S-maup: Statistic test to measure the sensitivity to the
Modiﬁable Areal Unit Problem
Juan C. Duque
Department of Mathematical Sciences (RiSE-group), Universidad EAFIT, Medell´ın, Colombia.
E-mail:jduquec1@eafit.edu.co
Henry Laniado
Department of Mathematical Sciences (RiSE-group), Universidad EAFIT, Medell´ın, Colombia.
E-mail:hlaniado@eafit.edu.co
Adriano Polo
Department of Economics, Universidad EAFIT, Medell´ın, Colombia.
E-mail:apololo@eafit.edu.co
Abstract:
This work presents a nonparametric statistical test, S-maup, to measure the sensitivity of a
spatially intensive variable to the eﬀects of the Modiﬁable Areal Unit Problem (MAUP). S-maup is
the ﬁrst statistic of its type and focuses on determining how much the distribution of the variable,
at its highest level of spatial disaggregation, will change when it is spatially aggregated. Through
a computational experiment, we obtain the basis for the design of the statistical test under the
null hypothesis of non-sensitivity to MAUP. We performed a simulation study for approaching
the empirical distribution of the statistical test, obtaining its critical values, and computing its
power and size. The results indicate that the power of the statistic is good if the sample (number
of areas) grows, and in general, the size decreases with increasing sample number. Finally, an
empirical application is made using the Mincer equation in South Africa.
Keywords: Modiﬁable Areal Unit Problem (MAUP), scale eﬀect, aggregation problem.
arXiv:1806.08433v1  [stat.AP]  21 Jun 2018

1
1
Introduction
Although spatial data are increasingly disaggregated, many socioeconomic studies require some
level of aggregation (e.g., neighborhoods, municipalities, states, districts, countries). Spatial aggre-
gation is useful for calculating rates and indexes, minimizing the inﬂuence of outliers, or preserving
conﬁdentiality Wise et al. (1997, 2001). Spatial aggregation is also useful for creating meaningful
units for analysis Yule and Kendall (1950); Duque et al. (2006), reducing computational complexity
Miller (1999), controlling for spurious spatial autocorrelation Bian and Butler (1999); Duque et al.
(2012), and comparing results at diﬀerent scales Holt et al. (1996a); Tagashira and Okabe (2002).
However, spatial aggregation triggers a problem known as the Modiﬁable Areal Unit Problem
(MAUP). The MAUP, introduced in the literature by Openshaw (1978) and Openshaw and Taylor
(1979), refers to the sensitivity of statistical results to changes in the spatial units of analysis. The
MAUP has two dimensions: the scale eﬀect and the zoning eﬀect. The scale eﬀect refers to changes
in the size of the spatial units, which implies a change in the number of spatial units, e.g., doing the
analysis at the state or county level. The zoning eﬀect refers to changes in the shape of the spatial
units preserving the number of units, e.g., aggregating USA counties into 50 states is merely one
of the many ways in which one can aggregate counties into 50 spatial units.
Although the literature on MAUP is extensive, to the best of our knowledge, there is no statisti-
cal tool that allows a practitioner to easily determine the level of sensitivity of a spatially intensive
variable to the MAUP. Hence, in this paper, we present S-maup, a nonparametric statistical test
to measure the sensitivity of a spatially intensive variable to the MAUP. Instead of looking at a
speciﬁc measure of central tendency or dispersion or at the coeﬃcient associated with the variable
in a speciﬁc regression, S-maup focuses on determining how much the distribution of the variable,
at its highest level of spatial disaggregation, will change when it is aggregated into a given number
of regions. For its calculation S-maup requires the number of areas, the ρ parameter, that measures
the degree of spatial correlation of the variable, and the number of regions in which the areas will
be aggregated. Under the null hypothesis of non-sensitivity to MAUP, S-maup would be useful
for determining the maximum level of aggregation that we can apply to a given variable before it
loses its distributional characteristics. S-maup could also be used to determine whether the results
obtained at a given scale (e.g., counties) hold for another scale (e.g., states).
The rest of this article is structured as follows. We begin with a literature review concerning
the primary research surrounding the MAUP. We then explore the eﬀects of the MAUP through a
computational experiment. Next, we propose a test statistic, S-maup, and its empirical distribution
under the null hypothesis of non-sensitivity to MAUP. Next, we establish the power and size of
the statistic under various levels of spatial autocorrelation and number of areas. We then present
a simple example of the use of the S-maup statistic. Last, we conclude and suggest avenues for
further investigation.
2
Literature Review
The eﬀects of aggregating spatial data have been a subject of study since the early 1930s and have
been referred to by diﬀerent names, such as aggregation eﬀects Gehlke and Biehl (1934), scale
problem Yule and Kendall (1950), ecological fallacy Robinson (1950), and Modiﬁable Areal Unit
Problem, MAUP, Openshaw and Taylor (1979). If one delves into the details, it can be argued
that these previous concepts are diﬀerent. However, these concepts possess as a common factor a
concern regarding the undesired eﬀects that result from working with aggregate data. Hereinafter,
we will refer to this problem as MAUP.

2
The literature on MAUP can be divided into three blocks: ﬁrst, deﬁnition of the problem
Openshaw (1977); Openshaw and Taylor (1979); Arbia (1989); second, measurement of its eﬀects on
statistics such as the mean Amrhein (1995); Steel and Holt (1996), median and standard deviation
Bian and Butler (1999), variance and covariance Amrhein and Reynolds (1996); Reynolds (1998),
and correlation coeﬃcient Gehlke and Biehl (1934); Yule and Kendall (1950); Openshaw and Taylor
(1979); Clark and Avery (1976); and last, potential ways to minimize the aggregation eﬀects Coulson
(1978); Fotheringham (1989); Arbia (1989); Fotheringham et al. (2000); Carrington et al. (2006).
It is well known that the impact of the MAUP on the mean can be considered negligible Arbia
(1989); Amrhein (1995); Amrhein and Reynolds (1996); Steel and Holt (1996). However, the MAUP
has a large impact on the variance, which decreases when the variable exhibit high values of spatial
autocorrelation Reynolds (1998). With respect to the statistical association, such as the covariance
and correlation coeﬃcient, Clark and Avery (1976), Openshaw and Taylor (1979) and Arbia (1989)
found that the sensitivity to MAUP increases as the level of spatial aggregation increases (scale
eﬀect), i.e., the correlation between variables X and Y will exhibit a wider variation if, for example,
USA counties are aggregated into 50 spatial units than if they were aggregated into 1,000 spatial
units.
The MAUP eﬀects have also been studied in OLS regressions Clark and Avery (1976); Open-
shaw (1978); Green and Flowerdew (1996); Tagashira and Okabe (2002), logit models Fotheringham
and Wong (1991), Poisson regression Flowerdew and Amrhein (1989), spatial interaction models
Arbia and Petrarca (2013), spatial econometrics models Arbia and Petrarca (2011), forecasts in
regional economy Miller (1998), and spatial autocorrelation statistics, such as the Moran’s coeﬃ-
cient, Geary’s Ratio, and G-Statistic Fotheringham and Wong (1991); Qi and Wu (1996); Jelinski
and Wu (1996). Other authors have studied the MAUP eﬀects in more sophisticated methods,
such as the factorial analysis Hunt and Boots (1996), spatial interpolation Cressie (1996), image
classiﬁcation Arbia et al. (1996), location and allocation models Goodchild (1979), and discrete
selection models Guo and Bhat (2004).
Although there is no solution to the MAUP because it is inherent to the use of spatial data, some
authors have proposed diﬀerent alternatives to minimize its eﬀects: the formulation of scale-robust
statistics King (1997), the design of optimal aggregations that minimize the loss of information
Moellering and Tobler (1972); Openshaw (1977); Nakaya (2000); Tagashira and Okabe (2002);
Duque et al. (2006), the use of a set of auxiliary or grouping variables together with variables at
the individual level Holt et al. (1996b); Wrigley et al. (1996), and the measurement of rates of
change through the concept of a fractal dimension Fotheringham (1989).
Most studies above required extensive computational experiments. Table 1 summarizes the main
characteristics of those experiments, including the covered dimensions (scale or zoning), studied
statistics (mean, variance, correlation, regression coeﬃcients, etc.), type of data (real or simulated),
studied variables (income, rates, random, etc.), and size of the experiment in terms of the number
of areas and regions (herein, we will refer to area as the smallest spatial unit of observation and
region as the spatial units that result from aggregating the areas into contiguous spatial units).
From this table, we can highlight the dominance of the use of real data over simulated data and
the evident increase in the size of the experiment as the computational capacity increases over
the years. As expected, the two driver parameters in these experiments are the number of areas
and the number of regions. Although it has been considered in a few experiment Openshaw and
Taylor (1979); Reynolds (1998); Bian and Butler (1999), the level of spatial autocorrelation of
the variables/attributes being aggregated plays an important role in the level of sensitivity of the
variable to the MAUP. Finally, the mean is signiﬁcantly highlighted by being the more common
grouping operator, i.e., if areas i and j, with attribute values Xi and Xj, are merged into a region,
the attribute value for the resulting region is calculated as the mean of Xi and Xj, which indicates

3
Figure 1. Instance of the experiment.
that all of the experiments use spatially intensive variables.
Based on the available literature, a practitioner can anticipate high(low) variation of its results
when the aggregation level is high(low) and the level of spatial autocorrelation of its variable is
low(high). However, there is no tool in the literature that allows the assignment of a speciﬁc number
and statistical signiﬁcance to that variation. The closest the research can get to that number would
require a computational experiment involving the calculation of the results for a large number of
random aggregations of the areas into a predeﬁned number of regions. This paper constitutes the
very ﬁrst attempt to formulate a nonparametric statistical test to easily measure the sensitivity of
a spatially intensive variable to the MAUP.
3
MAUP Eﬀects
In this section, we design a computational experiment to identify the key elements that should be
included in the construction of the statistical test. Following previous experiments in the literature
on the MAUP eﬀects (e.g., Amrhein and Reynolds (1996) and Arbia and Petrarca (2011)), we
consider the two main parameters involved in the exploration of scale and zoning eﬀects: number
of areas (N) and number of regions (K). As in Openshaw and Taylor (1979) and Reynolds (1998),
we also take into account diﬀerent levels of spatial autocorrelation, ρ.
Fig 1 summarizes the steps followed to generate an instance of the experiment: (1) yρ=0.9
is a random variable generated by a Spatial Autoregressive (SAR) process with autoregressive
parameter ρ = 0.9 and rook contiguity matrix. (2) The areas are randomly aggregated into K
spatially contiguous regions using a seed-based region growing algorithm proposed by Vickrey

4
Table 1. Computational experiments on MAUP.
Author
(Year)
Dimension
/
Eﬀect
on...
Grouping op-
erator
Data
Variable
Size
Gehlke
and
Biehl (1934)
Scale / rxy
Sum
Census
Tracts
in
Cleveland
Male
juvenile
delinquency
and
monthly income.
Agricultural
products and the number of farm-
ers
1) 252 areas into 200, 175, 150, 125, 100, 50, and
25 regions 2) 1,000 areas into 63, 40, 31, and 8
regions
Robinson
(1950)
Scale / rxy
Proportions
Nine geographic di-
visions of the USA in
1930
Race and illiteracy
97,272 individuals into 9 regions
Yule
and
Kendall (1950)
Scale / rxy
Mean
Agricultural counties
in England
Production of wheat and potatoes
per acre
48 areas into 24, 12, 6, and 3 regions
Clark and Av-
ery (1976)
Scale / rxy
Mean
Metropolitan area of
Los Angeles
Household income and education
level of the head of household
1,556 census tracts into 134 Welfare Planning
Council Study areas and 35 Regional Planning
Commission Statistical Areas
Openshaw and
Taylor (1979)
Scale - Zon-
ing / rxy
-
Counties in Iowa and
simulated data with
ρ+, ρ0, and ρ−
% of Republican votes and % pop-
ulation over 60 years.
99 areas into 6, 12, 18, 24, 30, 36, 42, 48, 54, 60,
66, and 72 regions
Arbia (1989)
Scale - Zon-
ing / µ, σ2,
σ, ρ
Mean
Quadrat in Hukuno
Town,
Japan
and
weights
of
wheat
plots of grain
Quadrat counts of houses and
weights of wheat plots of grain
1) Regular lattice of 32x32 into 16x16, 8x8, 4x4,
and 2x2 regions 2) Regular lattice of 25x20 cells
into 8x8, 4x4, and 2x2 regions
Fotheringham
and
Wong
(1991)
Scale - Zon-
ing / β′s
Mean and Pro-
portion
Metropolitan area of
Buﬀalo
Household income, % of popula-
tion per area, % of population over
65 years
871 areas into 800, 400, 200, 100, 50, and 25 re-
gions
Amrhein
(1995),
Steel
and
Holt
(1996)
Zoning
/
µ, σ2,
rxy y
β
Mean
and
weighted
aver-
age
Regular lattices
Simulated
data
with
Uniform,
Normal and Poisson distribution
10,000 areas into 10x10, 7x7, and 3x3 regions
Holt
et
al.
(1996a)
Zoning / σ2
Mean
City
of
Adelaide,
Australia
82 socioeconomic variables
917,000 people into 1,584 districts
Amrhein
and
Reynolds
(1996)
Zoning / σ2
Mean
Lancashire, UK
8 census variables
304 areas into 137, 122, 106, 91, 76, 61, 46, and
30 regions
Green
and
Flowerdew
(1996)
Scale - Zon-
ing / rxy, β
Mean
UK
Census variables and simulated
variables
Regular lattice of 120x120 into 1x1, 2x2, 3x3, 4x4,
and 5x5 regions
Qi
and
Wu
(1996)
Scale / I −
Moran
and
G−Statistic
Mean
Malasia
Biomass areas and elevation data
Regular Lattice of 220x188 into 2x2, 3x3, 4x4, ...,
and 20x20 regions
Jelinski
and
Wu (1996)
Scale / I −
Moran
and
G−Statistic
Mean
Manitoba, Canada
Normalized Diﬀerence Vegetation
Index (NDVI)
Regular lattice of 300x300 into 3x3, 5x5, 7x7, 9x9,
11x11, 13x13, and 15x15 areas
Reynolds
(1998)
Zoning / σ2,
rxy, and β
Mean
Regular lattices
Simulated variables with diﬀer-
ent levels of spatial autocorrela-
tion and variance
400 areas into 180, 160, 140, 120, 100, 80, 60, and
40 regions
Bian and But-
ler (1999)
Zoning / σ
Mean and Me-
dian
Regular lattices
Simulated data with diﬀerent lev-
els of spatial autocorrelation
Regular lattice of 512x512 into 3x3, 9x9, 11x11,
21x21, 31x31, 41x41, 51x51, 61x61, 71x71, and
81x81 pixel window sizes
Arbia and Pe-
trarca
(2011,
2013)
Scale / β
Mean
Regular lattice
Simulated data
Regular lattice of 64x64 into 32x32, 16x16, 8x8,
and 4x4.
rxy: Correlation, µ: Media, σ2: Variance, σ: Covariance, ρ: Spatial autocorrelation, β: Regression coeﬃcients.

5
(1961). The attribute value for each region is calculated as the mean value of the attribute values
of the areas assigned to the region. This random aggregation is repeated r = 30 times, so that
we generate 30 diﬀerent ways to aggregate N areas into K regions. (3) We calculate the mean
and variance of the original, disaggregated, variable as µo and σ2
o. (4) We calculate the mean and
variance of each one of the aggregated variables as µag and σ2
ag. (5) We calculate the relative
change in the mean (RCM), Eq (1), and the relative change in the variance (RCV ), Eq (2),
between the original variables and each of the 30 aggregated variables. (6) We summarize the eﬀect
of aggregating N areas into K regions as the mean RCM, RCM, and mean RCV , RCV , using Eq
(3) and (4). For each value of ρ considered in the experiment, we repeat steps (1) to (6) 50 times.
RCMµ,y
r
= |µo −µag
r |
µo
(1)
RCV σ2,y
r
=
σ2
o −σ2
ag,r

σ2o
(2)
RCM =
P30
r=1 RCMµ,y
r
30
(3)
RCV =
P30
r=1 RCV σ2,y
r
30
(4)
As we will show in the parametrization of the experiment, we generate instances of yρ for
diﬀerent levels of spatial autocorrelation (−0.9 < ρ < 0.9). If, for example, we generate two spatial
processes yρ=0.9 and yρ=−0.5, the observed aggregation eﬀects will have two sources, one that comes
from the change in the value of ρ, and one that comes from the diﬀerences in the values generated
by the random data generation process. To isolate the eﬀect that comes from the changes in ρ,
we generate the instances of yρ=0.0 by performing spatial permutations of the values obtained from
yρ=0.9.
As an example, Fig 2 summarizes the process that we implemented to generate yρ=0.0
from yρ=0.9: (1) Generate an SAR process yρ=0.9. (2) Generate a reference SAR process xρ=0.0.
(3) Generate yρ=0.0 by spatially redistributing the values of yρ=0.9 following the spatial pattern of
xρ=0.0, i.e., the highest value of yρ=0.9 goes to the area with the highest value of xρ=0.0; the second
highest value of yρ=0.9 goes to the area with the second highest value of xρ=0.0; and so forth. (4)
Estimate the true ρ value of yρ=0.0, if (0.0 −0.5) < ρ < (0.0 + 0.5) then keep yρ=0.0; otherwise,
repeat the process. Note that yρ=0.9 and yρ=0.0 have the same values and therefore the same mean
and variance, But due to the diﬀerences in the spatial distribution of the values, they have diﬀerent
ρ values.
Having clariﬁed the process that we follow at each instance and our strategy for generating the
yρ values, we present the parameters used in the computational experiment:
N =
Number of areas. N = {25, 100, 225, 400, 625, 900} ;
yρ
i =
SAR process with i = {1, . . . , 50} , and ρ = {±0.9, ±0.7, ±0.5, ±0.3, 0} ;
k =















for N=25, k = {3, 5, 10, 13, 15, 18, 20, 22, 24}
for N=100, k = {2, 4, 7, 12, 25, 40, 53, 67, 80, 90, 99}
for N=225, k = {3, 5, 10, 15, 30, 60, 90, 120, 150, 180, 200, 220}
for N=400, k = {4, 9, 18, 26, 50, 110, 160, 213, 267, 320, 360, 396}
for N=625, k = {4, 6, 14, 27, 43, 80, 170, 250, 333, 417, 500, 563, 618}
for N=900, k = {4, 9, 20, 40, 60, 120, 240, 360, 480, 600, 720, 810, 890}
r = 30
Number of random spatial aggregations.

6
Figure 2. Example of spatial autocorrelation generation.
Table 2. Eﬀect on mean - Proportion of signiﬁcant instances.
Number of areas
N = 25
N = 100
N = 225
N = 400
N = 625
N = 900
Proportion*
0
0.00063
0.00014
0.00041
0.00063
0.0012
Proportion of instances for which the two-sample t-test was rejected with α = 0.05. It includes instances with
k ≥10.
We implemented the experiment in Python 2.7.10. For the spatial aggregations, we use the
Python library ClusterPy 0.9.9 Duque et al. (2011). We ran the experiment in the supercomputer
APOLO, at the Center of Scientiﬁc Computation (Universidad EAFIT), equipped with a Dell Power
Egde 1950 III of 8 cores, 2.33 GHz Intel Xeon that executes Linux Rocks 6.1 to 64 bits.
Each box plot in Fig 3 summarizes the 50 values of RCM calculated for each value of ρ and
k. The maximum bounds value of the vertical axis in the ﬁgure show low relative changes in the
mean. To make sure that the mean eﬀect can be discarded, we calculate the two-sample t-test to
compare the mean of each original variable, µo, with the mean of each aggregated variable, µag. We
report in Table 2 the proportion of instances for which the two-sample t-test was rejected. From
this result, we can conclude that there is not a MAUP eﬀect on the mean, which is consistent with
those results found by Arbia (1989); Amrhein (1995) and Amrhein and Reynolds (1996).
Each box plot in Fig 4 summarizes the i = 50 values of RCV calculated for each value of ρ
and k. Unlike the case seen with the mean, the eﬀect of variance is considerably greater. The
box plots show that the eﬀect of MAUP on variance decreases for two reasons: an increase in the
level of spatial autocorrelation, ρ; and (2) an increase in the number of regions, k. These eﬀects on
variance are consistent with those found by Reynolds (1998).
To verify the MAUP eﬀect on variance, we use the Levene test for equality between the variance
of the original variable, σ2
o, with the variance of each aggregated variable, σ2
ag. Fig 5 shows the
percentage of instances for which the Levene test rejects the null hypothesis H0 : σ2
o = σ2
ag, with
α = 0.05. These results conﬁrm that the MAUP eﬀect decreases as either k or ρ increases.
Finally, in Fig 6 we present, for illustrative purposes, three instances with ρ = −0.9, ρ = 0.0
and ρ = 0.9 that aggregate N = 900 areas into k = 240 regions. These examples show how the
MAUP fades as ρ increases.

7
Figure 3. Relative change in mean - Average eﬀect. (a) N = 25; (b) N = 100; (c) N = 225; (d) N = 400; (e)
N = 625; (f) N = 900.

8
Figure 4. Relative change in variance - Average eﬀect. (a) N = 25; (b) N = 100; (c) N = 225; (d) N = 400; (e)
N = 625; (f) N = 900.

9
Figure 5. Proportion of instances for which the Levene test rejects the null hypothesis of equality of variance, with
a level of signiﬁcance α = 0.05. (a) N = 25; (b) N = 100; (c) N = 225; (d) N = 400; (e) N = 625; (f)
N = 900.

10
Figure 6. MAUP eﬀects at three levels of spatial autocorrelation, (a) ρ = −0.9, (b) ρ = 0, and (c) ρ = 0.9. Solid line:
original variable with N = 900; dashed lines: 30 aggregations with k = 240. The vertical lines indicate µo
and µag.
4
S-maup statistical test
Findings such as the eﬀect of MAUP on variance and how MAUP fades as ρ and k increase are useful
to ﬁnd the functional form of our statistical test, S-maup, for measuring the level of sensitivity
of a spatially distributed variable to the MAUP. We designed the test such that S-maup takes
values close to zero when the variable is not sensitive to the MAUP and values close to one when
the variable is highly sensitive to the MAUP. Furthermore, S-maup will be a univariate statistic
applicable to spatially expansive variables whose aggregated values result from the average of the
individual values.
4.1
S-maup
To ﬁnd the functional form of S-maup, it is necessary design an expression that describes the
distribution of the eﬀects of MAUP on the variance (RCV ). To summarize those eﬀects, we took
the median of each Box Plot in Fig 4. Fig 7 shows an example of those summarized eﬀects.
According to Fig 7, the mathematical expression of our test should take values close to one
when the variable under evaluation has high negative spatial autocorrelation, ρ and is aggregated
into a small number of regions, k. Conversely, the expression should take values close to zero when
the variable under evaluation has high positive spatial autocorrelation, ρ and is aggregated into a
large number of regions, k. Our expression should also be able to reproduce the way in which, for
a give k, the MAUP eﬀects decreases as ρ increases. Note that such a decrease is not the same for
all values of k: when k is large, the eﬀects of MAUP are low even for highly negative values of ρ;
therefore, for a high k, the reduction of the MAUP eﬀects, as ρ increases, are almost imperceptible.

11
Figure 7. Median RCM for N = 100.
Thus, as k increases, our expression should modify the speed and moment at which the MAUP
fades along ρ. Taking into account these diﬀerent conditions, we started the construction of our
S-maup statistic using an inverted logistic function Verhulst (1845), which is deﬁned by Eq (5).
M(ρ; L, η, τ) =
L
1 + ηeτx
,
(5)
where L determines the maximum value of the curve; η determines the moment at which the curve
begins to decline; and τ indicates the speed at which the curve declines. If we endogenize those
three parameters, we should be able to approximate any line of the type shown in Fig 7. This is
what we are going to develop in the rest of this subsection until we obtain an expression of M in
which parameters L, η and τ depend on ρ, k and N.
Starting with the parameter L, Fig 7 shows that the maximum value of each logistic curve
depends on the level of aggregation k. This aggregation can be deﬁned in relative terms as θ = k
N .
Therefore, the lower the level of aggregation (i.e., as θ approaches 1), the lower should be L. When
plotting each median RV C against θ, it depicts an inverted ”S” that could also be modeled as an
inverse logistic function with the expression presented in Eq (6), whose linear form is given by Eq
(7).
L(θ) =
1
1 + eb+mθ
,
(6)
Ln
1 −L
L

= b + mθ ,
(7)
where b and m are the parameters of the inverse logistic function. To estimate those parameters,
we used a robust linear regression model that minimizes the inﬂuence of outliers. The parameter

12
Figure 8. Adjustments of robust linear regression models: (a) Linearized logistic function (L); (b) Linearized power
function (η); (c) Linear function (τ).
associated with θ is signiﬁcant, and the adjusted R-squared = 86.7%. Fig 8(a) shows the robust
regression over the linearized logistic function.
Returning to the logistic curves in Fig 7, both the moment at which the curves begin to decrease,
η, and the speed of decreasing, τ, depend on k. Therefore, both parameters can be estimated as
function of θ =
k
N . For this, we adjusted an inverse logistic function for each curve of the type
presented in Fig 7. For each curve, the values of η and τ were calibrated using the optimized
module of Scipy Phyton Library Jones et al. (2001). With this process, we obtained a value for η
and τ for each value of θ. Then, we use a linearized power function, Eq (8), and a linear function,
Eq (9), to express η and τ as a function of θ.
η(θ) = pθa
(8)
τ(θ) = β0 + β1θ
(9)
The parameter associated with θ was signiﬁcant in both estimations and the adjusted R2, with
91.7% and 84.5% respectively. Fig 8b and 8c present the estimations.
Replacing the Eq (6), (8) and (9) in (5) we have the Eq (10).
M(ρ, θ) =
1
1+eb+mθ
1 + pθae(β0+β1θ)ρ
(10)
The results of the estimation of the parameters in the robust linear regression model for the
logistic function of L are as follows: m = 7.031 and b = −2.188. Considering that the model is

13
estimated with the linearized logistic function, these results were transformed by natural logarithm.
For the power function of η, the results are p = 0.516 and a = 1.287, because of the linearization
of the power function, we applied the natural logarithm to the parameter p. Finally, the results of
the linear function of τ are as follows: β0 = 5.319 and β1 = −5.532. Replacing in the equations
produces the following:
L(θ) =
1
1 + e−2.188+7.301θ
(11)
η(θ) = 0.516θ1.287
(12)
τ(θ) = 5.319 −5.532θ.
(13)
Thus, the expression of the S-maup statistic is the following:
M(ρ, θ) =
1
1+e−2.188+7.031θ
1 + [0.516θ1.287]e[5.319−5.532θ]ρ
(14)
Recall that S-maup statistic (M) is designed in such a way that for a bigger (smaller) sensitivity
of a variable to the MAUP, the larger (smaller) is the value of M. This characteristic allows us to
deﬁne a non-parametric unilateral statistical test, which is stated below:
H0: The variable yi is not signiﬁcantly aﬀected by the MAUP.
H1: The variable yi is signiﬁcantly aﬀected by the MAUP.
Where the statistic for the test is given by Eq (14), and therefore, H0 will be rejected if the statistic
value belongs to the rejection region (RR) deﬁned in Eq (15).
RR = {M|M > Mα;ρ,N}
(15)
Mα;ρ,N is the critical value given a signiﬁcance level α, a level of spatial autocorrelation (ρ), and
a number of areas (N). We implemented a Monte Carlo simulation to ﬁnd the empirical distribution
of the S-maup under the null hypothesis previously stated. The empirical distribution allows us to
obtain the critical values as well as the pseudo-value p to determine the proof signiﬁcance.
4.2
Critical values and p-value
To calculate the critical values, we performed an exhaustive simulation study based on non-
parametric statistic methodology. Recall that H0 means no sensitivity of a variable to MAUP,
which is equivalent to stating that, for a given k, the variance of the aggregated variable is statis-
tically equal to the variance of the original variable. For building the empirical distribution under
H0, we set a value for N and ρ and generated an SAR process with parameters (N, ρ). Then, we
randomly selected an integer value k such that 0.1N < k < N, thus yielding 30 random aggre-
gations of the variable into k regions. Next, we applied the Levene test for equality of variances
between the original variable and each one of the 30 aggregated variables. The SAR(N, ρ) variable
was kept if and only if the Levene test was not rejected in all 30 cases. If there was at least one
rejection, then we chose, at random, a new k and repeated the previous steps. This procedure was
repeated until we obtained 1,000 instances for each pair (N, ρ). We then calculated the S-maup

14
statistic for those instances using Eq (14) and generated the empirical distribution of the statistics
under H0. The critical values were obtained by calculating the 90%,
95%,
99% percentiles for
the empirical distribution. Table 3 presents the table of critical values. This Table implied the
generation of 54,000 instances.
Following the percentile approach utilized by Rey (2004), we can calculate a pseudo-p-value for
a given value of the S-maup test (M), using the Eq (16):
P(M) =
1
1, 000
1,000
X
j=1
Ψ ,
(16)
where Ψ = 1 if Mρ,N
j
> M, Ψ = 0 otherwise.
The vector Mρ,N
j
comes from the simulations
performed to produce Table 3.
Since those vectors are extremely computationally intensive to
produce (in some instances requiring months of supercomputer computation for completion), they
will be publicly available at http://www.
.edu, as well as the Python script to run the S-maup
statistic.
Table 4 presents some examples of the S-maup statistic for diﬀerent values of N and k. Note
that when the variable yi presents characteristics against the null hypothesis (H0), then the M value
of the S-maup should be greater than the critical value at some signiﬁcance level α, and therefore,
the pseudo-value p of the test must be smaller than the signiﬁcance level. If H0 is rejected, it can
be concluded that the variable yi is sensitive to the MAUP, and therefore, a MAUP eﬀect exists
when aggregating yi in k regions.
Note that when the spatial autocorrelation is highly positive (e.g., ρ = 0.801), the variable
allows high levels of aggregation. The results also conﬁrm that low levels of spatial aggregation do
not lead to the undesirable eﬀects of MAUP.
5
Power and Size
The power is a natural way of evaluating the test performance. It is deﬁned as the probability
of rejecting the null hypothesis, given that the null hypothesis is false. In other words, it is the
probability of not committing a type II error (β); thus, the power is (1 −β). In our context, the
power means the probability that suﬃcient statistical evidence exists in the sample to aﬃrm that
the variable yi is aﬀected by the MAUP, when in fact, the variable yi is aﬀected by the dimensions
of the MAUP. Hence, it is expected that the power of the test is close, or equal, to 1.
Since H1 implies that the variance of the original variable is diﬀerent from the variance of the
aggregate variable, we implemented the following simulation experiment to measure the power of our
statistical test: For each tuple (N, ρ), with N ∈{100, 400, 900} and ρ ∈{±0.9, ±0.7, ±0.5, ±0.3, 0}.
Given a tuple (N, ρ) we generate an SAR process and perform 30 random spatial aggregations
of the N areas into k regions such that k is selected at random as an integer value such that
0.1N < k < N. The SAR process is kept if and only if the Leven test between the original variable
and each one of the 30 aggregated variables is rejected. We repeat this process until we generate
1,000 valid instances for each tuple (N, ρ). Each entry in Table 5 reports the proportion of 1,000
instances for which our test rejects H0. Because most values are close to 1, we can argue that our
S-maup is highly eﬀective in identifying variables that are sensitive to the MAUP eﬀect.
Test size is also a way of evaluating the test performance. Test size is deﬁned as the probability
of rejecting the null hypothesis given that the null hypothesis is true. In other words, it is the
probability of committing a type I error (α). In our context, test size means the probability that
suﬃcient statistical evidence exists in the sample to aﬃrm that the variable yi is aﬀected by the

15
Table 3. Critical Values (Mα;ρ,N)
Number of areas (N)
ρ
α
25
100
225
400
625
900
-0.9
0.01
0.83702
0.09218
0.23808
0.05488
0.07218
0.02621
0.05
0.83699
0.08023
0.10962
0.04894
0.04641
0.02423
0.1
0.69331
0.06545
0.07858
0.04015
0.03374
0.02187
-0.7
0.01
0.83676
0.16134
0.13402
0.06737
0.05486
0.02858
0.05
0.83662
0.12492
0.08643
0.05900
0.04280
0.02459
0.1
0.79421
0.09566
0.06777
0.05058
0.03392
0.02272
-0.5
0.01
0.83597
0.16524
0.13446
0.06616
0.06247
0.02851
0.05
0.83578
0.13796
0.08679
0.05927
0.04260
0.02658
0.1
0.68900
0.10707
0.07039
0.05151
0.03609
0.02411
-0.3
0.01
0.83316
0.19276
0.13396
0.06330
0.06090
0.03696
0.05
0.78849
0.16932
0.08775
0.05464
0.04787
0.03042
0.1
0.73592
0.14282
0.07076
0.04649
0.04001
0.02614
0.0
0.01
0.82370
0.17925
0.15514
0.07732
0.07988
0.09301
0.05
0.81952
0.15746
0.11126
0.06961
0.06066
0.05234
0.1
0.71632
0.13621
0.08801
0.06112
0.04937
0.03759
0.3
0.01
0.76472
0.23404
0.24640
0.11588
0.10715
0.07070
0.05
0.70466
0.21088
0.15360
0.09766
0.07938
0.06461
0.1
0.63718
0.18239
0.12101
0.08324
0.06347
0.05549
0.5
0.01
0.67337
0.28921
0.25535
0.13992
0.12975
0.09856
0.05
0.59461
0.23497
0.18244
0.11682
0.10129
0.08860
0.1
0.46548
0.17541
0.14248
0.10008
0.08137
0.07701
0.7
0.01
0.52155
0.47399
0.29351
0.23923
0.20321
0.16250
0.05
0.48958
0.37226
0.22280
0.20540
0.16144
0.14123
0.1
0.34720
0.28774
0.18170
0.16442
0.13395
0.12354
0.9
0.01
0.28599
0.28938
0.43520
0.44060
0.34437
0.55967
0.05
0.21580
0.22532
0.27122
0.29043
0.23648
0.31424
0.1
0.17640
0.18835
0.21695
0.23031
0.19435
0.22411

16
Table 4. Example S-maup.
V ariable
N
k
ρ
M
Mα;ρ,n
Pseudo-v p
y1
i
1,000
400
0.007
0.24002
0.05234
0.0 ***
y2
i
1,000
600
0.007
0.05871
0.05234
0.034 **
y3
i
1,000
800
0.007
0.01187
0.05234
0.616
y4
i
500
100
-0.634
0.09237
0.05900
0.0 ***
y5
i
500
280
-0.634
0.05466
0.05900
0.078 *
y6
i
500
380
-0.634
0.00767
0.05900
0.852
y7
i
220
60
0.562
0.32197
0.18244
0.00 ***
y8
i
220
90
0.562
0.18513
0.18244
0.046 **
y9
i
220
150
0.562
0.04357
0.18244
0.443
y10
i
150
15
0.801
0.29201
0.22532
0.009 **
y11
i
150
50
0.801
0.08072
0.22532
0.366
y12
i
150
90
0.801
0.00997
0.22532
0.883
*** p < 0.01, ** p < 0.05, * p < 0.1.
Table 5. Estimated power of S-maup.
Number of areas (N)
ρ
N = 100
N = 400
N = 900
-0.9
0.989
0.985
0.997
-0.7
0.986
0.996
1.000
-0.5
0.981
0.998
1.000
-0.3
0.982
0.998
1.000
0.0
0.997
0.999
0.999
0.3
0.986
0.996
1.000
0.5
0.986
0.996
0.999
0.7
0.783
0.985
0.995
0.9
0.977
0.703
0.492
Level of signiﬁcance α = 0.05.

17
Table 6. Estimated size of S-maup.
Number of areas (N)
ρ
N = 100
N = 400
N = 900
-0.9
0.163
0.087
0.065
-0.7
0.080
0.037
0.080
-0.5
0.091
0.043
0.083
-0.3
0.073
0.097
0.136
0
0.102
0.066
0.026
0.3
0.081
0.057
0.038
0.5
0.098
0.062
0.032
0.7
0.043
0.032
0.045
0.9
0.110
0.024
0.009
Level of signiﬁcance α = 0.05.
MAUP, when in fact the variable yi is not. Hence, it is expected that the proportion of instances
for which our test commits type I error is close the theoretical signiﬁcance level (α).
The empirical test size is calculated following a similar procedure implemented to calculate the
power, but in this case, the tuple (N, ρ) is selected if and only if the Levene test is not rejected in
all 30 cases. Table 6 reports the size of our test, which show the best performance in scenarios of
positive spatial autocorrelation.
6
An illustrative application of the S-maup test
In this section, we present an empirical illustration within the context of a Mincer wage equation
Mincer (1974) that explains the salary based on schooling and experience. Eq (17) presents the
most basic version of the Mincer wage equation.
LNW = β0 + β1 ∗Y RSCHOOL + β2 ∗EXP + β3 ∗EXP 2 + ε,
(17)
where LNW is the natural logarithm of income (hourly wage), Y RSCHOOL years of schooling,
EXP years of potential labor market experience (calculated as the age in years minus years of
education plus 6), and ε is a mean zero residual.
It is important to clarify that this example
is merely illustrative. We use this equation because its simplicity allows us to present a simple
application of our test.
We use the 2011 census data from South Africa retrieved from the Integrated Public Use Mi-
crodata Series, International (IPUMS-International), at the Minnesota Population Center Center
(2015). The data include 688,310 individuals who were working at the time of the survey. We aggre-
gate the individual data into 206 municipalities using the weighted average of individual incomes,
years of schooling, and the potential work experience.
The 206 municipalities are our basic unit of analysis (i.e., our disaggregated variable). Other
administrative units in South Africa include 52 districts and 9 provinces. Table 7 shows some
descriptive statistics of our variables at the three administrative levels. Note how the standard de-
viation of the three variables narrows as the level of aggregation increases. The spatial distribution
of the variables is presented in Fig 9.

18
Figure 9. Municipalities: (a), (b) and (c). Districts: (d), (e) and (f). Provinces:(g), (h) and (i).
Table 7. Descriptive Statistics.
Municipalities
Variable
Obs.
Mean
Desv. Std.
M´ın.
M´ax.
LNW
206
10.51
0.35
9.64
11.73
YRSCHOOL
206
9.95
0.81
7.43
11.87
EXP
206
21.69
1.71
15.28
26.64
Districts
LNW
52
10.56
0.25
10.15
11.20
YRSCHOOL
52
10.06
0.61
8.28
10.99
EXP
52
21.59
1.21
18.66
24.24
Provinces
LNW
9
10.57
0.19
10.31
10.86
YRSCHOOL
9
10.00
0.44
9.35
10.59
EXP
9
21.77
0.77
20.54
23.09

19
Table 8. Mincer Model Estimate: South Africa.
LNW
Coef.
Desv. Std
p > |t|
Conﬁdence Interval at 95%
YRSCHOOL
0.3364
0.0259
0.000 ***
0.2852
0.3876
EXP
0.4008
0.1499
0.008 ***
0.1051
0.6965
EXP2
-0.0085
0.0034
0.016 **
-0.0153
-0.0016
CONST.
2.4796
1.6243
0.128
-0.7232
5.6825
Num. Obs. 206
F(3,202) = 68.84
R2 adjusted = 0.498
*** p < 0.01, ** p < 0.05, * p < 0.1.
Table 8 presents the estimation at the municipal level. The coeﬃcients of education and expe-
rience are signiﬁcant and exhibit the expected signs.
What would be the maximum level of spatial aggregation for which these results hold? Note that
here we are asking about the minimum value for k that preserves the distributional characteristics
of the variables; we are not aiming to evaluate a speciﬁc regionalization for a give value of k. We
can use our S-maup statistic to answer this question by identifying the minimum value of k for
which our test fails to reject the null hypothesis of no inﬂuence of the MAUP. In Table 9, we present
the results of our test for diﬀerent levels of spatial aggregations. For this, our test requires the level
of spatial autocorrelation of each variable (ρ) and the value of θ = k
N . Note that at k = 135, the
S-maup indicates that the variable LNW is aﬀected by the MAUP. This ﬁnding may imply that
the results obtained at municipal level (k = 206) may hold until an aggregation level of k = 136
that is the aggregation level at which all the variables involved in the regression do not lose their
distributional characteristics. Another conclusion from these results is that the results obtained at
the municipal level do not hold at district or province levels.
Fig 10 compares the coeﬃcients obtained at the municipal level (black and dashed vertical lines)
with the distribution of the coeﬃcients obtained by estimating the Mincer equation on 1,000 random
spatial aggregations of the k = 206 municipalities into k = 136 regions. Fig 10a, corresponding
to years of education, shows that 100% of the coeﬃcients estimated with k = 136 fall into the
95% conﬁdence intervals.
Fig 10b, corresponding to years of experience, shows that 98.8% of
the coeﬃcients estimated with k = 136 fall into the 95% conﬁdence intervals. Finally, Fig 10c,
corresponding to the squared years of experience, shows that 98.7% of the coeﬃcients estimated
with k = 136 fall into the 95% conﬁdence intervals.
Next, we estimated the Mincer model for k = 52 and compared it with the estimation for
k = 206 municipalities. As we did previously, we made 1,000 random aggregations and obtained
the distribution of the estimated coeﬃcients for K = 136 and k = 52. Fig 11 shows how the
estimations with k = 52 are more volatile and deviated than those with k = 136 regions. Note also
that the coeﬃcients for K = 206
7
Conclusions
This paper introduced the ﬁrst statistic of its kind for measuring the level of sensitivity of a spatially
expansive variable to the MAUP. The statistic is easy to implement because it only requires as input
parameters the level of aggregation θ = k
N and the level of spatial autocorrelation of the variable ρ.

20
Table 9. Estimator of the statistic S -maup: South Africa.
N = 206
LNW
YRSCHOOL
EXP
EXP2
ρ = 0.05
ρ = 0.25
ρ = 0.24
ρ = 0.40
k
M
Ps-v p
M
Ps-v p
M
Ps-v p
M
Ps-v p
200
0.011
0.806
0.011
0.820
0.011
0.819
0.011
0.833
180
0.022
0.589
0.021
0.619
0.021
0.619
0.020
0.656
150
0.057
0.242
0.052
0.330
0.053
0.327
0.048
0.414
136
0.087
0.101
0.079
0.197
0.079
0.194
0.072
0.302
135
0.091
0.094 *
0.081
0.187
0.082
0.185
0.073
0.295
134
0.093
0.089 *
0.083
0.181
0.084
0.179
0.076
0.290
132
0.099
0.077 *
0.088
0.166
0.089
0.166
0.079
0.273
124
0.124
0.036 **
0.111
0.115
0.112
0.114
0.099
0.208
122
0.131
0.032 **
0.117
0.107
0.118
0.104
0.104
0.186
120
0.139
0.025 **
0.123
0.094 *
0.125
0.091 *
0.110
0.167
118
0.147
0.019 **
0.131
0.081 *
0.132
0.080 *
0.142
0.101
110
0.182
0.003 **
0.161
0.043 **
0.163
0.042 **
0.149
0.093 *
108
0.192
0.001 **
0.169
0.034 **
0.172
0.033 **
0.149
0.093 *
52
0.584
0.000 ***
0.527
0.001 ***
0.533
0.001 ***
0.461
0.00 ***
9
0.863
0.000 ***
0.847
0.001 ***
0.849
0.001 ***
0.822
0.00 ***
*** p < 0.01, ** p < 0.05, * p < 0.1.
Figure 10. Distribution of coeﬃcients, k = 136: (a) YRSCHOOL; (b) EXP; (c) EXP2. horizontal black line: coeﬃ-
cient (206 municipalities), dashed lines are the respective conﬁdence intervals 95%.

21
Figure 11. Distribution of coeﬃcients.
line:k = 136, dotted line:k = 52: (a) YRSCHOOL; (b) EXP; (c) EXP2.
horizontal black line: coeﬃcient (206 municipalities). horizontal dotted line: coeﬃcient (52 districts).
The test exhibits good statistical power and size. We also provide the table of critical values and
a procedure to calculate the pseudo-p value of the test.
The empirical application shows the usefulness of the test for identifying the maximum level of
aggregation at which the original variable preserves its distributional characteristics. Additionally,
it can be useful to test whether two aggregation levels are comparable.
We recognize that the main properties of the S-maup were obtained from an empirical simulation
procedure, and they rely more heavily on hard experimental computation than theoretical methods.
However, the complexity of the question addressed in this paper may explain why this is the ﬁrst
attempt to answer it even though the MAUP has been in the literature since the late 1970s. We
hope that this ﬁrst attempt motivates other researchers to contribute other approaches to answer
the same question.
Acknowledgement
We thank Professor Andr´es Ram´ırez Hassan for his comments. We also thank the Cyberinfrastruc-
ture Service for High Performance Computing, Apolo, at Universidad EAFIT for letting us run our
computational experiments on their supercomputer.
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