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A Comparative Analysis of Accessibility Measures by the Two-Step Floating Catchment Area (2SFCA) Method

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The recent decade has witnessed a new wave of development in the place-based accessibility theory, revolving around the two-step floating catchment area (2SFCA) method. The 2SFCA method, initially serving to evaluate the spatial inequity of health care services, has been further applied to other urban planning and facility access issues. Among these applications, different distance decay functions have been incorporated in the thread of model development, but their applicability and limitations have not been thoroughly examined. To this end, the paper has employed a place-based accessibility framework to compare the performance of twenty-four 2SFCA models in a comprehensive manner. Two important conclusions are drawn from this analysis: on a small analysis scale (e.g., community level), the catchment size is the most critical model component; on a large analysis scale (e.g., statewide), the distance decay function is of elevated importance. In sum, this comparative analysis provides the theoretical support necessary to the choice of the catchment size and the distance decay function in the 2SFCA method. Justification of model parameters through empirical evidence (e.g., field surveys about local travel activities) and model validation through sensitivity analysis are needed in future 2SFCA applications for various urban planning, service delivery, and spatial equity scenarios.
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A Comparative Analysis of Accessibility Measures by the Two-Step Floating Catchment Area
(2SFCA) Method
Xiang Chen1, *, Pengfei Jia2
1. Department of Emergency Management, Arkansas Tech University, Russellville, Arkansas 72801,
USA Email: chenxiangpeter@gmail.com
2. Academy Information Center of Urban Planning, China Academy of Urban Planning and Design,
Beijing, 100044, China Email: j_pengfei@yahoo.com
*Corresponding author
This paper is the pre-print and may be different than the final version. Please download the
publisher’s version via the link below:
https://www.tandfonline.com/eprint/eji5g9iZI9j58W57qGzV/full?target=10.1080/13658816.20
19.1591415
A Comparative Analysis of Accessibility Measures by the Two-Step Floating
Catchment Area (2SFCA) Method
Abstract: The recent decade has witnessed a new wave of development in the place-based
accessibility theory, revolving around the two-step floating catchment area (2SFCA) method. The
2SFCA method, initially serving to evaluate the spatial inequity of health care services, has been
further applied to other urban planning and facility access issues. Among these applications,
different distance decay functions have been incorporated in the thread of model development,
but their applicability and limitations have not been thoroughly examined. To this end, the paper
has employed a place-based accessibility framework to compare the performance of twenty-four
2SFCA models in a comprehensive manner. Two important conclusions are drawn from this
analysis: on a small analysis scale (e.g., community level), the catchment size is the most critical
model component; on a large analysis scale (e.g., statewide), the distance decay function is of
elevated importance. In sum, this comparative analysis provides the theoretical support
necessary to the choice of the catchment size and the distance decay function in the 2SFCA
method. Justification of model parameters through empirical evidence (e.g., field surveys about
local travel activities) and model validation through sensitivity analysis are needed in future
2SFCA applications for various urban planning, service delivery, and spatial equity scenarios.
Keywords: 2SFCA; accessibility; sensitivity analysis; spatial interaction model; distance decay
1. Introduction
“Everything is related to everything else, but near things are more related than distant things
(Tobler 1970).” Tobler’s first law of geography has established the theoretical foundation of spatial
interaction (SI) models in the analysis of spatial associations between geographic entities (Miller 2004).
Modeling accessibility to urban facilities, as a proxy for the potential of space for interaction, has been
widely explored for two evaluative purposes: a planning purpose evaluating if land use or transportation
systems can cater to the need of urban population (Geurs and Wee 2004) and a corroborative purpose
identifying if the inequity of patronizing opportunities exists among different social groups (Kwan et al.
2003). Central to the discussion is how opportunities could be articulated and how the spatial separation
could be quantified to better capture the complexities of urban forms and individual mobilities (Kwan
1998).
The origin of accessibility studies can be traced back to Hansen’s (1959) empirical model for land
use planning, considering place-based accessibility as an integrated assessment of urban opportunities
(e.g., employment, shopping opportunities, residential activity). The placed-based accessibility
measures, mainly including cumulative-opportunity models and gravity-type models (Kwan 1998),
evaluate the degree of accessibility for a reference location (i.e., the demand location, such as a
community center) according to the density of or the proximity to surrounding facilities (Talen and
Anselin 1998; Neutens et al. 2010). The other thread of research has shifted the focus to individual-
based accessibility measures, relying on the construct of the space-time prism in time geography
(Hägerstraand 1970). Time-geographic accessibility metrics (i.e., number of opportunities within the
prism, utilities gained from the travel) have been proposed to represent individuals capabilities in
reaching opportunities given their motility and time constraints (Miller 2010, Neutens et al. 2010).
Despite the proliferating examples of place-based and individual-based accessibility measurements for
evaluating urban service delivery or justifying issues of social inequities (Kwan et al. 2003, Neutens et al.
2010), the recent decade has witnessed a new wave of development in the place-based accessibility
theory, revolving around the two-step floating catchment area (2SFCA) method (Luo and Wang 2003).
The 2SFCA method, as a special form of the gravity model (i.e., the major type of SI models), considers
accessibility to be mediated by not only the distance decay (“transferability in the parlance of SI theory)
but also the interactions between supply and demand (“complementarity”; Haynes and Fotheringham
1984). It overcomes the caveat of traditional place-based accessibility measures where the demand for
service is largely overlooked. The 2SFCA method, initially serving to evaluate the spatial inequity of
health care services (Wang 2012), has been further applied to other urban planning and facility access
issues, including green space (Dai 2011, Xing et al. 2018), job opportunities (Dai et al. 2018), food stores
(Chen 2017, Chen 2019, Dai and Wang 2011), and emergency shelters (Zhu et al. 2018). In addition, the
formation of the 2SFCA model has been further modified to accommodate other service scenarios, such
as measuring the crowdedness of the facility usage (Wang 2017).
The 2SFCA method as a prototype model has been further developed to capture the
complexities of real-world travel environments, travel behaviors, and diverse service needs. The thread
of development primarily concerns three distinct facets of the method: the catchment size, the regional
competition, and the distance decay (as reviewed in the next section). Specifically, various forms of the
distance decay function, such as the negative linear function (Schuurman et al. 2010), the inverse power
function (Schuurman et al. 2010), the exponential function (Jamtsho et al. 2015), the Gaussian function
(Wan et al. 2012a), and the kernel density function (Dai and Wang 2011), have been incorporated to
improve the 2SFCA method. Although there have been studies testing the sensitivity of the method,
primarily about the distance weight (Mcgrail 2012) and the distance impedance coefficient (Luo and
Wang 2003, Wan et al. 2012a), to date, there has been no comprehensive evaluation of the different
distance decay functions in the 2SFCA family. Furthermore, the discussion on the applicability and
limitations of different 2SFCA models has yet to be initiated. This research gap poses a considerable
challenge to the overarching understanding of the method.
To this end, the paper fills the gap by performing the sensitivity analysis of typical models in the
2SFCA family. First, we have conceptualized the analytic framework that includes twenty-four 2SFCA
models based on the place-based accessibility theory. Second, we have thoroughly compared and
contrasted these 2SFCA models by state-of-the-art statistical measures. Lastly, based on the analysis
results, we have offered suggestions for the choice of the model parameters. This comparative analysis
is the first to provide a comprehensive discussion on different 2SFCA models and could corroborate the
state of equity measures by the 2SFCA method in various planning scenarios.
2. Development of the 2SFCA method
The 2SFCA method, originated from a gravity model (Joseph and Bantock 1982) and further
inspired by a spatial decomposition method (Radke and Mu 2000), aims to evaluate the physical
accessibility to health care services based on the spatial interaction between the supply and demand
(Luo and Wang 2003). The operation of the 2SFCA method is a two-step search procedure. The first step
evaluates the supply-to-demand ratio Rj by dividing the capacity of a facility Sj (the supply) to all
population Pk (the demand) within a threshold distance d0 (the catchment size), as shown in Equation
(1). This step characterizes the capability of the supply in fulfilling the needs of all population in its
service radius. The second step calculates the accessibility index Ai for the demand point i as a
summation of the Rj derived from the first step, as shown in Equation (2). The summation is based on all
the service providers within the catchment, which conceptualizes the activity space of population living
at location i. The 2SFCA method contributes to the formation of the spatial interaction in previous place-
based accessibility measures, which have invariably overlooked the demand side of the spatial
interaction.
 (1)


 (2)
The original 2SFCA method has been further extended to capture the complex ways in which
the spatial interaction between the supply and demand is mediated. These improvements are primarily
towards three distinct but cohesive directions: the catchment size, the regional competition, and
distance decay functions. The thread of development, extended from a review article (Tao and Cheng
2016), is summarized below.
The first aspect of improvements considers the variation of the catchment size as a better
approximation of the maximum transferability.” The foundation of the 2SFCA method, rooted in the SI
economic theory, assumes that the benefit derived from the spatial interaction must outweigh the cost
of travel between the supply and demand. Thus, the catchment size in the 2SFCA method represents a
threshold distance, beyond which the interaction cannot be established. In the original 2SFCA model,
the search procedure in the first and the second steps is based on an arbitrarily defined catchment size.
In reality, the catchment size is beyond a simple delineation of the service area and is mostly dependent
on the utilization pattern of the service and the activity space of the consumer (Yang et al. 2006, Wang
2012). On the one hand, comparing to metropolitan areas, empirical evidence revealed that people
living in rural areas are more willing to travel long distances for services, in cases such as health care
utilization (Arcury et al. 2005; Mcgrail and Humphreys 2009) and food procurement (Mceachern and
Warnaby 2006). In this regard, the dynamic 2SFCA (D2SFCA) method was proposed to differentiate the
catchment size in areas with different population densities (Mcgrail and Humphreys 2014). On the other
hand, the service capacity of facilities varies and may be unable to fulfill the need of all the demand
within a predefined distance buffer. To address the issue, two separate modifications were
implemented, referred to as the variable 2SFCA (V2SFCA) method and the nearest-neighbor modified
2SFCA (NN-M2SFCA) method. The V2SFCA method considers that the supply catchment increases until
the capacity is exhausted by the population under coverage; similarly, the second step of the search is
terminated when the maximum supply-to-demand ratio (Rj) within the demand catchment is met (Luo
and Whippo 2012). The NN-M2SFCA method considers that people are more likely to seek services from
their nearest facilities, so the supply and the demand catchments should only include the nearest few
points (Jamtsho et al. 2015). The V2SFCA and the NN-M2SFCA methods were implemented beyond the
health care system, in cases where the capacity of facility largely dictates the service quality and
planning outcomes, such as the emergency shelter planning (Zhu et al. 2018). In addition, a spatial
disaggregation method was proposed to better estimate the catchment size (Bell et al. 2013). This
method uses a smaller geographic unit to measure the accessibility and then aggregates the accessibility
index to a larger geographic unit which contains the smaller units.
The second aspect of improvements considers the regional competition among comparable
service providers, a facet analogous to the concept of “intervening opportunities” in the SI theory
(Haynes and Fotheringham 1984). In reality, consumers are more likely to patronize a facility in their
immediate proximity in lieu of equalizing the visit possibility within an arbitrary distance. Thus, the 3-
step floating catchment area (3SFCA) method was developed by introducing a selection weight that
represents the probability of the demand point patronizing its nearby supply points (Wan et al. 2012b).
The generation of the selection weight in the 3SFCA method, however, is purely based on the relative
spatial proximity. In reality, the preference for service locations could be determined by not only the
proximity to but also the attractiveness of a facility. Thus, the Huff model (one type of SI models) was
incorporated in developing the selection weight in the 3SFCA method (Luo 2014). In a similar vein, a
facility might not be fully utilized because of the lack of functionality or the limited demand in its
proximity. Then, the modified 2SFCA (M2SFCA) method was proposed to consider the suboptimal
configuration of the supply (Delamater 2013). Although the 3SFCA method and its extensions add
another tier of consideration to estimate the facility utilization pattern, they have been criticized for
causing unnecessary complications in the model formation and therefore, posing considerable technical
challenges to the model implementation (Wang 2017).
The third aspect of improvements considers the form of the distance decay that intervenes in
the supply-demand interaction. To better represent the complex form of “transferability,” the distance
decay function f(d) (i.e., f(dij) and f(dkj)) is added to Equations (1) and (2), respectively, representing the
travel friction that contributes to the formation of Rj in the first step and subsequently Ai in the second
step, as shown in Equations (3) and (4), respectively. The distance decay function in the 2SFCA method
primarily takes three different forms: the binary decay, the continuous decay, and the hybrid decay, as
shown in Fig. 1 (adapted from Wang 2012). The binary decay function (Fig. 1a), primarily the original
2SFCA method (Luo and Wang 2003), is a constant within the catchment and is zero beyond. This
dichotomous approach may cause the edge effect, referring to the statistical bias that including or
excluding points near the boundary would significantly affect the outcome (Chen 2017, Van Meter et al.
2010). The continuous decay (Fig. 1b), such as the Gaussian form (Dai 2010) and the kernel density form
(Dai and Wang 2011), is under the premise that the function value is or converges to zero when
approaching the boundary. This form could alleviate the bias caused by the uncertainty of the
catchment size. The hybrid decay (Fig. 1c) is derived from the context that there is a sharp distinction
between urban and rural travel environments, because of the disaggregated settlement and service
patterns in rural regions (Arcury et al. 2005). For this reason, the enhanced 2SFCA (E2SFCA) method was
proposed by introducing a zone-based discrete distance weight Wr that estimates the distance decay in
different sub-zones (Luo and Qi 2009). Moreover, the hybrid decay could take the form of a piecewise
function composed of both binary and continuous decays, such as combining the binary form with a
negative linear form (Schuurman et al. 2010). To date, there has been no consensus about the best
formulation of the distance decay effect due to the lack of observation on the real-world travel and
service utilization patterns (Wang 2012).
 (3)




 (4)
Fig. 1. Three different forms of the distance decay function in the 2SFCA method: a) binary decay, b) continuous
decay, and 3) hybrid decay.
Although the 2SFCA family is originated from the SI models, the effect of the distance decay as
the foundation of the SI theory has not been thoroughly examined. There was a limited scope of work
testing the sensitivity of the 2SFCA method to the distance decay effect; and most of the discussion
revolved around the distance weight Wr (McGrail, 2012) or the distance impedance coefficient β (Luo
and Qi 2009, Luo and Wang 2003, Wan et al. 2012a) by a selected 2SFCA model. To date, no existing
research has scrutinized the full range of the distance decay functions in the 2SFCA family, eventually
posing considerable difficulty to the justification of the state of spatial inequity in various planning
scenarios where the 2SFCA method is applied.
3. Data collection
The research chooses the Supplemental Nutrition Assistance Program (SNAP) authorized food
retailers in the state of Arkansas, United States, as the subject of study. Although the 2SFCA method is
primarily designed for evaluating the accessibility to health care services, the paper chooses food
retailers for the following reasons: (1) comparing to health care centers, food retailers have a larger
quantity, a wider spatial distribution, and fewer spatial gaps, especially in the rural areas. This
consistency in the spatial pattern facilitates systematically testing different 2SFCA models; (2) the United
States Department of Agriculture (USDA) has published the Food Access Research Atlas that includes
distance standards for defining efficient food accessibility (USDA 2017), which can guide the evaluation
of the catchment size.
The SNAP food retailer data in the fiscal year of 2017 was collected from the Center on Budget
and Policy Priorities (CBPP 2017). This dataset includes food retailers (n = 2808) participating in the SNAP
program (formerly known as “food stamps,” a nutrition improvement program for low-income
populations in the United States). The supply weight (Sj) of the store was estimated by the distribution
of SNAP redemptions in dollar value by store type (USDA 2018). The definition of Sj is further articulated
in the supplemental data.
The demand weight (Pk) in the study was derived from the estimated SNAP recipient data (total
households received SNAP in the past 12 months) sourced from the United States Census Bureau (USCB)
2012-2016 American Community Survey 5-year estimates at the block group level as the most refined
unit (USCB 2016). The centroid of the block group was generated for deriving the accessibility index for
the block group. The store data, overlapped with the block groups (n = 2147), was geocoded in Esri
ArcGIS 10.4, as shown in Fig. 2.
Fig. 2. SNAP food stores overlapped with demand density (i.e., SNAP households per square miles) on the block
group scale.
4. Analytic framework
The comparative evaluation of the 2SFCA method is situated in the framework of placed-based
accessibility measures (Kwan 1998). Based on past examples in the 2SFCA applications, we have
proposed an analytic framework that includes six distance decay functions: the rectangular cumulative-
opportunity (CUMR), negative-linear cumulative-opportunity (CUML), inverse-power gravity-type
(POW), exponential gravity-type (EXP), and Gaussian gravity-type (GAUSS), and kernel density (KD)
models, generating a total of twenty-four measures, as shown in Table 1. These functions represent f(dij)
and f(dkj) in Equations (3) and (4). Examples drawn from these six models are illustrated in Fig. 3.
Fig. 3. Mathematical interpretation of different distance decay functions in the 2SFCA method.
Table 1. The analytic framework consisting of twenty-four distance decay functions in the 2SFCA method.

Examples
Cumulative-opportunity,
rectangular (CUMR)

CUMR05

N/A
Bell et al. 2013,
Ngui and
Apparicio 2011,
Luo and Wang
2003
CUMR1

CUMR10

CUMR20

Cumulative-opportunity,
negative linear (CUML)

CUML05
   
N/A
Schuurman,
Bérubé, and
Crooks 2010
CUML1

CUML10
 
CUML20
 
Gravity-type
inverse-power (POW)



POW05

2.0*
Yao, Murray, and
Agadjanian 2013
POW1

2.0*
POW10

2.0
POW20
  
1.5
Gravity-type
Exponential (EXP)
 

EXP05

9.2
Jamtsho, Corner,
and Dewan 2015
EXP1

4.6
EXP10

0.5
EXP20

0.2
Gravity-type Gaussian
(GAUSS)
 

GAUSS05

0.05
Dai 2010,
Wan et al. 2012a
GAUSS1

0.2
GAUSS10

21.7
GAUSS20
  
86.9
Kernel density (KD)
 
KD05

NA
Dai and Wang
2011
KD1

KD10

KD20

*If d0 is too small, the β in POW will be extremely large and will thus deviate from the realistic
interpretation of the travel friction (Guy 1983, Kwan 1998). Thus, β = 2.0 is designated for POW05 and
POW1.
Several notes about Table 1 should be mentioned:
(1) The binary decay in the original 2SFCA method can be regarded as the cumulative-
opportunity model (rectangular form) in the place-based accessibility measure (Kwan 1998), because
the demand/supply points within the catchment are fully counted and those beyond are completely
overlooked (i.e., f(d) = 1, if d d0; f(d) = 0, if d > d0).
(2) The kernel density function as an additional place-based proximity measure is introduced
into the analytic framework. The kernel function includes a bandwidth, which is equivalent to the
concept of the catchment size in the 2SFCA method (Dai and Wang 2011). The kernel density function
can be represented in different forms, such as Exponential, Gaussian, Quartic, Epanechnikov, Polynomial
of Order 5, and Constant. The Epanechnikov form, the shape of which shares a similarity with the
normal distribution (Gibin et al. 2007), is tested in this study.
(3) Although there are different forms of spatial separation (e.g., distance, time, monetary cost)
and health service access usually adopts travel time, this study employs the travel distance in terms of
the shortest path distance (in miles) based on the USDA’s standards for efficient food access using
distance measures. Specifically, USDA defines thresholds of efficient food access as half a mile and one
mile in urban areas and ten miles and twenty miles in rural areas (USDA 2017).
(4) The evaluation only includes the binary decay (CUMR) and the continuous decay (CUML,
POW, EXP, GAUSS, and KD). In cases of the continuous decay, the distance decay function yields a value
of 0 (CUML and KD) or 0.01 (POW, EXP, and GAUSS) at the boundary, where the distance impedance
coefficient β in each model is determined by d0, correspondingly. The choice of 0.01 is based on Wan et
al. (2012a): 0.01 is considered a critical value when the distance decay function converges to 0. The
hybrid decay is not discussed because it is a combination of the binary decay and/or continuous decay.
Exploration of the hybrid decay can be found in McGrail (2012) and Apparicio et al. (2017), where the
fast-step decay, the slow-step decay, and the continuous decay are compared.
5. Results
To implement the proposed analytic framework, we first generated the shortest path O-D cost
matrix between demand points and supply points using the Network Analysis module in ArcGIS 10.4.
Then we calculated the accessibility indices based on the twenty-four 2SFCA models with customized
Python scripts. Lastly, we visualized twenty-four sets of accessibility indices on the block group scale.
The mapping results are included in an online data repository (Chen and Jia 2019).
5.1 Statistical correlations between models
In order to identify the difference in the evaluation outcome between models, we performed
Pearson’s correlation analysis between every two of these twenty-four accessibility measures, as shown
in Table 2. All correlation coefficients (r) are significant at the level of 0.01.
The most remarkable effect is that the correlations between models with the same distance
decay function but different d0 remain significantly low, as can be seen from grey cells in Table 2. On the
other hand, the majority of high correlations (r 0.80) exist between models with the same d0, as can be
seen from the bold numbers in Table 2. The highest correlation (r 0.98) is found between CUML and
KD with the same d0. This consistency is very likely the result of their similarity in the mathematical
formulation, where CUML is a first-degree polynomial and KD is a second-degree polynomial. Other
groups of high correlations (r 0.97) exist between EXP and POW at d0 = 0.5 and 1 and between GAUSS
and EXP at d0 = 10 and 20. In addition, the models are ranked by the average of the correlation
coefficients with other models in descending order, as shown in Table 3. It can be seen from the table
that, POW20 is the highest in the average r (  ), and in contrast, the original model CUMR20 is
relatively low ( ). In general, the average correlations are higher for models at d0 = 10 and 20
than those at d0 = 0.5 and 1.
Table 2. Pearson’s correlation coefficient (r) between the 2SFCA models. Correlation between models with the same distance decay function is in grey
shading; high correlation (r 0.80) is in bold text.
CUMR05
1.00
CUMR1
0.27
1.00
CUMR10
0.14
0.20
1.00
CUMR20
0.11
0.16
0.55
1.00
CUML05
0.80
0.20
0.12
0.01
1.00
CUML1
0.33
0.93
0.20
0.15
0.25
1.00
CUML10
0.18
0.25
0.85
0.46
0.15
0.25
1.00
CUML20
0.18
0.23
0.83
0.80
0.15
0.22
0.78
1.00
POW05
0.95
0.24
0.13
0.11
0.88
0.30
0.17
0.17
1.00
POW1
0.34
0.92
0.20
0.15
0.29
0.97
0.25
0.22
0.35
1.00
POW10
0.35
0.41
0.41
0.23
0.31
0.46
0.55
0.40
0.36
0.50
1.00
POW20
0.43
0.45
0.58
0.40
0.36
0.47
0.72
0.65
0.42
0.49
0.88
1.00
EXP05
0.92
0.24
0.13
0.11
0.93
0.29
0.16
0.17
0.99
0.34
0.35
0.41
EXP1
0.33
0.86
0.19
0.14
0.26
0.96
0.23
0.21
0.31
0.98
0.50
0.47
EXP10
0.23
0.34
0.67
0.37
0.18
0.33
0.91
0.64
0.22
0.33
0.67
0.81
EXP20
0.22
0.29
0.84
0.62
0.18
0.28
0.92
0.93
0.21
0.28
0.53
0.79
GAUSS05
0.71
0.18
0.10
0.09
0.93
0.22
0.13
0.14
0.82
0.26
0.27
0.32
GAUSS1
0.25
0.69
0.15
0.11
0.20
0.85
0.19
0.16
0.23
0.87
0.47
0.41
GAUSS10
0.18
0.26
0.73
0.39
0.14
0.26
0.96
0.68
0.17
0.25
0.58
0.73
GAUSS20
0.19
0.25
0.90
0.65
0.16
0.24
0.90
0.96
0.18
0.24
0.47
0.71
KD05
0.85
0.21
0.12
0.10
1.00
0.27
0.15
0.16
0.92
0.31
0.32
0.38
KD1
0.32
0.95
0.20
0.15
0.24
1.00
0.25
0.22
0.29
0.96
0.45
0.47
KD10
0.16
0.23
0.90
0.48
0.13
0.23
0.99
0.80
0.15
0.23
0.51
0.68
KD20
0.16
0.21
0.78
0.87
0.13
0.19
0.68
0.98
0.15
0.20
0.34
0.57
CUMR05
CUMR1
CUMR10
CUMR20
CUML05
CUML1
CUML10
CUML20
POW05
POW1
POW10
POW20
Table 2. (Continued)
CUMR05
CUMR1
CUMR10
CUMR20
CUML05
CUML1
CUML10
CUML20
POW05
POW1
POW10
POW20
EXP05
1.00
EXP1
0.30
1.00
EXP10
0.21
0.32
1.00
EXP20
0.21
0.26
0.84
1.00
GAUSS05
0.89
0.23
0.16
0.17
1.00
GAUSS1
0.23
0.94
0.27
0.20
0.18
1.00
GAUSS10
0.16
0.24
0.97
0.86
0.13
0.20
1.00
GAUSS20
0.18
0.22
0.76
0.98
0.14
0.18
0.81
1.00
KD05
0.95
0.28
0.19
0.19
0.91
0.21
0.15
0.16
1.00
KD1
0.28
0.94
0.33
0.28
0.21
0.81
0.26
0.24
0.26
1.00
KD10
0.15
0.21
0.86
0.91
0.12
0.17
0.92
0.92
0.14
0.23
1.00
KD20
0.15
0.18
0.55
0.85
0.12
0.14
0.59
0.90
0.14
0.20
0.71
1.00
EXP05
EXP1
EXP10
EXP20
GAUSS05
GAUSS1
GAUSS10
GAUSS20
KD05
KD1
KD10
KD20
Table 3. Model ranking by the average of the correlation coefficients with other models (
).
Rank
Model

1
POW20
0.57
2
EXP20
0.54
3
GAUSS20
0.51
4
EXP10
0.51
5
CUML10
0.50
6
KD10
0.49
7
CUML20
0.49
8
GAUSS10
0.48
9
POW10
0.47
10
POW1
0.46
11
CUMR10
0.46
12
KD20
0.45
13
CUML1
0.44
14
EXP1
0.44
15
KD1
0.44
16
CUMR1
0.42
17
EXP05
0.41
18
POW05
0.41
19
CUMR05
0.40
20
KD05
0.39
21
GAUSS1
0.38
22
CUML05
0.38
23
GAUSS05
0.35
24
CUMR20
0.34
5.2 Accessibility by urban-rural status
The evaluation of place-based accessibility concerns the travel environment where the service is
delivered. This consideration stems from the distinction between urban and rural landscapes. In rural
areas, because health-related service facilities are limited in both quantity and density, rural residents
have a higher tolerance for long trips (Arcury et al. 2005, McGrail et al. 2015). The same observation
applies to food procurement (Mceachern and Warnaby 2006); and for this reason, USDA (2017) has
designated different criteria for efficient food access: urban access is defined under 0.5 miles or 1 mile;
rural access is defined under 10 miles or 20 miles. Similar to a comparative study that evaluated
accessibility indices by population size (McGrail 2012), we evaluated the 2SFCA models by urban-rural
status and drew attention to their regional applicability.
Census block groups and urban-rural areas are two sets of administrative units, and their
delineations are not overlapped. Thus, we first categorized the block groups into urban and rural units
by a reclassification method employed by the USDA (2017): A block group is labeled as urban if its
geographic centroid is within the USCB urbanized areas (with population of 50,000 or more) or urban
clusters (with population between 2,500 and 50,000); all other block groups are considered rural.
We then evaluated the results in urban and rural block groups, respectively. First, we selected
urban block groups and derived the average accessibility index for each urban unit i by the twelve
models with d0 = 0.5 and 1. Similarly, we selected rural block groups and derived the average for each
rural unit i by the other twelve models with d0 = 10 and 20. These two average accessibility indices (
)
served as the reference accessibility for urban and rural areas, respectively. We then employed the Root
Mean Square Error (RMSE) to evaluate the goodness of the fit between
and the modeling results, as
shown in Equation (5) and Table 4.
  
 (5)
Table 4. The evaluation of the 2SFCA modeling results by urban-rural status. The RMSE was derived between the
average urban/rural accessibility (
) and each 2SFCA model applied to urban or rural areas, respectively.
Urban areas
Rural areas
RMSE with urban
RMSE with rural
POW1
0.47
GAUSS10
0.84
CUML05
0.34
EXP10
0.54
GAUSS05
0.33
CUML10
0.49
KD05
0.33
POW10
0.46
EXP05
0.32
KD10
0.46
POW05
0.31
CUMR10
0.41
CUMR05
0.27
POW20
0.16
EXP1
0.20
EXP20
0.11
GAUSS1
0.17
GAUSS20
0.10
CUML1
0.14
CUML20
0.09
KD1
0.14
KD20
0.08
CUMR1
0.12
CUMR20
0.08
In urban areas, the POW1 has the best performance (RMSE = 0.47), followed by the CUML05
(RMSE = 0.34). In rural areas, GAUSS10 (RMSE = 0.84) performs significantly better than other models. In
contrast, CUMR1 and CUMR20 have the least level of correspondence with
in urban and rural areas,
respectively.
5.3 Variability analysis of the catchment size
The catchment size d0 dictates the modeling results to a great extent, in that a high correlation
exists between models with the same d0 (Section 5.1) and that different d0 should be applied to urban
and rural areas (Section 5.2). In existing studies, d0 is selected in a relatively arbitrary manner because
the lack of observation on the service area of the supply or the activity data of the demand. In this
regard, we would like to explore the effect of d0 in the 2SFCA method.
Coefficient of variation (CV), defined as the division of the standard deviation to the mean, is
explored in this study as the variability metric, as shown in Equation (6). The coefficient of variation was
originally a measure of price volatility in economic studies (Shiue 2002). It was also employed to test the
variability of landscape pattern in research on urbanization (Luck and Wu 2002). The spatial pattern
becomes consistent when CV becomes convergent. Thus, we examined if the spatial inequity interpreted
by the 2SFCA method could be relatively consistent as the d0 increases. Specifically, CV was calculated for
each model with a specific distance decay function. We then derived a series of CV by changing the d0 of
the model.



(6)
where xi is the accessibility index for a block group,  is the average accessibility index for all block
groups, and n is the total number of block groups (i.e., n = 2147).
We conducted our experiments based on a binary decay function (CUMR20) and five continuous
decay functions (i.e., CUML20, POW20, EXP20, GAUSS20, and KD20, as illustrated in Fig. 3). Specifically,
the distance decay function remains uniform in each set of analysis, while the inclusion-exclusion criteria
differ by d0. Fig. 4 is the results of Cv yielded by these six 2SFCA models with different d0 (ranging from
0.5 to 20 at an increment of 0.5). As Fig. 4 illustrates, all models have a large degree of variability with a
small d0; when d0 increases to a certain threshold, Cv becomes relatively convergent (i.e., d0 9.5). It is
also observed that POW20 has a different convergence pattern than other functions.
Fig. 4. Coefficients of variation (CV) yielded by six 2SFCA models as the catchment size (d0) increases.
6. Discussion
The comparative analysis of twenty-four 2SFCA models provides useful information for
understanding the intricacy of the model components in the 2SFCA family. Based on the different facets
of statistical analyses, we would like to offer suggestions for the choice of model parameters.
First, the catchment size d0 plays the most significant role in the assessment of accessibility. As
revealed by the correlation analysis in Table 2, there is a relatively high correlation between models with
the same d0, indicating that the accessibility pattern is relatively consistent across different models with
the same d0. This result further reveals that the change of d0 in the 2SFCA model may significantly affect
the evaluation outcome. The variability analysis in Fig. 4 further extends the observation: a small d0
could largely polarize the spatial pattern and introduces a considerable degree of uncertainty; a large d0,
on the contrary, may smooth the spatial pattern but may conceal local clusters of extreme values. This
paper confirms that this spatial smoothing effect (Luo and Wang 2003) exists in all types of 2SFCA
models regardless of the distance decay.
These results necessitate the careful selection of the catchment size in 2SFCA applications.
Specifically, on a small scale (e.g., community level), d0 must be carefully scrutinized because of the high
variability of the accessibility index. On a large scale (e.g., statewide), d0 must be set beyond a certain
threshold to derive a consistent spatial pattern (and this study suggests > 9.5 miles on the state scale). In
most 2SFCA models, the catchment size d0 and the distance impedance coefficient β are mathematically
dependent; a small catchment size d0 corresponds to a large β, signifying a high level of travel friction
that deters the spatial interaction (Luo and Wang 2003). Therefore, when d0 is determined, β must be
adjusted accordingly.
However, the methodological discussion about d0 is only based on the premise that the travel
behaviors are unknown. In reality, the catchment size represents the acceptable distance that people
are willing to travel (Luo and Wang 2003). Not only does the difference in the acceptable travel distance
exist across travel modes but also it differs by individuals. The specification of the catchment size, thus,
relies heavily on the evaluation of individuals’ activity space, which is fundamentally rooted in people’s
social discourses and identities, in aspects of income, car ownership, familiarity with the neighborhood,
and perception of safety (Chen and Kwan 2015). Although it is impossible to generalize these sets of
knowledge in the 2SFCA method as a place-based measurement, we recommend corroborating the
selection of the catchment size with field observations on people’s expected activity space. For example,
McGrail et al. (2015) conducted structured surveys to investigate the maximum time that people are
willing to travel to visit a doctor. Milakis et al. (2015) employed in-depth interviews to examine the
acceptable time for daily commuting. These empirical investigations can be deployed in a local study for
retrieving the realistic catchment size. When such observations are not available, we recommend using
distance thresholds in the literature (Allan 2014), federal definitions of efficient service areas (USDA
2017), and criteria in urban planning guidelines (Chen et al. 2013).
Second, the distance decay function in the 2SFCA method affects the level of accessibility to a
certain degree and is non-negligible under a large d0. One noticeable result is that CUMR20 has the
lowest average correlation with other models (Table 3). Also, CUMR 1 and CUMR 20 have the poorest
model performance in urban and rural areas, respectively (Table 4). Under the context that a high level
of correlation is the ground truth, this finding indicates that the binary decay 2SFCA model with a large
d0 may be relatively ineffective for accessibility evaluation. However, when a distance decay function is
incorporated, the correlation with other models becomes distinctly higher. Specifically, POW20, EXP20,
and GAUSS20 are the top three models in the correlation analysis (Table 3). We therefore recommend
EXP20 and GAUSS20 functions for assessment in areas with a mixed service landscape. We do not
recommend POW20 as it shows a different variability pattern than other models (Fig. 4). Also, we
recommend GAUSS10 for rural areas, as it performs significantly better than other functions (Table 4).
The distance decay function may not be uniform: it could be weighted as different binary decays
in sub-zones (Luo and Qi 2009) or could be a hybrid form combining both the binary and continuous
decays (Schuurman et al. 2010). As the 2SFCA is a special case of the gravity model, the precise
quantification of the distance decay should resort to the long-standing tradition of calibrating the
distance decay parameters in SI models; and a thorough understanding of the interaction data and the
spatial structure becomes a necessity (Fotheringham 1981). Also, the travel between origin and
destination is beyond a simple movement along road networks and is confounded by travel behaviors
and transport systems. Some recent 2SFCA models have attempted to incorporate these real-world
complexities, using trip-chaining (Fransen et al. 2015), public transit (Langford, Fry, and Higgs 2012), and
modal split (Mao and Nekorchuk 2013; Lin et al. 2018). Improvements on the model calibration and
behavioral formulation to demystify the effect of distance decay are still needed in future work.
Third, the last aspect of discussion is model validation. The majority of the 2SFCA model
development has resorted to the comparison with existing models for validation, primarily the
“milestone” models, such as 2SFCA (Luo and Wang 2003), E2SFCA (Luo and Qi 2009), and 3SFCA (Wan et
al. 2012b). Some studies have attempted to validate the 2SFCA results by comparing with existing
accessibility metrics published by federal agencies, such as the USDA Food Access Research Altas (Chen
2017, Chen 2019). The limited scope of validation efforts is very likely due to the lack of generally
accepted classifications on the accessibility index, especially for health-related services in small-area
estimation (McGrail et al. 2015). In the absence of a universal standard for accessibility assessment, we
urge alternative means to evaluate the robustness of the 2SFCA method: testing the sensitivity of the
model to different modeling parameters (Wan et al. 2012a) or different aggregation methods (Apparicio
et al. 2017) to ensure the consistency of the spatial pattern and identify sources of measurement
uncertainties. Only if the methodological validation is undertaken can the evaluative role of the 2SFCA
method for urban planning and equity measures be justified.
7. Conclusions
The paper has employed a place-based accessibility framework to compare the performance of
twenty-four 2SFCA models in a comprehensive manner. This comparison provides the theoretical
support necessary to the choice of model parameters in the 2SFCA’s applications for various urban
planning, service delivery, and spatial equity issues. Two important conclusions are drawn from this
paper. On the one hand, on a small analysis scale, the catchment size is the most critical model
component: variation in the catchment size can introduce a high degree of measurement uncertainties;
under this context, justification and sensitivity analysis of the catchment size become a necessity. On the
other hand, on a large analysis scale, the distance decay function is of elevated importance; using the
binary decay under a large catchment size will likely overestimate the supply-demand interaction and
thus obfuscate the inequity pattern of physical accessibility. Because these facets are scale-dependent,
we urge caution on the selections of analysis scale, aggregation method, and the proximity measure
between supply and demand (see Apparicio et al. 2017 for further discussion). Also, the fundamental
spatial structure of the data can influence the performance of the 2SFCA method to a meaningful
extent; thus, we suggest refining the statistical unit (e.g., Bell et al. 2013; Chen 2019) in future model
implementation.
This exploratory study also has limitations. First, the comparison of 2SFCA models is based on
the topical issue of food accessibility. The conclusions drawn from the study may not apply to other
service types, scales, or regions, because of the heterogeneity of the landscape across different study
areas. Second, the assessment of the accessibility by the 2SFCA method is purely place-based without
accounting for individual travel behaviors or mobilities. How to leverage the supply-demand interaction
articulated by the 2SFCA method for interpreting the level of accessibility actually experienced by
individuals on a daily basis, and reciprocally, using the empirical evidence to corroborate the place-
based accessibility metrics would be an important direction in future work.
Supplemental data. Definition of the supply weight (Sj) in the study.
Acknowledgements
This work was supported by National Key R&D Program of China (Grant No. 2018YFC0809900).
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... Although the modified methods identified areas where there are medical shortage more precisely, they still assumed that all populations in each travel time zone have equal access to health care services (Dai, 2011). In reality, residents are more likely to seek services in closest proximity to where they live rather than having a uniform probability of using services at any distance within the catchment area (Chen and Jia, 2019). To continuously account for the distance decay effect within a catchment area, Dai (2010) combined a Gaussian function with the 2SFCA (known as Gaussian 2SFCA) in order to measure health care accessibility. ...
... To try to identify the population demand more accurately, a variety of spatial units from small areas (e.g., census block groups) to large areas (e.g., census tracts), have been studied by researchers. The primary areal unit of spatial analysis used to measure accessibility is the census tract (e.g., Chen and Jia, 2019;Langford and Higgs, 2006;Luo and Wang, 2003;Wan et al., 2012) because it is the easiest residential area unit from which to obtain population data. In addition, other areal units, such as census block groups (e.g., Mao and Nekorchuk, 2013;Yang et al., 2006), ZIP codes (e.g., Dai, 2010;Delamater et al., 2012), and subdistricts (e.g., Cheng et al., 2016;Tao et al., 2018;Tao and Cheng, 2019), have been used to identify the potential population within that area. ...
... Furthermore, various measures have been adopted to estimate the spatial impedance between population demand and service supply locations, such as the Euclidean distance (i.e., a straight line) (Bell et al., 2012;McLafferty and Grady, 2004;Spencer and Angeles, 2007), the network distance (Chen and Jia, 2019;Dewulf et al., 2013), and the travel time (Dai, 2010;Langford and Higgs, 2006;Luo et al., 2018;Luo and Qi, 2009;Luo and Wang, 2003). The network distance and travel time respectively represent the shortest and most direct paths between two points, which better reflect the actual spatial barrier and provide more accurate results than the Euclidean distance (Apparicio et al., 2008). ...
Article
Health care accessibility is a vital indicator for evaluating areas where there are medical shortages. However, due to the lack of population data with a satisfactory spatial resolution, efforts to accurately measure health care accessibility among older individuals have been hampered to some extent. To address this issue, we attempt to measure accessibility to health care services for older bus passengers in Nanjing, China, using a finer spatial resolution. More specifically, based on one month's worth of bus smart card data, a framework for identifying the home stations (i.e., a passenger's preferred station near their residence) of older passengers is developed to measure the aggregate demand at the bus stop scale. On this basis, a measurement that integrates the Gaussian two-step floating catchment area (2SFCA) and the adjusted 2SFCA methods (referred to as the adjusted Gaussian 2SFCA method) is proposed to measure accessibility to health care services for older people. The results show that: (1) almost all home stations experience inflated demand, especially those located in the suburbs; (2) despite abundant health care resources, home stations in urban districts are rarely identified as high accessibility stations , due to high demand densities among the older population; and (3) more attention should be paid to two types of home stations-those with a medical institution and those with bed shortages, respectively. The first type is predominantly distributed in the periphery of the city, in the suburbs where the travel time required to access the nearest health care service by bus is longer. The second type is mostly located in the outskirts of urban districts and in the central area of one suburb. These findings could help policy makers to implement more appropriate measures to design and reallocate health care resources.
... Accessibility, defined as the physical access to healthcare services, depends on the geographical distribution of healthcare facilities, patients' residential locations, and their transportation resources (e.g., car ownership, access to public transit). Healthcare accessibility has been long evaluated using various spatial measures, including the density of services, proximity to healthcare centers, and more complex spatial interaction models [12]. While vaccination practice is underway, a promising research agenda is to explore the spatial complementarity between vaccine supply and demand from a geographical perspective, such as evaluating if the distribution of vaccination clinics and allocations of doses can accommodate the unequal disease landscape and if vaccination gaps exist in certain high-risk communities (e.g., neighborhoods of color). ...
... While vaccination practice is underway, a promising research agenda is to explore the spatial complementarity between vaccine supply and demand from a geographical perspective, such as evaluating if the distribution of vaccination clinics and allocations of doses can accommodate the unequal disease landscape and if vaccination gaps exist in certain high-risk communities (e.g., neighborhoods of color). This identification can rely on classical accessibility models, such as the two-step floating catchment area (2SFCA) method [12]. Understanding this spatial complementarity helps to identify regional pockets where vaccination coverage is devoid. ...
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Mass COVID-19 vaccination, as the last resort to bring society to a new normal, has been rapidly rolled out in the US. However, because of the lifting of international travel restrictions, amid the many uncertainties induced by the emerging B.1.1.529 variant, it remains unclear about the timeline of reaching herd immunity and when our daily life will return to normalcy. Since access to a vaccine is an important predicate to the achievement of herd immunity, we articulate the vaccine access issue as the degree of fit between patients and the healthcare system in five dimensions: availability, accessibility, accommodation, affordability, and acceptability. These five dimensions can be adopted in existing health practice and policy to elucidate effective strategies for raising COVID-19 vaccination rates and improving vaccine equity in the fight against the new variant.
... However, to our knowledge, there is no agreed-upon indicator of health need for spatial equity research on primary care, either internationally or in New Zealand. While the importance of selecting appropriate geographical units, 8 as well as appropriate accessibility criteria and thresholds 9 in spatial equity research has been established, the impact of using different health needs indicators is, to our knowledge, untested in the New Zealand context. In Canada, 10 2 different sets of indicators highlighted significant differences in need according to the indicator used. ...
... This supports previous work arguing that catchment size and distance decay function have the greatest impact on spatial accessibility results. 9 Therefore, the spatial equity of GP services was assessed using spatial autocorrelation and the Gini coefficient. Twenty-three AUs were identified as belonging to a cluster of high need across 3 or more different indicators. ...
Article
Background Populations with high needs often have poor health care access. Defining need is challenging, with no agreed-upon indicator of health need for primary care based spatial equity research in New Zealand. We examined seven potential indicators and tested for evidence of the Inverse Care Law in the Waikato region. Methods Indicators were identified through a systematic literature review and scored according to ten selection criteria. Indicators were mapped and analysed using GIS. Spearman’s correlations were calculated between indicators, and clusters of high need identified through spatial autocorrelation. The impact of incorporating indicator-based-weightings into an accessibility model was tested using ANOVA and Spearman’s correlations. GP service spatial equity was assessed by comparing clusters of access and need, and quantified through the Gini coefficient. Results While smoking rates met the most selection criteria, ambulatory sensitive hospitalisation (ASH) rates were significantly correlated with all indicators. Health needs were significantly clustered, but incorporating indicator weightings into the spatial accessibility analysis did not impact accessibility scores. A misalignment of access and need, and Gini coefficient of 0.281 suggests that services are not equitably distributed. Conclusions ASH rates seem a robust indicator of health need. However, data access issues may restrict their use. High need clusters vary spatially according to the indicator used. Key messages GIS techniques can identify ‘hot-spots’ of need, but these can be masked in accessibility models. Indicators should be carefully selected according to the research question.
... Healthcare accessibility is commonly defined as the ease of reaching healthcare services and facilities by overcoming certain travel impedances such as travel distance, time, and cost (Hewko et al. 2002;Higgs 2004;Mao and Nekorchuk 2013). Measures of healthcare accessibility include distance-based measures (Brabyn and Skelly 2002;Apparicio et al. 2008), gravity-based measures (Schuurman et al. 2010;Xiong et al. 2016), cumulative opportunity measures (Paez et al. 2010), the kernel density method (Guagliardo et al. 2004), and various methods based on the floating catchment area (FCA) (Luo and Wang 2003;Langford et al. 2016;Shah et al. 2016;Pan et al. 2018;Chen and Jia 2019). Apparicio et al. (2008) used minimum travel time and distance to the nearest hospital based on the road network to estimate the The Canadian Geographer/Le Géographe canadien 2021, 1-15 geographic accessibility of public hospitals in New Zealand. ...
... Essentially, while traditional gravitybased methods take a positive approach, the 2SFCA method takes a normative approach (Páez et al. 2012). Later, the 2SFCA method was improved to dynamically determine the supply and demand catchment sizes, which can help reveal more details in the spatial variation of accessibility compared to results based on catchments of fixed sizes (Luo and Whippo 2012;Chen and Jia 2019). The 2SFCA method now takes into consideration that patients may cross administrative boundaries for healthcare services and allows for nonadherence to administrative boundaries. ...
Article
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Public transit is indispensable for car-free households to access healthcare. Meanwhile, different households have unequal transit-based healthcare accessibility due to different socio-economic factors such as race/ethnicity and car ownership. Few studies have comprehensively explored the inequality in transit-based healthcare accessibility by integrating both racial/ethnic inequity and car ownership. This study fills the gap with an up-to-date analysis of transit-based healthcare accessibility in the Chicago Metropolitan Area at the census tract level. The results show that the percentage of car-free households is positively related to transit-based healthcare accessibility; while a higher percentage of minorities (i.e., Black/Hispanic) is negatively related to transit-based healthcare accessibility. Among all neighbourhoods with higher percentages of higher-than-average car-free households, Hispanic-majority neighbourhoods fare the worst; while White-majority neighbourhoods have much better healthcare accessibility than both Black-majority and Hispanic-majority neighbourhoods. © 2021 The Authors. The Canadian Geographer / Le Géographe canadien published by Wiley Periodicals LLC on behalf of Canadian Association of Geographers / l'Association canadienne des géographes
... Yang et al. (2006) claim that quantifying the healthcare accessibility not only measure the ratio of supply and demand but also should include the interaction between patients and the healthcare provider in term of the spatial framework. Chen & Jia (2019) state that placebased or spatial accessibility has become a new trend for development in urban facilities, one of the prominent approaches is the 2-Step Floating Catchment Analysis (2SFCA). ...
... This method considers the accessibility with distance decay and the interaction between supply and demand in healthcare facilities. Therefore, 2SFCA is a unique form of the gravity model (Chen & Jia, 2019). ...
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The presence of healthcare facilities is quite essential to provide good healthcare services in a particular area, however, the existence of healthcare facilities is not evenly distributed in Cianjur Regency. This condition leads to the disparities of healthcare facilities across the Cianjur Regency. In this paper, we aim to measure and map the spatial disparities of healthcare facilities using a Two-Step Floating Catchment Analysis (2SFCA). This method can calculate the magnitude of spatial accessibility for healthcare facilities by formulating the travel time threshold and the quality of healthcare facilities across the study area. This research shows the result that the spatial accessibility of healthcare facilities in the Cianjur Regency is not evenly distributed across the districts. The spatial accessibility value resulted from 2SFCA is ranging from 0- 3.97. A low value indicates low spatial accessibility, while a higher value shows good accessibility. The majority of districts in the Cianjur Regency have the spatial accessibility value 0-0.5 (86%). Meanwhile, only a few have the higher value; value 0.5-0.99 as much as 6.6%, 0.99-1.49 as 3.3%, and 3.48-3.97 has a percentage of 3.3%. Also, this analysis results in the cluster of good spatial accessibility in healthcare facilities, namely the Pagelaran District and Cipanas District. Interestingly, the downtown of Cianjur Regency has lower spatial accessibility compared to both areas.
... The second step aggregates all the physician-to-population ratios which are within reach of the population's travel cost. Following Luo and Wang's (2003) method, there have been several advancements in the 2SFCA, such as introducing distance decay functions (Dai, 2010) within the catchment areas; and applying varying sizes of catchment areas (Chen and Jia, 2019). ...
Article
Introduction Increasing attention has been paid to accessibility and equity during the last two decades. Yet, despite the proliferation of studies investigating accessibility and equity from the perspective of the younger generation, only modest progress has been made in embedding a temporal perspective and targeting health services for older adults. Currently, the number of people over 60s in London is growing rapidly and is projected to increase to approximately two million by 2035. Aims This research aims to examine walking accessibility to General Practitioners (GPs) for older people in Greater London, with a particular focus from a temporal perspective. Methods Three different datasets were used for this study, namely: GP services data and data on GP Opening Times data from NHS Choices; the London Lower Super Output Area (LSOA) atlas; and the road network derived from OpenStreetMap (OSM). This study uses on the cumulative method to calculate accessibility to GPs – and applies the vertical equity index to measure temporal equity. Results Our results show that opening times have a significant impact on accessibility to health services for older people in London. Overall accessibility peaked at midday when 15.88% of areas have a low degree of accessibility. Additionally, our study classifies local authorities into five groups based on their performance on accessibility and vertical equity measurements. We found several districts with inadequate and unequal accessibility that can be identified as vulnerable areas. Conclusions Gaining insights into the temporal variations in accessibility to GPs represents a key step towards providing optimal services. Our findings can be used to provide an evidence-based reference for transport planners and policymakers to promote age-friendly development and planning.
... In this study, the Gaussian-based two-step floating catchment area (2SFCA) method was used to evaluate the accessibility of urban parks by referring to existing studies (Guo et al., 2019). The 2SFCA method has been extensively employed to compute the park accessibility (Xiao et al., 2019;Lin et al., 2021) and has been proved to achieved better outcomes (Polzin et al., 2014) and capture the complexities of real world (Chen and Jia, 2019). In 2SFCA method, numerous different forms of the distance decay can be used, such as the negative linear function (Schuurman et al., 2010), the Gaussian function (Wan et al., 2012), the exponential function (Jamtsho et al., 2015), the inverse power function (Schuurman et al., 2010), and the kernel density function (Dai and Wang, 2011). ...
Article
COVID-19 has spread worldwide, leading to a significant impact on daily life. Numerous studies have confirmed that people have changed their travel to urban green spaces during the COVID-19 pandemic. However, in China, where COVID-19 has been effectively controlled, how the travel behavior of visitors to urban parks has changed under different risk levels (RLs) of COVID-19 is unclear. Faced with these gaps, we took a highly developed city, Wuhan, as a case study and a questionnaire survey was conducted with 3276 respondents to analyze the changes in park visitors' travel behaviors under different COVID-19 RLs. Using a stated preference (SP) survey method, four RLs were assigned: new cases in other provinces (RL1), Hubei province (RL2), Wuhan (RL3), and in the district of the park (RL4). The results indicated that visitors reduced their willingness to visit urban parks, with 78.39%, 37.97%, and 13.34% of visitors remaining under RL2, RL3, and RL4, respectively. Furthermore, the service radius of urban parks also shrank from 4230 m under no new cases of COVID-19 to approximately 3000 m under RL3. A higher impact was found for visitors using public transport, those with a higher income and higher education, and female visitors. Based on the modified travel behaviors, the Gaussian-based two-step floating catchment area (2SFCA) method was used to evaluate the accessibility and the Gini coefficient was calculated to represent the equality of the urban parks. A higher RL led to lower accessibility and greater inequitable access. The results should help the government guide residents’ travel behaviors after COVID-19.
... 28,29 This driving time threshold is also consistent with previous health care access studies using the 2SFCA method. 14,27,51 Limitations of the 30-min value are that it may not account for providers located further away but within what may be an acceptable driving time for some patients and that it may exclude those in rural areas. 30 Longer driving times can address these issues but limit the ability to detect small but meaningful high-or low-access areas. ...
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Pharmacogenetic (PGx) testing may be particularly beneficial in medically underserved populations by reducing the number of appointments required to optimize drug therapy and increasing the effectiveness of less expensive off-patent drugs. The objective of this study was to identify patient populations with poor health care access and assess prescribing trends for drugs with published PGx testing guidelines. We used electronic health record data from 67,753 University of Florida Health patients, geographic access scores calculated via the 2-step floating catchment area method, and a composite measure of socioeconomic status. Comparing the poorest (Q4) and greatest (Q1) access score quartiles, poor geographic access was significantly associated with fewer prescriber encounters (incidence rate ratio [IRR] 0.88, 95% confidence interval [CI] 0.86-0.91), fewer total unique drugs (IRR 0.92, 95% CI 0.9-0.95), and fewer PGx guideline drugs (IRR 0.94, 95% CI 0.9-0.99). After correcting for number of unique drugs, patients in low-access areas were prescribed a greater proportion of PGx guideline drugs (IRR 1.08, 95% CI 1.04-1.13). We detected significant interactions between Black race and access score. Compared to Q1, Black patients with Q4 access scores were disproportionately affected and had fewer encounters (IRR 0.76, 95% CI 0.7-0.82) and a higher proportion of PGx drugs (IRR 1.26, 95% CI 1.13-1.41), creating further disparity. Overall, these results suggest that improved geographic access to PGx testing may allow prescribers to make more efficient use of limited opportunities to optimize therapy for drugs with PGx testing guidelines.
Article
Equitable access to urban green spaces (UGS) is an important component of social justice and can be quantified using indices such as urban green space accessibility (UGSA). However, the spatiotemporal patterns and inequity of UGSA among cities with different developments during rapid urbanization are unclear, especially lack evidence at a macroscopic national scale during rapid urbanization. Therefore, we evaluated the UGSA in 366 cities of China during 1990–2015 by the Gaussian-based two-step floating catchment area method (Gaussian-based 2SFCA). Then, the inequity pattern of UGSA among cities with different economic developments was analyzed by the concentration curve and concentration index. Finally, the relationship between UGSA and urban spatial expansion was explored quantitatively by the spatial econometric model. The results showed that: (1) The overall UGSA in China declined significantly by nearly 57.23% during 1990–2015. From the regional perspective, the UGSA in the southeastern region was always lower than that in the northwestern region, the Eastern zone presented a downward trend. From the perspective of different sizes cities, the UGSA of the megacities kept decreasing during 1990–2015, while UGSA of the large, medium, and small cities had turned to increase since 2010. (2) During rapid urbanization, the equity of UGSA among the cities gradually improved, while the cities with low economic developments tended to have higher UGSA. (3) Urban spatial expansion led to the decrease of UGSA during 1990–2015, while the impact had spatiotemporal heterogeneity, and UGSA had a positive spatial spillover effect. Our research provides a comparative baseline for the improvement of UGSA from a macroscopic perspective for China's urbanization policy in the future and novel insights into the green justice issue. The results can be compared with the development of UGS in other countries at different urbanization stages to promote UGS design and policy.
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SARS-CoV-2 vaccine distribution is at risk of further propagating the inequities of COVID-19, which in the United States (US) has disproportionately impacted the elderly, people of color, and the medically vulnerable. We identify vaccine deserts - US Census tracts with localized, geographic barriers to vaccine-associated herd immunity - using a comprehensive supply database (VaccineFinder) and an empirically parameterized model of spatial access to essential resources. Incorporating high-resolution COVID-19 burden and time-willing-to-travel for vaccination, we show that early (February - March 2021) vaccine allocation disadvantaged rural and medically vulnerable populations. Data-driven vaccine distribution to vaccine deserts may improve immunization in the hesitant and control SARS-CoV-2.
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Two-step floating catchment area (2SFCA) methods that account for multiple transportation modes provide more realistic accessibility representation than single-mode methods. However, the use of the impedance coefficient in an impedance function (e.g., Gaussian function) introduces uncertainty to 2SFCA results. This paper proposes an enhancement to the multi-modal 2SFCA methods through incorporating the concept of a spatial access ratio (SPAR) for spatial access measurement. SPAR is the ratio of a given place’s access score to the mean of all access scores in the study area. An empirical study on spatial access to primary care physicians (PCPs) in the city of Albuquerque, NM, USA was conducted to evaluate the effectiveness of SPAR in addressing uncertainty introduced by the choice of the impedance coefficient in the classic Gaussian impedance function. We used ESRI StreetMap Premium and General Transit Specification Feed (GTFS) data to calculate the travel time to PCPs by car and bus. We first generated two spatial access scores—using different catchment sizes for car and bus, respectively—for each demanding population location: an accessibility score for car drivers and an accessibility score for bus riders. We then computed three corresponding spatial access ratios of the above scores for each population location. Sensitivity analysis results suggest that the spatial access scores vary significantly when using different impedance coefficients (p < 0.05); while SPAR remains stable (p = 1). Results from this paper suggest that a spatial access ratio can significantly reduce impedance coefficient-related uncertainties in multi-modal 2SFCA methods.
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As an important component of urban disaster prevention and mitigation systems, the balance and equity of emergency shelter distribution can be measured based on spatial accessibility utilizing the two-step floating catchment area (2SFCA) method. However, there are some issues in previous studies on emergency shelter accessibility evaluated by the 2SFCA method: (1) the high discretization of population distribution data and the travel cost being measured base on Euclidean distance; (2) ignoring the difference between shelter and population catchment sizes. To address these issues, we propose an improved 2SFCA method that computes the shelter and population catchments respectively to evaluate the emergency shelter accessibility in Changchun, China. We compare the proposed improved 2SFCA method to the original 2SFCA method. The results indicate that the catchment size and shelter accessibility calculated by the proposed method are more realistic and objective. The improved 2SFCA method is applicable method for evaluating the shelter accessibility and can provide advice for the planning and management of emergency shelters in the future.
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The popular two-step floating catchment area (2SFCA) method has been widely used in the literature to measure spatial accessibility of residents for a service. The 2SFCA method accounts for the ratio between the supply capacity and demand amount of the service as well as the complex spatial interaction between them. This article introduces an inverted two-step floating catchment area (i2SFCA) method that is derived from an extension of the classic Huff model and used to capture the “crowdedness” (scarcity of resource or intensity of competition) for facilities. The method is illustrated and validated by a case study of evaluating hospital inpatient services in Florida. Several possible uses of the measure are also discussed.
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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.
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两步移动搜寻法是公共服务设施空间可达性研究中的重要方法,在国内外公共服务设施布局研究中得到了广泛应用,且发展出了众多扩展形式。但国内研究中对两步移动搜寻法尤其是其扩展形式的应用还较为有限。本文对两步移动搜寻法的主要扩展形式进行系统梳理和总结,将国内外研究中提出的两步移动搜寻法扩展形式归纳为基于引入距离衰减函数的扩展、对搜寻半径的扩展、针对需求或供给竞争的扩展以及基于出行方式的扩展4 类,并分析了各种扩展形式的优缺点、适用情景以及未来可能改进方向。旨在为相关研究的方法选择提供参考,促进两步移动搜寻法及其扩展形式在国内相关领域的应用和发展。 Two- step floating catchment area (2SFCA) method is an important method of research on spatial accessibility to public services, which has been widely applied in studies on the spatial layout of public service facilities. Various extensions of 2SFCA have been developed. However, the application of the 2SFCA method and especially its extensions is still very limited in China. Thus, this study systematically summarizes the major extensions of the 2SFCA method. Such extensions found in existing literature can be classified into four categories. The first category focuses on the distance-decay function, replacing the dichotomous distance-decay form of the original 2SFCA by a multilevel discrete form as in the Enhanced 2SFCA, or continuous forms such as in the gravity-style, Gaussian-style, and Kernel-density-style distance-decay functions. The second category deals with the delimitation of catchment areas. The original 2SFCA form adopts a buffer- ring (BR) method delineating the catchment areas as concentric circles with certain radii of physical distance or travel time. A number of extensions have improved the BR method, including the Variable 2SFCA, the Dynamic 2SFCA, and the Multi Catchment Sizes 2SFCA. Moreover, a novel nearest- neighbor (NN) method for delineating the catchment areas by identifying a finite number of nearest facilities for each demand node has been proposed. The third category strives to improve the accuracy of 2SFCA by accounting for the competition among demand nodes or among supply nodes. The initial extension of this category is the Three-step Floating Catchment Area (3SFCA) method, following which other extensions such as the Modified 2SFCA and the Huff 2SFCA are proposed to make further improvements. The fourth category extends the travel means of the demand side, including the Multi-mode 2SFCA taking into account various potential transportation modes, and the Commuter-based 2SFCA integrating service visits and commuting behavior. The advantages and disadvantages, scenarios appropriate for their application, and potential improvements in the future of these extensions are also discussed. This study can contribute to the choice of method in relevant studies and promote the implementation and development of the 2SFCA method and extensions in relevant research fields in China.
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Geographic information systems (GIS) can be effectively utilized to carry out spatio-temporal analysis of spatial accessibility to primary healthcare services. Spatial accessibility to primary healthcare services is commonly measured using floating catchment area models which are generally defined with three variables; namely, an attractiveness component of the service centre, travel time or distance between the locations of the service centre and the population, and population demand for healthcare services. The nearest-neighbour modified two-step floating catchment area (NN-M2SFCA) model is proposed for computing spatial accessibility indices for the entire country. Accessibility values from 2010 to 2013 for Bhutan were analysed both spatially and temporally by producing accessibility ranking maps, plotting Lorenz curves, and conducting spatial clustering analysis. The spatial accessibility indices of the 205 sub-districts show great disparities in healthcare accessibility in the country. The mean-and median-based classification results indicate that, in 2013, 24 percent of Bhutan's population have poor access to primary healthcare services, 66 percent of the population have medium-level access, and 10 percent have good access.
Article
Accessibility is a fundamental but often neglected concept in transportation analysis and planning. Three complementary views of accessibility have evolved in the literature. The first is the constraints‐oriented approach, best implemented by Hägerstrand's space‐time prisms. The second perspective follows a spatial interaction framework and derives “attraction‐accessibility measures” that compare destinations' attractiveness with the travel costs required. A third approach measures the benefit provided to individuals by the transportation/land‐use system. This paper reconciles the three complementary approaches by deriving space‐time accessibility and benefit measures that are consistent with the rigorous Weibull axiomatic framework for accessibility measures. This research also develops computational procedures for calculating these measures within network structures. This provides realistic accessibility measures that reflect the locations, distances, and travel velocities allowed by an urban transportation network. Since their computational burdens are reasonable, they can be applied at the urban scale using a GIS.
Article
Limited accessibility measurements of park green spaces (PGSs) have given an account of multiple transportations when people enjoy the services. Considering the varying types and functions of PGSs, this study proposed a multi-mode method that is based on the traditional two-step floating catchment area method and relied on PGS classification to estimate spatial disparity between the supply of PGSs services and residents' demands in accessibility. Choosing the urban center of Wuhan as the study area, comparative analysis of space and statistics were conducted in accessibility between the traditional single-mode and the multi-mode methods. After a sensitive analysis of accessibility with varying travel time thresholds, underserved areas in the urban center were identified on a community level based on the PGS groups. Results indicated that the estimated accessibility average and standard deviation values by single-mode method were slimly higher than that of the multi-mode method. The average and standard deviation values of the accessibility among five time thresholds exhibited slight difference and an increasing trend. The further examination analysis of spatial accessibility in 25-min threshold demonstrated that accessibility presented spatial polarization and that most underserved areas distributed in the eastern and southwestern of the urban center in Wuhan. The findings may provide a more realistic estimation and further the knowledge of access to green spaces to help decision makers developing equal and effective planning policies and strategies.
Article
Diversity is a hot topic in geography and planning. Though a few studies have considered diversity in evaluating accessibility, diversity and the love of diversity have not been simultaneously included in one indicator. This study proposes a method to tackle this challenge, incorporating the constant-elasticity-of-substitution function, with a parameter on the preference for diversity (ρ), into the accessibility indicator of the two-step floating catchment area method (2SFCA). This new method (AC) is implemented to measure Beijing's job accessibility at the jiedao scale which is the basic administrative level in Chinese cities, and compared with the 2SFCA method (AG) and the 2SFCA method with an entropy component for job diversity (AE). All three indicators reveal a similar overall spatial pattern of job accessibility. However, measured by rank of accessibility score, there are some differences between them. Most jiedaos' ranks change when the AE method is used instead of the AG method, though most changes are within a small range. In contrast, when the AC method is used with the ρ-value close to 1, meaning that diversity is almost not valued at all, only a few jiedaos' ranks change. Sensitivity analysis indicates that the accessibility scores' absolute values with the AC method are highly sensitive to the ρ-value, but the overall spatial patterns seem not to be. However, with AC, a small decrease in ρ changes the jiedaos' ranks significantly. Overall, the results indicate that the AC method provides better understanding of accessibility for regions with different preferences on diversity of opportunities.
Article
Measures of geographic food access overlook an important source of statistical biases, termed the edge effect. The edge effect refers to the fallacy that events contributing to the spatial pattern of an analysis unit may be outside of that unit; thus merely summarizing events within the unit may lead to distortion of the estimation. Food procurement activities can happen beyond existing administrative boundaries. Delineating food access using unit-based metrics may misrepresent the true space within which food stores are accessible. To overcome this problem, this paper proposes a gravity-based accessibility measure to improve unit-based statistical approaches in food access research. In addition, this method accounts for the spatial interaction between food supply (e.g., food items in stock) and demand (e.g., population) as well as how this interaction is mediated by the spatiotemporal separation (e.g., travel time, modality). The method is applied to the case of Franklin County, OH and has revealed the food access inequity for African Americans by modes of transport, including walking, biking, and driving. The analysis of the correlation between mode-specific food access and socioeconomic status (SES) variables reveals that using a single modality in food access research may not fully capture the travel behavior and its relationship with local food environments. With modifications, the proposed method can help evaluate food access for a target population group, such as Supplemental Nutrition Assistance Program (SNAP) users or selected ethnic minorities who may face acute difficulties in procuring economically affordable and culturally appropriate foods.