The approaches to measuring the potential spatial access to urban health services revisited: distance types and aggregation-error issues
Apparicio et al. Int J Health Geogr
The approaches to measuring the potential spatial access to urban health services revisited: distance types and aggregation-error issues
Philippe Apparicio 0 3
Jérémy Gelb 0 3
Anne‑Sophie Dubé 2
Simon Kingham 1
Lise Gauvin 2
Éric Robitaille 2 4
0 Centre Urbanisation Culture Société, Institut National de la Recherche Scientifique , 385 Sherbrooke Street East, Montréal, QC H2X 1E3 , Canada
1 GeoHealth Laboratory, Department of Geography, University of Canterbury , Private Bag 4800, Christchurch 8140 , New Zealand
2 Depart‐ ment of Social and Preventive Medicine, Faculty of Medicine, University of Montréal , P.O. Box 6128 , Downtown Station , Montréal, QC H3C 3J7 , Canada
3 Centre Urbanisation Culture Société, Institut National de la Recherche Scien‐ tifique , 385 Sherbrooke Street East, Montréal, QC H2X 1E3 , Canada
4 Institut National de Santé Publique du Québec , 190 Boulevard Crémazie Est, Montréal, QC H2P 1E2 , Canada
Background: The potential spatial access to urban health services is an important issue in health geography, spatial epidemiology and public health. Computing geographical accessibility measures for residential areas (e.g. census tracts) depends on a type of distance, a method of aggregation, and a measure of accessibility. The aim of this paper is to compare discrepancies in results for the geographical accessibility of health services computed using six distance types (Euclidean and Manhattan distances; shortest network time on foot, by bicycle, by public transit, and by car), four aggregation methods, and fourteen accessibility measures. Methods: To explore variations in results according to the six types of distance and the aggregation methods, correlation analyses are performed. To measure how the assessment of potential spatial access varies according to three parameters (type of distance, aggregation method, and accessibility measure), sensitivity analysis (SA) and uncertainty analysis (UA) are conducted. Results: First, independently of the type of distance used except for shortest network time by public transit, the results are globally similar (correlation >0.90). However, important local variations in correlation between Cartesian and the four shortest network time distances are observed, notably in suburban areas where Cartesian distances are less precise. Second, the choice of the aggregation method is also important: compared with the most accurate aggregation method, accessibility measures computed from census tract centroids, though not inaccurate, yield important measurement errors for 10% of census tracts. Third, the SA results show that the evaluation of potential geographic access may vary a great deal depending on the accessibility measure and, to a lesser degree, the type of distance and aggregation method. Fourth, the UA results clearly indicate areas of strong uncertainty in suburban areas, whereas central neighbourhoods show lower levels of uncertainty. Conclusion: In order to accurately assess potential geographic access to health services in urban areas, it is particularly important to choose a precise type of distance and aggregation method. Then, depending on the research objectives, the choices of the type of network distance (according to the mode of transportation) and of a number of accessibility measures should be carefully considered and adequately justified.
Accessibility of health services; GIS; Sensitivity analysis; Uncertainty analysis; Cartesian distance; Network distances
The geographical accessibility of services (e.g. health
services, food stores, etc.) is an important issue in health
geography, spatial epidemiology and public health. Since
the 2000s, moreover, a growing number of articles have
been published on this topic (Fig. 1). In the wake of the
seminal article by Penchansky and Thomas [
], it has
generally been agreed that the concept of access is
multidimensional and can be defined in terms of affordability,
acceptability, availability and spatial accessibility. Other
scholars also note that this notion can be defined
according to two dimensions: potential or revealed, and spatial or
]. Potential accessibility considers the
probable utilization of services, given the population size and
its demographics, while revealed accessibility concerns the
actual use of services. Spatial access analyzes the
importance of spatial separation between supply and demand as
a barrier or a facilitator, and aspatial access focuses on
nongeographical barriers or facilitators [
the notion of access to health services encompasses four
major categories: revealed spatial access, revealed
aspatial access, potential spatial access, and potential aspatial
]. This study focuses on potential spatial access,
which refers to the ease with which residents of a given
area can reach services and facilities [
The deployment of potential spatial access measures
requires the specification of a set of four parameters,
namely: (1) a spatial unit of reference for the population,
i.e. a definition of residential areas (e.g. census tracts); (2)
an aggregation method, i.e. to account for the
distribution of population in the residential area; (3) a measure
of accessibility; and (4) a type of distance to be used in
computing the accessibility measures selected [
shown in a previous study [
], the choice of these
parameters is likely to generate different results, which could
potentially lead to significant measurement errors. For
example, for the Montreal metropolitan area, this study
has shown that potential spatial access varies a great deal
for 10% of census tracts, and mainly for those located in
suburban areas, according to the type of distance used,
and according to the aggregation method [
no study has attempted to simultaneously evaluate the
impact of these various parameters in order to identify
their respective importance in the evaluation of potential
spatial access; this is what we now propose to do, with
the help of sensitivity and uncertainty analyses.
Concretely speaking, the objective of this paper is to
revisit that previous study [
] by adding three important
improvements. It is first a matter of revisiting the
comparison of the types of distance by including four new
time-distances according to the mode of transportation
used: walking, cycling, public transit and car. Indeed,
since the advent of general transit feed specification
(GTFS) files, more and more studies on the access to
services have been based on shortest network time (by
public transit) [
]. Other recent research, although
rarer, also looks at bicycle accessibility [
]. The second
improvement involves evaluating aggregation errors by
including another aggregation method based on the
utilization of a land use map. Thirdly, other accessibility
measures that have been proposed in recent years, such
as the two-step floating catchment area (2SFCA) method
and its variants, have been added.
Evaluating potential spatial access to services and facilities in residential areas: specifying four parameters
Spatial unit of reference and aggregation methods
Selecting the appropriate spatial unit of analysis, i.e. the
operational definition for residential areas, is critical
for minimizing aggregation errors [
error arises from the distribution of individuals around
the centroid of spatial units . In the urban context,
as spatial units vary in size from smaller areas, such as
census blocks, to larger ones, such as census tracts, the
accessibility measured for smaller units is less subject to
aggregation error than that measured for larger spatial
As indicated by Hewko et al. [
], the census tract is
often selected for several reasons. First, detailed
socioeconomic, socio-demographic, and housing data are
available at the census tract level, which is not always the
case at finer levels such as dissemination areas or census
blocks. So, if we want to place accessibility measures in
relation to socioeconomic variables, by using either
classic or multilevel regressions, the census tract remains a
highly relevant choice. Secondly, census tracts include on
average about 5000 inhabitants and are relatively
homogeneous on the socioeconomic level and from the point
of view of housing. They consequently represent a
division at the neighbourhood level that is widely used by
urban planners and public health experts. Nonetheless,
using the census tract requires that we then apply
aggregation methods so as to limit errors in the measurement
of potential spatial access.
To evaluate the potential spatial access to a health
service for a population living in a residential area, e.g. a
census tract, several methods can be used [
6, 15, 16
The first method consists in computing the distance
between the centroid of the census tract and the
service (Fig. 2a). This method shows the inappropriateness
of ignoring the spatial distribution of the population
inside the census tract. The second method consists in
calculating the population-weighted mean centre of the
census tract (Eq. 1) and then evaluating the distance
between this new location and the service. Toward this
end, smaller spatial units entirely contained within the
census tracts can be used, such as dissemination areas,
census blocks, postal codes or buildings. This method
accounts for the spatial distribution of the population
inside the census tract in order to minimize aggregation
xi, yi =
b∈i wbxb ,
where wb represents the total population of spatial unit
b completely within census tract i (i.e. dissemination
area or census block or postal code) and xb, yb are the
Cartesian coordinates of the spatial unit b.
The third method consists in computing the distance
between the services and each centroid of spatial units
completely within census tracts, and then calculating the
average of these distances weighted by the total
population of each unit. For example, this operation is shown
based on dissemination areas and blocks contained
within census tracts in Fig. 2b, c.
The latter approach enhances the preceding one. It is
a matter of calculating the accessibility measures on the
level of blocks contained within census tracts, and then
computing the average weighted by the population.
However, the centroids of the blocks are first adjusted by using
dasymetric mapping methods designed to locate the
areas where the population in a given spatial unit lives
(e.g. census tract, dissemination area, block) [
This approach requires the use of either satellite images
, or land use or cover maps [
]. The basic principle
involves creating a binary mask separating residential
areas (1) from non-residential areas (0). For example, as
illustrated in Fig. 2d, a land use map was employed to
identify the residential portion of each block (the
category in yellow). A comparison of Fig. 2c, d shows that
the location of the block centroids is then more precise.
Compared with the previous methods, this latter method
is more accurate because it more exactly accounts for the
distribution of the population inside the census tract.
Since 2000, a number of literature reviews have been
published on the accessibility of health services [
]. They show that the five most commonly used
measures of the accessibility of health services are: (1) the
distance to the closest service [e.g. 23–25]; (2) the
number of services within n metres or minutes [e.g. 26, 27];
(3) the mean distance to the n closest services [e.g. 28];
(4) gravity models [e.g. 2, 29, 30]; and (5) two-step
floating catchment area (2SFCA) methods [
]. Table 1
synthesizes various approaches for conceptualizing
and measuring different dimensions of potential spatial
5, 28, 35
The first three measures are only based on the supply
of services. The most often used measure is clearly the
distance to the closest health service (e.g. nearest hospital
or medical clinic). It allows one to evaluate the
immediate proximity to the health services. If the most accurate
aggregation method detailed above is selected, i.e. an
aggregation method based on the population-weighted
mean of the accessibility measure for block centroids
(adjusted with the land use map) within census tracts,
this accessibility measure can be written as:
b∈i wb min dbs
where a weaker value of Aia implies better accessibility,
wb is the total population of census block b completely
within census tract i and dbs is the distance between
census block b and service s.
Immediate proximity or minimum travel time or distance
The distance between a location and the closest facility
Availability provided by the immediate surroundings or cumulative opportunity The number of facilities within a given distance from a point of origin
Average cost to reach all destinations
Average cost to reach diversity
Accessibility according to proximity and availability
The average distance between a location and all facilities
The average distance between a location and n facilities
Gravity models, two‑step floating catchment area (2SFCA) methods
The second accessibility measure—the number of
services within n metres or minutes—refers to the
cumulative opportunity, or, in other words, to the availability
provided by the immediate surroundings:
The 2SFCA method is a fairly recent one; it was
proposed in 2003 by Luo and Wang [
], based on the
work of Radke and Mu . As its name indicates, it
includes two steps. The first step assigns an initial ratio to
each health service, which takes the following form:
where a larger value of Aib implies better accessibility, wb
is defined as previously indicated, S represents all
services in the study area, and Sj is the number of services
within n metres or minutes of census block centroid b
(with Sj = 1 where dbs ≤ n and Sj = 0 where dbs > n).
To evaluate the average cost to reach diversity, the
mean distance to the n closest services is generally used.
For example, in a study on food deserts [
], the mean
distance to the three closest different chain-name
supermarkets is used as a proxy of variety in terms of food and
prices. For this measure, a weaker value implies better
where wb is defined as previously indicated, dbs represents
the distance between spatial unit centroid b and service s
(dbs is sorted in ascending order), and n is the number of
closest services to be included in the measure.
However, the three measures described above are only
based on the supply of services. Now, as mentioned by
several authors, as mentioned by several authors [
], the potential spatial access to health care depends on
both the location of the supply of health services and the
residential location of potential health users (demand).
Two types of measures allow for take these two
dimensions into account (supply and demand): i.e. gravity
models [e.g. 2, 29, 30] and two-step floating catchment area
(2SFCA) methods. In including an accurate aggregation
method, the gravity models can be written as:
where Adi is the mean value of potential gravity for census
tract i (a larger value implies better accessibility), n and s
are respectively the number of census blocks and of
services in the study area, Sj is the weight given to service s
such as its size (e.g. number of beds in a hospital)
(“supply side”), Vj is the potential population (“demand side”), α
represents the friction parameter (usually 1, 1.5 or 2), and,
finally, ni is the number of blocks within census tract i.
where Rj represents the supply-to-demand ratio within
catchment area d0, dkj is the distance between spatial
unit census tract centroid k and health service j, d0 is
the threshold distance or travel time (e.g. one kilometre
or 30 min), Sj and Pk are respectively the supply
capacity (e.g. number of medical clinics or number of beds in
a hospital) at location j and the demand at location k that
falls within catchment area j. Note that Pk/1000 could
also be used in order to obtain an initial ratio for 1000
inhabitants within the catchment area.
In the second step, for each demand location i (census
tract centroid), we search all supply locations j within the
threshold distance d0 from i and sum up the initial
supply-to-demand ratios Rj:
where a larger value of Aie implies better accessibility for
census tract i.
Many authors have suggested improvements to the
2SFCA method in order to remedy two limitations [
5, 8, 31, 33, 39–42
]. Firstly, in its initial form, the 2SFCA
method assumes that the population (Pk) inside the
catchment area (where dkj < d0) has the same
accessibility regardless of the distance separating this population
from the health service. Secondly, beyond the
threshold distance (d0), the accessibility is null. Luo and Qi
] have thus proposed the enhanced two-step floating
catchment area (E2SFCA) method, which is now widely
]. These authors then divide the catchment
area into three zones: 0–10 min (d1), 10–20 min (d2), and
20–30 min (d3) (Eqs. 8, 9). For each of these three zones,
it is then possible to apply a weighting (Wk) calculated by
using a Gaussian function:
where W1, W2, W3 = 1.00, 0.68 and 0.22 with a slow
stepdecay function or 1.00, 0.42 and 0.09 with a fast
stepdecay function. Also, some authors add a fourth zone
of 30–60 min, especially when the study area includes
rural areas [
]. The weightings are then: W1, W2, W3,
W4 = 1.00, 0.80, 0.55 and 0.15 with a slow step-decay
function or 1.00, 0.60, 0.25 and 0.05 with a fast
stepdecay function. Note that the values of the radii can be
modified according to the geographical context. For
example, Dewulf et al. [
]—who analyze the accessibility
of primary health care not in an urban context but on the
scale of an entire country (Belgium)—use radii of 1, 2, 5
and 10 km.
As mentioned by McGrail [
], some scholars
criticize the fact that the weightings are constant within each
radius and advocate using a continuous weighting
function: W = 1 for the first radius (0–10 min, for example);
W = 0 when the distance is >60 min; and W = ((60 − d)/
(60 − 10))1.5 for the radius of 10–60 min. Finally, to
reduce aggregation errors, Bell et al. [
a 3SFCA: it is a matter of calculating the 2SFCA or the
E2SFCA at a fine scale (e.g. dissemination areas and
blocks within census tracts), and then calculating the
mean per census tract. Consequently, by applying the
most accurate aggregation method, the E2SFCA can
thus be formulated with four radii or with a continuous
where ni is the number of blocks within the census tract;
W1, W2, W3, W4 = 1.00, 0.80, 0.55 and 0.15 with a slow
step-decay function or 1.00, 0.60, 0.25 and 0.05 with a
fast step-decay function; and Wbj is the weight for block
j with a continuous weighting function. This last
parameter can be calculated as follows:
611 and 139 inhabitants, as defined by Statistics
Canada. A total of 594 health services grouped into twelve
categories were integrated into geographic information
systems (ArcGis) (Figs. 4, 5). Note that a street address
can include several categories of health services. In the
end, this ultimately produces 535 geographical locations
if dij < 10 then Wbj = 1; if dij > 10 and dij ≤ 60 then
Wbj = ((60 − d)/(60 − 10))1.5; if dij > 60 then Wbj = 0.
Types of distance
Six types of distance can be used to calculate accessibility
measures: Euclidean distance (straight-line), Manhattan
distance (distance along two sides of a right-angled
triangle opposed to the hypotenuse), and shortest network
time distances according to the mode of transportation
used (on foot, by car, by bicycle, or by public transit)
(Fig. 3) [
In this paper, we investigate differences in results when
the geographical accessibility of selected health care
services for residential areas (census tracts) is computed
by using three parameters: (1) six types of distance, (2)
four aggregation methods, and (3) fourteen
accessibility measures. The specific objectives are to: (1) Compare
the types of distance; (2) Estimate aggregation errors for
several accessibility measures; and (3) Measure how the
assessment of potential spatial access varies according to
these three parameters.
Data and methods
Study area and health services
This study focuses on the territory served by the regional
transit authority for the Montreal area, which had a
population of about 3.8 million in 2011. The extent of this
territory is very similar to that of the Montreal census
metropolitan area (CMA). The study area is divided into
904 census tracts, 6167 dissemination areas and 27,126
blocks with respective average population sizes of 4170,
for these health services; they have all been precisely
geocoded from the centroid of the building. This spatial
dataset was provided by the Quebec Ministry of Health
and Social Services.
Computing the six types of distance
Euclidean and Manhattan distances can easily be
computed by using geographic coordinates:
(xi − xj)2 + (yi − yj)2,
dij = xi − xj + yi − yj ,
where xi, yi, xj, yj are the Cartesian coordinates of points i
and j with a plane projection.
Shortest network distance (by car)
To calculate trips as though made by car, we used the
Adresses Québec (AQ Directions) [
] road network—
which includes the speed limits and directions of traffic
for all road and street segments in the province of
Quebec. Based on the length of the road or street segment
and the speed limit on that road or street, the cost in
minutes to travel over each segment of the road network
can then be calculated [
Lft ∗ 60
Smph ∗ 5280
or Tmn =
Lm ∗ 60
Skmh ∗ 1000
where Tmn is the cost in minutes to travel over the road or
street segment, Lft and Lm are the length of the segment
in feet and metres respectively, and Smph and Skmh are the
speed limits in miles/h and km/h.
Shortest network distance (on foot)
The modeling of the network for travel on foot is also
based on the Adresses Québec road network. Compared
with the modeling of the network by car, a restriction was
added on segments of highway where pedestrians are not
allowed, whereas the direction of traffic was not used as
a restriction. Moreover, the elevation of each junction of
the network was extracted from a digital elevation model
at a resolution of 3 m. Based on these elevation data for
the junctions, it is then possible to calculate the walking
speed over the road or street segment (Wkmh) by using
the classic Tobler’s hiking function [
Wkmh = 6e−3.5 ddhx +0.05
where dh is the difference in elevation between the start
and end nodes of the road or street segment, and dx is
the segment’s length. When the slope is equal to 0 (flat
terrain), the walking speed is equal to 5 km/h (Fig. 6). By
applying Eqs. (14) and (15) as described above, it is then
possible to estimate the cost in minutes of foot travel for
each segment from the start node to the end node, and
Shortest network distance (by bicycle)
The cycling network was modeled by combining several
sources of data on bicycle paths obtained from the
municipalities of Montreal, Longueuil and Laval and
OpenCycleMap. This network was then merged with the
Fig. 6 Walking speed according to Tobler’s hiking function
Adresses Québec road network. Elevation data were
again used to calculate the slope for each segment of the
network. There is currently no consensus on cyclists’
average travel speed in urban areas. For example, Jensen
et al. [
] found an average speed of 14.5 km/h for cyclists
in Lyon (France), whereas Parkin and Rotheram [
Leeds (UK) obtained an average speed of 21.6 km/h.
However, other studies regularly suggest values of about
16 km/h [
], the average value used by Google
Maps,1 which was also selected for this study. Other
authors have shown that speed varies according to the
slope, length of segment, type of cycling infrastructure
and type of bicycle. With the help of a regression model,
Parkin and Rotheram  have thus estimated the impact
of slope on travel speeds: that is, 0.86 km/h for each
percentage of downhill gradient and −1.44 km/h for each
percentage of uphill gradient. El-Geneidy et al. [
looked at the impact of infrastructures on cyclists’ travel
speeds in Minneapolis (USA). They concluded that, all
other things being equal, only off-street bicycle paths
have a significant and positive impact on speed
(1.14 km/h) and that each kilometre of the segment
length is associated with a 0.32 km/h increase in speed.
Consequently, we modeled cyclists’ travel speeds (Ckmh)
on each segment as follows:
Ckmh = 16 +
(|si| ∗ −1.44) if si > 0
(|si| ∗ 0.86) if si < 0 + li ∗ 0.32
1.14 if ti = off street
where si is the percentage of slope on the segment from
the start to the end node or vice versa, li is the
segment length in kilometres, and ti is the type of cycling
Shortest network distance (by public transportation)
As done by Faber et al. [
] and Hadas [
transit feed specification (GTFS) files are used to calculate
travel times with public transit. GTFS data covering all of
our study area were obtained from the Agence
Métropolitaine de Transport (AMT) (regional transit authority for
the bus, metro and commuter train network). These data
were integrated into ArcGIS and combined with the
pedestrian network by using the Add GTFS to a Network
Dataset2 tool. Since travel times can vary according to the
time of departure, especially in outlying municipalities
where commuter trains and buses run far less frequently
than in central neighbourhoods, we calculated 13
distance matrices: that is, for Monday departures every
10 min from 7:00 a.m. to 9:00 a.m. We then selected the
minimum travel time for each of the 13 trips between
census spatial units (census tracts, dissemination areas
and blocks) and health services. This ensured that the
travel times would not be overestimated, especially for
trips to or from the suburbs.
Comparing distance types
To explore variations in results according to distance type
(Objective 1), we calculate the six distance
types—Euclidean, Manhattan, and shortest network time distances (on
foot, by car, by bicycle, or by public transit)—between
the 535 health services and the centroids of census tracts
(n = 904), dissemination areas (n = 6167) and blocks
(n = 27,126) and block centroids adjusted with a land use
map. In total, close to 197 million distances are computed
(Table 2), with a Python code for Euclidean and
Manhattan distances, and with the Network Analyst Extension of
ArcGIS (version 10.3) for the four shortest network time
Once these distance types are computed, correlation
analyses are performed globally and locally across all
the census tracts, dissemination areas and blocks
matrices. First, the global analysis, which yields one value for
the study area as a whole, allows us to assess the degree
a Euclidean, Manhattan, shortest network time (on foot), shortest network time (by car), shortest network time (by bicycle), and shortest network time (by public
of correlation between the four distance types. Then,
we examine correlations between the four distances for
each spatial unit centroid and the 535 health service
locations. This local analysis stage yields one mappable value
for each census tract, dissemination area and block and
allows us to identify spatial variation in the degree of
correlation between the six distance types.
Evaluating aggregation errors when measuring potential geographic access
The same approach, i.e. global and local analyses, was
used to evaluate aggregation errors for several
accessibility measures at the census tract level (Objective 2). To
do this, we calculated 14 accessibility measures (Table 3),
using six types of distance and four aggregation
methods, for a total of 336 measures. Although accessibility
was computed for each of the twelve categories of health
services, for purposes of conciseness, results are reported
only for the accessibility of general and specialized care
(i.e. hospitals; n = 62) for census tracts. It is worth noting
that similar patterns of correlation were observed for the
other health services.
The global analysis involves calculating correlations
between four aggregation methods: (1) the census tract
centroid (CTC); (2) the population-weighted mean of the
accessibility measure for dissemination areas within
census tracts (WDAC); (3) the population-weighted mean of
the accessibility measure for blocks within census tracts
(WBL1); and (4) the population-weighted mean of the
Table 3 List of measures of accessibility computed
1. Minimum distance
2. Average distance to all hospitals
3. Average distance to three closest hospitals
4. Average distance to five closest hospitals
5. Number of hospitals within 500 m or 10 min
6. Number of hospitals within 1000 m or 20 min
7. Number of hospitals within 2000 m or 30 min
8. Potential gravity model (friction parameter = 1)
9. Potential gravity model (friction parameter = 1.5)
10. Potential gravity model (friction parameter = 2)
11. Two‑step floating catchment area (2000 m or 30 min)
12. Enhanced two‑step floating catchment area with a slow step ‑
13. Enhanced two‑step floating catchment area with a fast step ‑ decay
14. Enhanced two‑step floating catchment area with a gradient func‑
a The catchment area is divided into three zones: 0–500 m or 0–10 min,
500–1000 m or 10–20 min, and 1000–2000 m or 20–30 min
accessibility measure for blocks (adjusted with the land
use map) within census tracts (WBL2).
The local analysis consists in simply calculating the
absolute differences for each of the 14 accessibility
measures obtained with the least accurate aggregation method
(CTC) and the most accurate method (WBL2). It is then
possible to calculate the univariate statistics and to map
Sensitivity analysis (SA) and uncertainty analysis (UA)
Sensitivity and uncertainty analyses are mainly used to
test the robustness of composite indicators [
During these analyses, a number of methodological
choices in fact intervene and modify the final index
indicator values. So it is important to determine how
sensitive the indicator is to these choices. A very unstable (i.e.
highly uncertain) indicator is problematic, as it can be
strongly influenced by specific methodological choices
(a particular weighting, for example). Conversely, a very
rigid indicator is not necessarily desirable either, because
methodological choices are supposed to help to construct
the index indicator, to give it meaning.
This type of analysis applies when one has a final score,
obtained with the help of a model, which itself depends
on several parameters. These parameters are called
uncertainty factors because they can take on several
different values that will alter the final score. This
description makes clear the parallel with our study. Indeed,
our final score is an indicator of potential spatial access,
obtained by using a model that includes three uncertainty
factors: the type of distance (6 choices), the aggregation
method (4 choices), and the accessibility measure (14
choices). This model can then take 336 different forms; in
other words, in the context of this study, there are 336
different ways of calculating a potential spatial access score
for each census tract. To our knowledge, this method has
never been used in this context, so that this is an original
application. Indeed, by using a sensitivity analysis (SA),
we can explain how each of the three parameters leads
to variation in the levels of accessibility for the entire
study area (Objective 3). Also, the use of an uncertainty
analysis (UA) allows us to identify and map census tracts
for which the 336 accessibility indicators vary the most
according to the three parameters.
Before performing these two analyses, the 336
indicators need to be transformed so that they are expressed
in the same units. The most common transformations
are normalization on a scale of 0–1 (Eq. 17), the z-score
standardization (Eq. 18), or the use of ranks (Eq. 19) [
To measure the uncertainty for each census tract, we
simply calculate the coefficient of variation (CV = STD/
Mean) of the 336 values of the previously transformed
accessibility indicators. So, for a census tract, the higher
the value of the CV is, the greater the uncertainty is, or, in
other words, the more the methodological choices locally
impact the assessment of potential geographic access.
For the sensitivity analysis, a reference indicator must
first be chosen from among the 336 indicators. We chose
the least complex indicator, calculated with the
following parameters: Euclidean distance, the CTC aggregation
method and the closest hospital as the accessibility
measure. For each of the 336 indicators, it is then possible
to calculate the average of the absolute differences with
respect to the reference indicator:
c=1 Ireference,c − Iq,c
where n is the number of census tracts.
One can then calculate the first order sensitivity indices
(Si), proposed by Sobol [
], i.e. the proportion of the
total variance attributable to each factor (distance types,
aggregation methods and accessibility measures). This
method is based on Sobol’s equation of variation
decomposition and is particularly suited for non-linear models.
In our case, the uncertainty factors do not have a
nonlinear, which justifies the use of this method :
V (Y )
Vxi (Ex−i (Y |Xi))
V (Y )
where Vi = variance explained by the uncertainty factor
Xi; V(Y) = total variance; X = uncertainty factors; Y = the
overall mean shift with the reference; Ex−i (Y|Xi) = the
expected value (mean) of Y for all combinations of the
indicator with factor Xi fixed to a particular modality;
Vxi (Ex−i (Y |Xi)) = the variance of these means for all
possible modalities of Xi.
One can then further decompose the variance by
adding second order indices that measure the proportion
of the variance attributable to interactions between two
V (Y )
Vxixj Ex−ij Y |Xi, Xj
− Vi − Vj
V (Y )
For example, one could evaluate the proportion of the
variance that is explained by the interaction of the type of
distance factor with the factor of the accessibility
measure. Finally, the total effect sensitivity index for a factor is
the sum of the first order and second order indices:
ST 1 = S1 + S12 + S13
ST 2 = S2 + S12 + S23
ST 3 = S3 + S13 + S23
where ST1, ST2, ST3 are the total effects for the type of
distance, the aggregation method and the accessibility
measure respectively. More detailed information on
sensitivity and uncertainty analyses can be found especially
in the work of Sobol [
], Nardo et al. [
] and Saisana
et al. [
Correlations between the six types of distance
Before exploring the correlations, it is relevant to analyze
a few statistics for the different types of distance
calculated between the 535 destinations and the 904 census
tracts (n = 483,640) (Table 4). For Cartesian distances,
the mean values are 19.7 km for Euclidean distance
compared with 25.2 km for Manhattan distance, i.e. a
significant difference of 5.5 km (P = 0.01) (Fig. 7). Since
Manhattan distance is the length of the two sides of a
right-angled triangle opposed to the hypotenuse—with
the latter representing Euclidean distance—(Fig. 3a), it
is therefore evident that all the univariate statistics are
higher for Manhattan distance.
Regarding the shortest network times, it is no
surprise that the statistics show that the means of the
trips are greater on foot (mean = 292 min), followed
by trips by bicycle (mean = 95 min), public transport
(mean = 79 min) and car (mean = 23 min) (Table 4;
Fig. 7). In other words, compared with trips by car, the
trips are, on average, 12.6 times longer on foot, 4.1 times
longer by bicycle and 3.4 times longer by public transit.
Another interesting result is that the value of the 10th
percentile for bicycle travel times is lower than the value
for public transit (27.20 vs. 32.57 min). This means that,
for 10% of the fastest trips, the bicycle is 5 min faster than
public transport. To put it another way, the bicycle is a
very good alternative to public transit for short trips.
Table 5 presents results for global correlation coefficients
between the six types of distance computed for all health
service locations (n = 535). From the correlation
matrices, three main results can be highlighted. First, at the
metropolitan scale, independently of the type of distance
used except for shortest network time by public transit,
the results are globally similar as indicated by high
correlation coefficient values (>0.90). Second, in
comparison with Manhattan distance, Euclidean distance is most
strongly correlated with all the shortest network time
distances. This means that if it is impossible to compute
network distances in a study focusing on geographical
accessibility in the Montreal CMA, Euclidean distance
seems preferable to Manhattan distance. Third, the
correlations between the three shortest network times—
on foot, by car, by bicycle—are very high (>0.95), but
the correlations of the shortest network time (by public
transit) with all other types of distance are much weaker
(between 0.76 and 0.82).
Although the global correlations are high, they are not
perfect (the values differ from one). For this reason, local
variations at the intra-metropolitan scale must exist and
should be examined in detail. Local Pearson correlations
have been calculated from the centroids of census tracts,
dissemination areas, and blocks. For purposes of
simplification, we are only presenting the results for census
tracts. Note that the results show similar spatial patterns
for the three spatial scales.
Firstly, Fig. 8a–d presents local Pearson coefficients
between Euclidean distance and the four shortest
network time distances (on foot, by car, by bicycle and by
public transit). The maps show that with increasing
distance from the central business district, local
correlations are reduced between Euclidean distance and the
four shortest network time distances. For all spatial units
in the centre of the Island of Montreal, the correlations
are higher. For spatial units located on the periphery of
the CMA, notably on the North and South shores, which
are characterized by suburban areas, the correlations are
weaker. It is not surprising that these results are in line
with those of the previous study [
]. They also show that
the local correlations between Euclidean distance and
the shortest network time distance by public transit are
much lower. Indeed, the strongest local correlations are
mainly found in the central portion of the Island of
Montreal, where public transit is much more highly
developed, especially due to the presence of the metro lines.
Secondly, it is possible to analyze the local
correlations between the four shortest network time distances
(Figs. 8e, f, 9a–d). The local correlations are generally
fairly strong between the shortest network time
distances by car, on foot and by bicycle. On the other hand,
the local correlations are much weaker with distances by
public transit (Fig. 9a, c, d).
In sum, the results of the local correlations allow us to
highlight two important findings. On the one hand, for a
study covering the entire Montreal region, it is preferable to
use network distances because Cartesian distances
(especially Euclidean distances) are less accurate in suburban
areas (the North and South shores). On the other hand, the
distance by public transport is very different from the other
types of distance (on foot, by bicycle, by car), particularly
in parts of the region where the public transit system is less
dense (in the eastern and western portions of the Island of
Montreal and on the North and South shores).
The global analysis of aggregation errors is performed by
means of Spearman’s rank correlations between the four
methods of aggregation used to calculate 14 accessibility
measures at the census tract level (Table 6). Due to lack
of space, we only report the correlation for two types of
distance (Euclidean distance and shortest network time
on foot). Note that similar patterns of correlation are
observed for the other types of distance.
Correlations between the four aggregation
methods are high (>0.9) for all accessibility measures except
for the number of hospitals within 500 and 1000 m or
within 10 and 20 min. For example, correlation between
the least and most accurate aggregation methods (CTC
and WBL2) is 0.704 for the number of hospitals within
500 m and 0.740 for those within 10 min on foot. This
means that if we want to assess service provision in a
close-proximity area around a census tract, it is
preferable to use an aggregation method that precisely accounts
for the distribution of population within it; if not, the risk
of error may be considerable.
A second stage of comparison of aggregation methods
consists in assessing the absolute difference between the
geographical accessibility results obtained with the CTC
and WBL2 aggregation methods. The descriptive
statistics for local errors are reported in Table 7 for hospitals.
Not surprisingly, the local errors are on the whole quite
Potential gravity model (friction parameter = 1)
Potential gravity model (friction parameter = 1.5)
Potential gravity model (friction parameter = 2)
Two‑step floating catchment area (2000 m or 30 min)
a Aggregation method based on census tract centroid (the least accurate method)
b Aggregation method based on the population-weighted mean of the accessibility measure for dissemination areas within census tracts
c Aggregation method based on the population-weighted mean of the accessibility measure for blocks within census tracts
d Aggregation method based on the population-weighted mean of the accessibility measure for block centroids (adjusted with the land use map) within census
tracts (the most accurate method)
small, though not insignificant: for example, compared
with the most accurate method, the census tract centroid
method misestimates the distance to the closest hospital
by an average of 236 m (Euclidean distance) and 4.17 min
(on foot). Up to the third quartile (75%), the local errors
are still quite small: for 75% of census tracts, the error
associated with the census tract centroid approach is
<218 m or 3.65 min. However, in 10% of cases, the error
is >649 m and 10.22 min, and in 5% of census tracts the
error is >1.1 km and 17 min (Table 7). Despite the high
correlations, significant errors in the measurement of
geographical accessibility can occur in a small number of
Absolute differences between aggregation methods for
the closest hospital computed using Euclidean distance
and shortest network time (on foot) are further mapped
in Fig. 10a, b. Again, stronger absolute aggregation errors
are observed in suburban areas on the South and North
shores of the CMA; errors remain smaller in central areas
of the Island of Montreal. Moreover, the use of the local
Getis–Ord Gi* statistic clearly shows that hot spots in
aggregation errors are located on the North and South
shores (Fig. 10c, d). This shows that, in suburban areas,
where the surface area of census tracts is greater than in
central neighbourhoods, it is preferable to use an
accurate aggregation method to prevent significant
Sensitivity analysis (SA) and uncertainty analysis (UA)
The sensitivity analysis was performed by using three
transformations (z-score standardization,
normalization on a scale of 0–1, and use of ranks). Table 8 reveals
several interesting findings. First, the uncertainty factor
generating the most variance is the accessibility
measure, with 74–86% of the total variance, depending on the
transformation used (first order sensitivity index). When
placed in interaction with the type of distance, the
accessibility measures also explain 10–20% of the total
variance (second order sensitivity index) for a total sensitivity
index of over 90.
The second most important uncertainty factor is the
type of distance, with 3–6% of the total variance (first
order sensitivity index); 10–20% of the variance when
placed in interaction with the accessibility measure
(second order sensitivity index); and a value for the total
sensitivity index of 13.55–23.12. Thirdly, the impact of the
aggregation method is much more limited: <1% for the
first order sensitivity index; and a value for the total
sensitivity index of between 0.18 and 2.40.
The UA results, mapped in Fig. 11a, b, clearly
indicate areas of strong uncertainty on the North and
South shores, whereas central neighbourhoods show
lower levels of uncertainty. In other words, the choices
made regarding the three parameters—distance types,
accessibility measures and aggregation methods—have
Absolute difference between accessibility
measure obtained from CTCa
and WBL2b aggregation methods
a Aggregation method based on census tract centroid (the least accurate method)
b Aggregation method based on the population-weighted mean of the accessibility measure for block centroids (adjusted with the land use map) within census
tracts (the most accurate method)
relatively little impact in the development of the
assessment of potential geographic access in central
neighbourhoods, unlike the case on the North and South
The SA results allowed us to show that the evaluation
of potential geographic access may vary a great deal
depending on the accessibility measure and, to a lesser
degree, the type of distance and aggregation method. This
is not surprising, as the 14 accessibility measures selected
refer to conceptualizations of potential geographic access
that are very different from one another. The choice of
this parameter should thus be given considerable
attention and have a specific justification as it helps to make
the results vary substantially. It is then relevant to
calculate several measures that enable potential geographic
access to be described in all its complexity. For example,
measures based on both supply and demand dimensions
(2SFCA and gravity models) are better adapted to
general, large-size services (e.g. hospitals), whereas
measures of immediate proximity or cumulative opportunities
are more suited to describing less common, specialized
services (specialized centres). Moreover, these
cumulative opportunity measures are especially well adapted to
describing the supply of services within an immediate
environment (within one mile, for example). That is why
they are often used in health studies on food deserts [e.g.
28, 69, 70] or the food environments around schools [e.g.
71–73]. It should also be remembered that these
accessibility measures produce variables that may be either
continuous or discrete, which may result in a particular
one being chosen in keeping with the study design.
Although the choice of the type of distance has less
impact on the results obtained, it clearly interacts to
a certain extent with the measure of accessibility. The
Table 8 Results of sensitivity analysis (Sobol’s indexes)
interactions between these two sources of uncertainty
generate more variation than the method of aggregation
alone. For example, in 2SFCA methods, the catchment
areas may be either larger or smaller depending on the
type of distance chosen (car travelers can go further than
pedestrians), and the same goes for measures that count
the number of services within a specific radius. So this
choice also requires careful consideration, in taking into
account possible interactions with the latter measure.
Moreover, although Cartesian distances (Euclidean and
Manhattan) are strongly correlated with the four
network distances, local variations are nonetheless observed,
notably in suburban areas. Given that it has become
much easier to calculate network distances—because of
free access to geographical data and highly effective tools
(GIS or online services)—the use of Cartesian distances
in urban areas is no longer preferable today. Indeed, the
time required for the computation of numerous network
distances is no longer a limitation.
Another aspect should be mentioned concerning the
comparison of the types of distance. The correlations
have been shown to be weaker between public transit
and other network distances. This can be explained by
the unequal distribution of public transport (especially
the subway) across the study area. The same remark also
applies, although to a lesser extent, to cycling
infrastructure. We also found strong correlations between network
distances on foot, by bicycle and by car, which might lead
one to believe that using one or the other of these types
of distance comes down to applying a simple multiplying
factor, which is not the case. The impact of topography on
pedestrians’ or cyclists’ speeds is much greater than for
car drivers. In a city with a more pronounced topography
(e.g. San Francisco, La Paz), these correlations would
certainly have been weaker. Finally, the correlations with
Cartesian distances were also weaker for peripheral areas
than for central areas. In the Montreal context, this is in
part explained by the presence of the bridges that link the
Island to the North and South shores. In comparison, if
we had conducted the study in the New York City area,
the local correlations between Cartesian versus network
types of distance would probably have been very high for
census tracts on the island of Manhattan, and weaker for
those located in Brooklyn, Queens, Staten Island and
Jersey City because of the bridges.
Finally, although the influence of the aggregation
method is fairly marginal globally, we have
nonetheless shown that errors in accuracy caused by the lack of
an aggregation method can be important locally,
especially in suburban areas, where census tracts mostly
have lower population densities and where land use
is largely non-residential. Because the accessibility of
health services may be more problematic in suburban
areas than in more central urban areas,
geographical accessibility studies should be based on the most
accurate aggregation method. The question of the use
of an aggregation method is especially important when
accessibility measures calculated on the level of census
tracts are introduced as dependent variables into
models for predicting health outcomes. Consider the classic
example of a multilevel model with individual
variables (level 1), socioeconomic variables and measures
of the accessibility of health services or health-related
resources at the census tract level (level 2). If the
accessibility measures are not calculated by using an
aggregation method—in other words, if they are obtained by
only using the census tract centroids—that could lead
to errors or lack of precision in the estimation of the
impact of the accessibility of health services or
healthrelated resources on health.
This article evaluates the potential geographic access to
urban health services using 14 accessibility measures, six
types of distance and four aggregation methods. Based
on these three parameters, 336 indicators of geographic
access at the census tract level have been obtained. A
sensitivity analysis has shown that the parameters that
create the greatest variation in the evaluation of potential
geographic access are, in descending order: the
accessibility measures and, to a far lesser extent, the type of
distance and the aggregation method used. An uncertainty
analysis also made it possible to show that inaccuracies in
the evaluation of geographic access are much greater in
the suburbs than in central neighbourhoods.
In sum, in order to accurately assess potential
geographic access to health services in urban areas, it is
particularly important to choose a precise type of distance
and aggregation method so as to limit inaccuracies in
measurements. Then, depending on the research
question and/or research objectives, the choices of the type of
network distance (according to the mode of
transportation) and of a number of accessibility measures should be
carefully considered and adequately justified.
2SFCA: two‑step floating catchment area; E2SFCA: enhanced two ‑step float ‑
ing catchment area; 3SFCA: three‑step floating catchment area; CMA: census
metropolitan area; CTC: census tract centroid; GIS: geographic information
systems; GTFS: general transit feed specification; SA: sensitivity analysis; UA:
uncertainty analysis; WBL1: population‑ weighted mean of the accessibility
measure for blocks within census tracts (WBL1); WBL2: population‑ weighted
mean of the accessibility measure for blocks (adjusted with the land use map)
within census tracts; WDAC: population‑ weighted mean of the accessibility
measure for dissemination areas within census tracts.
PA and JG are the principal investigators of the study. They carried out the GIS,
statistical and mapping analyses. ASD participated in the literature review. All
authors jointly drafted and critically revised the paper. All authors read and
approved the final manuscript.
The authors would like to thank the three anonymous reviewers for their
careful reading of our manuscript and their many insightful comments and
The authors declare that they have no competing interests.
Availability of data and materials
Consent for publication
Ethics approval and consent to participate
The authors are grateful for the financial support provided by the Canada
Research Chair in Environmental Equity.
Springer Nature remains neutral with regard to jurisdictional claims in pub‑
lished maps and institutional affiliations.
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