3.1. Data and study setting

This study focuses on Pennsylvania, U.S. The state’s geographic diversity introduces substantial variation in healthcare infrastructure and physical accessibility, shaped by natural barriers, population density gradients, and uneven hospital distribution. These spatial disparities intersect with demographic and structural inequities: rural communities face aging populations and mobility constraints, while wealth disparities and environmental health risks disproportionately burden underserved urban and peri-urban neighborhoods. Together, these dynamics make Pennsylvania a compelling case study for analyzing spatial accessibility in a way that reflects broader national trends.

The analysis integrates multiple data sources across three spatial scales: block groups (n = 9,740), counties (n = 67), and hospitals (n = 191). Demographic and socioeconomic data, including population and mean household income (S1 File), were obtained from the U.S. Census Bureau’s American Survey (ACS) 5-Year Estimates for 2018–2022. These data provide the demand-side context for block groups and support county-level aggregation for validation against public health outcomes [27]. Hospital data were acquired from the Commonwealth of Pennsylvania Department of Health (S2 File), including facility names, addresses, and the number of licensed beds as a proxy for service capacity [28]. The dataset encompasses a full spectrum of hospital types, including general acute care, specialty, and federal hospitals. Health outcome data, specifically the percentage of the population reporting poor or fair health, were sourced from the County Health Rankings and Roadmaps (S3 File) [29], allowing for validation of accessibility scores at the county level.

To capture empirical travel behavior, the study utilizes SafeGraph’s (now Advan patterns) anonymized human mobility dataset (S4 File), which includes over 3.5 million hospital visits across Pennsylvania during calendar year 2023 [30]. Visits were aggregated at the census block group level and linked to hospitals using address matching. To accurately link mobility visits to hospitals, we implemented a structured address-matching process. The hospital dataset provided facility names and county information, which we used to query the Google Maps API and obtain standardized addresses. We then used the Placekey platform [31] to find the unique Placekey for each hospital, which separates spatial location (“where”) from descriptive attributes (“what”). To ensure consistent spatial referencing, we extracted the “where” component of each Placekey and compared it to those in the mobility dataset. For each match, we applied fuzzy string matching [32] to hospital names to identify the most likely corresponding record. This approach enabled reliable linkage between hospital locations and mobility data, even when address formats or facility names varied slightly. In total, 191 hospitals were successfully matched and included in the final analysis. Geodesic distances, measured in miles between centroids of origins and destinations, were used to estimate travel distances in all modeling steps.

3.2. Evaluate the bounded catchment areas assumption

A central methodological debate in spatial accessibility modeling concerns whether to constrain patient–provider interactions within predefined catchment areas or to adopt an open, boundary-free approach. Catchment-based models, such as the E2SFCA, impose geographic limits (typically based on travel time or distance), around healthcare providers and their surrounding populations. While this structure simplifies computation and aligns with administrative planning, it risks oversimplifying real-world mobility, as patients may bypass nearby facilities in favor of more specialized or preferred providers farther away. In contrast, open models remove arbitrary thresholds and can capture long-distance travel patterns, but they risk overestimating accessibility by assuming that all populations, including those in remote areas, have equal access to distant hospitals, potentially introducing noise from unrealistic interactions.

This assumption test empirically evaluates whether catchment areas reflect actual human mobility behaviors and seeks to clarify whether spatial accessibility frameworks should impose bounded catchment areas or adopt a more flexible, boundary-free approach grounded in empirical mobility data.

From the supplier (hospital) perspective, we compute:

1. 1. Hospital coverage ratio:

where is the number of unique block groups visiting hospital and is the total number of block groups.

1. 2. Relative visit-weighted distance for hospitals (HosRWD)

where is the number of visits from block group to hospital , is the distance between them and is set of block groups with non-zero visits to hospital .

From the demand (population at block group level) perspective, we compute:

1. 1. Hospital Choice Ratio (PopChoiceRatio)

where is the number of unique hospitals visited by block group and is the total number of hospitals

1. 2. Relative visit-weighted distance for block groups (PopRWD)

where is set of hospitals visited by block group .

Lower values in these metrics (especially the relative distance ratios) indicate more localized, catchment-like behavior, supporting the use of geographic boundaries in accessibility models. Higher values suggest broader or more dispersed travel behavior, consistent with an open, boundary-free model.

3.3. Evaluate the fixed catchment sizes assumption

A key critique of fixed catchment sizes is their failure to account for variations in travel behavior, particularly the tendency of rural populations to travel longer distances for care compared to urban residents. Numerous studies argue that catchment boundaries should reflect observed mobility patterns rather than rely on arbitrary thresholds. To address this limitation, our study introduces a visit-weighted average distance metric for both population origins and hospital destinations.

For each block group , the visit-weighted average distance to utilized hospitals is calculated as:

where is the number of visits from block group to hospital , and is the distance between them.

Similarly for each hospital the average distance from its visitors is:

This metric captures how far individuals actually travel by weighting each distance according to the number of visits to a given hospital.

This approach offers three distinct advantages. First, it enhances behavioral realism by prioritizing observed travel behavior over theoretical assumptions. Second, it incorporates visit intensity, ensuring that hospitals frequently visited by patients exert a proportionally greater influence on the calculated average, thus capturing disparities in healthcare utilization. Third, it accounts for heterogeneity in both provider and population behavior, recognizing that certain hospitals (e.g., specialized or regional centers) may attract patients from greater distances, while some communities routinely travel farther for care.

3.4. Distance decay consideration

The E2SFCA method incorporates distance decay in two sequential steps: 1) hospitals attract fewer visitors as the distance to surrounding neighborhoods increases, and 2) residents reduce their utilization of hospitals as travel distance grows. This dual-decay framework assumes a symmetrical decline in interaction likelihood between supply and demand as spatial separation increases. To evaluate this assumption, we fit both power-law and linear (negative exponential in log-linear form) distance decay models to empirical hospital–neighborhood visit data. The power-law model takes the form while the exponential model is expressed as . For each hospital, we estimated model parameters and assessed fit quality using two criteria: coefficient of determination (R²) and Akaike Information Criterion (AIC). We then classified the best-fitting model using a systematic rule-based approach.

A hospital was classified as following a power-law decay if the power-law model outperformed the linear model on both R² and AIC metrics, and its distance coefficient was significantly negative (β < 0, p < 0.05). Conversely, it was classified as following a linear decay if the linear model showed superior fit by the same criteria and also had a significantly negative coefficient. If neither model met these conditions, i.e., if the better-fitting model lacked a statistically significant negative coefficient, the hospital was categorized as exhibiting no clear decay.

3.5. Evaluate the uniform distance decay assumption

The E2SFCA method assumes a uniform distance decay function applies identically to all suppliers and neighbors, disregarding potential variability in spatial interaction behavior. To test this assumption, we estimate hospital-specific decay parameters using a power-law model (). For each hospital, we fit the model to its visitation data to derive a unique decay parameter (), reflecting how rapidly visitation frequency declines with distance. This approach allows us to quantify heterogeneity through comparing decay rates across suppliers and evaluate uniformity to assess whether a single decay parameter is sufficient. This test is expected to provide new insights on whether the uniform decay assumption aligns with empirical patterns or obscures critical variations in healthcare-seeking behavior, advocating for context-sensitive decay functions in spatial accessibility modeling.

3.6. Development of revised method

Following the independent evaluation of the four core assumptions underlying the E2SFCA method, we develop a revised spatial accessibility framework (detailed in Section 4.5) that integrates the methodological insights derived from the preceding analyses. The proposed framework incorporates key modifications to the original E2SFCA structure. These adjustments are designed to enhance the model’s capacity to represent observed patterns of healthcare-seeking behavior and spatial interaction.

4.1. Bounded catchment areas

This section empirically evaluates a core assumption underpinning spatial accessibility models like the E2SFCA method: that healthcare interactions are geographically constrained and typically occur within bounded catchment areas. While this assumption is often taken for granted in the design of such models, few studies have systematically quantified the extent to which real-world healthcare interactions align with this theoretical framework.

As shown in Table 1, our analysis reveals a pronounced localization in hospital–patient interactions. From the hospital perspective, the average visit-weighted distance to patients (HosRWD) is only 33.53 miles, in contrast to the 130.9-mile average distance across all block groups. On average, each hospital serves only 340 out of 9,740 block groups, or about 3.5% of the possible population base. This indicates a sharply bounded spatial reach for most hospitals. From the block group perspective, the average visit-weighted distance to utilized hospitals (PopRWD) is just 14.7 miles, compared to 130.9 miles if all hospitals were considered equally. Furthermore, each block group engages with only 5 out of 191 hospitals on average (2.6%), reinforcing the conclusion that residents overwhelmingly seek care from a small, nearby subset of providers.

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Table 1. Summary of interaction patterns between hospitals and block groups.

https://doi.org/10.1371/journal.pone.0346009.t001

Although the original E2SFCA method implicitly assumes localized interactions via predefined catchments and distance decay functions, it does not specify whether these assumptions match empirical behavior. The significant differences between actual travel distances and theoretical full-network distances validate that spatial interaction is indeed bounded in practice. This spatial concentration of interactions justifies the continued use of catchment areas in spatial accessibility models.

4.2. Fixed catchment sizes

While the previous section demonstrated that patient-provider interactions are geographically bounded, this section focuses on how those bounds are represented in accessibility models. Specifically, it critiques the common use of fixed-distance catchment sizes, showing that actual travel behavior varies significantly across contexts and cannot be captured by a one-size-fits-all threshold.

To more accurately reflect real-world travel behavior, we employed a visit-weighted average distance metric, which adjusts travel distances based on the frequency of visits to each hospital. Unlike arbitrary static thresholds (e.g., a fixed 30-mile catchment), this metric captures the actual spatial extent of interactions, offering a more behaviorally grounded approach to defining access. The average VWAD for all hospitals was 33.5 miles (median = 28.9 miles), while residents traveled an average of 14.7 miles (median = 11.14 miles) to access care. These findings suggest that a one-size-fits-all catchment, such as 30 miles, may either overestimate or underestimate access depending on local conditions, provider type, and population characteristics.

Fig 1 illustrates the variability of catchment sizes by mapping dynamic catchment areas for hospitals, where the spatial reach expands or contracts based on actual visitation patterns. Fig 2 shows catchment sizes for block groups. The underlying data reveal similarly uneven patterns of hospital utilization. These findings suggest that visit-weighted distance metrics offer a more flexible and empirically grounded alternative to fixed catchment sizes, allowing spatial accessibility models to better reflect how far people actually travel for care.

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Fig 1. Hospitals’ dynamic catchment area.

https://doi.org/10.1371/journal.pone.0346009.g001

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Fig 2. Block groups’ dynamic catchment sizes.

https://doi.org/10.1371/journal.pone.0346009.g002

4.3. Distance decay consideration

The analysis of distance decay is critical for understanding how spatial friction influences healthcare interactions. To capture this empirically, we evaluated the visitation patterns for all hospitals in the study area. Using a rigorous model selection process governed by the AIC and the coefficient of determination (R²), we assessed whether hospital-specific distance decay behavior was best described by a power-law or an exponential function. A threshold of ΔAIC> 2 was utilized to indicate a meaningful statistical improvement of one model over the other.

The calibration results reveal significant heterogeneity in healthcare-seeking behavior across the modeled hospitals. As summarized in Table 2, the power-law model provided a superior fit for the majority of facilities (hospitals, 68.3%). This indicates a heavy penalty for short distances with a long tail, representing steep drop-offs in patient volume beyond the immediate neighborhoods surrounding the hospital. Conversely, the exponential function was the optimal fit for 31.7% of hospitals, representing a smoother, more gradual decline in visitation over distance. Across all facilities, the empirical fitting yielded an average R² of 0.331 and a median parameter of –0.26 (S5 Table).

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Table 2. Summary of distance decay model calibration.

https://doi.org/10.1371/journal.pone.0346009.t002

Fig 3 illustrates these two distinct distance decay patterns derived from the mobility data. Fig 3a shows a hospital catchment best characterized by a power-law decay, where visitation drops sharply near the facility. Fig 3b depicts a hospital conforming to an exponential decay, where patient origins are dispersed more gradually and evenly across the region.

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Fig 3. Two samples of the distribution of visits over distance comparing the fitted power-law and exponential models.

a) Example of Power-Law distance decay. b) Example of Exponential distance decay.

https://doi.org/10.1371/journal.pone.0346009.g003

These findings highlight a critical limitation in traditional E2SFCA applications, which almost universally assume a static, uniform distance decay function (e.g., a single value or a single functional form applied equally to all facilities). The empirical mobility data proves that hospital catchments are highly heterogeneous. By dynamically assigning the statistically optimal functional form (power-law or exponential) and a unique parameter to each specific hospital, our mobility-driven framework captures the true complexity of real-world healthcare access far better than static global assumptions.

4.4. Uniform distance decay

To evaluate the assumption of uniform distance decay across hospitals, we modeled visitation patterns using a power-law function: , where α represents baseline visitation (i.e., interaction at short distances), and denotes the decay rate. Each hospital was fitted with its own set of parameters, allowing us to capture provider-specific distance decay behavior.

The median decay rate () across hospitals was –0.26, indicating a relatively gradual decline in visitation with increasing distance for most providers. The mean , at –0.24, closely aligned with the median, suggesting a relatively symmetric distribution with limited influence from outliers. This pattern points to greater consistency in distance decay behavior across providers than previously observed. In contrast, the baseline visitation parameter () exhibited more variation. The median was 38.8, reflecting strong local interaction for a typical hospital, while the mean α was higher at 56.1, indicating a moderate right-skew driven by hospitals experiencing unusually high visitation from nearby populations. This wide variability in both decay and baseline interaction reveals the limitations of using a uniform distance decay function across all providers. These findings support the need for hospital-specific decay parameters to more accurately reflect the diverse spatial behaviors in healthcare utilization.

4.5. Revised E2SFCA method

Building on the empirical evaluation of the four foundational assumptions of the E2SFCA method, we developed and implemented a revised spatial accessibility model that reflects observed patterns in healthcare-seeking behavior. The two steps of the revised method are:

Step 1: Estimate the distance-decay-adjusted supply-to-demand ratio for provider using an empirically fitted power-law decay function.

where the index uniquely represents a specific healthcare supply location (an individual hospital), and the index represents a specific population demand location (a Census Block Group). Accordingly, represents the supply capacity at hospital , is the population size at location , and is the network distance between the population at and the hospital at . Furthermore, and are the hospital-specific distance decay parameters calibrated from mobility data, and is the dynamic catchment area specifically serving hospital , defined by a radius equal to its Visit-Weighted Average Distance .

Step 2: For each population location , define its dynamic catchment , based on its . Then, aggregate all accessible providers within this radius to compute the accessibility score.

This section provides a reproducible description of the proposed mobility-driven revision of the E2SFCA method. Building on the empirical tests of the four E2SFCA assumptions, the revised method (i) defines dynamic catchment radii using visit-weighted average travel distance derived from observed visits, and (ii) applies hospital-specific distance-decay functions estimated from visitation flows. The complete computational workflow of the revised E2SFCA method is detailed in Algorithm 1.

Algorithm 1: Mobility-Driven Revised E2SFCA

Input

B: Set of all population locations (block groups), i ∈ B

H: Set of all healthcare providers (hospitals), j ∈ H

P_i: Population at location i

S_j: Supply capacity at provider j

d_ij: Geographic distance between location i and provider j

w_ij: Observed visit count from location i to provider j (mobility data)

Output

A_i: Spatial accessibility score for each population location i

Phase 1: Empirical Parameter Estimation

For each provider j ∈ H do:

V_j ← {i ∈ B | w_ij > 0}

VWAD_j = (Σ (w_ij· d_ij)) / (Σ w_ij)

Fit power-law decay model:

w_ij = α_j· d_ij^−β_j

Extract hospital-specific parameters: α_j, β_j

For each location i ∈ B do:

V_i ← {j ∈ H | w_ij > 0}

VWAD_i = (Σ (w_ij· d_ij)) / (Σ w_ij)

Phase 2: Step 1 – Compute Provider-to-Population Ratio

For each provider j ∈ H do:

C_j ← {i ∈ B | d_ij ≤ VWAD_j}

WeightedDemand_j = 0

For each location i ∈ C_j do:

W_ij = α_j· d_ij^−β_j

WeightedDemand_j = WeightedDemand_j + (P_i· W_ij)

R_j = S_j / WeightedDemand_j

Phase 3: Step 2 – Compute Spatial Accessibility

For each location i ∈ B do:

C_i ← {j ∈ H | d_ij ≤ VWAD_i}

A_i = 0

For each provider j ∈ C_i do:

w_ij = α_j· d_ij^−β_j

A_i = A_i + R_j

Return A = {A_i | i ∈ B}

4.6. Accessibility maps

The study concludes with a comparative analysis of healthcare accessibility in Pennsylvania, contrasting the original E2SFCA method with default parameters from Drake et al. (2021) with the revised E2SFCA framework empirically calibrated using human mobility data [7]. To ensure comparability, the accessibility scores from both approaches were normalized using a min–max scaler, rescaling values between 0 and 1. Fig 4 presents accessibility scores derived from the original E2SFCA method. As expected, these scores exhibit a distance decay dominated pattern: neighborhoods located near hospitals receive uniformly high accessibility ratings, while access steadily declines with increasing distance.

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Fig 4. Accessibility map based on E2SFCA and default parameters.

https://doi.org/10.1371/journal.pone.0346009.g004

This pattern reflects the underlying assumptions of fixed catchment sizes and uniform distance decay. In contrast, Fig 5 maps accessibility using the revised E2SFCA model, which incorporates dynamic catchment areas, hospital-specific distance decay rates, and empirically derived attractiveness indices. This approach produces a more nuanced accessibility landscape, revealing disparities that are obscured by the traditional model, such as hospitals with high regional pull or underserved areas near densely located but overburdened providers. By visualizing and comparing both approaches, this analysis demonstrates how empirically grounded refinements can improve the realism and equity of spatial accessibility assessments.

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Fig 5. Accessibility map based on revised E2SFCA.

https://doi.org/10.1371/journal.pone.0346009.g005

Beyond the Pennsylvania case study, this scalable mobility-driven framework empowers policymakers and planners to identify true behavioral healthcare deserts for equitable resource allocation and to optimize dynamic emergency response routing during patient surges. Furthermore, this approach can be adapted to national contexts and other critical infrastructure, ensuring that future spatial planning aligns with actual human behavior rather than rigid geographic assumptions.

5.1. Correlation with health outcomes

The first validation compares accessibility scores from both the traditional and revised E2SFCA models to public health outcomes—specifically, the percentage of the population reporting poor or fair health across Pennsylvania counties. Because the health outcome data is available at the county level, we aggregated our block-group-level accessibility scores to match this spatial resolution. Fig 6 shows the association of accessibility models with county-level health status.

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Fig 6. Comparative correlation of traditional and revised accessibility models with health outcomes.

a) The E2SFCA with default values. b) Revised E2SFCA.

https://doi.org/10.1371/journal.pone.0346009.g006

As shown in Fig 6a, the traditional E2SFCA method exhibits a weak positive correlation (r = 0.17, p = 0.172) with poor or fair health outcomes, suggesting higher accessibility is paradoxically associated with worse health, which contradicts real-world expectations. In contrast, the revised method (Fig 6b) shows a slightly negative correlation (r = –0.12, p = 0.35), which aligns more plausibly with the expectation that improved accessibility is associated with better health. While neither result is statistically significant, an outcome expected given the many confounding factors influencing health, the revised model demonstrates a directionally more realistic relationship.

5.2. Correlation with household income

The second validation compares accessibility scores to median household income, under the premise that wealthier populations typically enjoy better spatial access to healthcare [33,34]. Fig 7 represents comparative relationship between accessibility models and household income.

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Fig 7. Accessibility and income: validating spatial models against socioeconomic indicators.

a) The E2SFCA with default values. b) Revised E2SFCA.

https://doi.org/10.1371/journal.pone.0346009.g007

As shown in Fig 7a, the traditional E2SFCA method yields a weak, non-significant correlation (r = 0.14, p = 0.27). However, the revised model (Fig 7b) shows a stronger and significant correlation (r = 0.31, p = 0.011), indicating that the revised scores better reflect socioeconomic patterns known to affect access.

5.3. Multivariate Validation and Spatial Autocorrelation

While the bivariate relationships demonstrate directional improvements, evaluating spatial models against complex health outcomes requires controlling for socioeconomic confounders. To provide a more robust validation, we conducted a multivariate Ordinary Least Squares (OLS) regression predicting ‘poor or fair health’ using the revised accessibility scores while controlling for household income.

The multivariate regression (R-squared = 0.053) revealed that while income had the expected negative association with poor health (p = 0.065), the spatial accessibility score was not a statistically significant independent predictor (p = 0.494). Hospitals are heavily agglomerated in dense urban cores, yielding high spatial accessibility scores, yet these same environments frequently suffer from systemic, localized social determinants of health (e.g., extreme poverty, poor environmental quality) that override physical proximity to care. Furthermore, comparing highly granular block-group spatial access against aggregated county-level health outcomes likely introduces an ecological fallacy, washing out neighborhood-level effects. Finally, we assessed the distributional realism of the models using the Gini index and Global Moran’s I (Table 3).

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Table 3. Quantitative comparison of spatial inequality and clustering.

https://doi.org/10.1371/journal.pone.0346009.t003

The traditional model produced a highly smoothed accessibility landscape with low spatial inequality (Gini = 0.318) and extreme artificial clustering (Moran's I = 0.919). In contrast, the revised mobility-driven model unmasks significant disparities in actual healthcare accessibility (Gini = 0.630). While accessibility remains spatially clustered (Moran's I = 0.426), the clustering is far less artificial. These robust metrics confirm that the revised framework captures a much more granular and realistic landscape of healthcare access.

To further contextualize these validation results and explore the underlying mechanisms driving the massive variations observed in hospital catchment areas, we propose three distinct hypotheses for future research. First, facility specialization likely dictates catchment size: highly specialized facilities will exhibit expansive, exponential catchments, whereas general community hospitals will exhibit sharply bounded, power-law catchments. Second, the urban-rural spatial divide heavily influences mobility: rural hospitals will demonstrate geographically larger catchment areas due to the scarcity of alternative providers, whereas urban catchments will be smaller but more densely concentrated due to high facility competition. Finally, socioeconomic mobility barriers constrict travel behavior: catchment areas in low-income neighborhoods will exhibit steeper decay rates (restricted bounds) due to transportation inequities and limited vehicle access, compared to affluent areas where patients possess the resources to bypass local facilities for preferred care networks. Testing these hypotheses in future studies will be critical for unpacking the exact behavioral and structural determinants of spatial access.

6.1. Conceptual limitations

First, while the revised E2SFCA model improves upon traditional assumptions by leveraging empirical mobility patterns, it does not fully address several non-spatial factors that critically shape healthcare access. These include financial and insurance-related barriers, linguistic or cultural compatibility, healthcare needs varying by population subgroups, and system-level constraints such as provider capacity, appointment availability, or wait times. Moreover, like most spatial models, the analysis assumes that individuals have perfect knowledge of all providers within their catchment area and base decisions primarily on distance, which oversimplifies the complexity of healthcare-seeking behavior. Although observed mobility patterns implicitly capture some aspects of patient preference, such as the tendency to bypass nearby facilities, this approach does not fully account for the nuanced reasons behind such choices. Modeling these dimensions would require detailed behavioral, demographic, and institutional data beyond the scope of this study.

Additionally, the model is cross-sectional in nature and does not account for temporal dynamics in healthcare accessibility. While the mobility data offers some time granularity, the supply-side data such as hospital capacity is static. This limits the ability to assess how accessibility changes in response to factors like seasonal demand surges, temporary hospital closures, or staffing fluctuations.

6.2. Data limitations

The SafeGraph (Advan) mobility data, while providing unprecedented spatial granularity, has inherent limitations regarding sample representativeness. It represents an anonymized panel of mobile devices covering approximately 7.5% of the population [10]. As demonstrated in recent comprehensive quantitative bias analyses, which evaluated distributional differences between SafeGraph mobility samples and ACS Census populations, the sampling is spatially and demographically uneven. Specifically, device location data systematically under-samples older adults (the elderly), low-income, low-education, and Hispanic populations [10]. Because these vulnerable demographic groups often face the highest barriers to healthcare access, their underrepresentation introduces potential biases and may lead to an overestimation of true accessibility in disadvantaged neighborhoods.

Furthermore, privacy protections and data aggregation result in the loss of granularity: routes are not traceable, and data are reported as aggregated flows or counts rather than individual-level trips. These limitations reduce the ability to infer exact travel paths, calculate precise travel times, or identify multimodal transport use (e.g., public transit versus driving), which is especially relevant for evaluating spatial access among underserved, transit-dependent populations.