
1960
INSET A - COLORAOO SPRINGS CITY AND ADJACENT AREA
C'PMI' Trw;!.t
Figure 2
S1
Table 2
Social Area Index Values
Colorado Springs Census Tracts
Tract
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Mean
Standard
Deviation
52
Social Rank Index
88.5
83.4
77.6
81 .7
87.4
86.6
87.8
88.7
84.5
86.7
74.0
76.9
66.7
68.5
73.3
85.1
76.5
83 .6
64.1
84.2
69.6
75.5
77.4
56.0
76.2
69.4
70.5
68.8
66.2
84.2
78.34
8.147
Urbanization Index
21 .5
25.2
33.7
36.1
30.3
33.4
24.7
50.1
45.2
52.3
36.4
34.0
40.2
38.2
39.1
70.6
40.1
46.1
29.2
19.3
31.0
47.6
59.9
33.4
33.4
27.5
30.3
24.7
28.8
42.7
34.26
11.165
ACTUAL CONCENTRIC
•
~
D
1
to >3 s . d.
0
to
1 s.d .
-2
to
0 s.d.
20
0
""
2.
tbl._ h piU" ~.t t.M n""rt ~
"""w' t Up''"''"'!!''''''
eM' "por\- MtJI - H
1*
--- - - _.
INSET A· COLORADO SPRINGS CITY AND ADJACENT AREA
CrpytreHf
Figure 3
53
4
ACTUAL SECTOR
..
0
(Nwl
-1
0-3
at "'!!!leU.
I.""""t.rlllen,.r"~
til!
"M' . ' In
".'.1>1[\ ""'}.19
r...,.. trw"
INSET A - COLORADO PRINGS CITY AND ADJACENT AREA
Figure 4
54
groups of Colorado Springs are represented by relatively small populations.
In order to have some basis for areally classifying or regionalizing the
social area data, the mean and standard
deviation (s.d.) for social rank and urbanization were calculated . Data for
social rank ranged from - 3 s.d. to +2
s.d. and were arbitrarily grouped into
three categories (0 to +2 s.d.; - 1 to 0
s.d.; and -3 to -1 s.d.). The urban
rank index ranged from - 2 s.d. to > 3
s.d. and was also grouped into three
categories (1 to > 3 s.d.; 0 to + 1 s.d.;
and -2 to 0 s.d.). The census tracts
for both indexes were then mapped in
terms of standard deviation units (Figures 3 and 4) .
In order to ascertain whether significant differences in locations of the
weighted mean areal centers existed,
t-tests were run on both the theoretical and Colorado Springs data. A test
for equality of variances was also performed on the standard distances. The
results of these analyses appear in
Table 3.
test for the pattern of concentric zonation by first calculating the weighted
mean areal centers and the standard
distances. If both of the above stipulated conditions are met, that is, 10cationally similar weighted mean areal
centers and progressively increasing
and significantly different standard distances, then the spatial ordering of the
units should be as described by concentric zonation. Alternatively, if the
weighted mean areal centers are significantly different in location while
the standard distances are approximately the same, then sector ordering
should be present.
To assess the practical usefulness of
the two test statistics, the Colorado
Springs metropolitan area was selected
for analysis. 16 Thi rty census tracts contained within the corporate limits of
Colorado Springs, with a population of
70,194 were classified according to
the Shevky-Bell indexes of social and
urban rank (Figure 2 and Table 2). The
segregation index was not used in this
study because the non-white ethnic
Table 3
Results of t-Test and Equality of Variances Test
For both the Theoretical and Actual Models
CONCENTRIC
t-Test
SECTOR
Equality of
Variances
t-Test
Equality of
Variances
THEORETICAL
HOI
H02
HOa
Ml
MI
M2
Ma
Ma
Ml
Ml
M2
M2
Ma
Ma
Mz
.00026
.039
.590
* .436
* .273
* .630
*3 2.148
*29.259
*2 8.000
.992
.923
.933
1.606
1.606
1.750
* 5.690
* 3.970
* 3.037
1.726
1.541
.892
ACTUAL
HOI
Ho z
HOa
1.780
.031
1.876
·Stati stically Significant Diffe rence at the .05 level
55
Results of Testing
The results of the t-tests and tests for
equality of variances for the theoretical model are summariz~d as follows:
(a) the null hypothesis is retained for
the weighted mean areal centers of
the theoretical concentric model ; (b)
the null hypothesis is rejected for the
weighted mean areal centers of the
theoretical sectorial model ; (c) the null
hypothesis is rejected for the standard
distances from the concentric model ;
and (d) the null hypothesis is retained
for the sectorial standard distances. As
noted previously in the examination of
Table 1, these results were anticipated.
In the analysis of the Colorado Springs
data the results are summarized as follows: (a) the null hypothesis is retained
for the concentric weighted mean
areal centers of the Shevky-Bell urban
rank index ; (b) the null hypothesis is
rejected for the sectorial WMAC's of
the Shevky-Bell social rank index ; and
(c) the null hypothesis is retained for
both the concentric and sectorial
standard distances.
Conclusions
It would appear that further testing
of the weighted mean areal center and
standard distance statistics relative to
a variety of other theoretical models
may well produce a satisfactory test
statis tic for similar problems of spatial
ordering in social area analysis. If the
potential viability of the technique
employed herein is accepted, then certain conclusions can also be noted with
respect to the distributive properties
of the Shevky-Bell urban social indexes.
Thi s analysis suggests that in Colorado Springs, urban rank, or family
statu s, at the trace level is spatially distributed in a concentric manner ;
whereas social rank, or economic status, organizes itself in a sectorial manner. However, the results are not entirely conclusive. This is illustrated by
the fact that in the case of the Colorado Springs data, the null hypothesis
for the weighted mean areal centers
was retained ; furthermore, the null hypothesis for the standard distances was
Table 4
Weighted Mean Areal Centers and Standard
Distances for Actual Models
No . of Tracts
Weighted MAC
(x-y coordinates)
Standard Oistance
16
5.90, 5.69
4.387
8
4.13, 5.90
2.730
6
5.67, 5.60
1.560
-3to - 1s.d.
8
7.27, 3.90
5.436
-1 to
0 s.d.
8
4.40, 6.20
3.149
o to
2 s.d.
14
6.56, 6.65
3.527
Model
CONCENTRIC
-2 to 0 s.d.
o to 1 s.d.
1 to 3 s.d.
SECTOR
56
not rejected . In order to reject the null
hypothes is for the sta ndard di sta nces
of our actual data, especia ll y with such
a small N-size, substanti al differences
in this measure would have to exist. As
can be noted in Table 4, the standard
di stances do increase appreciably for
Colorado Springs, thus suggesting that
the hypothesis fa concentric zonatio n
is viable. On the other hand, in the
case of sector ordering, the null hypothesis is rejected for the we ighted
mean area l centers and the null hypothesi s is retained for the sta ndard distances. Th ese results support a sectoria l arrangement for the social rank
index.
1') For definitions and procedural steps in calculating
the mean areal cente r see: Hart, john F., " Central
Tendency in Areal Distributions," Economic Geograph y, Vol. 30, No.1 (1954), pp . 48-59; and Warntz ,
William and David Nell, "Contributions to a
5tatistical Methodology for Areal Dist ribution s,"
lournal 01 Regional Science, Vol. 2, No.1 ('19601.
pp . 47-60. Weighting the mean areal cent er in this
study is accomplished by adding social area index
values to the x-y coordin ates.
(2 ) A definition and method for calcul ati ng th e sta ndard
distance is found in : Bachi , Robe rto, " Standard Distance Measures and Related Methods for Spatial
Analysis," Papers 01 the Regional Science Association , Vol. 10 (19621, pp . 83-132 .
t3) Burgess, Ernest W ., " The Growth of the City: An
In troduction to a Research Project ," Publications
01 the American Sociological Society, Vol. 18 (1923),
pp. 85-97. Also reprinted in : The Urban Vision: Se-
uses factor analysis to show that, in the case of Los
Angeles and San Francisco, the census measures
selected to construct the indices formed a structure consistent with Shevky's formulations.
These studies are found in th e following works :
Berry, Brian j . L. and Philip H. Rees, " The Factorial
Eocology of Calculla ," Th e American lournal 01
Sociology, Vol. 74 , NO . 3 (1969), pp. 445-491; and
Mu rd ie , Robert A., F,lctorial Ecology 01 Metropolitan Toronto, 1951-1961 : An Essay on the Social Geography 01 the City (Ch icago: Universily of Chicago,
Department of Geography, 1969).
Anderson , Theodore R., and janice A. Egeland,
" Spatial Aspects of Social Area Analysi s," American
Sociological Review. Vol . 26 (19611. pp. 392-398.
Ibid .. p. 395.
Denis C. McElrath , " The Social Areas of Rome : A
comparative AnalysiS," American Sociological Review, Vol. 27, NO.8 (1 962) pp. 276-291.
Murd ie, op . cit.
Berry and Re es, op . cit.
Ibid ., p. 459.
lected Interpretations 01 the Modern American
City , edited by jack Tager and Park Dixon Goist
(Homewood, Illinois: The Dorsey Press, 1970) , pp.
105-115.
(4 ) Hoyt, Homer, The Structure and Growth 01 Residential Neighborhoods in American Cities (Washi ngton,
D.C.: United States Government Printin g Office,
1939).
IS ) Shevky, Eshref and Marilyn Wi ll iams, Th e Social
Areas 01 Los Angeles (Berke ley and Los Angeles: The
University of California Press , 1949) .
t6) This short but monumental work which put forth
the fo rm at for several subsequen t studies dealing
with socia l area typologies is found in : Eshref
Shevky and Wendell Bell , Social Area Analysis:
Theory, Illustrative Applications and Computational
Procedures (Stanford : Stanford University Press,
(7 )
1955) .
Bell , Wendell , "Economic, Fam ily, and Ethnic Status: An Empirical Test ," American Sociological Review, Vol. 20 (1955), pp. 45-52. In this study Bell
(8 )
I' )
(10 )
I ll )
1'2)
1'3 )
(14 )
(15) The computer program entitled : " Two and Three
Dimension Pallern Stati sti cs Program with Weighting," calcu lales distances between points (x-y
coordinates) and orders these distances in order to
solve for the WMAC and standard distance. This
program is capable of handling a larger number of
data and computing spalial stalistics more rapidly
than the method of using a grid syste m with base
lines or working with various mathematical and
statistical formulas (see notes 1 and 2) .
(16) A social area analysis was conducted on 1960 census Colorado Springs data, see : U. S. Bureau of
Census, U. S. Census 01 Population and Housing :
1960 Census Tracts, Final Report PHC (1)-29 (Washington , D.C.: U. S. Government Printing Office,
1961). The population of the Colorado Sp ri ngs SMSA
in 1960 was 143,742.
57