The Modifiable Areal Unit Problem in a Repression Model
Whose Independent Variable Is a Distance from a
Predetermined Point
Naoto Tagashira, Atsuyuki Okabe
Geographical Analysis, Volume 34, Number 1, January 2002, pp. 1-20 (Article)
Published by The Ohio State University Press
DOI: https://doi.org/10.1353/geo.2002.0006
For additional information about this article
https://muse.jhu.edu/article/12090
Accessed 31 Jul 2017 23:36 GMT
Naoto Tagashira
and Atsuyuki Okabe
The Modifiable Areal Unit Problem in a Regression
Model Whose Independent Variable Is a Distance
from a Predetermined Point
This paper deals with the modifiable areal unit problem in the context of a regression
model where a dependent variable is an attribute value (say, income) of an atomic
data unit (say, a household) and an independent variable is a distance from a predetermined point (say, a central business district) to the atomic data unit (a disaggregated model). We apply this disaggregated model to spatially aggregated data in
which the dependent variable is the average income over a spatial unit and the independent variable is the average distance from each household in a spatial unit to the
predetermined point (an aggregated model). First, estimating the slope coefficient by
the least squares method, we prove that the variance of the estimator for the slope coefficient in the aggregated model is larger than that in the disaggregated model. Second, focusing on variations in the variance of the estimator for the slope coefficient in
the aggregated model with respect to the number of zones, we obtain the number of
zones in which the variance is close to that in the disaggregated model. Third, we obtain the zoning system that has the minimum variance for a fixed number of zones. We
also calculate the maximum variance in order to examine the range of the variance.
When we investigate urban phenomena with a regression model, we often use a
distance variable as an independent variable. For example, let us consider the analysis of an income distribution in a city. A possible hypothesis is that the income of a
household varies with respect to the distance from the central business district
(CBD) to the household. To test this hypothesis, we need the data of each income
and the distance from each household to the CBD. However, individual income data
is usually unavailable. Thus we often use aggregated data with respect to zones, say,
census tracts.
This aggregation is problematic, because the results calculated from aggregate
level data are unlikely to correspond to those from individual level data. In addition,
the results from aggregate level data might vary according to the degree of aggregation. The former problem is known as the ecological fallacy and the latter as the
Naoto Tagashira is a research engineer at the Central Research Institute of Electric Power
Industry. E-mail: [email protected] Atsuyuki Okabe is a professor in the Center for
Spatial Information Science, University of Tokyo. E-mail: [email protected]
Geographical Analysis, Vol. 34, No. 1 (January 2002) The Ohio State University
Submitted: 9/09/99. Revised version accepted: 3/22/01.
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modifiable areal unit problem. The modifiable areal unit problem contains two problems: the scale problem and the aggregation problem (Openshaw 1984b). The scale
problem results from the difference in the number of zones partitioning a region.
The aggregation problem results from the particular configuration for a fixed number
of zones.
This paper deals with the ecological fallacy problem, the scale problem, and the aggregation problem in the context of a regression model where a dependent variable is
an attribute value (say, income) of an atomic data unit (say, a household) and an independent variable is a distance from a predetermined point (say, the CBD) to the
atomic data unit (we call this model a disaggregated model). We apply this disaggregated model to spatially aggregated data in which the dependent variable is the average income over a spatial unit and the independent variable is the average distance
from each household in a spatial unit to the predetermined point (we call this model
an aggregated model).
Okabe and Tagashira (1996) discuss the difference between the expected value of
the estimator for the slope coefficient in the aggregated model and that in the disaggregated model. In this paper, we focus on the difference in the variance of the estimator for the slope coefficient in the aggregated model and that in the disaggregated
model. The first objective of this paper is to examine which model has the smaller
variance. If the answer is the disaggregated model, the second objective is to obtain
the number of zones and the configuration of zones in which the variance of the estimator for the slope coefficient in the aggregated model is as close as possible to that
in the disaggregated model. This result would be very useful in practice, because it
would show under what condition we can safely use the aggregated model.
Related studies are many. Gehlke and Biehl (1934), Robinson (1950), and Yule
and Kendall (1950) focus on variations in a correlation coefficient with respect to
the number of zones. Among them, Robinson (1950) particularly gives the empirical evidence that the ecological fallacy problem can in fact occur. Their studies are
extended to variations in estimators for coefficients of a regression model (for example, Blalock 1964; Sawicki 1973; Clark and Avery 1976; Openshaw 1978, 1984a,
1984b; Arbia 1989; and Fotheringham and Wong 1991). In their models, however,
variables are mostly aspatial attribute values and a distance variable is not always
explicitly treated. The distance variable is explicitly considered in spatial interaction models (for example, Curry 1972; Johnston 1973, 1975, 1976; Cliff, Martin,
and Ord 1974, 1975; Curry, Griffith, and Sheppard 1975; Masser and Brown 1975,
1978; Okabe 1977; Openshaw 1977; Slater 1985; Batty and Sikdar 1982a, b, c, d;
and Amrhein and Flowerdew 1992). These studies, however, cannot be applied to
the regression model which we are going to deal with in this paper. The most relevant studies to this paper are Openshaw (1978, 1984a, 1984b). He proposed some
zone-design criteria (Openshaw 1978) and examined the ecological fallacy through
empirical experiments (Openshaw 1984a). In addition, he showed how the goodness of fit was affected by the choice of zones (Openshaw 1984b). This paper elaborates his papers.
This paper consists of five sections including this introductory section. In the next
section, we prove that the variance of the estimator for the slope coefficient in the aggregated model is larger than that in the disaggregated model. In section 2, focusing
on the scale problem, we examine the number of zones for which the variance of the
estimator for the slope coefficient in the aggregated model is close to that in the disaggregated model. In section 3, we turn to the aggregation problem, and obtain the
zoning system that has the minimum variance for a fixed number of zones. We also
calculate the maximum variance in order to examine the range of the variance. In the
last section, we summarize the major results of this paper.
Naoto Tagashira and Atsuyuki Okabe
/
3
1. MODEL
Suppose that there are n atomic units (say, households) in a region S. Let yi be an
attribute value (say, income) of the ith atomic unit, and xi be the distance between the
ith atomic unit and one predetermined center (say, the CBD).
We assume that the relationship between yi and xi is truly represented by a general
regression model,
y i f ( x i ) ε i , i 1,..., n ,
(1)
E(ε i ) 0, i 1,..., n .
(2)
Suppose next that the region S is divided into m subregions, S1, ..., Sm, which are
mutually exclusive and collectively exhaustive. Let Ik be the set of indices of atomic
units in a subregion, Sk, and let
y( k) 1
nk
∑
i ∈I k
y i , x ( k) 1
nk
∑
i ∈I k
x i , ε ( k) 1
nk
∑
i ∈I k
εi ,
(3)
which are the average values of yi, xi, and εi across discrete data units in the subregion
Sk, respectively.
When the above model is applied to spatially aggregated data, one might expect
that the equation
y( k) f ( x ( k) ) ε ( k) , k 1,..., m ,
(4)
also holds. However, it is well known that the above equation holds if and only if the
general model is specified as a linear regression model. For example, Okabe and
Tagashira (1996) provide the proof. Thus when we want to deal with a nonlinear
model, we should transform the nonlinear model into a linear model through an appropriate transformation.
Since we are concerned with a model that satisfies equation (4), we assume from
now on that the relationship between yi and xi is truly given by the ordinary linear regression model, that is,
Model 1 (Disaggregated Model):
y i β 0 β1 x i ε i , i 1,..., n ,
(5)
E(ε i ) 0, i 1,..., n ,
(6)
Var (ε i ) σ 2 , i 1,..., n ,
(7)
Cov (ε i , ε j ) 0, i j, i, j 1,..., n .
(8)
We notice, of course, that the assumptions (7) and (8) are sometimes violated in actual application. In particular, aggregation bias with spatial autocorrelation is discussed in depth (Cliff and Ord 1973; Anselin 1988). Yet we assume the ordinary linear
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regression model here because we want to focus on the variance of the estimator for the
slope coefficient without increasing complexity brought by relaxing these assumptions.
Then, using the ordinary least squares method, we can get unbiased estimators.
Aggregating this disaggregated model with respect to households in each subregion Sk, we obtain
Model 2 (Aggregated Model):
y( k) β 0 β1 x ( k) ε ( k) , k 1,..., m ,
E(ε ( k) ) 1
nk
Var (ε ( k) ) ∑
i ∈I k
1
n k2
E(ε i ) 0, k 1,..., m ,
∑
i ∈I k
Cov (ε ( k) , ε ( h) ) (9)
Var(ε i ) 1
(10)
σ2
, k 1,..., m ,
nk
∑ ∑
n k2 n h2 i ∈I k j ∈I h
(11)
Cov(ε i , ε j ) 0, k h, k, h 1,..., m .
(12)
Note that the aggregated model has the same coefficients as the disaggregated
model. Also note that the disaggregated model has homoskedasticity [equation (7)],
whereas the aggregated model has heteroskedasticity [equation (11)]. Thus the aggregated model is not the ordinary linear regression model. However, if we apply the
weighted least squares method with weights given by Var(ε(k))1, that is, nk in the aggregated model, to the estimation of coefficients, we can obtain unbiased estimators.
Therefore, the expected coefficients in the aggregated model are the same as those in
the disaggregated model. From now on, in order to distinguish the coefficients in the
disaggregated model from those in the aggregated model, we denote the coefficients
in the disaggregated model by β′0 and β′1.
1.1 Comparison of the Variance of the Estimator for the Slope Coefficient in the
Aggregated Model with That in the Disaggregated Model
Let us next obtain the variance of the estimator for the slope coefficient in equation (9). Since the aggregated model has heteroskedasticity [equation (11)], we apply
the weighted least squares method with weights given by nk to the estimation of β1.
The estimator, β̂1, of β1 is
∑ k 1 n k ( x ( k) x )( y( k) y)
,
βˆ 1 m
2
(
)
n
x
x
∑ k 1 k ( k)
m
(13)
where x– and –y are the averages of x(k) and y(k) in the region S, respectively. From this
equation, the variance of β̂1 is given by
σ2
Var(βˆ 1 ) ∑x
n
m
∑ n k x (2k)
k 1
i 1
n
i
2
σ2
n
1
m
∑
k 1
p k x (2k)
,
( AD)
2
(14)
Naoto Tagashira and Atsuyuki Okabe
/
5
where pk denotes the ratio of the number of atomic units in Sk to the number of
atomic units in S, that is, nk/n, and AD is the average distance between the predetermined point and all atomic units. Since the above equation contains nk or x(k), we notice that the variance of the estimator is influenced by the choice of subregions.
In order to compare the variance of the estimator for the slope coefficient in the
aggregated model to that in the disaggregated model, let us next calculate the variance of the estimator in the disaggregated model. Using the ordinary least squares
method, we obtain
σ2
Var(βˆ 1′ ) n
∑ x i2 n
∑ xi
i 1
2
σ2
1
,
n AD 2 ( AD)2
(15)
n
i 1
where AD2 is the average of the squared distance from the predetermined point to all
atomic data units. Since equation (14) can be rewritten as
Var(βˆ 1 ) σ2
n
1
1 m
∑
n k 1
∑
i ∈I k
xi
nk
,
2
( AD)2
we can examine which model has the smaller variance by comparing
with
∑
(16)
∑
m
k 1
(Σ
i ∈I k x i
)
2
nk
n
x 2.
i 1 i
Using the Cauchy-Schwarz inequality, we notice that the following inequality holds
for k,
(Σ
i ∈I k x i
)
2
nk
∑
i ∈I k
x i2 .
(17)
The equality is satisfied when nk 1 or all atomic units in Sk have the same distance.
Aggregating the above equation with respect to Sk, we obtain
m
∑
k 1
(Σ
i ∈I k x i
nk
)
2
n
∑ x i2 .
i 1
(18)
Therefore, the variance of the estimator for the slope coefficient in the disaggregated
model is smaller than that in the aggregated model except in special cases in which
each subregion Sk has one atomic unit or all atomic units in Sk have the same distance
[this result corresponds with Cramer (1964)].
Thus the aggregated model is no better than the disaggregated model. In practice,
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however, if the difference between them is small, we may use the aggregated model.
In the next section, we attempt to find the number of zones for which the variance of
the estimator for the slope coefficient in the aggregated model is close to that in the
disaggregated model. Then we use the relative efficiency to compare the variances of
the estimators defined by
Var (βˆ 1 )
Re (βˆ 1 / βˆ 1′ ) Var (βˆ 1′ )
AD 2 ( AD)2
m
∑
p k x (2k)
k 1
.
( AD)
(19)
2
2. THE SCALE PROBLEM
In this chapter, we examine variations in the variance of the estimator for the slope
coefficient in the aggregated model with respect to the number of zones and obtain the
number of zones for which the variance is close to that in the disaggregated model.
2.1 Concentric Zones
When we analyze urban phenomena, we often use concentric zones. Historically,
this zoning stems from Burgess’s (1925) concentric zone theory, followed by factorial
ecologists (Herbert and Johnston 1976). Clark (1951) also assumed concentric zones
centered at the CBD when he examined the population density in a city. It is hence
worth examining the variance when subregions are given by concentric zones.
Suppose that S is represented by a circle. Then we can assume that the radius of
the circle is 1 without loss of generality. The reason is that both the denominator and
the numerator of equation (19) contain the terms of the same power of the distance.
Thus, although the relative distance value affects the relative efficiency, the absolute
distance value does not. We also deal with variations in the variance with the number
of zones, but the absolute distance does not matter to the trend of these variations.
Suppose next that Sk is represented by a ring centered in a predetermined center
whose inner boundary is given by the circle with radius rk1 and whose outer boundary is given by the circle with radius rk (note that S1 is given by a disk with radius r1)
and that the width of a ring is the same for k, that is, rk k/m. Let (x, θ) be the polar
coordinates of a point in S where the origin is placed at the predetermined center, O,
and let g(x) be the distribution function of atomic units at (x, θ) (note that the distribution function is assumed to be isotropic from O). Then p(k), x(k), AD, and AD2 are calculated as follows.
pk ∫ ∫ S k xg( x ) dθdx
x ( k) ∫ ∫ S xg( x ) dθdx
rk
∫ ∫ S k x 2 g( x ) dθdx
∫ ∫ S k xg( x ) dθdx
∫r
xg( x )dx
k1
1
∫0 xg( x )dx
rk
∫r
k1
rk
∫r
k1
(20)
,
x 2 g( x )dx
,
(21)
xg( x )dx
1
2
∫ ∫ x 2 g( x ) dθdx ∫0 x g( x )dx
AD S
,
1
∫ ∫ S xg( x ) dθdx
xg( x )dx
∫0
(22)
Naoto Tagashira and Atsuyuki Okabe
/
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1
3
∫ ∫ S x 3 g( x ) dθdx ∫0 x g( x )dx
AD 2 .
1
∫ ∫ S xg( x ) dθdx
xg( x )dx
(23)
∫0
2.1.1 Uniform Distribution Function. First, we consider the case in which atomic
units are uniformly distributed over S. Substituting rk k/m into equations (20), (21),
and (22), we obtain
pk 1
m2
(2k 1) ,
(24)
x ( k) 2(3k 2 3k 1)
,
3m (2k 1)
(25)
AD 2
.
3
(26)
Then we obtain the variance of β̂1 as
Var(βˆ 1 ) σ2
72
,
n {4 h(m )}
(27)
where
h(m ) 6
m
2
ψ (m 1 / 2)
m
4
γ log 4
m4
,
(28)
where ψ(x) is the Di gamma function, and γ is the Euler number.
We next prove the following theorem.
THEOREM 1: Assume that a region S consists of equal-width concentric zones;
atomic units are uniformly distributed over S. Then, if the disaggregated model is expressed by the ordinary regression model whose independent variable is a distance
from the center, the variance of the estimator for the slope coefficient in the aggregated model decreases with an increase in the number of zones.
PROOF. From equation (27), it is obvious that if h(m) increases as m increases,
the variance of β̂1 decreases as m increases. Hence we wish to prove that ∆h h(m 1) h(m) is positive for m 1. Using the property of the Di gamma function
given by
ψ m
3
1
1
ψ m
,
1
2
2
1
∆h is written as
m
2
(29)
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
1
1
1 
1


 1
ψ m
∆h 
6
γ log 4 




2
2
2
m 2 
(m 1)2 

 (m 1)
 m
2
(m 1) (2m 1)
4
.
(30)
From the definition of the Di gamma function, the relation
ψ m
m
1
1
γ log 4 2 ∑
2m
k
1
2
2
k 1
(31)
holds for m 1. Thus
1
1 
2
 m

 1
∆h 3
 0 (32)
 2 
2
2
2
4
m  (m 1) (2m 1)
 (m 1)
 m
(m 1) 
holds for m 1. This completes the proof. Having understood that the estimator variance decreases with an increase in the
number of zones, and that the disaggregated model has the smaller variance than the
aggregated model, we now wish to know the number of zones for which the variance
of the estimator for the slope coefficient in the aggregated model is close to that in
the disaggregated model.
Under the hypotheses introduced in this section, the variance of the estimator for
the slope coefficient in the disaggregated model that we obtain is
Var(βˆ 1 ) 18σ 2
.
n
(33)
Thus the relative efficiency is written as
Re (βˆ 1 / βˆ 1′ ) 4m 2
(
)
4 m 4 6 m 2 ψ m 1 γ log 4
2
.
(34)
We show the numerical values of Re(β̂1/β̂1′ ) in Table 1. For three zones, for example,
the relative efficiency is 1.1865. This means that the variance of the estimator for the
slope coefficient in the aggregated model is larger than that in the disaggregated
model by 18.65 percent. For example, if we require the relative efficiency to be less
than 1.05, we should prepare six zones or more. If we want to make the efficiency
smaller, we need a larger number of zones. In order to make it less than 1.01, we need
to have thirteen zones or more.
2.1.2 Negative Exponential Distribution Function. Up to now we have assumed
that the atomic data units were distributed uniformly. This assumption, however, is
sometimes unrealistic. Thus we next assume a more realistic distribution.
Naoto Tagashira and Atsuyuki Okabe
/
9
TABLE 1
Variation in Re(β̂1/β̂1′ ) with the Number of Concentric Zones Where Atomic Units Are Distributed Uniformly
Number of zones
Re(β̂1/β̂1′ )
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.1865
1.0995
1.0622
1.0427
1.0311
1.0237
1.0187
1.0151
1.0125
1.0105
1.0089
1.0077
1.0067
1.0059
1.0052
1.0046
1.0042
1.0038
When atomic units refer to households, the spatial distribution is likely to follow
Clark’s (1951) law given by
g( x ) a exp(bx ), a, b 0, ( x , θ) ∈ S .
(35)
From equation (19), since the absolute value of the number of the atomic units does
not affect the relative efficiency, we can put a 1 without loss of generality.
Using equations (20) and (21), the values of pk and x(k) are written as
pk (
exp b
k
m
){ (1 b ) exp( )(1 b )}
k
m
k1
m
b
m
(36)
exp(b)(1 b) 1
and
( ) b
exp
x ( k) b
m
2
2 ( k1)
m2
2b
{ ( )( b
b exp
b
m
( k1)
m
2 
(
) (b
k1
1
m
k2
m2
k
m
b2 2 b
)}
k
m
),
2
(37)
1
respectively. Since Var(β̂1) becomes complicated, we numerically investigate the values
of Var(β̂1) with b 1, 2, 4, 6. Note that b 1, 2, 4, 6 implies that the densities at 1/2,
which is a half of the radius of the region, are about 61 percent, 37 percent, 14 percent,
5 percent of the density of the center and that the densities at 1, which is the fringe of
the region, are about 37 percent, 14 percent, 2 percent, 0.2 percent of the density of the
center, respectively. We think b 6 is large enough to represent a significant decrease
in the distribution of atomic data units that we encounter in actual application.
The values of Re(β̂1/β̂1′ ) are shown in Table 2. We notice from this table that the
values of the relative efficiency decrease with an increase in m for all values of b.
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TABLE 2
Variation in Re(β̂1/β̂1′ ) with the Number of Concentric Zones Where Atomic Units Are Distributed According to g(x, θ) exp (bx) Where b 1.0, 2.0, 4.0, 6.0
Re(β̂1/β̂1′ )
Number of zones
b 1.0
b 2.0
b 4.0
b 6.0
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.1637
1.0881
1.0554
1.0381
1.0278
1.0212
1.0168
1.0136
1.0112
1.0094
1.0080
1.0069
1.0060
1.0053
1.0047
1.0042
1.0037
1.0034
1.1526
1.0828
1.0523
1.0361
1.0264
1.0202
1.0159
1.0129
1.0107
1.0090
1.0076
1.0066
1.0057
1.0050
1.0045
1.0040
1.0036
1.0032
1.1623
1.0888
1.0565
1.0392
1.0288
1.0221
1.0175
1.0142
1.0117
1.0099
1.0084
1.0073
1.0063
1.0056
1.0049
1.0044
1.0040
1.0036
1.2176
1.1178
1.0749
1.0521
1.0384
1.0295
1.0234
1.0190
1.0157
1.0133
1.0113
1.0098
1.0085
1.0075
1.0067
1.0059
1.0053
1.0048
Since the variance of β̂1′ is constant, we also notice that the variance of β̂1 decreases
with an increase in m.
Table 2 also shows that for b 1, 2, 4, if the number of zones is the same, the
values of the relative efficiency are smaller than those for the uniform function. This
means that in the case of the negative exponential function whose concavity is not so
large, we obtain the variance closer to the disaggregated model than the uniform
function.
For b 6 (the density declines quite rapidly), however, the relative efficiency is
larger than that for the uniform function. Thus, we need a larger number of zones if
we want to make the variance as close to the disaggregated model as the uniform distribution. For example, in order to make the relative efficiency less than 1.01, we
have to prepare fourteen zones or more.
2.2 Square Grid Zones
We next assume that a region is square shaped and that subregions are given by
square grid zones (see Figure 1). As we stated in the case of the concentric zones,
since the absolute distance value does not affect the following discussion, we assume
that a half of the side length of the region is 1 without loss of generality.
2.2.1 Uniform Distribution Function. First, suppose that atomic data units are distributed uniformly. In this case, we obtain
pk 1
.
m
(38)
The value of x(k) is given by
x ( k) L(k)
,
3 m
where the detailed explanation of L(k) can be found in Appendix 1.
(39)
Naoto Tagashira and Atsuyuki Okabe
/
11
FIG. 1. Square Grid Zones
Substitution of these equations into equation (14) gives
Var(βˆ 1 ) σ2
n
1
m2
2
Σm
k 1 L(k) 4
{
9
)}
(
2 log
2 1
2
.
(40)
Using equation (15), we also obtain the variance of the estimator for the slope coefficient in the disaggregated model as
σ
Var(βˆ 1′ ) n
2
9

4 4 2 2 log

(
)
{
(
2 1 log
)}
2 1
2
.
(41)


The values of Re(β̂1/β̂1′ ) are shown in Table 3. We notice from this table that Var(β̂1)
decreases with an increase in m. If we have only sixteen zones, the variance of the
estimator for the slope coefficient in the aggregated model is larger than that in
the disaggregated model by 33.81 percent. If we require a smaller variance, a larger
number of zones is needed. For example, 484 zones lead to the relative efficiency of
less than 1.01.
Now, let us compare the results of Table 3 with those of Table 1. We first reconsider the square grid zones by taking Figure 1, for instance. There are thirty-six zones
in Figure 1. However, there are some zones that have the same distance variable and
the same number of atomic units. For example, there are seven other zones that have
the same distance variable and the same number of atomic units as the Sk shown in
Figure 1 (the diagonally shaded zones). When we use the weighted least squares
method, if the different zones have the same distance and the same number of
atomic units, their zones correspond to one independent variable. Considering this
matter, we notice that Figure 1 has thirty-six zones, but that it corresponds to only six
independent variables.
Thus let us compare the result of the six zones in Table 1 with that of thirty-six
zones in Table 3. We can find that the relative efficiency of the Table 3 is larger than
that of Table 1. This finding is true for other numbers of zones. This means that the
square grid zones are not better than the equal-width concentric zones in the case of
the same number of independent variables when we estimate the slope coefficient in
the aggregated model whose independent variable is the distance from the center.
12
/ Geographical Analysis
TABLE 3
Variation in Re(β̂1/β̂1′ ) with the Number of Square Grid Zones Where Atomic Units Are Distributed Uniformly
Number of zones
Re(β̂1/β̂1′ )
16
36
64
100
144
196
256
324
400
484
576
676
784
900
1024
1156
1296
1444
1600
1.3381
1.1274
1.0681
1.0426
1.0293
1.0213
1.0163
1.0128
1.0104
1.0085
1.0072
1.0061
1.0053
1.0046
1.0040
1.0036
1.0032
1.0029
1.0026
Let us next consider why the square grid zones are not better than the equal-width
concentric zones. We can transform equation (14) as
Var(βˆ 1 ) σ2
n
1
∑
h k
p k p h ( x ( k) x ( h) )2
.
(42)
Then we notice that if we want to lessen the variance, we need to enlarge the difference between the distance variables with weights given by the number of zones. In
the case of the square grid zones, zones are not divided on the basis of the distances
from the center. Thus, even if the ith atomic unit has a larger distance than the jth
atomic unit, the ith unit is sometimes assigned to the Sk that has a smaller average distance than the Sh to which the jth unit is assigned (we call this the assignment loss in
this paper). Due to this assignment loss, the difference between x(k) and x(h) is smaller
than the difference in a case where we divide the area containing Sk and Sh so as not
to cause the assignment loss. On the other hand, since the concentric zones are made
on the basis of the distances from the center, the assignment loss does not occur. We,
of course, notice that since the values of pk in the square grid zones are different from
those of the equal width concentric zones, the influence of the weights is not clear.
However, Table 1 and Table 3 imply that we need a larger number of independent
variables in the case of the square grid zones than in the equal-width concentric
zones to obtain the same relative efficiency due to the assignment loss.
2.2.1 Negative Exponential Distribution Function. Let us next assume that the distribution of the atomic units is represented by Clark’s law, that is, the negative exponential function. Using the method shown in Okabe and Tagashira (1996), we can
calculate the values of pk and x(k). Then we can obtain the relative efficiency of the estimator for the slope coefficient in the aggregated model to that in the disaggregated
model
in the case of b 1, 2, 4. Note that b 1, 2, 4 implies that the densities at
–
√2/2, which is the half of the distance from the center to the farthest point of this region, are about 49 percent, 24 percent, 6 percent of the density of the center and that
Naoto Tagashira and Atsuyuki Okabe
/
13
–
the densities at √2, which is the farthest point of this region, are about 24 percent, 6
percent, 0.3 percent of the density of the center, respectively. We think that b 4 is
large enough to represent a significant decrease in the distribution of atomic data
units in the case of the square grid zones.
Table 4 shows that the variance of the estimator for the slope coefficient in the aggregated model decreases with an increase in the number of zones. Table 4 also
shows that for b 1, 2, if the number of zones is the same, the values of the relative
efficiency are smaller than those in the uniform function. For b 4, however, the relative efficiency is larger than that in the uniform density. If we want to make the efficiency less than 1.01, for example, we have to prepare 576 zones. Comparing the
results of Table 4 with those of Table 2, we can find that the relative efficiency in the
square grid zones is larger than that in the concentric zones for the corresponding
number of zones. The reason would be similar to the case of the uniform function.
3. THE AGGREGATION PROBLEM
In this section, we investigate the effects of different zoning systems for a fixed
number of zones. We first calculate the zoning system that has the minimum variance, which is referred to as the min-var zoning system in this paper. This zoning is
obtained by solving the following minimization problem:
min
pk
subject to
Var (βˆ 1 )
(43)
∑ k 1 p k 1 .
(44)
m
For comparative purposes, we compute the zoning system that has the maximum
variance, which is referred to as the max-var zoning system. This zoning system is
obtained by maximizing Var(β̂1) under the constraint (44). In addition, in the case of
the concentric zones, we also compare the equal-width concentric zones with the
TABLE 4
Variation in Re(β̂1/β̂1′ ) with the Number of Square Grid Zones Where Atomic Units Are Distributed According to g(x, θ) exp (bx) Where b 1.0, 2.0, 4.0
Re(β̂1/β̂1′ )
Number of zones
b 1.0
b 2.0
b 4.0
16
36
64
100
144
196
256
324
400
484
576
676
784
900
1024
1156
1296
1444
1600
1.3178
1.1198
1.0641
1.0401
1.0275
1.0201
1.0153
1.0121
1.0098
1.0081
1.0068
1.0058
1.0050
1.0043
1.0038
1.0034
1.0030
1.0027
1.0024
1.3224
1.1208
1.0645
1.0404
1.0277
1.0202
1.0154
1.0122
1.0098
1.0081
1.0068
1.0058
1.0050
1.0043
1.0038
1.0034
1.0030
1.0027
1.0024
1.4231
1.1535
1.0809
1.0503
1.0344
1.0251
1.0191
1.0150
1.0121
1.0100
1.0084
1.0071
1.0061
1.0053
1.0047
1.0041
1.0037
1.0033
1.0030
14
/ Geographical Analysis
min-var and the max-var systems. In the case of the grid zones, we compare the
square grid zones.
3.1 Concentric Zones
We deal with the uniform and the negative exponential functions as the distribution functions of the atomic data units. Then we can write the values of pk and x(k) in
terms of the radius of Sk, that is, rk. In the case of the uniform distribution,
p k rk2 rk21
(45)
and
x ( k) 2 rk2 rkrk1 rk21
.
rk rk1
3
(46)
For the negative exponential distribution,
pk { exp(brk )(1 brk ) exp(brk1 )(1 brk1 )}
exp(b)(1 b) 1
(47)
and
x ( k) exp(brk1 )(rk21 2rk1 / b 2 / b 2 ) exp(brk )(rk2 2rk / b 2 / b 2 )
. (48)
exp(brk1 )(rk1 1 / b) exp(brk )(rk 1 / b)
Hence, we use rk as a variable of the minimization problem instead of pk. When we
solve the minimization problem, we first calculate the values of Var(β̂1) by changing
rk(k m) at the intervals of 0.01 under the constraint of rk1 rk. For example, for
three zones, the values of rk change such as r1 0.01,0.02,…,0.98, r2 r1 0.01,
r1 0.02,…, 0.99, and r3 1.00. Comparing all these variances, we find the combination of rk that leads to the minimum variance.
Similarly, we employ rk as a variable of the maximization problem and detect the
maximum variance. However, in the case of the maximization problem, the widths of
some zones, that is, the values of rk rk1, in the solution become 0.01. Since we
think this is sometimes too small in actual situations, we also obtain the solution at the
intervals of 0.1.
We calculate the min-var and the max-var zoning systems for m 3, 4, 5; for b 0
(uniform), 1, 2, 4, 6 and show the min-var systems in Table 5. Before we focus on the results of Table 5, let us reconsider the min-var systems. As we mentioned before, if we
want to lessen the variance, we need to enlarge the difference between the distance
variables with weights given by the number of atomic units contained in the zones. If
the difference were not weighted by the number of zones, for example, the min-var systems for m 3 would bring that r1 0.01 and r2 0.99 and the numbers of atomic
units contained in S1 and S3 would be very small. However, since the difference is
weighted by the number of zones, the numbers of atomic units contained in zones are
not so small. For example, for m 3, b 0, the values of rk are r1 0.46 and r2 0.75.
These values of rk decrease with an increase in b. The reason is that since more atomic
Naoto Tagashira and Atsuyuki Okabe
/
15
TABLE 5
Min-var Zoning Systems in Concentric Zones Where Atomic Units Are Distributed According to g(x, θ) exp (bx) Where b 0.0, 1.0, 2.0, 4.0, 6.0
Number of zones
b
r1
r2
r3
r4
3
0.0
1.0
2.0
4.0
6.0
0.46
0.43
0.39
0.33
0.27
0.75
0.72
0.69
0.62
0.54
1.00
1.00
1.00
1.00
1.00
4
0.0
1.0
2.0
4.0
6.0
0.38
0.34
0.31
0.25
0.21
0.61
0.57
0.53
0.46
0.39
0.81
0.79
0.76
0.70
0.62
1.00
1.00
1.00
1.00
1.00
5
0.0
1.0
2.0
4.0
6.0
0.32
0.29
0.26
0.21
0.17
0.52
0.48
0.45
0.37
0.31
0.69
0.66
0.63
0.55
0.47
0.85
0.83
0.81
0.75
0.68
r5
1.00
1.00
1.00
1.00
1.00
units are located near the center with an increase in b, in order to have each zone contain the certain number of atomic units, we need to make the values of rk(k m)
smaller with an increase in b. This trend is similar to other number of zones.
As for the max-var zoning systems, each zone width becomes the interval value except for one zone. For example, for m 5, b 0, the values of rk are r1 0.01, r2 0.02, r3 0.03, and r4 0.04 under the intervals of 0.01. This is because in contrast
to the min-var systems, the max-var systems decrease the difference between the distance variables.
We next show the relative efficiency of the estimator for the slope coefficient in the
aggregated model to that in the disaggregated model in Table 6. Let us first look at
the relative efficiency of the max-var zoning system. Under the interval of 0.01, the
values of the relative efficiency of the max-var systems are much larger than those of
the min-var systems. Although the values of the efficiency under the intervals of 0.1
are also larger than those of the min-var systems, they are much lower than those
under the intervals of 0.01. In addition, Table 6 shows that the values of the relative
efficiency under the intervals of 0.01 in the negative exponential function are smaller
than those in the uniform function. Thus we notice that we have to be more careful
about the aggregation problem in the case of the uniform function than that in the
exponential function. However, the relative efficiency under the intervals of 0.1 decreases with an increase in b until b 2, or 4 turns to an increase. The value under
the intervals of 0.01 might also increase if we examine a larger value of b. Hence, note
that if we deal with the negative exponential function with a significant decrease,
there is the possibility that the relative efficiency becomes large.
Let us next move on to the relative efficiency in the case of the equal width concentric zones. The values of the relative efficiency are much smaller than those of the
max-var systems. Especially, for b 2,4, the values are close to those of the min-var
systems. The reason is that for b 2, 4, the min-var zoning systems are very similar
to the equal-width zones (see Table 5).
3.2 Rectangular Grid Zones
Next suppose that a region is square shaped and it is divided into m rectangles (see
Figure 2). To make calculation simpler, we assume that zones are determined by r1,r2,
16
/ Geographical Analysis
TABLE 6
Re(β̂1/β̂1′ ) for the Min-var Zoning, the Max-var Zoning, and Equal-Width Zones in Concentric Zones Where
Atomic Units Are Distributed According to g(x, θ) exp (bx) Where b 0.0, 1.0, 2.0, 4.0, 6.0
Max-var zoning
Interval
Number of zones
b
Min-var zoning
0.01
0.1
Equal-width zones
3
0.0
1.0
2.0
4.0
6.0
1.1480
1.1445
1.1446
1.1561
1.1785
325.24
233.76
174.22
104.94
71.505
4.6642
3.8017
3.2055
3.0478
4.1526
1.1865
1.1637
1.1526
1.1623
1.2176
4
0.0
1.0
2.0
4.0
6.0
1.0800
1.0780
1.0780
1.0842
1.0967
147.47
106.85
80.319
49.268
41.348
2.5347
2.1806
1.9301
2.0315
2.5718
1.0995
1.0881
1.0828
1.0888
1.1178
5
0.0
1.0
2.0
4.0
6.0
1.0504
1.0492
1.0492
1.0530
1.0610
84.619
61.807
46.844
29.237
28.895
1.7487
1.5736
1.4629
1.5571
1.8353
1.0622
1.0554
1.0523
1.0565
1.0749
…,r√––m/2 as shown in Figure 2. Then, since the values of pk and x(k) can be written in
terms of rk, we can solve the minimization and the maximization problems with the
same way we used in the case of the concentric zones.
Table 7 shows the min-var systems for m 16, 36, 64, 100; b 0 (uniform density), 1, 2, 4. The values of the rk(k m) decrease with an increase in b. In addition,
the min-var zoning systems are close to the square grid zones for b 1, 2. As for the
max-var zoning system, each zone width becomes the interval value except for one
zone. These findings are similar to those in the case of the concentric zones.
The values of the relative efficiency are shown in Table 8. Looking at the max-var
systems, we notice that the values vary according to the interval values and that under
the intervals of 0.01 the values of the relative efficiency are much larger than those of
the min-var systems. We also notice that the relative efficiency of the max-var systems
decreases with an increase in b until b 1, or 2 turns to an increase. Thus we have to
deal with the aggregation problem carefully in the case of the uniform function and
the exponential function with a significant decrease.
Focusing on the square grid zones, we notice that the relative efficiency is much
smaller than that of the max-var systems. Especially, b 1,2 lead to the values that
are fairly close to those of the min-var systems. The reason is that the min-var zoning
systems are very similar to the square grid zones for b 1, 2 (see Table 7).
Now let us compare the values of the relative efficiency of the min-var and the
max-var systems in Table 8 to those in Table 6. We have already explained that thirtysix zones in the grid zones corresponds to six independent variables in terms of the
weighted least squares method. In the same way, sixteen zones corresponds to three
variables. Then we compare the results of sixteen grid zones with those of three concentric zones. We have also mentioned that the square grid zones are not better than
the equal-width concentric zones in the case of the same number of independent
variables. Similarly, the relative efficiency of the min-var systems is larger than that in
the concentric zones. This finding is true of the max-var systems under the intervals
of 0.1. However, the values of the max-var systems under the intervals of 0.01 in grid
zones are smaller than those in the concentric zones. The reason is that although sixteen grid zones corresponds to three independent variables, sixteen zones have only
one variable r1 for the maximization problem. Since the degree of freedom is too
small for the maximization, we can not lessen the difference between the distance
FIG. 2. Rectangular Grid Zones
TABLE 7
Min-var Zoning Systems in Rectangular Grid Zones Where Atomic Units Are Distributed According to
g(x, θ) exp (bx) Where b 0.0, 1.0, 2.0, 4.0
Number of zones
b
r1
r2
r3
r4
16
0.0
1.0
2.0
4.0
0.56
0.52
0.49
0.40
1.00
1.00
1.00
1.00
36
0.0
1.0
2.0
4.0
0.40
0.37
0.34
0.27
0.71
0.68
0.65
0.57
1.00
1.00
1.00
1.00
64
0.0
1.0
2.0
4.0
0.33
0.29
0.26
0.21
0.57
0.53
0.49
0.41
0.79
0.76
0.73
0.66
1.00
1.00
1.00
1.00
100
0.0
1.0
2.0
4.0
0.27
0.24
0.22
0.17
0.47
0.43
0.40
0.33
0.65
0.62
0.58
0.51
0.83
0.81
0.78
0.72
r5
1.00
1.00
1.00
1.00
18
/ Geographical Analysis
TABLE 8
Re(β̂1/β̂1′ ) for the Min-var Zoning, the Max-var Zoning, and Square Grid Zones in Rectangular Grid Zones
Where Atomic Units Are Distributed According to g(x, θ) exp (bx) Where b 0.0, 1.0, 2.0, 4.0
Max-var zoning
Interval
Number of zones
b
Min-var zoning
0.01
0.1
Square grid zones
16
0.0
1.0
2.0
4.0
1.3159
1.3145
1.3213
1.3578
56.943
47.129
39.654
64.105
5.3220
4.5333
4.1041
6.2631
1.3381
1.3178
1.3224
1.4231
36
0.0
1.0
2.0
4.0
1.1176
1.1171
1.1198
1.1346
28.146
23.347
19.782
31.978
2.6247
2.3179
2.2238
3.0781
1.1274
1.1198
1.1208
1.1535
64
0.0
1.0
2.0
4.0
1.0625
1.0622
1.0637
1.0718
18.572
15.446
13.229
21.254
1.7946
1.6374
1.6261
2.0490
1.0681
1.0641
1.0645
1.0809
100
0.0
1.0
2.0
4.0
1.0389
1.0388
1.0397
1.0448
13.801
11.512
9.9839
15.891
1.4180
1.3320
1.3363
1.5623
1.0426
1.0401
1.0404
1.0503
variables extremely. On the other hand, since the concentric zones have the same
number of maximization variables as the independent variables, we can make the difference between the distance variables small significantly.
4. CONCLUSIONS
The problem considered in this paper is the modifiable areal unit problem in the
regression model where the dependent variable is the attribute value of the atomic
data unit and the independent variable is the distance variable from the center of the
region to the atomic data unit. We applied this model to spatially aggregated data,
and examined the effects of zones used for aggregating data on the variance of the estimator for the slope coefficient in the aggregated model. In this last section, we discuss the implications of the results we have obtained.
If we could use the individual level data, it would be better to estimate the coefficient in the disaggregated model. However, individual level data are usually unavailable. Thus we need to estimate the coefficient using the aggregated model. But, as we
mentioned in section 1, the variance of the estimator for the slope coefficient in the
aggregated model is larger than that in the disaggregated model. Hence we calculated the relative efficiency of the estimator for the slope coefficient in the aggregated
model to that in the disaggregated model with respect to the number of zones. We
first dealt with the equal-width concentric zones. When atomic data units are distributed uniformly, the relative efficiency decreases with an increase in the number of
zones (Theorem 1). For example, six zones or more lead to the relative efficiency becoming less than 1.05 (Table 1). This means that if we prepare six or more zones, the
variance of the estimator for the slope coefficient in the aggregated model is larger
than that in the disaggregated model by less than 5 percent. If we want to make the
relative efficiency closer to 1.0, thirteen or more zones achieve the relative efficiency
of less than 1.01. When the atomic units are distributed following Clark’s law, or the
negative exponential function, the relative efficiency varies according to the degree of
concavity of the function. If the concavity is not large, the relative efficiency in the
negative exponential function is smaller than that in the uniform function. However,
Naoto Tagashira and Atsuyuki Okabe
/
19
when the concavity is very large, the relative efficiency is larger than that in the uniform density. These values of the relative efficiency are shown in Table 2. In the case
of the square grid zones with the uniform function, in order to achieve the relative efficiency of less than 1.01, we need at least 484 zones (Table 3). In the case of the negative exponential function with small concavity, the relative efficiency is smaller than
that in the uniform function. But when the concavity is very large, we need larger
zones in order to obtain the same efficiency as the uniform function. These values of
the relative efficiency are shown in Table 4.
In section 3, we obtained the zoning systems that achieve the minimum and the
maximum estimator variance. The maximum estimator variance is much larger than
the minimum variance, especially in the uniform distribution or the exponential distribution with large concavity. However, the variance in the equal-width concentric
zones or the square grid zones is much closer to the minimum variance than the maximum variance.
In practical situations, it is difficult to obtain the zoning system that yields the minimum variance. Thus it is the alternative measure to use the equal-width concentric
zones or the square grid zones. Furthermore, as the number of zones increases, the
variance of the estimator for the slope coefficient in the aggregated model approaches that in the disaggregated model. And we can use Tables 1–4 in order to relate the relative efficiency to the number of zones.
APPENDIX 1
To simplify the explanation of L(k), we first assume that a subregion Sk is located on
an upper right quarter of the region such as Figure 1. If the Sk is located in the Xk 1
column and in the Yk 1 row counted from the origin (the center), L(k) in equation
(39) is given by
L(k) X k3 log
Yk Tk
Y 1 Wk
Tk X k
(X k 1)3 log k
(Yk )3 log
Yk 1 U k
Yk Vk
Vk X k 1
(Yk 1)3 log
Wk 1 X k
2{X kU k (Yk 1) Tk X kYk Yk Vk (X k 1)
Uk Xk
Wk (X k 1)(Yk 1)} ,
where Tk X k2 Yk2 , U k X k2 (Yk 1)2 , Vk (X k 1)2 Yk2 ,
Wk (X k 1)2 (Yk 1)2 .
Since subregions located in the upper right quarter of the region and those in other
quarters are symmetrical with respect to the horizontal axis, the vertical axis, or the
center, by moving the subregions in other quarters to corresponding locations on the
upper right quarter, we can calculate the values of L(k) of subregions in other quarters
similarly.
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