Spatial inequality in school physical education resources in Shaanxi, China (2021–2024): patterns, determinants, and policy implications

Abstract

Equity in school physical education (PE) resources is widely emphasized, yet county-level evidence rarely distinguishes demand-driven differences from true under-provision or examines how human, material, and financial inputs jointly shape spatial inequality. We develop an entropy-weighted, demand-adjusted PE resource adequacy index for 107 counties in Shaanxi, China (2021–2024) and combine a student-density-based supply–demand typology with spatial diagnostics (standard deviational ellipse; Global and Local Moran’s I), Dagum Gini decomposition, and GeoDetector. Demand-adjusted adequacy shows a persistent core-periphery gradient: in 2024, 75 of 107 counties fall into the Very Low and Low tiers, and extensive Low-Low regimes persist in peripheral areas. Overall inequality peaks in 2022 (Gini = 0.472) and then declines to 0.424 in 2024, but remains structured mainly across prefecture-level cities (≈ 50% of total inequality), with within-city variation small (≈ 9%). Determinants shift over time, and nonlinear interaction enhancement indicates that finance, staffing, and infrastructure operate as complementary inputs rather than independent levers. These findings support jurisdiction-targeted and cluster-oriented intervention packages to reduce demand-adjusted shortfalls.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-42848-7.

Introduction

Educational equity has become a central goal of education policy worldwide, and equitable access to educational resources is widely viewed as essential for social inclusion and human capital development^1,2. In China, balanced development in basic public education is an explicit policy priority aimed at narrowing regional gaps and improving equality of opportunity³. However, equity is ultimately realized through measurable inputs at local administrative levels, where allocation decisions shape students’ day-to-day learning conditions—and where equity assessments can be sensitive to both spatial scale and how demand is accounted for.

Importantly, macro-level indicators of “balanced” education development typically summarize aggregate inputs (e.g., overall per-student spending or general teacher staffing) and implicitly treat resources as fungible across subjects. From an education production function perspective, curriculum domains draw on different combinations of labor and capital, with varying degrees of substitutability⁴. Physical education (PE) is particularly dependent on non-fungible and spatially fixed inputs—specialist teachers, dedicated facilities/space, and equipment—so equalization in aggregate education resources does not necessarily translate into equitable opportunities for quality PE^2,5,6. Moreover, key PE inputs are often indivisible and constrained by space and safety standards, meaning that generic increases in education spending or staffing may not relieve PE-specific bottlenecks. Because PE simultaneously serves educational and public-health goals, inequities in PE inputs can translate into unequal opportunities for health-enhancing physical activity, strengthening the case for PE-specific equity diagnostics^7,8.

Against this backdrop, resources devoted to school physical education (PE) warrant distinct analytical treatment. PE is closely linked to students’ physical literacy, health, and lifelong engagement in physical activity—outcomes increasingly emphasized in global education and health agendas^2,8,9. Quality Physical Education (QPE) guidelines frame equitable PE provision as an input-enabled condition, emphasizing (i) a sufficient and qualified teacher workforce, (ii) adequate facilities and equipment, and (iii) sustained and protected investment^2,5,10. Yet whether PE inputs are adequate relative to local demand, and how such adequacy is distributed at the county level, remains insufficiently documented—and analyses based on coarse units (e.g., provinces or prefecture-level cities) or single indicators may mask localized inadequacy¹¹.

Accordingly, we ask whether spatial hierarchies and clusters in PE inputs persist when provision is evaluated as demand-adjusted adequacy rather than raw volume. Because totals of teachers, facilities, and funding mechanically covary with student population size, equity claims based on volume alone risk conflating scale with adequacy. We therefore operationalize PE resource equity as a demand-adjusted adequacy construct. In addition, PE inputs are complementary and tend to co-locate—specialist teachers, usable facilities/space, and financial support often rise or fall together—so disparities can reflect bundled constraints rather than isolated shortages. Such bundling can be reinforced by interacting mechanisms, including cumulative advantage in core areas, spatial dependence across neighboring counties, and administrative-scale structuring^12–16. A design that combines demand-adjusted measurement with spatial diagnostics, multi-scale inequality decomposition, and interaction testing can therefore move beyond descriptive mapping and assess mechanism-consistent evidence on persistent inequality^12–16.

We examine these issues in Shaanxi Province, an informative case in northwestern China characterized by pronounced within-province heterogeneity: a Xi’an-centered metropolitan core alongside less-developed peripheral areas in the north and south. This “provincial-capital growth pole + peripheral counties” configuration is not unique to Shaanxi; within-province core-periphery gradients in economic capacity and public-service provision are widely documented across China’s inland provinces under uneven development^17,18. Shaanxi also operates under the standard prefecture-level city–county administrative hierarchy used nationwide; thus, the county-level allocation dynamics, scale effects, and spatial dependence mechanisms examined here are directly comparable to those in other provinces with similar governance structures. In addition, as part of China’s northwest macro-region, Shaanxi shares broadly similar development constraints with several neighboring provinces, where provincial-capital primacy coexists with resource-constrained peripheral counties.

This internal gradient, under a shared provincial policy and administrative setting, provides a useful context to assess how demand pressure and provisioning capacity co-evolve across counties. We analyze all 107 counties in the province over 2021–2024, enabling structural comparison within a common institutional setting. The purpose of selecting Shaanxi is to test the proposed demand-adjusted and mechanism-oriented framework under a shared institutional setting with pronounced within-province heterogeneity, rather than to claim novelty from the case selection itself. Accordingly, while empirical magnitudes are context-specific, our intended inference is analytic generalization: the demand-adjusted adequacy framework and spatial diagnostics are designed to be transferable to other regions characterized by similar core-periphery developmental structures. In particular, transferability is most plausible for settings that exhibit (i) capital-city primacy in fiscal capacity and specialist-teacher concentration, (ii) substantial county-level heterogeneity in public-service capacity, and (iii) adjacency-based spatial dependence across neighboring counties. National policy has also emphasized strengthening school PE and improving basic conditions for provision, making sub-provincial adequacy diagnostics particularly policy-relevant¹⁹.

Methodologically, we develop a multidimensional demand-adjusted adequacy index aligned with QPE input foundations (human, material, and financial inputs) using administrative statistics, and we evaluate provision primarily with per-student measures and ratios. We then integrate spatial diagnostics, multi-scale inequality decomposition, and determinant/interaction testing, and we assess supply–demand matching using student density as a proxy for demand intensity^12–16.

This study is motivated by four gaps in the literature. First, PE inputs are specialized and cannot be inferred from general education resources alone². Second, analyses at coarse spatial units can mask substantial county-level disparities¹¹. Third, reliance on single proxies underrepresents QPE’s multidimensional adequacy concept. Fourth, determinant analyses often overlook interaction mechanisms and nonlinear enhancement across complementary inputs^15,16. To address these gaps, we provide county-level evidence for all 107 counties in Shaanxi (2021–2024), identify the policy-relevant administrative scale structuring inequality using Dagum’s Gini decomposition¹⁴, and test key determinants and their interaction/enhancement patterns using GeoDetector^15,16, alongside spatial diagnostics of clustering and spatial dependence^12,13. Together, these elements motivate the conceptual framework and hypotheses presented in the next section.

Literature review

Education resource inequality

Uneven allocation of educational resources is widely documented in China and internationally, with advantages often concentrating in economically developed and urbanized areas while peripheral regions face persistent constraints^17,20. A key methodological concern is scale sensitivity: analyses at coarse administrative levels can “average out” local shortages and mask within-province heterogeneity, producing misleading inferences about equity¹¹. In China, multidimensional indices and national-level evidence suggest some gaps may be narrowing under targeted policies^18,21. However, county-level studies indicate that localized disparities can remain large and persistent even when broader regional gaps moderate, implying that broad labels (region or urban-rural) are insufficient to explain distributional patterns²².

Internationally, similar patterns appear. School-finance reforms can reduce disparities unevenly as local fiscal capacity continues to shape effective spending and implementation capacity^20,23,24. Implementation constraints can leave schools without essential staffing, facilities, or equipment needed to deliver PE curricula, particularly in under-resourced communities²⁵. European comparative evidence further documents cross-system differences in PE provision and implementation conditions at school level²⁶. Together, these findings support a common interpretation: spatial inequality reflects both structural development gradients and local service-capacity constraints, making sub-provincial diagnosis analytically necessary^11,18.

A remaining debate is how equity should be assessed when demand differs substantially across places. Because raw resource totals mechanically covary with student population, equity claims based on volume alone risk conflating scale with adequacy. This motivates demand-adjusted and fine-grained (county-level) measurement to distinguish true under-provision from differences driven primarily by population distribution^23,27.

Concept of school physical education (PE) resources

School PE resources are commonly conceptualized as a bundle of enabling inputs that support PE provision, typically including human resources (PE teachers and staffing), material resources (facilities and equipment), and financial resources (budget inputs and funding streams)²⁸. This classification is widely used in Chinese PE resource assessments and is consistent with international QPE guidance emphasizing qualified teachers, adequate facilities/equipment, and protected investment as foundational inputs for equitable provision^2,5,8,10.

A persistent measurement challenge is the reliance on single proxies (e.g., student–teacher ratio or per-student funding). Because QPE emphasizes complementarity across multiple inputs, multidimensional measurement is better aligned with the concept of adequacy^2,29. Two additional considerations reinforce a PE-specific analytic focus. First, PE provision is sensitive to local labor-market and infrastructure constraints (e.g., recruitment and retention of specialist teachers; maintenance of fields and equipment), which can vary sharply within provinces^6,30. Second, key inputs are complementary: financial effort is more likely to translate into higher adequacy when staffing and school infrastructure can absorb and operationalize investment into usable facilities, equipment, and program delivery^2,6. Together, these features motivate demand-adjusted, multidimensional measurement and the explicit testing of interaction mechanisms rather than treating inputs as independent or fully substitutable.

Existing spatial and inequality methods

Empirical work on school sports/PE resources has employed both qualitative and quantitative approaches, linking spatial distribution to governance arrangements, development conditions, and policy inputs^22,25. In education-equity research, commonly used quantitative tools can be grouped into four functions: (i) descriptive diagnosis of spatial patterns (GIS mapping and related visualization), (ii) spatial dependence testing (Global Moran’s I and Local Moran’s I/LISA), (iii) inequality quantification (Gini coefficients and decomposition methods), and (iv) determinant assessment using regression-based approaches, spatial econometrics, or stratified heterogeneity tools such as GeoDetector^12–16.

A key limitation is that these tools are often applied separately. Mapping can identify where disparities concentrate, inequality indices quantify how much dispersion exists, and determinant analyses explore potential drivers, but fewer studies connect these components into a coherent mechanism-oriented explanation that links “where-how-why” in a single analytical pipeline^11,31. In particular, many determinant studies emphasize additive effects and overlook interaction and nonlinear enhancement, despite strong theoretical reasons to expect that PE staffing, facilities, and earmarked funding operate as complementary inputs^2,6. This motivates integrated designs that jointly test spatial patterning, multi-scale inequality structure, and interaction mechanisms—especially in settings with pronounced core-periphery gradients and persistent low-resource clusters^14–16.

Conceptual framework and research hypotheses

Building on the Quality Physical Education (QPE) input perspective, this study conceptualizes county-level PE resource adequacy as a demand-adjusted outcome produced by the joint action of structural context, provisioning capacity, and demand pressure^2,32. The framework emphasizes that adequacy depends on multidimensional PE inputs (human, material, and financial) and is shaped by spatial and administrative mechanisms operating across neighboring counties and jurisdictional scales^11–13.

Figure 1 illustrates this conceptual logic. As shown in the framework, three sets of county-level conditions form the primary explanatory structure. First, structural context—including urbanization, economic capacity, and local governance—captures broader development conditions and institutional environments that may generate cumulative advantage in metropolitan core areas^20,23. Second, supply capacity, reflected in the PE teacher workforce, facilities, and dedicated PE funding, represents the immediate ability of local systems to provide PE services^2,6. Third, demand pressure, proxied by student scale, student density, and access constraints, reflects the intensity of educational demand that PE resources must accommodate^15,23. These three dimensions jointly feed into school PE resource adequacy at the county level, operationalized as a multidimensional capacity encompassing human, material, and financial inputs, and interpreted explicitly from a supply-demand matching perspective^2,15.

Fig. 1.

Conceptual framework and research hypotheses. The framework conceptualizes county-level PE resource adequacy as a demand-adjusted outcome produced by the joint action of structural context, provisioning capacity, and demand pressure. It summarizes the hypothesized core-periphery gradient, localized spatial patterning, administrative-scale inequality structure, and determinant/interaction mechanisms tested in the empirical analysis.

Based on this framework, four testable hypotheses guide the empirical analysis:

H1 (Core-periphery gradient under demand adjustment)

Even after demand adjustment, PE resource adequacy shows a core-periphery gradient, with higher adequacy in metropolitan core areas and lower adequacy in peripheral counties.

H2 (Localized spatial clustering)

Demand-adjusted adequacy exhibits positive spatial autocorrelation, forming High-High clusters in core areas and Low-Low clusters in peripheral areas, alongside a smaller number of outliers (HL and LH).

H3 (Administrative-scale inequality structure)

Between-city differences and cross-city overlap account for a larger share of total inequality than within-city differences, indicating that inequality is structured primarily across prefecture-level jurisdictions.

H4 (Determinants and interaction effects)

Development and capacity factors (urbanization, PE funding effort, teacher supply, school provisioning, and demand pressure) are key determinants of demand-adjusted adequacy. For major determinants, pairwise interactions exhibit nonlinear enhancement, such that for some factor pairs, q(X_i⨂Xⱼ) > max[q(X_i), q(Xⱼ)], indicating synergistic effects.

Methods

Data and study area

Study area. Shaanxi Province, located in northwestern China, comprises ten prefecture-level cities and 107 county-level units (districts and counties; hereafter “counties”). It exhibits marked heterogeneity in development level, population distribution, and terrain across three macro-regions (Guanzhong, Northern Shaanxi, and Southern Shaanxi). Figure 2 shows Shaanxi’s location within China, the three macro-regions, and the counties used in the analysis; the locator map in panel (a) is based on Natural Earth base data³³.

Fig. 2.

Location of Shaanxi Province within China and analytical units (107 counties). (a) Locator map showing Shaanxi within China. (b) The three macro-regions of Shaanxi (Guanzhong region, Northern Shaanxi, and Southern Shaanxi). (c) Prefecture-level cities and the 107 county-level units (counties) used in the analysis.

Source: Panel (a) is based on the Natural Earth low-resolution country boundaries dataset (public domain). Panels (b)–(c) are derived from the administrative boundary base layer used in this study’s GIS mapping.

Because all 107 counties are included, the analysis is population-based rather than sample-based. Shaanxi is an informative case because it combines a Xi’an-centered metropolitan core with less-developed peripheral areas in the north and south, enabling a clear assessment of how PE resource adequacy varies within a shared institutional setting. This setting is also policy-relevant, as Shaanxi is part of China’s western region and a focus of regional equalization initiatives (e.g., the Western Development Strategy)³⁴. Although empirical magnitudes are specific to Shaanxi, the analytical framework is applicable to other regions with similar core-periphery and urban–rural structures.

Data sources. Data were compiled from the China City Statistical Yearbook, the Shaanxi Statistical Yearbook, and prefecture-level city statistical yearbooks, which together provide annual series needed to construct the PE resource adequacy index and explanatory factors for 2021–2024^35,36. Variables include enrollments, PE teacher counts, education expenditures, and facilities-related indicators. PE-related administrative information was supplemented with Ministry of Education annual reports and relevant policy documents to support indicator definitions and cross-validation. All variables were harmonized at the county level and cross-checked across sources for consistency; no counties were excluded from the final dataset.

Macro-regions. Guanzhong region (Central Shaanxi) includes Xi’an, Xianyang, Baoji, Tongchuan, and Weinan; Northern Shaanxi includes Yulin and Yan’an; Southern Shaanxi includes Hanzhong, Ankang, and Shangluo.

Methodology

To examine the spatial distribution of school PE resource adequacy and its determinants, we implemented a four-step analytical workflow integrating index construction, spatial diagnosis, inequality decomposition, and determinant/interaction testing (Fig. 3)². Guided by the conceptual framework (Sect. 2.4; Fig. 1), each step addresses a complementary question: (i) adequacy measurement, (ii) spatial clustering, (iii) inequality structure across scales, and (iv) dominant drivers and interactions^2,12–16.

Fig. 3.

Analytical workflow for PE resource adequacy analysis in Shaanxi Province. The workflow summarizes the four-step design: (1) construction of the demand-adjusted, multidimensional PE resource adequacy index; (2) spatial dependence diagnosis; (3) inequality measurement and decomposition across administrative scales; and (4) determinant and interaction testing, with robustness checks.

Step 1: Composite index construction and descriptive assessment (demand-adjusted adequacy). We constructed a demand-adjusted, multidimensional PE resource adequacy index based on the indicator system described in Sect. 3.3 (Table 1), using entropy weighting estimated from pooled 2021–2024 data and applied consistently across years^37,38. We then mapped the index using choropleth visualization (Jenks natural breaks) and summarized spatial concentration using a standard deviational ellipse^39,40. Supply-demand matching was further assessed by combining the adequacy index with student density as a demand proxy²³.

Table 1.

School Physical Education Resource Adequacy Index System.

Pillar (pillar weight)	Indicator (indicator weight)	Measurement method	Unit	Direction
Human Resources (44.87%)	Student-teacher ratio (6.47%)	Total number of students/total number of PE teachers	students per teacher	-
	Full-time teachers ratio (21.03%)	Number of full-time PE teachers/number of part-time PE teachers	Ratio	+
	Teacher training participation rate (2.57%)	Teachers attending in-service training/total PE teachers	%	+
	New teacher ratio (13.76%)	Newly hired PE teachers/total PE teachers	%	+
	Teacher vacancy ratio (1.04%)	(Required PE teachers − actual PE teachers)/required PE teachers	%	-
Material Resources (16.91%)	Per-student sports field area (14.41%)	Total area of sports fields/total students	m² per student	+
	Equipment compliance rate (2.50%)	Schools meeting minimum equipment standards/total schools	%	+
Financial Resources (38.22%)	Per-student total PE expenditure (6.10%)	Total PE expenditure/total students	yuan per student	+
	Per-student facility expenditure (5.55%)	Field construction & maintenance expenditure/total students	yuan per student	+
	Per-student equipment expenditure (15.86%)	Equipment expenditure/total students	yuan per student	+
	Per-student sports program funding (10.71%)	Sports programs/activities funding/total students	yuan per student	+

“+” denotes a positive indicator (higher is better); “−” denotes a negative indicator (lower is better). For the equipment compliance indicator, operational standards follow China’s official school sports equipment and facility regulations, while the conceptual rationale aligns with international QPE guidance². Specifically, we reference the national standard GB/T 19851.1–2022 on sports equipment and playground requirements for primary and secondary schools and the national “trial basic standards” for school sports and health conditions^43,44.

Step 2: Spatial dependence diagnosis (where inequities concentrate). Spatial dependence was tested using Global Moran’s I and Local Moran’s I (LISA) based on a first-order queen contiguity, row-standardized spatial weights matrix; statistical significance was assessed via permutation tests (999 permutations)^12,13,41.

Step 3: Inequality quantification and decomposition (how inequality is structured). We measured overall inequality using Dagum’s Gini coefficient and decomposed it into within-group, between-group, and transvariation (overlap) components¹⁴. Decomposition was conducted for both macro-regions (Guanzhong region, Northern Shaanxi, and Southern Shaanxi) and prefecture-level cities to identify the dominant administrative scale of inequality.

Step 4: Determinants and interaction effects (why disparities form). We applied GeoDetector to estimate each factor’s explanatory power (q statistic) and to test nonlinear enhancement through interaction effects^15,16. Predictors were discretized into strata prior to estimation, and analyses were conducted for 2021 and 2024 to examine temporal shifts in dominant drivers and interaction configurations. For the discretization and parameter considerations in GeoDetector applications, we followed published guidance and robustness approaches^40,42.

Robustness checks. Robustness and sensitivity analyses were conducted to assess the influence of weighting schemes, outliers, spatial weight specifications, map classification, Dagum grouping choices, and GeoDetector discretization^13,37,41. Detailed procedures and results are reported in Sect. 4.5.

Evaluation index system

Quality Physical Education (QPE) refers to a planned, progressive, and inclusive learning experience embedded in the school curriculum that aims to develop students’ physical literacy and lifelong engagement in physical activity². International guidance consistently emphasizes that equitable QPE provision depends on three enabling inputs: (i) an adequate and qualified teacher workforce, (ii) sufficient facilities and equipment, and (iii) sustained and protected investment^2,5,8,10. Accordingly, we measure the input foundations of QPE rather than instructional processes or student outcomes, which are not directly observable from available administrative statistics.

Because absolute PE resource totals covary with student population size, we evaluate provision as demand-adjusted adequacy rather than raw supply volume^1,23. Most indicators are expressed as per-student measures or ratios (e.g., student-teacher ratio, per-student expenditure, and per-student facility area), so the composite index reflects adequacy relative to educational demand^1,2. We also include a teacher vacancy ratio to capture unmet staffing needs (required vs. actual PE teachers)^2,6.

Indicator selection followed established composite-indicator principles, including conceptual relevance, measurability, cross-unit comparability, and data availability³⁷. Consistent with equity principles, demand-adjusted indicators were used whenever applicable (primarily per-student measures)¹. The system comprises three pillars—Human Resources, Material Resources, and Financial Resources—with 11 indicators (Table 1). Detailed indicator definitions, data sources, and key references supporting each indicator’s inclusion are provided in Supplementary Table S1.

To ensure temporal comparability across 2021–2024, entropy weights were estimated using the pooled dataset (2021–2024) and then applied consistently to compute the composite PE resource adequacy index for each county-year, so that temporal changes are not driven by year-specific reweighting^37,38. Prior to entropy weighting, all indicators were standardized following established guidance for composite indicators³⁷. Positive indicators were normalized so that higher standardized values indicate better provision, whereas negative indicators were normalized so that lower raw values correspond to better provision³⁷. The entropy-derived weights indicate that Human Resources contribute the largest share of the composite index (44.87%), followed by Financial Resources (38.22%) and Material Resources (16.91%) (Table 1).

Results

Supply-demand adequacy and spatial distribution

To incorporate demand conditions, we used student density (students per km²) as a proxy for demand intensity and examined its alignment with the demand-adjusted PE resource adequacy index²³. Using provincial medians, counties were classified into four supply–demand types: High-supply/High-demand (HS-HD), High-supply/Low-demand (HS-LD), Low-supply/High-demand (LS-HD), and Low-supply/Low-demand (LS-LD). Overall, 24 counties (22.43%) were classified as LS-HD, indicating the most critical mismatch (high demand pressure but low adequacy). The remaining counties were distributed across HS-HD (n = 30, 28.04%), HS-LD (n = 24, 22.43%), and LS-LD (n = 29, 27.10%). This typology separates demand pressure from adequacy: LS-HD counties indicate demand-adjusted shortfalls under high demand intensity and thus represent the highest-priority equity concern, whereas LS-LD counties may reflect low demand pressure rather than severe mismatch. We therefore interpret spatial gradients and local patterns in light of both demand (student density) and supply capacity (teachers, schools, and PE investment). Figure 4(a) maps the spatial distribution of the four supply–demand types.

Fig. 4.

Spatial distribution of PE resource adequacy and supply–demand typology in Shaanxi Province (2024). (a) Supply–demand typology based on provincial medians of student density (demand proxy) and the demand-adjusted PE resource adequacy index (adequacy proxy): High-supply/High-demand (HS-HD), High-supply/Low-demand (HS-LD), Low-supply/High-demand (LS-HD), and Low-supply/Low-demand (LS-LD). (b) Demand-adjusted PE resource adequacy in 2024, classified into five tiers (Very Low, Low, Medium, High, and Very High) using Jenks natural breaks. Mapped values are the entropy-weighted composite adequacy index derived from demand-adjusted indicators (human, material, and financial inputs).

Based on the entropy-weighted composite index (demand-adjusted indicators), PE resource adequacy in 2024 shows a clear core-periphery gradient. Guanzhong region—particularly Xi’an and nearby Xianyang—had higher adequacy, whereas many counties in the northern and southern peripheries lagged behind. Using Jenks natural breaks³⁹, counties were classified into five tiers (Very Low, Low, Medium, High, and Very High). In 2024, most counties fell into the Very Low and Low tiers (75 of 107), while relatively few reached High and Very High (10 of 107). Figure 4(b) presents the Jenks-classified spatial distribution of PE resource adequacy in 2024.

A standard deviational ellipse (SDE) analysis further summarizes spatial concentration and its temporal stability (Table 2; Fig. 5)⁴⁰. Across 2021–2024, the mean center remained within the Guanzhong region, indicating no major relocation of the distributional core. In 2022, the centroid shifted northward toward the Tongchuan area (Yaozhou vicinity), and then moved southward toward the Xianyang corridor in 2023–2024 (Fig. 5). The ellipses are consistently elongated, with semi-major axes of approximately 227–239 km and semi-minor axes of approximately 104–107 km (Table 2), and eccentricity remains stable (≈ 0.887–0.900). Ellipse area varies modestly from about 74,596 km² to 80,502 km², suggesting that the overall spatial concentration of PE resource adequacy was broadly stable over 2021–2024.

Table 2.

Standard deviational ellipse parameters of school PE resource adequacy in Shaanxi Province (2021–2024).

Year	Mean center longitude (°E)	Mean center latitude (°N)	Semi-minor axis (km)	Semi-major axis (km)	Orientation angle (°)	Perimeter (km)	Area (km²)	Eccentricity
2021	108.783	34.946	104.247	239.014	15.440	1120.302	78268.051	0.900
2022	108.843	34.963	103.807	235.788	13.035	1107.498	76886.190	0.898
2023	108.784	34.767	104.623	226.980	18.312	1077.487	74596.281	0.887
2024	108.734	34.775	107.341	238.748	19.022	1126.772	80501.864	0.893

Semi-major/minor axes and perimeter are reported in kilometers (km) and area in square kilometers (km²), converted from the projected-coordinate SDE outputs (m and m²). The orientation angle follows the default definition of the GIS standard deviational ellipse tool used in this study (degrees).

Fig. 5.

Standard deviational ellipse (SDE) and mean-center trajectory of PE resource adequacy in Shaanxi Province, 2021–2024. (a) Annual SDE overlays showing the dispersion and orientation of county-level adequacy each year. (b) Interannual trajectory of the mean center, illustrating changes in the spatial center of the adequacy distribution over time.

Because the index is demand-adjusted (per-student/ratio measures), the observed gradient cannot be attributed to population distribution alone. Core areas retain higher adequacy even after accounting for demand, supporting H1. The next section tests whether this spatial pattern corresponds to statistically significant spatial dependence using Global Moran’s I and Local Moran’s I (LISA)^12,13.

Spatial autocorrelation and clustering trends

Global Moran’s I. Global spatial autocorrelation shows a clear weakening trend over time. As reported in Table 3, the global Moran’s I for the demand-adjusted PE resource adequacy index was 0.290 in 2021 (p = 0.001), indicating a moderate positive spatial autocorrelation¹². The statistic declined to 0.170 in 2022 (p = 0.004) and was 0.203 in 2023 (p = 0.003). By 2024, Moran’s I had fallen to 0.105 (p = 0.004), suggesting weak but still statistically significant spatial dependence. Overall, the magnitude of global spatial dependence more than halved from 2021 to 2024, implying that province-wide clustering of similar adequacy levels attenuated and the spatial pattern became more diffuse. Nevertheless, Moran’s I remained positive and significant throughout the study period (Table 3), indicating that spatial dependence persisted even under demand-adjusted measurement.

Table 3.

Global Moran’s I statistics for school PE resource adequacy in Shaanxi Province (2021–2024).

Year	Moran’s I	E(I)	SD(I)	Z-value	P-value
2021	0.290	−0.009	0.061	2.978	0.001
2022	0.170	−0.009	0.060	2.725	0.004
2023	0.203	−0.009	0.061	2.895	0.003
2024	0.105	−0.009	0.060	2.697	0.004

Moran’s I measures global spatial autocorrelation of the PE resource adequacy index. E(I) = − 1/(n − 1) under spatial randomness (n = 107). Z-values and P-values are based on permutation tests (999 permutations).

Local LISA clusters (2021–2024). Despite the decline in global spatial autocorrelation, Local Moran’s I (LISA) indicates that localized spatial patterning persisted over 2021–2024. Figure 6 maps the spatial distribution of LISA cluster types for each year, distinguishing High-High (HH), Low-Low (LL), and spatial outliers (High-Low, HL; Low-High, LH) at the county level.

Fig. 6.

Local Moran’s I (LISA) cluster types of PE resource adequacy in Shaanxi Province, 2021–2024. Maps show the county-level LISA cluster/outlier types (quadrant classification) for each year. High-High (HH) and Low-Low (LL) indicate counties with high/low adequacy surrounded by neighbors with similarly high/low adequacy. High-Low (HL) and Low-High (LH) indicate spatial outliers where a county’s adequacy differs from that of its neighbors. Blue indicates HH, orange indicates LL, grey indicates HL, and red indicates LH.

As shown in Fig. 6 and summarized in Table 4, Low-Low (LL) clusters were the dominant category in every year, accounting for 42.99% of counties in 2021, increasing to 51.40% in 2022, and remaining high at 47.66% in both 2023 and 2024. This indicates that nearly half of all counties consistently belonged to extensive contiguous low-adequacy clusters, primarily located in peripheral areas. In contrast, High-High (HH) clusters were comparatively limited and core-concentrated, accounting for 25.23% in 2021, decreasing to 20.56% in 2022, rebounding to 25.23% in 2023, and then falling to 19.63% in 2024 (Table 4), suggesting that high-adequacy counties remained spatially concentrated around central urban cores.

Table 4.

LISA cluster types of school PE resource adequacy in Shaanxi Province (2021–2024).

Year	High-High (HH) n (%)	Low-High (LH) n (%)	Low-Low (LL) n (%)	High-Low (HL) n (%)
2021	27 (25.23%)	19 (17.76%)	46 (42.99%)	15 (14.02%)
2022	22 (20.56%)	15 (14.02%)	55 (51.40%)	15 (14.02%)
2023	27 (25.23%)	19 (17.76%)	51 (47.66%)	10 (9.35%)
2024	21 (19.63%)	22 (20.56%)	51 (47.66%)	13 (12.15%)

Percentages are calculated over 107 counties each year. Cluster types are derived from the LISA quadrant categories in the LISA output.

Outlier patterns changed modestly over time. Low-High (LH) outliers accounted for 17.76% of counties in 2021, 14.02% in 2022, 17.76% in 2023, and increased to 20.56% in 2024, implying sharper contrasts at some core-periphery or urban-rural margins in the latest year. High-Low (HL) outliers accounted for 14.02% in both 2021 and 2022, declined to 9.35% in 2023, and rose to 12.15% in 2024 (Table 4).

Taken together, declining global Moran’s I but persistent local clusters indicate that spatial dependence became weaker at the province-wide scale while remaining pronounced in localized pockets. The persistence of extensive LL clusters and the concentration of HH clusters around central cores provide strong evidence of localized spatial structuring in PE resource adequacy, supporting H2. This contrast frames the discussion in Sect. 5.2 on province-wide spatial diffusion versus local lock-in.

Sources of inequality in PE resource adequacy (2021–2024)

Regional Perspective (Three Macro-Regions). Overall inequality of the PE resource adequacy index (Dagum Gini), decomposed by Shaanxi’s three macro-regions (Guanzhong region, Northern Shaanxi, and Southern Shaanxi), peaked in 2022 and then declined modestly. The overall Gini increased from 0.420 (2021) to 0.472 (2022), before falling to 0.446 (2023) and 0.424 (2024) (Table 5, Panel A). Dagum decomposition shows that transvariation (Gt), which captures distributional overlap across groups, was the largest contributor in most years. In 2021, overlap accounted for 42.38% (Gt = 0.178), exceeding within-region inequality (34.95%, Gw = 0.147) and between-region differences (22.67%, Gb = 0.095). In 2022, overlap further increased to 48.70% (Gt = 0.230), while the between-region share declined to 16.35% (Gb = 0.077). In 2023, the decomposition structure shifted temporarily: within-region inequality became the largest component (38.78%, Gw = 0.173), while between-region differences rose to 30.03% (Gb = 0.134) and overlap declined to 31.19% (Gt = 0.139). By 2024, overlap again dominated, accounting for 43.07% (Gt = 0.183), followed by within-region inequality (37.31%, Gw = 0.158) and between-region differences (19.62%, Gb = 0.083). Overall, the between-region component remained comparatively smaller (16.35–30.03%), indicating substantial within-region heterogeneity and distributional overlap across macro-regions. This temporary 2023 shift further suggests that macro-regional inequality can be driven by overlap or within-region dispersion depending on the year, rather than by stable differences in regional means alone.

Table 5.

Dagum Gini decomposition of school PE resource adequacy in Shaanxi Province (2021–2024).

Panel A. Three macro-regions.
Year	Overall Gini (G)	Within-group Gini (Gw)	Between-group Gini (Gb)	Transvariation Gini (Gt)	Contribution of Gw (%)	Contribution of Gb (%)	Contribution of Gt (%)
2021	0.420	0.147	0.095	0.178	34.95%	22.67%	42.38%
2022	0.472	0.165	0.077	0.230	34.96%	16.35%	48.70%
2023	0.446	0.173	0.134	0.139	38.78%	30.03%	31.19%
2024	0.424	0.158	0.083	0.183	37.31%	19.62%	43.07%
Panel B. Ten prefecture-level cities.
2021	0.420	0.039	0.220	0.161	9.39%	52.33%	38.29%
2022	0.472	0.042	0.204	0.226	8.92%	43.30%	47.78%
2023	0.446	0.040	0.224	0.182	8.96%	50.25%	40.79%
2024	0.424	0.039	0.212	0.173	9.14%	50.00%	40.86%

Overall Gini (G) is calculated using all counties and is identical across grouping schemes. Differences between Panel A and Panel B reflect alternative group structures (macro-regions vs. prefecture-level cities) and therefore affect only the decomposition into within-group (Gw), between-group (Gb), and transvariation (Gt) components. Components and contribution shares may not sum exactly due to rounding. Macro-regions are defined as in Sect. 3.1.

City-level perspective (ten prefecture-level cities). Decomposition by prefecture-level city shows a stronger administrative-scale structuring of inequality (Table 5, Panel B). While the overall Gini follows the same temporal pattern, inequality is driven mainly by between-city differences (Gb) and cross-city overlap (Gt), whereas within-city inequality (Gw) remains consistently small (≈ 9%). In 2021, between-city differences accounted for 52.33% (Gb = 0.220), overlap accounted for 38.29% (Gt = 0.161), and within-city inequality contributed 9.39% (Gw = 0.039). In 2022, the overlap component (47.78%) slightly exceeded the between-city component (43.30%), reflecting a higher degree of cross-city overlap in that year. By 2023–2024, between-city inequality again became the largest component (50.25% in 2023; 50.00% in 2024), while within-city inequality remained stable. Across 2021–2024, the within-city component remains around 9%, whereas between-city differences consistently account for roughly half of total inequality (Table 5, Panel B). Together, these results indicate that PE resource adequacy inequality is structured primarily across prefecture-level jurisdictions rather than within them, supporting H3.

Overall, Table 5 highlights a clear scale contrast: at the macro-regional level, inequality is driven mainly by within-region heterogeneity and overlap, whereas at the prefecture-city level, inequality is structured primarily by between-jurisdiction differences and cross-city overlap, with within-city variation consistently small. Importantly, the decomposition shows that the dominant inequality structure differs by grouping scheme (macro-regions vs. prefecture-level cities), indicating that the policy-relevant scale is an empirical result rather than a presumption.

Determinants and interaction effects

Using the GeoDetector model, we assessed how eight factors (X₁-X₈) explain spatial variation in school PE resource adequacy in 2021 and 2024 (Table 6). The ranking of determinants shifted substantially over time. In 2021, GDP per capita (X₁) showed the strongest explanatory power (q = 0.224), followed by school PE performance level (X₇; q = 0.185) and urbanization (X₈; q = 0.173). Teacher supply (X₅) exhibited moderate explanatory power (q = 0.164), whereas student density (X₃; q = 0.106), population mobility (X₄; q = 0.118), and school supply capacity (X₆; q = 0.092) were weaker. PE funding effort (X₂) was the weakest factor in 2021 (q = 0.053) (Table 6). By 2024, PE funding effort (X₂) became the dominant determinant (q = 0.353), followed by urbanization (X₈; q = 0.317) and student density (X₃; q = 0.298) (Table 6). Teacher supply (X₅) and school supply capacity (X₆) also increased markedly (q = 0.252 and 0.243, respectively). In contrast, GDP per capita (X₁) remained nearly unchanged (q = 0.226) but declined in rank as other factors strengthened. Overall, the largest increases were observed for PE funding effort (Δq = 0.300), student density (Δq = 0.192), school supply capacity (Δq = 0.151), and urbanization (Δq = 0.144) (Table 6). This temporal re-ranking is discussed in relation to relevant policy developments during 2021–2024 in Sect. 5.4^45–47.

Table 6.

GeoDetector factor detection results and temporal changes (2021 vs. 2024).

No	Factor	Measurement	q (2021)	q (2024)	Δq (2024 − 2021)
X1	Economic development level	GDP per capita	0.224	0.226	0.002
X2	Investment in physical education	Proportion of educational funds dedicated to PE	0.053	0.353	0.300
X3	Resource demand intensity	Student density (students per km²)	0.106	0.298	0.192
X4	Population mobility	Proportion of floating (migrant) population	0.118	0.102	−0.016
X5	Teacher supply level	Number of PE teachers in primary & secondary schools	0.164	0.252	0.088
X6	Resource supply capacity	Number of primary & secondary schools	0.092	0.243	0.151
X7	School PE performance level	School PE work assessment grade	0.185	0.196	0.011
X8	Urbanization level	Urbanization rate	0.173	0.317	0.144

q-values indicate the explanatory power of each factor for the spatial distribution of PE resource adequacy. Δq denotes the change from 2021 to 2024. Statistical significance was assessed using permutation tests (999 permutations); all reported q-values passed the significance test at p < 0.05.

Interaction effects. Interaction detection indicates nonlinear enhancement for key determinants, with several factor pairs yielding q-values substantially higher than individual effects (Fig. 7). In 2021, the strongest interaction was GDP per capita × teacher supply (X₁⨂ X₅; q = 0.417). By 2024, the strongest interactions increased (maximum q rising to 0.626), and the interaction structure reconfigured, with GDP per capita × student density (X₁⨂ X₃; q = 0.626) emerging as the largest interaction. In addition, PE funding effort interacted strongly with school supply capacity (X₂⨂ X₆; q = 0.607) and teacher supply (X₂⨂ X₅; q = 0.588) (Fig. 7). These interaction patterns are consistent with the complementarity of QPE input foundations (teachers-facilities-protected investment), which motivates testing nonlinear enhancement rather than purely additive effects (Sect. 2.2). Overall, these results show that spatial variation in PE resource adequacy is associated with combinations of development context, demand pressure, and provisioning capacity rather than single factors alone, supporting H4.

Fig. 7.

GeoDetector interaction heatmaps and their temporal changes. (a) Interaction effects among explanatory factors in 2021. (b) Interaction effects among explanatory factors in 2024. (c) Changes in interaction effects between 2021 and 2024 (Δq), calculated as q₂₀₂₄ - q₂₀₂₁.

Robustness checks

We conducted robustness checks to assess whether the main findings are sensitive to weighting choices, outliers, spatial specifications, and GeoDetector settings. Across all tests, the demand-adjusted core-periphery gradient (H1), persistent localized clustering patterns (H2), the prefecture-level (between-city) structuring of inequality (H3), and the leading determinants and interaction effects identified by GeoDetector (H4) remained stable.

Index weighting. We recalculated the composite PE resource adequacy index using equal weights and PCA-based weights instead of entropy weights. Spatial distributions, Global Moran’s I, LISA cluster patterns, and the main conclusions were highly consistent across weighting schemes, indicating low sensitivity to the index construction approach.

Outliers and influential units. We applied 1% winsorization and performed a leave-one-city-out analysis (excluding one prefecture-level city at a time). Key statistics shifted only marginally and the direction and significance of main results were preserved, suggesting findings are not driven by extreme values or any single prefecture/county.

Spatial weights matrix. We tested alternative neighbor definitions (queen vs. rook contiguity; k-nearest neighbors with k = 4, 6, 8; inverse-distance weights). The central High-High cluster and widespread peripheral Low-Low clusters persisted across specifications, and Global Moran’s I remained positive (with modest variation in magnitude), indicating that spatial dependence is not an artifact of a particular spatial weights choice.

Map classification. We varied class breaks (e.g., quantiles and different numbers of classes). While some boundary cases shifted visually, the underlying core-periphery pattern remained, indicating that mapping choices did not drive the substantive interpretation.

Inequality decomposition groupings. We tested alternative grouping schemes in Dagum decomposition (e.g., East-West splits and alternative city groupings). The prefecture-level decomposition continued to show that between-city differences and cross-city overlap account for the dominant shares of total inequality, while within-city inequality remained small, reinforcing the conclusion that inequality is structured mainly across prefecture-level jurisdictions.

GeoDetector settings. We varied discretization methods for continuous predictors (equal intervals, quantiles, natural breaks) under the same number of strata. The top determinants (e.g., urbanization, PE funding effort, student density, teacher supply, and school supply capacity in 2024) and strong interaction pairs (e.g., X₂⨂X₆, X₂⨂X₅, X₁⨂X₃) remained consistent, indicating robustness to binning choices.

Overall, findings were robust across specifications, supporting the reliability of the main inferences.

Discussion

This study evaluates sub-provincial inequality in demand-adjusted PE resource adequacy in Shaanxi (2021–2024) using a multidimensional index aligned with QPE input foundations and an integrated spatial-inequality-determinant workflow^1,2,32. Four findings are central: a persistent core-periphery gradient after demand adjustment (H1), durable localized clustering despite weakening global dependence (H2), a dominant prefecture-level structuring of inequality (H3), and strong nonlinear interaction enhancement consistent with configuration-based mechanisms (H4)^2,13,14,16.

Core-periphery inequality under demand-adjusted measurement (H1)

A key contribution of the adequacy perspective is that it separates population-scale effects from true under-provision¹. Because the composite index is built primarily from per-student measures and ratios, the higher adequacy observed in the Xi’an-centered core reflects differences in local capacity to supply qualified PE teachers, maintain facilities, and sustain PE investment, rather than differences in student concentration alone^2,6,10. This demand-adjusted gradient aligns with the broader education-equity literature showing that service capacity and public resources concentrate in economically developed and urbanized areas, while peripheral counties face persistent constraints^17,20.

Importantly, the supply-demand typology (HS-HD/HS-LD/LS-HD/LS-LD) strengthens the equity interpretation by distinguishing “low supply under low demand” from demand-adjusted shortfalls under high demand. In substantive terms, LS-HD counties represent the highest-priority equity concern because they combine high demand pressure with low adequacy, whereas LS-LD counties may reflect low demand pressure rather than severe mismatch. This directly addresses the concern that observed spatial inequality may partly reflect heterogeneous demand rather than inequitable provision¹.

Localized spatial dependence and persistent clusters (H2)

The combination of weakening global Moran’s I and persistent LISA clusters suggests that Shaanxi’s spatial dynamics are increasingly shaped by localized regimes rather than a single province-wide gradient¹³. While global autocorrelation declines over time, Low-Low clusters remain extensive and contiguous, and High-High clusters remain concentrated around the metropolitan core (Table 4; Fig. 6). This pattern is consistent with the QPE input logic: PE adequacy depends on co-located, complementary inputs—specialist teachers, usable facilities/equipment, and sustained investment—that tend to cluster spatially rather than disperse evenly^2,8.

Importantly, a lower global Moran’s I should not be read as uniform improvement across counties. It indicates weaker province-wide sorting of similar adequacy levels, but it remains compatible with persistent local concentration of disadvantage. In our results, the rise in Low-High (LH) outliers in 2024 points to sharper contrasts at some boundaries between low-adequacy counties and better-resourced neighbors (Table 4). Taken together, province-scale “diffusion” can coexist with localized persistence of low-adequacy regimes.

From this perspective, the stability of peripheral Low-Low regimes is consistent with the risk of a local low-level trap. Fiscal commitment, specialist-teacher attraction/retention, and facility constraints often move together across adjacent counties. When these constraints reinforce one another, disadvantage becomes geographically self-reinforcing rather than idiosyncratic^13,41. This reading also matches the interaction patterns in Sect. 4.4. Funding effort shows its strongest associations when paired with teacher supply and school provisioning capacity, consistent with finance operating primarily through complementary inputs rather than as a stand-alone lever. The policy implication is that raising average adequacy is not sufficient if large Low-Low regimes remain intact. Interventions are more likely to weaken lock-in when they are designed as coordinated packages aligned with input complementarity—for example, targeted transfers tied to PE-specific budget items, cross-county teacher mobility and allowance schemes, and shared procurement and maintenance arrangements for facilities and equipment—implemented at a scale that matches the geography of persistent clusters^2,8. Thus, the declining global Moran’s I should not be interpreted as province-wide improvement; rather, it is consistent with weaker province-wide sorting alongside persistent local lock-in in extensive Low-Low regimes.

Administrative-scale structuring of inequality (H3)

Dagum decomposition provides clear evidence that inequality is structured primarily across prefecture-level jurisdictions, with within-city inequality remaining consistently small¹⁴. This diagnostic is policy-relevant because it identifies the scale at which equalization mechanisms are most likely to be effective. When education finance and implementation capacity are organized at an intermediate administrative tier, prefecture-level differences in fiscal health and governance capability can set persistent “ceilings and floors” for counties within each jurisdiction, making prefecture affiliation highly predictive of adequacy outcomes^1,23,24.

The dominance of between-city inequality also helps interpret why within-city equalization alone has limited effects on province-wide convergence. If internal dispersion is already relatively constrained, reallocations within a prefecture can reduce within-jurisdiction inequality but cannot close the larger gaps across structurally advantaged and disadvantaged prefectures. This mechanism-oriented reading strengthens H3 by linking the inequality structure to a governance channel rather than treating decomposition as a descriptive add-on.

Dynamic drivers and interaction mechanisms in PE adequacy (H4)

GeoDetector results show a substantive shift in what differentiates adequacy outcomes between 2021 and 2024¹⁶. By 2024, PE funding effort, urbanization, and demand pressure (student density) explain substantially more spatial variation than in 2021. Over the same period, teacher supply and school provisioning capacity also become more salient. Together, this pattern suggests that beyond baseline development differences, local prioritization of PE within education spending and the ability to respond to concentrated demand increasingly differentiate adequacy outcomes over time^2,5,32.

Policy developments during 2021–2024 provide an institutional context for interpreting the rising salience of finance. Two features are especially relevant: (i) mandated expansion of after-school sport services and (ii) the specification of funding-guarantee arrangements. The 2021 “Double Reduction” policy required schools to expand after-school services. It also instructed provinces to establish funding-guarantee mechanisms (e.g., fiscal subsidies and regulated service charges) for participating staff⁴⁵. A 2022 joint notice promoted higher-quality after-school sports services and encouraged mobilizing external providers (e.g., sports schools/clubs and part-time coaches). It further specified that subsidies for part-time coaches should be covered through the after-school service funding mechanism⁴⁶. In 2023, central finance raised the national per-student public funding baseline for compulsory education in connection with Double Reduction implementation and after-school service provision⁴⁷. At the provincial level, Shaanxi’s sports–education integration implementation opinion reinforced PE and after-school training/competition priorities and recognized teachers’ after-school coaching workload. A subsequent four-ministry notice then standardized after-school services and reiterated requirements for stable funding arrangements^48,49. Taken together, these documents clarify financing channels and define recurrent expenditure items (e.g., staff subsidies and contracted services). They also imply capital-related needs (e.g., facility and equipment upgrading). This context highlights the relevance of budget earmarking capacity for PE implementation. Given that X₂ is defined as the share of education funds dedicated to PE, this policy context is consistent with the observed rise in the explanatory power of X₂ in later years.

The interaction detection results further support a configuration-based mechanism consistent with QPE complementarity. Strong nonlinear enhancement implies that financial effort is most strongly associated with higher adequacy when paired with absorptive capacity—especially teacher supply and school infrastructure—rather than operating as an isolated lever. This interpretation aligns with established evidence that specialist staffing and professional capacity are central constraints in many education systems, and that capacity-building and professional development shape whether investments translate into effective provision^6,50–52. At the same time, GeoDetector identifies stratified heterogeneity and interaction enhancement rather than causal effects; thus, the value lies in identifying robust, policy-relevant configurations that merit targeted intervention and causal evaluation in future work¹⁶.

Generalizable patterns beyond Shaanxi and implications

Although the empirical setting is Shaanxi, the findings support general propositions relevant to other regions where metropolitan areas concentrate fiscal resources and specialist labor and where subnational governments bear responsibility for education service provision¹. First, demand-adjusted adequacy is a more policy-relevant equity metric than raw input totals because it distinguishes population-driven scale effects from true under-provision^1,23. Second, persistent localized clustering indicates that adequacy is shaped by geographically shared labor markets and infrastructure constraints, so interventions are more durable when designed at a cluster/area level rather than as isolated local fixes¹³. Third, when financing and implementation capacity are organized at an intermediate administrative tier, inequality tends to be structured across jurisdictions, implying that cross-jurisdiction equalization mechanisms are critical^14,23,24. Fourth, strong interaction enhancement supports an absorptive-capacity principle: targeted investment improves adequacy most when paired with staffing and infrastructure capacity that converts funding into usable PE inputs^1,2,8.

These implications also sharpen the academic contribution. By aligning measurement with multidimensional QPE input foundations and integrating spatial diagnosis, multi-scale inequality decomposition, and interaction testing, the study addresses fragmented indicator practices and moves from pattern description toward a mechanism-oriented explanation of persistent spatial inequality^16,29,37.

Limitations and future research

Several limitations qualify interpretation without undermining the central conclusions. First, the 2021–2024 window is short; longer series are needed to distinguish sustained convergence from temporary fluctuation. Second, the analysis focuses on one province; cross-province comparisons would test generality under different fiscal and demographic contexts. Third, the adequacy index captures input foundations rather than instructional processes or student outcomes; this is consistent with QPE guidance but should be complemented by outcome-linked analyses in future work^2,8. Fourth, composite indicators depend on administrative reporting quality; robustness checks reduce concerns that key findings are artifacts of weighting, outliers, or spatial specifications, but further work using finer quality measures (e.g., facility usability and teacher qualification structure) would strengthen inference³⁷. Finally, GeoDetector identifies associations and interaction enhancement rather than causal effects; quasi-experimental evaluations of PE funding reforms and teacher-allocation policies would help establish causal pathways and quantify policy effectiveness¹⁶.

Overall, improving PE equity requires treating inequality as a demand-adjusted, multi-scale, and configuration-driven challenge shaped jointly by development gradients, spatial dependence, and jurisdictional capacity.

Conclusions and policy implications

Conclusions

This study examined sub-provincial inequality in demand-adjusted PE resource adequacy in Shaanxi (2021–2024) using an integrated framework combining spatial diagnostics, Dagum Gini decomposition, and determinant/interaction analysis. Four conclusions follow:

(1) A resilient core-periphery hierarchy persists after demand adjustment. The Xi’an-centered core maintains higher adequacy than peripheral counties even when measured relative to student demand, indicating that fiscal capacity, human-resource availability, and institutional capability translate into unequal QPE input foundations.

(2) High and low adequacy remain spatially clustered. Adequacy forms High-High clusters in central areas and extensive Low-Low clusters in peripheral regions, indicating spatial dependence and the persistence of geographically contiguous disadvantage.

(3) Inter-city disparities dominate intra-city disparities. Dagum decomposition shows that overall inequality is driven primarily by between-prefecture differences and cross-prefecture overlap, while within-prefecture inequality remains small, identifying the prefecture level as a critical scale structuring equity outcomes.

(4) Adequacy is shaped by multiple drivers and their interactions. By 2024, policy effort and demand pressure (together with urbanization and provisioning capacity) show stronger explanatory power, and interaction enhancement indicates that adequacy is produced by complementary bundles of investment and absorptive capacity rather than by single factors in isolation.

While the estimates are province-specific, the analytical framework and the identified mechanisms are transferable to other regions with metropolitan dominance and decentralized service provision, where demand-adjusted adequacy, spatial dependence, and jurisdictional capacity jointly shape local equity outcomes².

Policy implications

The findings suggest several policy directions to improve PE equity and adequacy. Rather than treating inequality as a uniform “more input” problem, the results point to three actionable design principles: (i) inequality is structured primarily across prefecture-level jurisdictions, (ii) low-adequacy regimes persist as spatial clusters and demand-adjusted shortfalls (LS-HD) represent the most consequential mismatches, and (iii) determinants operate through interaction enhancement, implying limited returns to single-lever interventions.

(1) Target equalization at the prefecture level, guided by adequacy gaps. Because inequality is dominated by between-prefecture differences, a practical priority is to strengthen cross-prefecture equalization mechanisms that raise the baseline capacity of lagging jurisdictions¹⁴. Rather than uniform transfers, allocations can be rule-based and adequacy-guided, for example increasing support where prefecture-level adequacy is systematically lower and where LS-HD shortfalls are more prevalent, with earmarking tied to PE-specific capacity building (staffing and basic facility maintenance).

(2) Use the LS-HD classification as an explicit prioritization rule. The supply-demand typology provides an operational way to distinguish low adequacy under low demand (LS-LD) from demand-adjusted shortfalls under high demand pressure (LS-HD). Policy targeting can therefore prioritize LS-HD counties for immediate capacity support, while treating LS-LD counties differently (e.g., monitoring and incremental improvement rather than emergency-type interventions), reducing the risk of misallocating scarce resources to low-demand contexts.

(3) Design interventions as bundles that match interaction enhancement (investment × capacity). The interaction results indicate that financial effort is most strongly associated with higher adequacy when paired with provisioning capacity (teacher supply and school capacity). Accordingly, policy options should emphasize bundled packages—for example, pairing PE funding increases with (i) teacher deployment/recruitment and (ii) facility/equipment maintenance capacity—rather than single-factor actions. This bundle logic aligns with QPE input complementarity and with the interaction enhancement identified in our analysis^2,15,16.

(4) Adopt area-based strategies in persistent Low-Low regimes. The persistence of contiguous Low-Low clusters suggests that county-level improvements are interdependent. In such settings, cluster-oriented delivery (e.g., shared training hubs, circuit-teacher arrangements, inter-county resource sharing, and coordinated facility access) is a practical option to weaken geographic lock-in and improve durability compared with isolated county-by-county initiatives.

(5) Institutionalize monitoring with adequacy- and cluster-based indicators. Given the relative stability of spatial patterns over 2021–2024, progress is likely to be gradual without sustained governance. A feasible approach is to embed the adequacy index and a small set of spatial indicators (e.g., LS-HD counts; Low-Low prevalence; prefecture-level gaps) into routine dashboards and to set time-bound targets for narrowing prefecture gaps and reducing persistent low-adequacy regimes. This supports iterative adjustment rather than one-off campaigns.

In other settings, the same policy logic can be mapped to the relevant intermediate governance tier (e.g., state/region/province) and local units (e.g., county-level units or municipalities), with demand proxies and adequacy indicators defined according to locally available administrative statistics. In sum, improving the geography of PE provision requires treating inequality as a systemic, multi-scale challenge: cross-jurisdiction equalization, local absorptive-capacity building, bundled intervention packages, and spatially targeted strategies are complementary and most effective when implemented in a coordinated manner.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

C.X. wrote the original draft. B.S. reviewed and edited the manuscript. Both authors approved the final version.

Funding

This work was supported by the National Social Science Foundation of China (Grant No. 19BTY084), the Shaanxi Provincial Social Science Fund Project (Grant No. 2025QBQ002), the Philosophy and Social Science Foundation of Shaanxi Province, China (Grant No. 2025QN0518), and the Scientific Research Program of the Shaanxi Provincial Department of Education, China (Grant No. 24JK0111).

Data availability

The raw indicators supporting this article are drawn from publicly available statistical yearbooks and official government reports cited in the References. Aggregated datasets and analysis code that support the findings of this study are available from the corresponding author upon reasonable request. Basemaps used to generate Figs. 2 and 4-6 were derived from the Standard Map Service of the Ministry of Natural Resources of the People’s Republic of China (http://bzdt.ch.mnr.gov.cn/), based on the standard map with review number GS(2024)0650 (accessed 22 Oct 2025). The maps are used for academic, non-commercial purposes, and national boundaries/base layers were not modified.

Code availability

Custom code used for data processing and analysis is available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Ethics approval and consent to participate

Not applicable. This study did not involve human participants, human data or tissue, nor animals, and no personally identifiable information was collected; therefore ethics committee approval and informed consent were not required.