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METHODOLOGYOpen Access

Estimating spatial accessibility to facilities on the

regional scale: an extended commuting-based

interaction potential model

Paul Salze1,2*, Arnaud Banos3, Jean-Michel Oppert4,5, Hélène Charreire4,6, Romain Casey7, Chantal Simon7,

Basile Chaix8,9, Dominique Badariotti1,2, Christiane Weber1,2

Abstract

Background: There is growing interest in the study of the relationships between individual health-related

behaviours (e.g. food intake and physical activity) and measurements of spatial accessibility to the associated

facilities (e.g. food outlets and sport facilities). The aim of this study is to propose measurements of spatial

accessibility to facilities on the regional scale, using aggregated data. We first used a potential accessibility model

that partly makes it possible to overcome the limitations of the most frequently used indices such as the count of

opportunities within a given neighbourhood. We then propose an extended model in order to take into account

both home and work-based accessibility for a commuting population.

Results: Potential accessibility estimation provides a very different picture of the accessibility levels experienced by

the population than the more classical “number of opportunities per census tract” index. The extended model for

commuters increases the overall accessibility levels but this increase differs according to the urbanisation level.

Strongest increases are observed in some rural municipalities with initial low accessibility levels. Distance to major

urban poles seems to play an essential role.

Conclusions: Accessibility is a multi-dimensional concept that should integrate some aspects of travel behaviour.

Our work supports the evidence that the choice of appropriate accessibility indices including both residential and

non-residential environmental features is necessary. Such models have potential implications for providing relevant

information to policy-makers in the field of public health.

Background

Measuring spatial accessibility

Accessibility is a major issue for many types of stake-

holders in policy making in the fields of transport,

urban planning, marketing and public health. Because it

may encompass more dimensions than the spatial one

(e.g. temporal, social, economic), there is no single

established definition of accessibility. Several literature

reviews provide a global and historical overview of exist-

ing definitions and associated measures, as well as some

developments and examples of applications [1-7]. A use-

ful classification of the existing operational accessibility

measures has been proposed by Geurs and van Wee [7].

The authors distinguish four broad categories of

measurements. “Infrastructure-based” measurements are

used to assess the efficiency of the transport network

(e.g. traffic congestion, mean travel speed). “Location-

based” measurements deal with the spatial distribution

of opportunities (e.g. distance to the nearest opportu-

nity, number of available facilities within a neighbour-

hood), generally at an aggregated level. “Person-based”

measurements refer to disaggregated space-time accessi-

bility measurements at the individual level. “Utility-

based” measurements are based on benefits assessment

and utility maximisation theory for both individuals and

population groups. Whatever the category, specifying

the measurement makes it necessary to define some

interrelated elements: the degree and type of disaggrega-

tion, origins and destinations, attractiveness and travel

impedance [6].

* Correspondence: paul.salze@live-cnrs.unistra.fr

1Université de Strasbourg; Image, Ville, Environnement, Strasbourg, France

Full list of author information is available at the end of the article

Salze et al. International Journal of Health Geographics 2011, 10:2

http://www.ij-healthgeographics.com/content/10/1/2

INTERNATIONAL JOURNAL

OF HEALTH GEOGRAPHICS

© 2011 Salze et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons

Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

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In the public health domain, there is growing interest

in the study of the relationships between individual

health behaviours (e.g. food intake, physical activity)

and measurements of spatial accessibility to the related

opportunities (e.g. food outlets, sport facilities) [8,9].

One important aim is to assess whether social depriva-

tion is associated with specific spatial accessibility

levels to certain types of facilities, contributing to an

amplification of the social disparities in unhealthy

behaviours [10].

In a recent methodological review, we noted that in

most previous studies, spatial accessibility to a given

type of facilities was measured either as the distance to

the nearest opportunity or as a count or density of

opportunities within a neighbourhood (administrative

unit or time/distance buffer) [11]. Although these “clas-

sical” measurements are very useful due to their simpli-

city (both to understand and to compute), they present

some limitations. Indeed, by ignoring some aspects of

travel behaviour, they only provide a “one-dimensional”

biased view of accessibility [12].

The limits of “classical” indices

The nearest opportunity measurement assumes that sur-

rounding opportunities other than the nearest one are

not included in the possible destinations that individuals

may choose. Handy and Niemeier [6] have shown that

this is an unrealistic assumption. Indeed, in two com-

munities in the San Francisco Bay Area (CA, USA), they

found that more than 80% of the residents used to visit

more than one supermarket in a month.

The count of facilities within a neighbourhood, also

known as a container index [12], overcomes this limita-

tion by considering all available opportunities within a

neighbourhood. However, it assumes that an opportu-

nity situated just beyond the limit of the neighbourhood

will not be accessible and that all the opportunities

within a neighbourhood are equally accessible, which is

questionable with respect to spatial barriers or the per-

ception of the distances.

In order to address this last question, the use of kernel

density estimation (KDE) [13] and of an enhanced two-

step floating catchment area method (E2SFCA) [14]

have been proposed to assess accessibility to health care

[14-16] or food stores/physical activity facilities [17-19].

The main idea of such methods is to take into account

both the demand (population) and the supply (health

practitioners) side and to partly include travel impe-

dance specification (frictional effect of space: more

weight is given to opportunities near to the origin).

Nevertheless, most of these studies used the distance

weighting function provided by the available GIS soft-

ware without addressing that specific point.

The delimitation of the neighbourhood [20] in con-

tainer index, KDE and E2SFCA method is another criti-

cal point. Using circular or network-based buffers

instead of administrative units may be more appropriate

because it frees the study from administrative bound-

aries. Unfortunately this approach does not solve the

issue of “clear-cut neighbourhood boundaries” and the

choice of the size and shape of the buffer remains pro-

blematic [21]. This last point about neighbourhood deli-

mitation and the fact that accessibility to facilities can

be seen as environmental exposure [22] naturally lead

us to the broader question of how the environment is to

be defined.

Defining the environment

By focusing on residential neighbourhoods and ignoring

potential exposure that occurs around other activity

places (e.g. workplace or school), most studies in the

health literature have fallen into what has been called

the “residential trap” [23]. In some studies, exposure or

accessibility levels have been assessed around schools

[24-27] or both homes and workplaces [28]. In this

study, origins (i.e. workplace and home) were considered

separately when assessing relationships between accessi-

bility and health outcomes and it would have been inter-

esting to focus on cumulative exposure. While the

“residential trap” is no longer relevant (because not only

homes are taken into account) it could be more appro-

priate to see this problem as the absence of a dynamic

dimension ("motionless trap”).

Including the dynamic dimension of accessibility

It appears to be necessary to consider both residential

and non-residential environmental influences on health

behaviours, which implies including spatial or spatial-

temporal dynamics of individuals and populations (i.e.

mobility). In that sense, “person-based” or disaggregated

individual-space-time measurements of accessibility [29]

are totally relevant but the results may be difficult to

interpret for population-wide studies. For example, they

make it possible to evaluate accessibility levels over a

whole day in regard to location and duration of activ-

ities according to individual characteristics (e.g. gender)

[30]. In health studies, Kestens and colleagues [31] used

individual experienced activity spaces to measure acces-

sibility to different kinds of food stores in each location

visited during a weekday by different categories of popu-

lation (according to age and income). Such methods

require large sets of very detailed data which are not

always available, especially for large study zones. That

point is discussed by the authors who used data from a

very large mobility survey. However, because of limited

information on time use, they were unable to integrate

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temporal constraints (i.e. time-budgets) in their estima-

tions of accessibility levels.

Aim of the study

The general aim of the work was to propose measure-

ments of accessibility to a set of facilities (three types of

food outlet: hyper/supermarkets, grocery stores, bak-

eries) on the regional scale. In a context of generalised

car-owning and thus increasing accessibility levels, more

and more people chose to live in suburban and rural

areas (growing suburbanisation process) which are asso-

ciated with better living environments. The functioning

of urban, suburban and rural areas cannot be discon-

nected from each other and have then to be seen as a

whole system, making the regional scale a level of parti-

cular interest.

Because of the aggregated nature of available data and

in order to overcome some of the limitations of the

measurements mentioned above (nearest opportunity,

container index, KDE), we chose to use a potential

accessibility index [32]. Accessibility is defined as a

potential for spatial interaction (i.e. an intensity of

possible destinations) that makes it possible to take

account of a global aspect of travel behaviour.

In the first part of this work, we present some histori-

cal and theoretical considerations and then provide a

complete example of application of a potential model

including a detailed calibration process. The second part

of the paper is dedicated to the presentation and appli-

cation of an extended potential model for a commuting

population.

Methods

Study zone

Our study territory was the Bas-Rhin département,

an administrative region of about 4800 km2situated in

Eastern France. Greater Strasbourg (Strasbourg city and

surrounding municipalities) is the main regional

metropolis, accounting for about 50% of the population

of the département. Built upon land-use, demographic

and employment data, Figure 1 provides a general

overview of the extent of urban, suburban and rural

areas in the département [33]. Regarding urbanisation

levels, it is important to note that our study area is fairly

Figure 1 Bas-Rhin département: urbanisation level and population. This map shows the distribution of population and urbanisation levels in

the Bas-Rhin département. About half of the population of the département lives in Greater Strasbourg (i.e. Strasbourg city and 27 surrounding

municipalities). Other important cities are Haguenau (32,000 inhabitants) and Sélestat (17,000 inhabitants). Most of the département exhibits low

urbanisation levels: 446 out of 526 municipalities have less than 2000 inhabitants.

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heterogeneous, resulting in marked spatial disparities in

the distribution of population, facilities, services and

work opportunities, and thus in expected accessibility

levels.

Data on distribution of facilities

Data on the distribution of facilities were obtained from

the French National Institute of Statistics and Economic

Studies (INSEE). Two censuses, the Municipal Census

(Inventaire Communal, 1998) and the Businesses Census

(SIRENE, 2000) provide the number of available oppor-

tunities for each administrative unit (526 municipalities

in the Bas-Rhin département). In this work, we have

chosen to focus on three types of food outlet (bakeries,

grocery stores and hyper/supermarkets) that may pre-

sent contrasting situations. Bakeries and grocery stores

can be seen as proximity services and both grocery

stores and hyper/supermarkets sell general food items,

but have retail floor areas of less and more than 120 m2

respectively (1998 Municipal Census classification).

Table 1 shows that variability in the distribution of food

outlets across the territory was high and that the shop-

ping behaviours of French households differed according

to the type of food outlet [34].

The 1998 Municipal Census provides additional data

on travel behaviour: for each type of opportunity, if it is

not available in a given municipality, the destination (i.e.

the municipality) chosen by the majority of inhabitants is

provided. Even if it is incomplete because of the absence

of data for intra-municipality and extra-département

trips, that information makes it possible to approach the

distribution of spatial interactions (Figure 2).

The potential model as an accessibility index

It is generally recognized that the use of potential

models as accessibility indices was first introduced by

Hansen [32]. The potential model belongs to the family

of gravity-based or spatial interaction models. These

models are based on social physics and assume some

analogies between physical (e.g. Newton’s Law of Gravi-

tation) and social phenomena such as migration [35],

retailing [36] or population distribution [37].

The concept of potential first appeared in Stewart [37]

who noted that the influence of population between two

places was inversely proportional to the distance

between them (inverse distance weighting function).

Since those early years, many other forms of distance

weighting functions have been proposed for spatial

interaction models in general, and for potential models

more specifically (e.g. inverse power, negative exponen-

tial). As stated by Pooler [[38], p. 276], “virtually any

function which is monotonically decreasing with

increasing dijis a candidate for inclusion in a potential

equation”. A general formulation of the potential model

can then be written as:

Φi

s

j

s

ij

j

n

O f d

(

=⋅

=∑

)

1

(1)

where Φi

opportunity s, Oj

tance (or time) between i and j and f(dij) is an impedance

travel (or distance decay) function for travel between i

and j. It can refer to travel behaviour or “frictional effect

of space” and can be seen as people’s willingness to travel

according to trip purpose (e.g. to go shopping for food or

to go to a fitness centre for performing physical activity),

demographic characteristics (e.g. age, gender) or destina-

tion attractiveness.

) =

(dij) = exp(-a·dij) are the most commonly used functions.

According to Handy and Niemeier [[6], p.1177], the latter

is “the most closely tied to travel behaviour theory”.

These functions have been implemented in the Accessi-

bility Analyst extension for the desktop GIS software

sis the potential at point i for a given type of

sis an opportunity at j, dijis the dis-

Inverse power f dd

ij

ij

(

−and negative exponential f

Table 1 Distribution of food outlets among municipalities of the Bas-Rhin département (France) according to number

and type of food outlets

Number of food outletsType of food outlet

BakeriesGrocery storesHyper/supermarkets

0250 (47.5)

219 (41.6)

41 (7.8)

5 (1.0)

7 (1.3)

4 (0.8)

65

3.7

344 (65.4)

166 (31.6)

11 (2.1)

3 (0.6)

1 (0.2)

1 (0.2)

14

1.8

456 (86.7)

49 (9.3)

18 (3.4)

2 (0.4)

0 (0.0)

1 (0.2)

83

2.0

1 - 2

3 - 5

6 - 7

8 - 10

More than 10

% of households shopping at least once a week

Mean number of visits in a week

In the upper part of the table, values are number of municipalities with percentages in parentheses. In the lower part, values given for grocery stores include

little supermarkets (retail floor area between 120 and 300 m2).

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package Arc View 3 [39]. However, we will show below

that the use of other functions may be relevant.

It is important to note that some analogy exists

between KDE and potential model. In both cases,

opportunities are weighted according to a function of

the distance (gravity-based models). The main difference

between the two methods lies in the mathematical prop-

erties of the function. For KDE, the kernel is a function

that integrates to one while it is not necessarily the case

for the potential model. This allows a greater degree of

flexibility in the definition of the type of function and of

the associated parameters.

Model calibration

In our work, the calibration process consisted in defin-

ing two elements of the model specification: the travel

impedance and the set of potential destinations. In

order to specify travel impedance, three steps were

necessary: 1) choosing the distance metric dij(e.g. Eucli-

dean distance, travel time), 2) defining the travel impe-

dance function f(dij) (e.g. inverse power, negative

exponential) and 3) setting the parameters of this func-

tion (e.g. the constant). These stages are presented in

the next three sections. The specification of the set of

potential destinations (i.e. neighbourhood delimitation)

is presented in a fourth section.

Choosing the distance metric

One critical point when estimating spatial accessibility

concerns the definition of the distance measurement

[40]. Many different “distances” may be used including

Euclidean distance, Manhattan (or rectangular) distance,

network distance, time-distance or economic cost.

Because using sophisticated and more precise measures

such as travel times may introduce computational diffi-

culties, we sought to verify whether simpler Euclidean

distances would be very different from travel times or

network distances on the regional scale.

We first built and validated a model in order to esti-

mate travel times between all the municipalities in the

département (see Appendix 1 for details). Pearson corre-

lation coefficients were then calculated in order to assess

the strength of the associations between Euclidean dis-

tance, network distance and time-distance of observed

commuting trips (N = 4690). Data were log-transformed

because of data distribution skewness. Results showed

that the associations between all three measures were

very strong (correlations above 0.97), so that we could

conclude that Euclidean distances were a good approxi-

mation of the two other more specific distances for the

regional scale. We therefore decided to keep our models

as simple as possible by using only Euclidean distances.

Defining a travel impedance function

In order to define the travel impedance function, Tay-

lor’s proposal [41] was to find a linear relation between

trip length and volume of interactions. For this purpose,

he suggested transforming data according to the Goux

typology of distance decay functions (i.e. square-root

exponential, exponential, normal, Pareto and log-

normal) and to find which of these transformations

provides the best fit (i.e. least squares).

We applied this method to available data on trip

lengths for shopping purposes (Figure 2). Because none

of the transformations allowed us to get an acceptable

linear pattern, we adopted a probabilistic approach

[[42,43] cited by Ingram [1]]. This led us to represent

data differently (Figure 3) and we found that the dis-

tance weighting function would belong to a generalised

negative exponential functions family. This family of

functions is defined as:

f d

(

d

ijij

)exp()

= − ⋅

(2)

where b is a distance exponent to be determined. The

Gaussian function is a particular case for which the dis-

tance exponent equals two. Smoothing properties (con-

vexo-concave shape) of this family of functions make it

“related to empirical results referring both to the per-

ception of space and to the mobility of populations”

[[44], p.14].

Figure 2 Distribution of trip length for shopping purposes. The

graph shows population counts by trip length (1 km intervals) for

observed travels to hyper/supermarkets (green), grocery stores

(orange) and bakeries (blue). Trip lengths are Euclidean distances

between administrative centres of municipalities. Dots represent

observed values; lines are Gaussian probability density functions.

Trip lengths distribution shows a bell-shaped pattern with an under-

representation of population counts for short distances. This can be

due to the absence of data for intra-urban travels and a possible

spatial structure effect (i.e. relatively few centres of municipalities

situated less than 2 km away one from the other).

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Setting the parameters for the travel impedance function

Once the type of function was defined, the next step

consisted in setting the associated parameters (i.e. a and

b) that would produce the best fit to the observed data

[41]. That can be done by performing either linear or

non-linear regression analysis with transformed or non-

transformed data respectively. In the case of linear

regression, the idea is to conduct the analysis for differ-

ent values of the distance exponent b.

Both methods were tested (see Appendix 2) and pro-

duced equivalent goodness of fit (R2= 0.99; p < 0.001).

Nevertheless, by plotting observed data and the pre-

dicted values of the regression model, some differences

were found (results not shown): the non-linear regres-

sion model tightly fit observed data for intermediate dis-

tances but seemed to underestimate probabilities for

shorter and longer trips. In contrast, the linear regres-

sion model seemed to better estimate probabilities for

longer trips but clearly underestimated values for short

and intermediate distances.

That was particularly true in the case of hyper/super-

markets and our conclusion for this point was that

probability values predicted by the linear regression

model were closer to the idea we have about the phe-

nomenon (e. g. for hyper/supermarkets, a probability

value for spatial interaction that tends towards zero

below 10 km does not seem to be realistic, see Figure 2).

Consequently, we decided to calibrate the travel impe-

dance function for each type of food outlet using the

values of the parameters derived from the linear regres-

sion analyses (Table 2).

Specifying the set of potential destinations

Because no spatial interactions are observed beyond

a certain distance threshold for a given purpose (e.g.

20 km for hyper/supermarkets, see Figure 2) and

because of the asymptotic nature of the exponential

function (i.e. every opportunity of the study would con-

tribute to the potential value calculation), it may be

necessary to define a maximum distance Dij (or span)

above which opportunities would not be included in the

calculation (i.e. defining neighbourhood limits).

Because the negative exponential function is short-

tailed, long distances have limited effects on the

accessibility estimation, and function truncation thus

does not lead to an important loss of information

[44]. New formulation of the potential can then be

written as:

Φi

s

j

s

ij

j

n

ijij

O f d

(

dD

=

⋅≤

⎧

⎪⎪

⎪

⎪

⎨

⎩

=∑

0

)

1

if

otherwise

(3)

Figure 3 Distribution of probabilities for spatial interaction at a given distance. These graphs show cumulated per cent of trips that are

greater than a given distance for hyper/supermarkets (left) and grocery stores (right). Data for bakeries are not shown. Plotted values can also

be seen as the probability for interaction at a given distance. It allowed us to have an idea of the type of distance decay function.

Table 2 Parameters values retained from the calibration

process

Parameter value

a

Bakeries2.14.10-6

grocery stores9.333.10-7

Hyper/supermarkets 1.156.10-6

Span (in m)

b

1.6

1.65

1.6

12 000

14 000

19 000

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Obviously, this new formulation is to be related to the

container approach (rectangular distance decay function)

[5] but it overcomes its limitation (a rough threshold) if

the impedance travel function is correctly calibrated and

tends towards zero when dijis close to the span value

Dij. In the present work, spans were defined according

to the distribution of trip lengths (Figure 2) as the maxi-

mum travel distance observed rounded to the higher

integer value in kilometres (see Table 2).

Results

Potential accessibility estimation: application of the

“original” potential model

Using the calibrated model, we estimated potential

values for each type of food outlet in the study zone.

The models were implemented in the XLISP-STAT pro-

gramming environment [45,46] and ArcGIS 9.2 (ESRI,

Redlands, California) was used for mapping.

One of the advantages of the potential model is that

it can be used as a spatial smoothing technique [42]

and allows producing pseudo-continuous (raster data)

surface maps when applied to a regular grid of points.

We applied this transformation from discrete to

pseudo-continuous by estimating potential accessibility

values for the whole département (with a 5 km margin

for border effect correction) on a regular grid of more

than 1,000,000 of points (spatial resolution: 100 m).

This was the finest spatial resolution we could process

with reasonable computing times. Once the accessibil-

ity values had been estimated for each type of food

outlet, outputs of the models were mapped by convert-

ing point values into raster data with the 100 m spatial

resolution.

Results are presented on Figures 4 and 5. Producing

pseudo-continuous surface maps presents at least two

main advantages. First, by getting data that are free

from administrative boundaries, it allows approaching a

more realistic and precise estimation of accessibility

levels. For example, map comparison clearly shows that

there are no areas with null accessibility even though

facilities are not locally available (i.e. no grocery stores

in the municipality) (Figure 4). Second, it proves its use-

fulness for emphasising different kinds of retailing stra-

tegies. For example, it appears that while accessibility to

bakeries is quite good all over the département, hyper/

supermarkets are concentrated in most populated areas

(Figure 5).

Comparing count of facilities data and potential

accessibility values

In order to compare the output of the first model with

original data, potential accessibility values were esti-

mated for all municipalities administrative centres

(N = 526). Graphical comparisons were conducted

between potential accessibility values and number of

food outlets and between ranking of municipalities

Figure 4 Maps of number of grocery stores (left) and potential accessibility surface (right). These maps show the distribution of grocery

stores by municipality (left) and smoothed surface of potential accessibility (right). Potential accessibility was estimated with an exponential-

shaped function and a 14 km span. Class limits were defined manually for visualisation and comparison purposes.

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(first rank for the highest values) according to poten-

tial accessibility values and count of facilities data. In

both cases, results showed a very high variability in the

outputs of the model: except for Strasbourg city, which

continued to occupy the first place, ranking of munici-

palities appeared completely modified, even for impor-

tant cities such as Haguenau and Sélestat (Figure 6).

Correlation coefficients were estimated in order to

assess the importance of these variations. Because of

the extreme skewness of the original data distribution

(see Table 1), we estimated Spearman’s rank correla-

tion coefficients. All associations were statistically sig-

nificant at the 0.01 level. Results indicate a poor

relationship between the rankings of municipalities for

the count of facilities and potential accessibility values

(0.342 for grocery stores, 0.355 for hyper/supermarkets

and 0.523 for bakeries).

An extended potential model for commuters

The approach developed previously allowed us to esti-

mate accessibility levels, but in a very limited and static

way. Indeed, so far we did not take into account popula-

tion movements across space while the presence of

opportunities both in the residential neighbourhood and

around activity places (e.g. workplaces) may impact

population accessibility levels.

One basic idea is therefore to extend the potential

model in order to take account of the case of commu-

ters: people have access to opportunities not only in the

area i where they reside, but also in the area k where

they work.

In that case, we can estimate a cumulative potential

accessibility to a type of service s such as:

ΦΦΦ

Cik

s

Cik i

s

Cik k

s

=+

,,

(4)

where ΦCik i

accessibility in i and k for commuters living in i and

working in k.

This first approach does not consider trip length dik

between i (home) and k (work) and hence that increasing

travel time will reduce available time (or time-budget) for

a given purpose (e.g. shopping). Because time-budget is

not unlimited, if commuting time becomes too long,

commuters will not have access to services neither in

i nor k. It is then necessary to choose a threshold value

for trip length beyond which facilities will not be accessi-

ble anymore. This threshold max(dik) can be defined in

an empirical way (e.g. using commuting data), assuming

for example that the time-budget for activities is null

when commuting time or distance reaches the threshold

value. The available time-budget for a given activity in

s

,and ΦCik k

s

,

are respectively the potential

Figure 5 Potential accessibility surfaces for hyper/supermarkets (left) and bakeries (right). These maps show smoothed surfaces of

potential accessibility to hyper/supermarkets (left) and bakeries (right). Potential accessibility was estimated with an exponential-shaped function

and a 19 km span for hyper/supermarkets and a 12 km span for bakeries. Class limits were defined manually for visualisation and comparison

purposes.

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i and k will be reduced according to a trip length weight-

ing function:

()

max()

,max()

d

d

d

dd

ik

ik

ik

ik ik

=⎛

⎝

⎜

⎞

⎟

⎠

≤

(5)

where θ is a parameter reflecting the relationship

between the commuting time or distance and the avail-

able time-budget. Standardisation by max(dik) ensures

that g(dik) ranges from zero when dikequals zero (i.e.

living and working in the same place) to one when dik

equals max(dik).

The potential accessibility ΦCik

commuters living in i and working in k can then be

written as:

s

to a service s for

ΦΦΦ

Cik

s

Cik i

s

Cik k

s

ik

d

=+⋅−

() (( ))

,,

1

(6)

where (1 - g(dik)) is the trip length weighting term

which ranges from zero (when no time is available for

the given activity, i.e. the travel time is too long) to one

(when full time-budget is available, i.e. no travel time).

then theoretically ranges from 0 to 2× ΦCik i

when i and k coincide (i.e. living and working in the

same place).

Because commuters of a given municipality may have

different destinations, we introduce the proportion of

commuters for each destination in the previous equation

(Equation 6). A global cumulative potential accessibility

for commuters living in a municipality i can then be

written as:

ΦCik

s

s

,

ΦΦΦ

Ci

s

ik

i

Cik i

s

Cik k

s

ik

k

n

C

C

d

=+⋅−

=∑

( ) (( ))

,,

1

1

(7)

where Cikis the number of commuters living in i and

working in each destination k and Ciis the total number

of commuters. ΦCik i

the potential model (Equation 3).

s

,and ΦCik k

s

,

are estimated using

Estimating potential accessibility for a commuting

population

The analysis conducted for the choice of the distance

metric (see section “Methods: choosing the distance

metric”) also allowed us to calibrate the θ parameter in

the trip length weighting function g(dik) (Equation 5):

the high correlation coefficient value between time and

distance led us to conclude that the time-budget should

decrease in a linear way as the Euclidean distance

increases and θ was then set to 1. The threshold dis-

tance for which potential accessibility value around

home and work reaches zero was defined as the

maximum travel distance observed for all commuters

(65 km). We estimated commuting-based potential

accessibility to hyper/supermarkets, grocery stores and

bakeries and compared it to the outputs of the potential

model. Models were calibrated using the same para-

meters values as previously (see Table 2).

Cartographic outputs for hyper/supermarkets and bak-

eries are presented in Figures 7 and 8. Because our

application is based upon municipality-level data (com-

muting data) accessibility values are then estimated for

administrative centres but mapped for municipality

Figure 6 Graphical comparisons between original data and model’s output. The graphs show the differences between frequencies of

hyper/supermarkets and potential accessibility values (on the left) and ranks (on the right). Line equation is y = x.

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Page 10

Figure 7 Maps of potential accessibility to hyper/supermarkets for residents (left) and commuters (right). These maps make it possible

to compare potential accessibility to hyper/supermarkets between residents (left) and commuters (right). Potential accessibility was estimated

with an exponential-shaped function and a 19 km span. In the case of commuters, potential accessibility is cumulated in municipalities of both

residence and workplace and weighted according to the inverse distance between them and the number of commuters. Class limits are defined

according to quantiles.

Figure 8 Maps of potential accessibility to bakeries for residents (left) and commuters (right). These maps make it possible to compare

potential accessibility to hyper/supermarkets between residents (left) and commuters (right). Potential accessibility was estimated with an

exponential-shaped function and a 12 km span. In the case of commuters, potential accessibility is cumulated in municipalities of both residence

and workplace and weighted according to the inverse distance between them and the number of commuters. Class limits are defined

according to quantiles.

Salze et al. International Journal of Health Geographics 2011, 10:2

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Page 11

areas. Map analysis shows that the introduction of jour-

ney to work into the model dramatically increases the

overall accessibility levels. The impact of Strasbourg city

as the biggest employment centre is particularly striking.

Indeed in both cases, values seem to follow a clear

inverse gradient as the distance to Strasbourg city

increases.

Comparing potential accessibility between residents and

commuters

Graphical comparisons reinforce the previous observa-

tions of an overall increase in accessibility levels either

for supermarkets (Figure 9, left), bakeries (Figure 10,

left) or grocery stores (results not shown). In all cases,

the increase is high in major urban poles and suburban

municipalities. The situation is more contrasted in the

case of rural municipalities under urban influence: the

increase is very strong for some of them while others

have similar levels as rural municipalities out of urban

influence. Graphical comparison of ranks confirms that

observation (Figure 9 and 10, right) and Spearman’s

rank correlation coefficients show that the ranking

variability is slightly higher for bakeries (0.665) and

grocery stores (0.674) than for hyper/supermarkets

(0.778). These results can be related to the previous

map analysis: distance to urban poles plays a major

role in the estimation of accessibility levels. That

seems to be especially true in rural remote areas where

accessibility levels for both residents and commuters

are the lowest.

Discussion

The objective of this study was to propose measures of

spatial accessibility to a set of facilities on the regional

scale, with the aim of applying the methodology to the

field of health behaviours and neighbourhood depriva-

tion in future work. For this purpose, we first chose to

use a potential accessibility index, and we proposed to

improve it by introducing the dynamics of a commuting

population.

Data availability

Our developments have mainly been driven by available

data. Because commuting data were only available at an

aggregated level, it has not been possible to conduct the

analysis below the municipality level. Furthermore, as

we decided to work with a fairly old database, we were

unable to assess data quality (see for example [47], [48]

and [49] for methodological discussions on this point).

In the present work, we chose to apply our models to

three different kinds of food outlets for illustrative pur-

poses, and it is important to emphasise here that the

method could be applied to any type of facility (e.g. pub-

lic services, sport facilities) and using disaggregate data

if available.

The potential model: introducing travel behaviour

Our work was based on the observation that most stu-

dies that attempted to link health behaviours and spatial

accessibility used measurements that do not take into

account travel behaviour (nearest opportunity, container

Figure 9 Potential accessibility to hyper/supermarkets for residents and commuters: graphical comparisons. The graphs show the

differences between values (on the left) and ranks (on the right) of potential accessibility to hyper/supermarkets according to urbanisation levels.

Major urban poles are plotted in red, secondary poles in brown, suburban municipalities in orange and rural municipalities under or out of

urban influence respectively in yellow and green. Line equation is y = x.

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Page 12

index) [11]. We exposed a complete example of applica-

tion (including calibration process) of a potential model

which is relatively easy to implement and for which only

few data are needed for computation. The potential

model is very close to the KDE both in a theoretical

(gravity-based measure) and practical (spatial smoothing

technique) perspective. The main advantage of our

method is that every part of the model specification can

be controlled, especially the definition of the travel

impedance function. This advantage is nevertheless very

relative as it is widely accepted that in the case of KDE,

compared to the type of function used, the size of the

span is much more crucial (i.e. neighbourhood delimita-

tion) [13].

The resulting potential accessibility index includes

some aspects of travel behaviour and is partly free from

administrative boundaries. It may thus provide a more

accurate picture of accessibility levels experienced by

the population. In our study zone, it appeared for exam-

ple that areas with null or very weak accessibility were

almost inexistent, reflecting a good global accessibility

level to food outlets. These results obviously need to be

put in relation with the transport mode chosen and thus

with car ownership and transportation possibilities. But

because 85% of the households of the département

owned at least one car (1999 Population Census,

INSEE), we hypothesise that our resulting index is valid

on this scale of analysis and for this specific study zone.

A classical limitation of such aggregated models is

that it assumes that all individuals in a municipality

experience the same accessibility level, thus the same

“frictional effect of space”. Indeed, the model was

calibrated using trip length data, assuming that spatial

interaction distributions were only due to travel beha-

viours, the distance decay being constant for the whole

zone studied and the population homogeneous. The

model thus did not take into account possible local spa-

tial variations of the distance decay which can result

either from behavioural discrepancies or from the spa-

tial structure effect [50].

Another limitation is that of the lack of data that may

have resulted in inaccurate measurements. Results

showed that areas with the lowest accessibility levels

were mainly distributed near the borders of our study

zone (Figure 4). This observation has nevertheless to be

interpreted carefully because of the absence of informa-

tion about facilities provision outside the département.

Indeed it has been shown that edge effects may strongly

impact gravity-based accessibility measurements (espe-

cially when using large distances) [51].

Integrating the spatial dynamics of a commuting

population

Our proposition was to extend the potential model to a

commuting population. The resulting index reflects an

accessibility level by car that takes into account the

cumulated spatial distribution of facilities around both

home and workplace. This model resulted in an overall

increase in the accessibility levels. The increase was

nevertheless not uniform, and the results highlighted the

role of distance to major urban poles. Some rural muni-

cipalities with low levels of accessibility ultimately

enjoyed better accessibility levels than some secondary

urban poles. These observations emphasise the fact that

Figure 10 Potential accessibility to bakeries for residents and commuters: graphical comparisons. The graphs show the differences

between values (on the left) and ranks (on the right) of potential accessibility to bakeries according to urbanisation levels. Major urban poles are

plotted in red, secondary poles in brown, suburban municipalities in orange and rural municipalities under or out of urban influence respectively

in yellow and green. Line equation is y = x.

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Page 13

considering populations as static “objects” could lead to

biased results and may partly explain why published stu-

dies on the relationships between individual behaviours

and environmental determinants exhibit inconsistent

findings [52]. This important result also supports the

calls for the developments of new measurements taking

into account both residential and non-residential expo-

sure [22,53,23,31].

The other strength of this extended model is that it

introduces a temporal constraint, although in a very

coarse way, through the notion of time-budget. In

their study on foodscape exposure, Kestens and collea-

gues identified this point as an important one in mea-

suring the influence of food stores in a space-time

perspective [31].

Several limitations are associated with the develop-

ment of our extended potential model. First, the dis-

tance and time-budget weighting functions were based

on Euclidean distances between municipalities because

our analysis showed a very strong correlation between

travel-times, network distances and Euclidean distances

on the regional scale. Nevertheless, the travel-time

model used (see Appendix 1) did not take account of

road traffic data, which may strongly impact trip dura-

tions (especially during peak hours). Further investiga-

tion will therefore be necessary to refine the travel-time

model and more generally, to address this question of

the time-budget weighting function. It may be indeed

relevant to disaggregate our index and to calibrate the

weighting functions for different segments of population,

e.g. according to median income, age and structures of

households [31,54], motorisation rates [55] or travel

modes [56].

A second limitation is that the model only considers

accessibility for origins (municipality of residence) and

destinations (workplace) and thus leads to the underes-

timation of potential accessibility levels because facilities

present along the daily space-time path are not taken

into account. Obviously, the extended model we pro-

pose can be refined to include these intervening oppor-

tunities. However, we believe with other authors [57,31]

that a methodological shift towards individual-based

and activity-based models would be necessary to

address this question in health studies. In the meantime,

given actual data availability, the use of aggregated

models still remains necessary when dealing with large

study zones.

Conclusions

The aim of this study was to propose measurements of

spatial accessibility to a set of facilities on a regional

scale. The measurements provided different pictures of

accessibility levels. We first applied a potential model

which overcomes some of the limitations of more

simple accessibility indices by partly encompassing the

multidimensional aspect of the accessibility estimation

issue [12]. Then, we proposed an extension to the origi-

nal potential model. That allowed us to get very differ-

ent results by integrating the dynamics of commuting.

Our work supports existing evidence of the importance

of the inclusion of specific questions about the location

of non-residential activities in health surveys and of the

choice of appropriate accessibility indices for analyses

dealing with 1) socio-spatial disparities in facilities

related to health behaviours and 2) relationships

between individual behaviours and accessibility to speci-

fic facilities. Such extension of spatial interaction models

has potential implications for improving our under-

standing of environmental influences on health out-

comes on the regional scale, and then for providing

relevant information to policy-makers in the field of

public health.

Appendix 1 - Building and validating the travel

time model

Using a road network database (Georoute 2002, Insti-

tut Géographique National, France) and the Network

Analyst extension for ArcGIS 9.2 (Environmental Sys-

tems Research Institute, Redlands, CA), we estimated

network distances and travel times between all the

municipalities in the département. Speed limits were

assigned to each segment of the road network follow-

ing previous work of Hilal [58], according to land-use

type (urban vs. rural areas), elevation (hill areas vs.

plain areas) and road hierarchy (highways, primary

roads, secondary roads).

This “travel-time” model was validated for 100 ran-

domly selected journeys, by comparing calculated travel

times with travel times obtained from two different on-

line mapping/itinerary providers (©Mappy and©ViaMi-

chelin). Student’s paired t-tests were used to assess the

differences between travel times. Results showed that

the differences between travel times were statistically

significant for each of the three tested pairs (p < 0.001).

Interestingly, we observed that, compared to the travel

times provided by our model, data from the first opera-

tor tended to overestimate and those from the second

operator tended to underestimate travel times. A Stu-

dent’s paired t-test was then applied to assess the differ-

ences between the travel times of the model and the

mean of the two travel times provided by the operators.

Even though very close to the threshold value, the test

did not reach statistical significance (N = 100; p =

0.052). Furthermore, 95% of the travel times difference

absolute values were lower than 7.4 minutes (mean:

2.71; SD: 2.32) and the largest travel times differences

(above 5 minutes) were observed for the larger trip

lengths (above 50 minutes) (Figure 11).

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Page 14

Appendix 2 - Setting the parameters of the

distance weighting function

In a first step, data were log-transformed and linear

regression analyses performed for several distance

exponents in order to find the best fit (i.e. the minimal

standard error of estimate) (Figure 12). Because of our

probabilistic approach (probability reaches 1 at null

distance), the constant term was not included in the

regression model which then tooktheform

log ( ) Pd

i ij

= − ⋅

where Pi is the probability for

interaction at distance dij.

For non-linear regression analyses, the model was of

the form Pd

i ij

= − ⋅

exp (

used in the iterative process have been set to values

relatively close to expected ones (i.e. rounded values of

parameters derived from the linear regression analysis).

)

and the initial parameters

Acknowledgements

This work is part of the ELIANE (Environmental LInks to physical Activity,

Nutrition and hEalth) study. ELIANE is a project supported by the French

National Research Agency (Agence Nationale de la Recherche, ANR-07-

PNRA-004). JMO is the coordinator of the ELIANE study; CS, BC and CW are

the principal investigators in the ELIANE study. The authors would like to

thank F. Magnin-Feyssot (Image, Ville, Environnement, Strasbourg, France) for

technical assistance.

Author details

1Université de Strasbourg; Image, Ville, Environnement, Strasbourg, France.

2CNRS, ERL 7230, Strasbourg, France.3UMR 8504, CNRS/Université Paris 1,

Géographie-Cités, Paris, France.4UREN, INSERM U557/INRA U1125/CNAM/

Université Paris 13/CRNH Ile-de-France, Bobigny, France.5Université Pierre et

Marie Curie-Paris6; Service de Nutrition, Groupe Hospitalier Pitié-Salpêtrière

(AP-HP), Centre de Recherche en Nutrition Humaine Ile-de-France (CRNH

IdF), Paris, France.6Lab-Urba, Institut d’Urbanisme de Paris, Université Paris

Est-Créteil, France.7Université de Lyon, INSERM/U870 INRA/U1235, CRNH

Rhône-Alpes, Hospices Civils de Lyon, Oullins, France.8INSERM U707, Paris,

France.9Université Pierre et Marie Curie-Paris6, UMR-S 707, Paris, France.

Authors’ contributions

PS and AB conceived and designed the study. PS performed analysis and

interpretation of data, parts of the programming and drafted the

manuscript. AB performed the main part of the programming and helped to

draft the manuscript. JMO, HC, CS, BC, RC, DB, and CW participated in the

critical revision of manuscript. All authors have read and approved the final

manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 30 September 2010 Accepted: 10 January 2011

Published: 10 January 2011

References

1.Ingram D: The concept of accessibility: A search for an operational form.

Regional Studies 1971, 5:101-107.

2.Vickerman RW: Accessibility, attraction, and potential: a review of some

concepts and their use in determining mobility. Environment and

Planning A 1974, 6:675-691.

3. Morris JM, Dumble PL, Wigan MR: Accessibility indicators for transport

planning. Transportation Research Part A: General 1979, 13:91-109.

4.Pirie GH: Measuring accessibility: a review and proposal. Environment and

Planning A 1979, 11:299-312.

5.Koenig JG: Indicators of urban accessibility: Theory and application.

Transportation 1980, 9:145-172.

6. Handy SL, Niemeier DA: Measuring accessibility: an exploration of issues

and alternatives. Environment and Planning A 1997, 29:1175-1194.

7.Geurs KT, van Wee B: Accessibility evaluation of land-use and transport

strategies: review and research directions. Journal of Transport Geography

2004, 12:127-140.

8.Brownson RC, Hoehner CM, Day K, Forsyth A, Sallis JF: Measuring the Built

Environment for Physical Activity: State of the Science. American Journal

of Preventive Medicine 2009, 36:S99-S123.

9.Lytle LA: Measuring the Food Environment: State of the Science.

American Journal of Preventive Medicine 2009, 36:S134-S144.

Figure 11 Differences of travel times between model and

operators according to trip length. This graph shows differences

between travel times estimated by our model and on-line

operators. Difference in minutes is plotted for each pair of

municipalities exchanging commuters according to trip duration.

Figure 12 Standard error of estimates of linear regression

model for hyper/supermarkets according to distance exponent.

This graph shows standard error of estimates resulting from the

linear regression model according to several distance exponents.

Salze et al. International Journal of Health Geographics 2011, 10:2

http://www.ij-healthgeographics.com/content/10/1/2

Page 14 of 16

Page 15

10.Gordon-Larsen P, Nelson MC, Page P, Popkin BM: Inequality in the Built

Environment Underlies Key Health Disparities in Physical Activity and

Obesity. Pediatrics 2006, 117:417-424.

Charreire H, Casey R, Salze P, Simon C, Chaix B, Banos A, Badariotti D,

Weber C, Oppert JM: Measuring the Food Environment Using

Geographical Information Systems: A Methodological Review. Public

Health Nutrition 2010, 1-13.

Talen E, Anselin L: Assessing spatial equity: an evaluation of measures of

accessibility to public playgrounds. Environment and Planning A 1998,

30:595-613.

Silverman B: Density estimation for statistics and data analysis London:

Chapman and Hall; 1986.

Luo W, Qi Y: An enhanced two-step floating catchment area (E2SFCA)

method for measuring spatial accessibility to primary care physicians.

Health & Place 2009, 15:1100-1107.

Guagliardo M: Spatial accessibility of primary care: concepts, methods

and challenges. International Journal of Health Geographics 2004, 3:3.

Yang D, Goerge R, Mullner R: Comparing GIS-Based Methods of

Measuring Spatial Accessibility to Health Services. Journal of Medical

Systems 2006, 30:23-32.

Diez Roux AV, Evenson KR, McGinn AP, Brown DG, Moore LV, Brines S,

Jacobs DR: Availability of Recreational Resources and Physical Activity in

Adults. American Journal of Public Health 2007, 97:493-499.

Moore LV, Diez Roux AV, Brines S: Comparing Perception-Based

and Geographic Information System (GIS)-Based Characterizations

of the Local Food Environment. Journal of Urban Health 2008,

85:206-216.

Moore LV, Diez Roux AV, Nettleton JA, Jacobs DR: Associations of the

Local Food Environment with Diet Quality–A Comparison of

Assessments based on Surveys and Geographic Information Systems.

American Journal of Epidemiology 2008, 167:917-924.

Riva M, Apparicio P, Gauvin L, Brodeur J: Establishing the soundness of

administrative spatial units for operationalising the active living

potential of residential environments: an exemplar for designing

optimal zones. International Journal of Health Geographics 2008, 7:43.

Chaix B, Merlo J, Evans D, Leal C, Havard S: Neighbourhoods in eco-

epidemiologic research: Delimiting personal exposure areas. A response

to Riva, Gauvin, Apparicio and Brodeur. Social Science & Medicine 2009,

69:1306-1310.

Ball K, Timperio A, Crawford D: Understanding environmental influences

on nutrition and physical activity behaviours: where should we look and

what should we count? International Journal of Behavioral Nutrition and

Physical Activity 2006, 3:33.

Chaix B: Geographic Life Environments and Coronary Heart Disease: A

Literature Review, Theoretical Contributions, Methodological Updates,

and a Research Agenda. Annual Review of Public Health 2009, 30:81-105.

Frank LD, Glanz K, McCarron M, Sallis JF, Saelens BE, Chapman J: The

Spatial Distribution of Food Outlet Type and Quality around Schools in

Differing Built Environment and Demographic Contexts. Berkeley Planning

Journal 2006, 19:79-95.

Pearce J, Blakely T, Witten K, Bartie P: Neighborhood Deprivation and

Access to Fast-Food Retailing: A National Study. American Journal of

Preventive Medicine 2007, 32:375-382.

Zenk SN, Powell LM: US secondary schools and food outlets. Health &

Place 2008, 14:336-346.

Kestens Y, Daniel M: Social Inequalities in Food Exposure Around Schools

in an Urban Area. American Journal of Preventive Medicine 2010, 39:33-40.

Jeffery R, Baxter J, McGuire M, Linde J: Are fast food restaurants an

environmental risk factor for obesity? International Journal of Behavioral

Nutrition and Physical Activity 2006, 3:2.

Kwan MP: Space-Time and Integral Measures of Individual Accessibility:

A Comparative Analysis Using a Point-based Framework. Geographical

Analysis 1998, 30:191-216.

Kwan MP: Gender and individual access to urban opportunities: a study

using space-time measures. Professional Geographer 1999, 51:210-227.

Kestens Y, Lebel A, Daniel M, Thériault M, Pampalon R: Using experienced

activity spaces to measure foodscape exposure. Health & Place 2010,

16:1094-1103.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.Hansen WG: How Accessibility Shapes Land Use. Journal of the American

Institute of Planners 1959, 25:73.

Salze P, Badariotti D, Banos A, Charreire H, Oppert JM, Casey R, Simon C,

Chaix B, Weber C: Entre dasymétrique et choroplète: typologie de

l’espace adaptée à l’étude des relations entre santé et environnement.

In Actes des Neuvièmes Rencontres de Théo Quant: 4-6 March 2009; Besançon,

France Edited by: Foltête J-C .

Eymard I: De la grande surface au marché: à chacun ses habitudes. Insee

Première 1999, 636.

Ravenstein EG: The Laws of Migration. Journal of the Statistical Society of

London 1885, 48:167-235.

Reilly WJ: The law of retail gravitation New York; 1931.

Stewart JQ: An Inverse Distance Variation For Certain Social Influences.

Science 1941, 93:89-90.

Pooler J: Measuring geographical accessibility: a review of current

approaches and problems in the use of population potentials. Geoforum

1987, 18:269-289.

Liu S, Zhu X: Accessibility Analyst: an integrated GIS tool for accessibility

analysis in urban transportation planning. Environment and Planning B:

Planning and Design 2004, 31:105-124.

Apparicio P, Abdelmajid M, Riva M, Shearmur R: Comparing alternative

approaches to measuring the geographical accessibility of urban health

services: Distance types and aggregation-error issues. International

Journal of Health Geographics 2008, 7:7.

Taylor P: Distance decay in spatial interactions. Concepts and Techniques in

Modern Geography 1975, N°2.

Echenique M, Crowther D, Lindsay W: A spatial model of urban stock and

activity. Regional Studies 1969, 3:281-312.

MTARTS: Metropolitan Toronto And Region Transportation Study Province of

Ontario, Toronto; 1966.

Grasland C, Mathian H, Vincent J: Multiscalar analysis and map

generalisation of discrete social phenomena: Statistical problems and

political consequences. Statistical Journal of the United Nations Economic

Commission for Europe 2000, 17:157-188.

Betz D: XLISP: An experimental object-oriented programming language.

1988.

Tierney L: Xlisp-Stat: A Statistical Environment Based on the XLISP

Language. 1989.

Paquet C, Daniel M, Kestens Y, Leger K, Gauvin L: Field validation of

listings of food stores and commercial physical activity establishments

from secondary data. International Journal of Behavioral Nutrition and

Physical Activity 2008, 5:58.

Cummins S, Macintyre S: Are secondary data sources on the

neighbourhood food environment accurate? Case-study in Glasgow, UK.

Preventive Medicine 2009, 49:527-528.

Lake AA, Burgoine T, Greenhalgh F, Stamp E, Tyrrell R: The foodscape:

Classification and field validation of secondary data sources. Health &

Place 2010, 16:666-673.

Fotheringham AS: Spatial Structure and Distance-Decay

Parameters. Annals of the Association of American Geographers

1981, 71:425-436.

Van Meter EM, Lawson AB, Colabianchi N, Nichols M, Hibbert J, Porter DE,

Liese AD: An evaluation of edge effects in nutritional accessibility and

availability measures: a simulation study. International Journal of Health

Geographics 2010, 9:40.

Wendel-Vos W, Droomers M, Kremers S, Brug J, Lenthe FV: Potential

environmental determinants of physical activity in adults: a systematic

review. Obesity Reviews 2007, 8:425-440.

Inagami S, Cohen DA, Finch BK: Non-residential neighborhood exposures

suppress neighborhood effects on self-rated health. Social Science &

Medicine 2007, 65:1779-1791.

Páez A, Gertes Mercado R, Farber S, Morency C, Roorda M: Relative

Accessibility Deprivation Indicators for Urban Settings: Definitions and

Application to Food Deserts in Montreal. Urban Studies 2010,

47:1415-1438.

Bertrand L, Thérien F, Cloutier MS: Measuring and Mapping Disparities in

Access to Fresh Fruits and Vegetables in Montréal. Canadian Journal of

Public Health 2008, 99:6-11.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

Salze et al. International Journal of Health Geographics 2011, 10:2

http://www.ij-healthgeographics.com/content/10/1/2

Page 15 of 16