Table 1 Methodological developments in the 2SFCA family

Distance Decay within Catchment Area

Introduction of distance decay within the catchment area

First study:

• E2SFCA Luo & Qi, 2009 [12]

Development within category:

• McGrail & Humphreys, 2009 [72]

• Dai, 2010 [67]

• Schuurman et al., 2010 [32]

• Plachkinova et al., 2018 [73]

• Jin et al., 2019 [33]

• Tao et al., 2020 [34]

\({R}_{j}=\frac{{S}_{j}}{{\sum }_{r}{\sum }_{i \in \{{d}_{ij}\le {d}_{r}\}}{P}_{i}*{W}_{r}}\)

\({A}_{i}^{*}= {\sum }_{r}{\sum }_{j \in \{{d}_{ij}\le {d}_{r}\}}{R}_{j}*{W}_{r}\)

Approximating Reality

Moderate increase of modeling complexity

No additional data required

Variable Catchment Area Sizes

Variable catchment size definition

First study:

• V2SFCA Luo & Whippo, 2012 [13]

Development within category:

• McGrail & Humphreys, 2014 [35]

• Jamtsho et al., 2015 [68]

• Ni et al., 2015 [74]

• Kim et al., 2018 [36]

• Tao et al., 2018 [37]

• Bozorgi et al., 2021 [75]

\({R}_{j}=\frac{{S}_{j}}{{\sum }_{i \in \{{d}_{ij}\le {d}_{x}({P}_{i})\}}{P}_{i}*f({d}_{ij},\beta )}\)

\({A}_{i}^{*}={\sum }_{j \in \{{d}_{ij}\le {d}_{x}({R}_{j})\}}{R}_{j}*f({d}_{ij},\beta )\)

Approximating Reality

Slight increase of modeling complexity

May require additional data

Outcome Unit Modification

Outcome unit modification to relative terms

First study:

• SPAR Wan, Zhan et al., 2012 [38]

Development within category:

–

\({R}_{j}=\frac{{S}_{j}}{{\sum }_{i \in \{{d}_{ij}\le {d}_{0}\}}{P}_{i}*f({d}_{ij},\beta )}\)

\({A}_{i}^{*}= {\sum }_{j \in \{{d}_{ij}\le {d}_{0}\}}{R}_{j}*f({d}_{ij},\beta )\)

\({A}_{i}^{SPAR}= \frac{{A}_{i}^{*}}{{A}_{\varnothing }}\)

Correcting Methodology

No increase of modeling complexity

No additional data required

Provider Competition

Correcting demand overestimation by introducing a provider competition-based selection weight

First study:

• 3SFCA Wan, Zou et al., 2012 [39]

Development within category:

• Luo, 2014 [40]

• Tang et al., 2017 [43]

• Paez et al., 2019 [44]

• Matthews et al., 2020 [76]

• Jang, 2021 [41]

• Shen et al., 2021 [42]

\({Prob}_{ij}= \frac{f\left({d}_{ij}, \beta \right)}{{\sum }_{i \in \{{d}_{ij}\le {d}_{0}\}}f\left({d}_{ij}, \beta \right)}\)

\({R}_{j}=\frac{{S}_{j}}{{\sum }_{i \in \{{d}_{ij}\le {d}_{0}\}}{P}_{i}*f\left({d}_{ij},\beta \right)*{Prob}_{ij}}\)

\({A}_{i}^{*}= {\sum }_{j \in \{{d}_{ij}\le {d}_{0}\}}{R}_{j}*f({d}_{ij},\beta )\)

Correcting Methodology

High increase of modeling complexity

No additional data required

Local & Global Distance Decay

Correcting for sub-optimally configured healthcare system by modeling local and global distance decay

First study:

• M2SFCA Delamater, 2013 [45]

Development within category:

• Bauer & Groneberg, 2016 [46]

\({R}_{j}=\frac{{S}_{j}*f({d}_{ij},\beta )}{{\sum }_{i \in \{{d}_{ij}\le {d}_{0}\}}{P}_{i}*f({d}_{ij},\beta )}\)

\({A}_{i}^{*}= {\sum }_{j \in \{{d}_{ij}\le {d}_{0}\}}{R}_{j}*f({d}_{ij},\beta )\)

Correcting Methodology

Slight increase of modeling complexity

No additional data required

Subgroup-Specific Access

Subgroup-specific access measure for selective population-provider pairing

First study:

• Subgroup-specific 2SFCA Wang, 2007 [47]

Development within category:

• Xiao et al., 2021 [48]

• Yang et al., 2021 [69]

• Shao & Luo, 2022 [49]

\({R}_{j}=\frac{{S}_{j}}{{\sum }_{i \in \{{d}_{ij}\le {d}_{0}\}}{P}_{i}*f({d}_{ij},\beta )}\)

\({A}_{i}^{*}= {\sum }_{j \in \{{d}_{ij}\le {d}_{0}\}}{R}_{j}*f({d}_{ij},\beta )\)

\({AG}_{i}^{*}={A}_{i}^{*}\)

\(*\frac{{\sum }_{j \in \{{d}_{ij}\le {d}_{0}\}}{SG}_{j}*f({d}_{ij},\beta )}{{\sum }_{j \in \{{d}_{ij}\le {d}_{0}\}}{S}_{j}*f({d}_{ij},\beta )}/\frac{{\sum }_{i \in \{{d}_{ij}\le {d}_{0}\}}{PG}_{i}*f({d}_{ij},\beta )}{{\sum }_{i \in \{{d}_{ij}\le {d}_{0}\}}{P}_{i}*f({d}_{ij},\beta )}\)

Fitting Context

High increase of modeling complexity

Additional data on subgroup-specific provider and population shares required

Multiple Transportation Modes

Multiple transportation modes

First study:

• MM-2SFCA Mao & Nekorchuk, 2013 [14]

Development within category:

• Polzin et al., 2014 [77]

• Langford et al., 2016 [50]

• Ni et al., 2019 [78]

• Tao & Cheng, 2019 [79]

• Zhou et al., 2020 [51]

• Xing & Ng, 2022 [80]

\({R}_{j}=\frac{{S}_{j}}{{\sum }_{n}{\sum }_{\begin{array}{c}i \in \{{d}_{ij\left({M}_{n}\right)}\\ \le {d}_{0{(M}_{n})}\}\end{array}}{P}_{i({M}_{n})}*f({d}_{ij},\beta )}\)

\({A}_{i}^{*}={\sum }_{n}\frac{{P}_{i\left({M}_{n}\right)}}{{P}_{i}}{\sum }_{\begin{array}{c}j \in \{{d}_{ij\left({M}_{n}\right)}\\ \le {d}_{0{(M}_{n})}\}\end{array}}{R}_{j}*f({d}_{ij},\beta )\)

Approximating Reality

High increase of modeling complexity

Additional data on mode-specific distances and mode-specific user shares required

Time-Dependent Access

Dynamic time-varying access measure

First study:

• Spatio-Temporal 2SFCA

• Ma et al., 2018 [52]

Development within category:

• Song et al. 2018 [53]

• Paul & Edwards, 2019 [55]

• Xia et al., 2019 [54]

\({R}_{jt}=\frac{{S}_{j}}{{\sum }_{i \in \{{d}_{ij({T}_{t})}\le {d}_{0}\}}{P}_{i}*f({d}_{ij({T}_{t})},\beta )}\)

\({A}_{it}^{*}= {\sum }_{j \in \{{d}_{ij({T}_{t})}\le {d}_{0}\}}{R}_{j}*f({d}_{ij({T}_{t})},\beta )\)

Approximating Reality

No increase of modeling complexity

Additional data on time-varying travel time, provider supply or population size required