2.1 Evaluation methods for higher education efficiency

Higher education efficiency has consistently been a primary concern in educational economics, with an emphasis on evaluating the relationship between scarce resource inputs and corresponding outputs. Methodologically, efficiency assessment approaches are broadly categorized into parametric and non-parametric techniques [20], with Stochastic Frontier Analysis (SFA) being the most widely applied among parametric methods [21–26], and DEA dominating the non-parametric sphere [27–28].

SFA involves constructing a cost or production function and estimating frontier parameters before application. However, determining a suitable functional form remains challenging, as assumptions about the selected functional form can substantially impact estimation results, leading to variability and potential inconsistencies in outcomes. At present, no definitive method or standard exists for identifying the “correct” or “appropriate” functional form [29]. Additionally, SFA is generally constrained to single-output scenarios, complicating its application in contexts with multiple inputs and outputs. This limitation is pertinent in higher education, where universities typically operate with complex, multi-dimensional input-output relationships, making the selection of an appropriate production function a non-trivial and often ambiguous task [30]. Therefore, while SFA serves as a valuable methodological tool, its applicability in multi-output systems, coupled with the complexities of selecting an appropriate functional form, continues to pose significant challenges.

In contrast, DEA circumvents the need for stringent assumptions regarding the functional form between inputs and outputs, while also avoids the subjective weighting of these variables [31–34]. Moreover, DEA’s dimensional invariance renders it particularly suitable for analyzing systems characterized by multiple inputs and outputs [35–40]. Based on the aforementioned features, DEA has been widely employed as a benchmarking tool across various sectors, including banking, healthcare, agriculture, transportation, and education [41]. Building on the groundbreaking work of Charnes et al. [27], DEA’s utility in evaluating educational systems, especially in higher education, has been extensively validated [42–55]. DEA method offers two key advantages in evaluating efficiency of HEIs. It allows for a multidimensional assessment of university performance, revealing critical insights into various aspects of efficiency [13]. Besides, DEA’s flexibility, derived from its non-reliance on a predefined production function, enables the pinpointing of areas for enhancement and the exploration of developmental potential.

DEA has developed numerous extended models, with the three-stage DEA model gaining recognition in higher education and other academic domains in recent years. This model’s enhanced capability to account for environmental variables and random noise generates efficiency estimates that reflect real-world conditions more accurately. Fuentes et al. [56] employed the three-stage DEA model to assess the technical efficiency of the learning-teaching process in higher education, finding contextual variables exerting a significant influence. Wu et al. [29] adopted the same model to measure the performance of HEIs across 31 provinces in China, effectively addressing measurement errors and environmental variables. Xue et al. [57] combined the three-stage DEA and the Malmquist index to measure the efficiency of scientific research in Chinese universities under the Ministry of Education, while incorporating external environmental factors. Their findings revealed that the three-stage DEA model provided more precise efficiency measurements for scientific research inputs and outputs than traditional DEA methods, with scale optimization identified as the main internal driver of efficiency enhancement. Beyond the education sector, the three-stage DEA has found widespread use in efficiency evaluations across fields, including energy efficiency [58], industrial eco-efficiency [59], green productivity growth in manufacturing [60], and cultural industry investment efficiency [61]. Across these studies, environmental variables consistently demonstrated significant influences on efficiency outcomes, underscoring the three-stage DEA model’s value for robust and accurate efficiency assessments.

2.2 Analytical units for higher education efficiency

Current research on higher education efficiency using the DEA method predominantly focuses on two main analytical units. The first, most prevalent approach adopts a micro-level perspective, treating universities or institutions as DMUs to examine the efficiency of teaching, research, and other institutional functions. This approach compares performance either across different universities or among various departments within individual institutions. The second approach takes a macro-level perspective, considering countries or regions as DMUs. This research compares HEI efficiency across different nations or regions within a single country, offering broader insights into national or regional educational systems.

Research considering universities as DMUs: Athanassopoulos and Shale [62] conceptualized cost and outcome efficiency to provide deeper understanding of university operations. The UK has been a pioneer in research on higher education efficiency, amassing numerous studies. Flegg et al. [63] examined the technical efficiency of 45 UK universities from 1980/81–1992/93, revealing that improvements in technical efficiency were largely driven by gains in pure technical efficiency, while scale efficiency had a smaller impact. Johnes [64] analyzed the efficiency of over 100 HEIs in the UK, finding that the average levels of both technical and scale efficiency were considerable, though marked disparities existed between the most and least efficient institutions. Casu and Thanassoulis [65] focused on cost efficiency within UK universities’ central administration, highlighting challenges in defining assessment units and input-output relationships. Glass et al. [66] calculated efficiency scores to inform policy evaluation and potential funding allocation within the higher education system in the UK. Bradley et al. [67] evaluated the efficiency and productivity changes across approximately 200 further education institution in the UK from 1999 to 2003. Beyond the UK, research on higher education efficiency extends to European countries, Australia, the United States, Latin America, and several Asian regions. In Germany, Kempkes and Pohl [30] utilized DEA and SFA to examine the efficiency of 72 public universities, concluding that eastern German universities outperformed their western counterparts in TFP change, despite lower average efficiency scores. Başkaya and Klumpp [68] conducted a comparative analysis of the input-output efficiency between public and private universities in Germany. In Greece, Katharaki and Katharakis [69] evaluated the efficiency of 20 public universities using performance indicators, DEA, and econometric techniques, identifying inefficiency in human resource management. In Poland, Nazarko and Šaparauskas [13] used a CCR-CRS output-oriented DEA model to analyze potential inputs, outputs, and environmental factors in 19 universities of technology, finding that institutional outcomes contributed more to enhancing output efficiency than the volume of resources available. In Italy, Guccio et al. [70] applied bootstrapped DEA algorithms to evaluate the effects of university reform on educational efficiency, showing progressive efficiency gains. In Australia, some scholars [71–72] utilized DEA to measure the relative, technical, and scale efficiency of universities, reporting relatively high overall efficiency levels. In the United States, Hirao [73] measured efficiency of the top 50 public and private business schools in 2006, noting that while both types demonstrated high technical efficiency, public institutions lagged in scale and overall efficiency. In Colombia, Visbal-Cadavid et al. [74] applied CCR, BCC, and SBM output-oriented DEA models to examine technical, pure technical, scale, and mix efficiency across public universities, further examining productivity changes from 2011 to 2012 using the Malmquist index. Navas et al. [75] investigated the efficiency of Colombian HEIs using a cross-efficiency DEA method, finding that certain institutions demonstrated high efficiency in teaching or employment, whereas others excelled in research. In Thailand, Kantabutra and Tang [76] employed DEA models focused on teaching and research efficiency to assess faculty-level efficiency at public universities, comparing performance differences between government-run and autonomous universities.

Some studies target the efficiency evaluation of various faculties and academic departments within universities. For instance, Leitner et al. [77] assessed the efficiency of natural sciences and engineering departments at Austrian HEIs with two input variables and 12 output variables, finding that both scale and pure technical effects contributed to the economic efficiency. Gimenez and Martinez [78] proposed a cost efficiency analysis model within the DEA framework and applied it to assess the efficiency of various departments at the Autonomous University of Barcelona.

Research considering countries or regions as DMUs: Studies of this type are relatively scarce, with the majority focusing on cross-national comparative analyses. Agasisti and Pohl [79] performed a comparative evaluation of the efficiency of public universities in Italy and Germany during the 2001–2007 period, finding that while German universities tended to be more efficient overall, Italian institutions exhibited significant efficiency improvements during this period. Wolszczak-Derlacz [80] evaluated the efficiency of HEIs in Europe and the United States, accounting for national-level heterogeneity. Agasisti et al. [81] conducted a meta-frontier analysis to compare the productivity between elite universities in China and Europe, revealing that Chinese universities demonstrated more rapid productivity growth than their European peers. Dincă et al. [82] assessed the educational efficiency of 28 European Union member states, concluding that the older member states consistently outperformed newer ones. Chen et al. [83] conducted a comprehensive assessment of educational efficiency and its dynamic changes across 29 major countries using the super-SBM model and the Malmquist index, highlighting substantial disparities in average educational efficiency, with Japan achieving the highest ranking and Norway the lowest. Moreover, the study identified technological progress as the key contributor to the growth of TFP in education. Jean-Pascal and Nicolas [84] measured the efficiency of educational diffusion and research productivity in the United States, considering geographical disparities between urban and rural regions, alongside distinctions between public and private universities. Their results indicated that urban universities placed greater emphasis on educational quality, while public universities demonstrated higher educational efficiency but lower research efficiency. Liao et al. [85] assessed the efficiency of educational resource use, productivity dynamics, and regional technological disparities across 35 European countries and four regions, using DEA Super-SBM, meta-frontier analysis, and the Malmquist productivity index.

In studies on China, several analyses have evaluated the efficiency of provincial-level HEIs and university research performance. Zhang et al. [86] performed a provincial-level analysis of dynamic changes in Chinese HEIs, proposing a DEA-based framework for optimizing the distribution of enrollment quotas. Sun et al. [16] undertook a dynamic evaluation of higher education efficiency across 30 provinces of mainland China, employing a meta-frontier Super-SBM model to explore regional disparities. The findings indicated that efficiency change was the primary driver of performance, with significant technology gap variations among the eastern, central, and western regions. In light of these disparities, the study recommended that the Chinese government should prioritize reducing regional imbalances in higher education. Liu et al. [87] applied the DEA super-SBM model, the Malmquist productivity index, and meta-frontier analysis to evaluate the efficiency, productivity changes, and technology gaps within the provincial higher education systems of China. Their findings indicated that provinces with lower literacy levels had higher efficiency than middle and high-literacy provinces, and efficiency change was the primary driver of productivity growth in higher education rather than technological advancements. Chen et al. [88] adopted a super-efficiency DEA model alongside the Malmquist index to assess the resource allocation efficiency of Chinese agricultural universities, also capturing dynamic evolution and identifying efficiency drivers.

Research on China’s higher education places a pronounced emphasis on research efficiency. For instance, Johnes and Yu [89] investigated the relative efficiency of research output across 109 regular universities in 2003 and 2004. Their study indicated that average research efficiency was greater in comprehensive universities than in specialized institutions, and in universities situated in coastal regions as opposed to those located in western China. Xiong et al. [90] and Ng & Li [91] measured the research performance of Chinese HEIs in humanities and social sciences, further proposing an efficiency decomposition by Malmquist productivity index. Wang et al. [92] adopted DEA-BCC model to evaluate and contrast scientific input-output efficiency of research-oriented universities in the Chengdu-Chongqing urban agglomeration, finding a marginal increase in average research efficiency. However, the study noted that the influence of scale on overall efficiency remained minimal, and there was a prominent misalignment between research themes, funding allocation, and human resources in the sampled universities. Wang and Hu [93] examined the overall and stage-specific research efficiencies of 18 HEIs in Jiangsu province.

2.3 Factors influencing higher education efficiency

With respect to methodologies for identifying the determinants of educational efficiency, scholars have used Ordinary Least Squares (OLS) [94], bootstrapped truncated regression [95–97], and the Tobit model [30,98]. Of these, the Tobit model is the most commonly adopted and considered highly robust, particularly advantageous for handling censored data and providing straightforward estimation results.

The input-out efficiency of HEIs is influenced by external and internal factors, broadly divided into external environmental influences and internal structural determinants [99–101]. Key external factors include regional economic development [30] and financial resources [102], while the primary internal factor is human capital [103–104]. Together, these factors critically affect HEIs’ operational efficiency and provide insights into institutional performance variations.

Regional economic development: Wolszczak-Derlacz [80] highlighted regional heterogeneity in the allocation of higher education resources across the United States and Europe, with affluent regions showing notably higher efficiency in resource use. Berbegal-Mirabent et al. [105] measured the resource allocation efficiency of Spanish universities and highlighted regional economic activity as a key determinant. Similarly, Kempkes and Pohl [30] found a positive correlation between per capita GDP and higher education efficiency in German HEIs. Chen et al. [106] further demonstrated a significant positive relationship between regional economic environment and both TFP and technical efficiency in China’s “double first-class” universities. Consistent with these findings, some studies argued that regional economic development provided strong financial support for enhancing university efficiency [107–108]. Nevertheless, other studies suggest that neither per capita GDP nor regional economic strength exerts a significant or clearly synchronous impact on higher education efficiency [101,109]. Moreover, when resource allocation structure is irrational, further regional economic development may exacerbate issues such as redundant inputs or inadequate outputs [110–111].

Financial resources: Grounded in public goods theory, universities are recognized as key nonprofit institutions, with government financial support serving as a critical material foundation to alleviate resource constraints and foster institutional development [112–113]. Robst [114], utilizing the SFA model in his analysis of higher education cost-efficiency, identified a significant positive relationship between government appropriations and cost efficiency. Similarly, Klumpp [115], through the application of the DEA–Malmquist model, confirmed that increased funding inputs positively influenced the international university ranking. Wolszczak-Derlacz and Parteka [101] demonstrated that a higher proportion of external funding significantly enhanced institutional efficiency in their study of 259 public HEIs across seven European countries. Conversely, Chen et al. [106] reported an inverse relationship between regional financial support and university TFP. Fadda et al. [116] further argued that government funding often followed a differentiated approach, which might result in opportunistic behavior or the Matthew effect, undermining HEIs’ incentives to optimize resource allocation efficiency.

Human capital: The faculty title structure is a widely recognized indicator for assessing the operational and developmental status of HEIs [117]. Chen and Yang [118] argued that teachers with senior academic titles significantly contributed to talent cultivation and scientific research due to their advanced expertise and extensive research resources, thereby enhancing university efficiency. In the same vein, Salas-Velasco [97], through a two-stage semi-parametric approach with single and double bootstrap procedures, found that universities with a higher proportion of professors among their academic staff exhibited greater efficiency. Chen and Shu [55], using the super-efficiency DEA model, Malmquist index, and Tobit-DEA model, analyzed the determinants of scientific and technological innovation efficiency. Their findings suggested that a higher proportion of senior full-time faculty, an optimized resource allocation structure, as well as government support positively influenced innovation efficiency. In the Chinese academic community, numerous studies have confirmed that the proportion of full-time faculty with senior ranks significantly enhanced the technical efficiency of HEIs [103,119]. However, studies focusing on China’s western regions indicate that the proportion of employees with tertiary education or above, as a proxy for human capital, negatively affects the effectiveness of public investment and the higher education efficiency in these regions [120–121].

Other factors: Oliviera and Santos [95] analyzed the efficiency of 42 public schools in Portugal and found that regional unemployment rate exerted a negative influence on school efficiency. Wolszczak-Derlacz and Parteka [101] concluded that a higher proportion of female faculty members significantly enhanced institutional efficiency. Hendin [122] suggested that policy implementation was essential in facilitating university resource acquisition. The government utilizes policy instruments to bolster higher education by redistributing resources to universities in less developed regions and particular kinds of institutions [123]. However, Nisar [124] argued that universities’ responses to policies often followed a nonlinear and unpredictable pattern, making the concrete impact of policy enactment difficult to predict. Additionally, several Chinese studies demonstrate that indicators such as the proportion of operating expenses to total educational expenditures, regional industrial development levels, and per-student education funding exhibit significant positive or negative correlations with the overall efficiency of higher education [119,125,126].

Drawing on prior research, this study offers supplementary contributions, improves methodologies, and provides novel insights from a research perspective. (1) Expanding the research perspective. The majority of previous studies concentrates on macro- or micro-level analyses of higher education efficiency. Macro-level research typically compares efficiency disparities across different countries or regions, while micro-level studies concentrate on efficiency differences among universities or departments. However, there is a notable scarcity of research at the meso-level, particularly regarding the higher education efficiency across the four major regions of China (Eastern, Central, Western, and Northeastern). Consequently, this study fills the gap in meso-level research by undertaking a thorough analysis of the input-output efficiency of higher education in western China, further uncovering internal disparities and their underlying causes. This analysis aims to provide policy recommendations to mitigate the deficiencies in China’s higher education system and promote more balanced development. (2) Enhancing the research methodology. Most existing studies employ traditional DEA models to evaluate higher education efficiency. In contrast, the three-stage DEA model offers significant advantages over traditional models by effectively eliminating the influence of environmental factors and random noise on efficiency, thereby yielding more accurate efficiency estimates. However, the application of the three-stage DEA method in the context of regional higher education research in China has been insufficient. Therefore, this study utilizes the three-stage DEA model to improve the precision of efficiency estimates for higher education in western China, ultimately providing more accurate, effective, and targeted recommendations for regional higher education management. (3) Supplementing dynamic efficiency research. Current literature predominantly focuses on static efficiency in higher education, with limited attention to dynamic changes in efficiency. As Parteka and Wolszczak-Derlacz [15] pointed out, dynamic changes in HEIs’ productivity have only been marginally analyzed. Because the DEA-CCR, DEA-BCC, and network DEA models are predominantly designed for assessing static efficiency, they are ill-suited for capturing the dynamics of efficiency changes over time [127–128]. This insurmountable shortcoming renders singular models, such as SFA or DEA, insufficient to fully address the increasingly complicated and precise computational demands in contemporary higher education [106]. As a result, scholars have begun to augment these models by incorporating Malmquist index models [129–130]. This study advances the literature by incorporating the global Malmquist index to systematically analyze efficiency changes over time, which provides a thorough grasp of the dynamic trajectories and underlying drivers of higher education efficiency in western China. (4) Providing new evidence on discrepancies in efficiency influencing factors. There is considerable divergence in the academic discourse surrounding the factors influencing higher education efficiency. Existing studies primarily focus on factors such as regional economic development levels, government financial support, and the internal human capital structure of universities, often in the context of developed countries or regions. However, for the relatively underdeveloped and resource-constrained western China, it is crucial to investigate whether the influencing factors and mechanisms discussed in existing literature are consistent with those in developed regions. This study conducts a multi-faceted and comprehensive analysis of the factors affecting higher education efficiency, taking into account the unique characteristics of western China, thereby providing new evidence regarding the determinants of higher education efficiency in western China.

3.1 Three-stage DEA model

Data Envelopment Analysis (DEA) is proposed by American operations researchers Charnes et al. in 1978, aimed at quantifying the relative efficiency of DMUs with multiple inputs and outputs of the same type [27]. This non-parametric approach uses linear programming techniques to systematically assess the relative efficiency of DMUs [131]. The DEA model spatially maps input-output points, establishing boundaries based on minimal input or maximal output, known as the best practice production frontier. This framework functions as a benchmark for evaluating the distance of other points from the frontier, thereby embodying the concept of relative efficiency by measuring the proximity of input-output points to the best practice production frontier [97].

However, the traditional DEA model has limitations in efficiency evaluation, such as its inability to account for external environmental factors, random noise, and its reliance on static analysis [29]. Fried et al. [17] explain that the efficiency of DMUs can be affected by three distinct phenomena: the efficiency with which production activities are organized by management, the characteristics of the environment in which production activities are carried out, and the stochastic impacts of luck, omitted variables, and related phenomena, which can be captured in a random error term in a regression-based evaluation of DMUs’ efficiency. To accurately disentangle the three influences—managerial inefficiencies, environmental effects, and statistical noise—comprehensive data on environmental characteristics, inputs, and outputs are essential. Moreover, capturing the stochastic nature of luck requires the use of stochastic modeling approaches; however, it is noteworthy that most DEA models, particularly those adopted in previous studies, are deterministic, thus failing to adequately address the inherent stochastic complexities of DMU efficiency assessments. In light of this, Fried et al. [17] put forward a three-stage DEA model in 2002, building upon the traditional DEA framework.

Fried’s approach consists of three stages: In the first stage, the traditional DEA model is applied to calculate the initial efficiency scores of DMUs along with various input slacks. The second stage establishes a SFA regression model [21–22], where input slacks are the dependent variables and external environmental factors are the independent variables. This model adjusts the inputs from the first stage to account for environmental influences and random errors, ensuring that the adjusted DMUs are subjected to identical external environment and stochastic components. In the third stage, the adjusted inputs from the second stage, along with the original outputs, are utilized to recompute the efficiency scores of each DMU using the traditional DEA model. This re-evaluation provides improved metrics of managerial efficiency, as the second stage has effectively removed environmental influences and statistical noise from the input data of DMUs. The primary advantage of this methodology lies in its capacity to eliminate the effects of non-managerial factor—namely, environmental influences and random noise—on efficiency, while preserving the managerial inefficiencies. This ensures that the final efficiency scores accurately reflect the internal managerial efficiency of the DMUs.

3.1.1 First stage: Initial DEA efficiency evaluation.

In general, DEA models include the Charnes–Cooper–Rhodes (CCR) model and the Banker–Charnes–Cooper (BCC) model, with the former assuming constant returns to scale (CRS) [27], and the latter assuming variable returns to scale (VRS) [132–133]. The CCR model measures overall technical efficiency (TE), incorporating both scale efficiency (SE) and pure technical efficiency (PTE) simultaneously, with TE = SE × PTE, while the BCC model focuses solely on PTE evaluation.

In period t (t = 1, ⋯, T), there are I DMUs (DMU_i^t, i = 1, ⋯, I). Each DMU has N inputs and M outputs, where x^t and y^t represent the input and output vectors, respectively. These are defined as , and . The production possibility set for a given DMU during period t is defined as T^t, and traditional DEA production technology is defined as follows:

(1)

In equation (1), λ = (λ₁^t, λ₂^t, ⋯, λ_I^t) represents the I-dimensional weight vector employed in assessing TE. Equation (1) demonstrates that traditional DEA measures production efficiency of DMUs based only on the production technology set T^t corresponding to the current period. Traditional DEA constructs the production technology set T^t using input-output data of cross-sectional DMUs for a single period. Consequently, when calculating the efficiency of DMUs across different periods (t = 1, 2, ⋯, T), distinct technological reference sets (T = T¹, T², ⋯, T^T) are employed, which renders the efficiency values of DMUs across different periods non-comparable.

To address this limitation and enhance the comparability of efficiency across different periods, while also facilitating a longitudinal analysis of efficiency variations for each DMU, Pastor and Lovell [134] introduce a frontier construction method based on global benchmark technology, which forms the basis of global DEA. It utilizes panel data on inputs and outputs of all DMUs across all periods to establish a single production possibility set, serving as a common reference for production technology that encompasses all contemporaneous benchmark technologies [135]. Efficiency outcomes for different DMUs across various periods are then computed relative to this common frontier, thus forming a global DEA framework.

In this framework, T^G represents the global production technology set and T^t denotes the production technology set for each individual period. The construction of the global production technology set is as follows:

(2)

The linear programming formulation of the input-oriented global DEA method based on this global production technology set is presented below:

(3)

Here, denotes the weight variable. Equation (3) is based on CRS. When the constraint is added, the model corresponds to VRS.

Another issue that needs clarification here is the number of DMUs. Generally, the number of DMUs should not be less than the product of the number of input and output indicators, nor should it be less than three times the sum of these indicators [136]. However, this is a rough guideline, and the actual number of DMUs should be assessed based on the results of the DEA analysis [137]. In this study, the number of DMUs appears to be 12 (provinces). However, when considering the time dimension (13 years) and the resulting panel data, the number of DMUs increases to 156 (12 × 13) in the global DEA model. Taking the technical efficiency of higher education input-output as an example, three sets of first-stage DEA results are presented in S7-S9 Tables. S7 Table matches the 12 provinces with the 13 years, numbering the matching results from 1 to 156. Therefore, the number of DMUs included in the model is 156, and TE values are calculated using the traditional DEA model. S8 Table uses the 12 provinces as DMUs and calculates TE values using the global benchmark technology (global DEA model). S9 Table also uses the 12 provinces as DMUs and calculates TE values using the traditional DEA model. It can be observed that the TE values in S7 Table and S8 Table are identical, and the distinction between the TE values is much greater than in S9 Table, indicating a more even distribution of TE values. Therefore, it can be concluded that in this study, the global DEA method is more robust and superior to the traditional DEA, and when global DEA is applied, the number of DMUs is expanded from 12 to 156 (provinces × years). Consequently, the final number of DMUs in this study is 156.

3.1.2 Second stage: Utilizing SFA to decompose initial input slacks.

The second-stage analysis aims to disaggregate the input slacks from the first stage into three distinct components: environmental influences, managerial inefficiencies, and statistical noise resulting from measurement errors in data used to derive the initial slacks. Thus, in the second stage, each input slack is treated as a dependent variable, with K observable environmental variables Z_i serving as independent variables. For each input slack, an individual SFA regression equation is constructed, resulting in a total of N regression equations. By performing regression analysis, external environmental influences and stochastic factors are filtered out, isolating the input redundancies attributable solely to managerial inefficiencies for each DMU. The regression equation for the n-th input slack is as follows:

(4)

where S_ni denotes the first-stage slack for the 𝑖-th DMU on the n-th input, representing the discrepancy between the optimal and actual input levels. Z_i denotes the environmental variables, where Z_i = [Z_1i, ⋯, Z_Ki]. βⁿ is the parameter vector associated with these environmental variables. fⁿ(Z_i;βⁿ) represents the deterministic feasible slack frontiers, capturing the influence of environmental factors on S_ni, and is typically defined as fⁿ(Z_i;βⁿ) = Z_iβⁿ. However, this relationship contains noise. [fⁿ(Z_i;βⁿ) + v_ni] represents the stochastic feasible slack frontier, indicating the minimum achievable slack under the noise−influenced environment described by variables (Z_i, v_ni) and parameters (β_n, σ_un²). Any slack exceeding [fⁿ(Z_i;βⁿ) + v_ni] is attributed to managerial inefficiency. The composite error term (v_ni+ u_ni) comprises v_ni, representing statistical noise, assumed to follow a normal distribution v_ni~ N(0, σ_vn²), and u_ni, representing managerial inefficiency, with u_ni ≥ 0, following a truncated normal distribution u_ni ~ N⁺(μ_n, σ_un²). The terms v_ni, u_ni and Z_i are assumed to be mutually independent. Each of the N regressions in equation (4) can be estimated by maximum likelihood techniques.

Subsequently, the results of the regression analysis derived from the SFA model are employed to further adjust the input variables of the DMUs. This adjustment eliminates the influence of environmental and stochastic factors on efficiency, thereby aligning all DMUs within a common external environment. Below is the adjustment formula:

(5)

Here, and X_ni represent the adjusted inputs and pre-adjustment inputs, respectively, accounts for adjustments to environmental factors, signifies the adjustment of all DMUs to a standardized operating environment, specifically the least favorable environmental context identified within the sample. Meanwhile, represents standardizing all DMUs to a uniform condition of “nature”, corresponding to the unluckiest situation observed in the dataset. Consequently, DMUs operating under relatively unfavorable environments and/or bad luck receive upward input adjustments of a smaller magnitude, while those in more favorable environments and/or enjoying good luck experience larger upward adjustments to their inputs. This ensures that all DMUs are adjusted to a uniform environmental level. addresses adjustments for stochastic error components, thus positioning all DMUs on an equivalent level of stochastic influence.

The SFA model is typically analyzed using Frontier 4.1 software, and the output file from the regression results contains estimated values for parameters such as βⁿ, σ² and γ corresponding to each slack variable, from which σ_vn and σ_un can be derived based on equation (6).

(6)

The parameter γ represents the proportion of total variance attributable to managerial inefficiency variance.When γ approaches 1, the influence of managerial inefficiency predominates, while a γ value approaching 0 indicates that statistical noise is the dominant influence.

To implement equation (5), v_ni must be obtained for each DMU by separating statistical noise from managerial inefficiency in the residuals of equation (4). Following the methods proposed by Luo [138] and Kumbhakar and Lovell [139], the managerial inefficiency u_ni can be isolated, as expressed by the following equation:

(7)

where σ_*² = σ_un²σ_vn²/ σ², λ = σ_un/ σ_vn, and ε = v_ni+ u_ni is the composite error term. The function φ (·) and Ф (·) denote the density function and cumulative distribution function of the standard normal distribution, respectively.

After isolating the inefficiency component u_ni, the random disturbance v_ni can be obtained, calculated as follows:

(8)

Finally, by substituting v_ni into equation (5), the adjusted input variables are obtained.

3.2 Global Malmquist productivity index

Total Factor Productivity (TFP) is commonly used to assess the contribution of the overall efficiency of input factors to output growth. In the analysis of TFP changes, the Malmquist Total Factor Productivity Index (MI) is decomposed using the DEA method. The concept of the MI was originally introduced by Swedish economist and statistician Sten Malmquist in 1953, and it was subsequently developed by Caves, Färe, and others. Caves et al. [140] proposed employing the MI to compare the performance of a DMU with itself across different periods of time. Färe et al. [141] used DEA models to compute the MI, subsequently decomposing it into two principal components: technical efficiency change (EC) and technological change (TC). Since then, DEA-based MI has been extensively utilized for measuring changes in productivity. This approach employs a distance function to represent the production process with multiple inputs and outputs, thereby circumventing problems associated with subjective weight assumptions, since it is not dependent on a particular specification of the production function [140,142,143].

The MI for each DMU is defined as the ratio of its efficiency score at time t + 1 to that at time t, with both scores being computed relative to a designated benchmark technology [144]. However, when considering the production possibility set, the MI uses a contemporaneous production possibility set based on data observed within the same time period, which lacks circularity. Its adjacent period components can yield inconsistent assessments of productivity change. To address the limitations of non-circularity and disparate measurements, Pastor and Lovell [134] introduced a global Malmquist productivity index (GMI), where the production possibility set (or production frontier) is constructed using data from all DMUs across all periods. By adopting a common reference technology that spans all periods, the technical changes calculated between period t and t + 1 are benchmarked against the production technology of all periods, thereby ensuring the comparability of inter-temporal performance changes across the evaluated DMUs. This GMI satisfies circularity, offers a unified measure of productivity change, and is immune to the issue of infeasibility [19,134,145]. Moreover, technological change or frontier shift can be regarded as a global phenomenon, driven by factors such as changes in economic conditions or the introduction of new, more efficient technologies. In many cases, it is reasonable to assume that these factors are either uniform or, at the very least, highly analogous across all observations in an analysis. As a result, a single metric can be employed to represent the shift in the frontier for all DMUs [146]. Furthermore, the GMI expands the set of DMUs incorporated into the analysis, improving the precision of the frontier, particularly when the number of DMUs is limited. This enhancement increases the reliability and stability of results. In addition, by incorporating a global benchmark technology in the three-stage DEA model to construct the frontier, the use of the GMI becomes indispensable for accurately assessing productivity changes.

According to the definitions and formulas provided by Pastor and other scholars [134,147], consider a panel dataset with = 1, ⋯, I DMUs observed over t = 1, ⋯, T time periods. DMUs utilize inputs to generate outputs . The production technology at time t is represented by .

A global Malmquist productivity index is defined on as:

(9)

Here, the output distance functions . D^G (x^t+1, y^t+1) represents the DEA efficiency score of (x^t+1, y^t+1) when benchmarked against the global frontier, and D^G (x^t, y^t) denotes the DEA efficiency score of (x^t, y^t), similarly benchmarked against the global frontier. Since the GMI employs a single global benchmark technology, constructed from all periods as the reference frontier for all sample observations, the calculation does not require the use of a geometric mean [134,148].

By further decomposing the equation (9), the GMI can be separated into the EC and TC indices.

(10)

(11)

(12)

Specifically, D^G(x^t+1, y^t+1)/D^t+1(x^t+1, y^t+1) quantifies the distance between the period t + 1 frontier and the global frontier, where a greater ratio denotes a closer alignment between the t + 1 period frontier and the global frontier. Similarly, D^G(x^t, y^t)/D^t(x^t, y^t) represents the distance between the frontier at period t and the global frontier, with a higher ratio suggesting greater proximity of the period t frontier to the global frontier. The ratio of these two terms captures the relative shift of the period t + 1 frontier compared to the period t frontier. TC refers to the advancement of production technology, which shifts the production frontier outward, allowing for higher output with the same input levels, and reflects impacts relevant to the higher education system as a whole. This suggests that technological progress plays a significant role in improving production efficiency. EC captures variations in the relative efficiency of a DMU, reflecting the extent to which a DMU approaches the production frontier, and it can be further disaggregated into two components: pure technical efficiency change (PEC) and scale efficiency change (SEC), as expressed below:

(13)

GMI is finally expressed as GMI = PECSECTC. When the GMI > 1, total factor productivity improves, indicating increased output efficiency under the same input conditions. when the GMI < 1, total factor productivity declines, reflecting reduced production efficiency. When GM = 1, TFP remains constant.

3.3 Tobit model

The Tobit model is specifically developed to address the issue of censored dependent variables. Parameters in this model are estimated through maximum likelihood estimation, enabling it to effectively handle observations where efficiency scores are either 0 or 1, thereby ensuring the robustness and consistency of the estimation results. In this study, the dependent variable is the DEA efficiency score, adjusted in the third stage, ranging between (0, 1]. Employing the ordinary least squares method for parameter estimation may lead to distortions in the results [149–150]. Given the bounded nature of the dependent variable, the Tobit model is used to address the censoring of efficiency scores, allowing for precise estimation of factors influencing efficiency and ensuring unbiased, consistent estimates. It is assumed that the latent efficiency score y_it^* is an unrestricted continuous variable, while the observed efficiency score y_it is a truncated variable. The Tobit model is represented in the following form [151]:

(14)

where x_it represents the vector of independent variables, encompassing various factors influencing efficiency; β denotes the regression coefficients to be estimated; and ε_i is the error term, with ε_i ~ N (0, σ²).

4.1. Input–output variables

Following the overarching theoretical framework for efficiency measurement, this study develops an evaluation index system for the input-output efficiency of higher education from both input and output dimensions. Widely recognized in academia, the input indicators encompass financial, human, and physical resource inputs [152], while the output indicators are primarily structured around the three core functions of higher education: talent cultivation, research outcomes, and social service [88]. Building on prior research [87,119,153–155], as detailed in S1 Table, this study develops a comprehensive evaluation index system for higher education input-output efficiency, comprising seven input and seven output indicators, as presented in Table 1.

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Table 1. Indicators of Input-Output in Higher Education.

https://doi.org/10.1371/journal.pone.0325901.t001

4.2 External environmental variables

In the second phase of the three-stage DEA, to obtain more accurate and true efficiency values, an SFA regression on input slack is required to eliminate the effects of external environmental factors and statistical noise on the efficiency scores obtained in the first stage. According to the model of Battese and Coelli [156–157], these environmental variables denote external factors that impact SFA technical efficiency and are incorporated into the variance of the SFA technical inefficiency distribution. Therefore, for the external environmental factors entered into the SFA regression model, this study selects those variables that significantly impact input slack in higher education, specifically: per capita Gross Regional Product (GRP), the proportion of government fiscal expenditure on education, urbanization rate, density of highways and railways, and the proportion of the tertiary industry in GRP. These variables comprehensively assess the influence of the external environment on input slack (managerial inefficiency) in higher education from the perspectives of macroeconomic conditions, government investment, urbanization, infrastructure, and labor market demand.

4.3 Determinant variables in the Tobit model

In Section 2.3, “Factors influencing higher education efficiency”, a systematic review of the factors affecting input-output efficiency in higher education has been conducted. Grounded in the theoretical foundations of existing literature and considering the unique context of western China, this study constructs an analytical framework from three dimensions: the regional socio-economic environment, human capital, and educational funding.

In the dimension of the regional socio-economic environment, the relatively slow economic development in western China means that local government fiscal revenue directly impacts its capacity to support higher education. Therefore, the proportion of regional fiscal revenue to gross regional product (RFR/GRP) is selected as an indicator to assess the level of local economic development and its support for higher education.

In the dimension of human capital, given the relative lag in the recruitment and development of high-level talent within western HEIs, the structure of academic titles serves as a direct indicator of the quality of internal human capital. Accordingly, this study selects the proportion of full-time faculty with senior and associate senior academic titles (SAS/FTF) as a representative indicator of internal human capital level in HEIs, reflecting the overall academic and research capacity of the faculty. At the same time, the proportion of employees with college education or above (PECE) is used to assess the overall level of external human capital in western China, indicating the capacity of the regional education system to supply qualified labor to the workforce.

In the dimension of educational funding, per-student education expenditure (PSEE) serves as a key indicator of regional educational resource investment intensity, directly reflecting the allocation of resources. This metric is particularly significant in western China, where economic underdevelopment and uneven resource distribution prevail. For HEIs in the region, which operate under relatively constrained resources, efficient allocation of funding is essential for improving institutional efficiency. Within this context, the proportion of higher education operating expenses to total education expenditure (HEOE/TEE) further reveals the structural characteristics of funding allocation, assessing the efficiency of financial resources directed toward routine teaching and research activities.

4.4 Data description

The study spans from 2010 to 2022 and includes 12 provinces, municipalities, and autonomous regions in China’s western region, specifically: Chongqing Municipality, Sichuan, Yunnan, Guizhou, Shaanxi, Gansu, Qinghai, Guangxi Zhuang Autonomous Region, Tibet Autonomous Region, Ningxia Hui Autonomous Region, Xinjiang Uygur Autonomous Region, and Inner Mongolia Autonomous Region. Data on higher education input and output variables are sourced from Educational Statistics Yearbook of China, China Statistical Yearbook on Science and Technology, China Educational Finance Statistical Yearbook, and Compilation of Science and Technology Statistics of Higher Education Institutions, spanning from 2010 to 2023. Specifically, data for X₃, X₆, X₇, Y₁, Y₂, and Y₃ are derived from the Educational Statistics Yearbook of China, X₂, X₄, X₅, Y₅, and Y₆ from the China Statistical Yearbook on Science and Technology, Y₄ and Y₇ from the Compilation of Science and Technology Statistics of Higher Education Institutions, and X₁ from the China Educational Finance Statistical Yearbook.

The data for external environmental variables and influencing factors are sourced from various yearbooks spanning from 2010 to 2023. Regional fiscal revenue, GRP, per capita GRP, urbanization rate, proportion of tertiary industry in GRP, and highway and railway mileage are obtained from the China Statistical Yearbook; data on the proportion of government fiscal expenditure on education and the proportion of regional fiscal revenue to GRP come from the Finance Yearbook of China; per-student education expenditure and the proportion of higher education operating expenses to total education expenditure are sourced from China Educational Finance Statistical Yearbook; the proportion of full-time faculty with senior and associate senior academic titles is obtained from the Educational Statistics Yearbook of China; and the proportion of employees with college education or above is derived from the China Labor Statistical Yearbook.

To obtain a deeper insight into the distributional characteristics and fundamental statistical properties of the higher education input and output variables, this study conducts descriptive statistical analysis on seven input variables(X₁–X₇) and seven output variables (Y₁–Y₇). The analysis includes sample size, mean, standard deviation, minimum, maximum, and coefficient of variation. The descriptive statistics for each relevant variable are presented in Table 2.

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Table 2. Descriptive Statistics of Higher Education Input-output Variables.

https://doi.org/10.1371/journal.pone.0325901.t002

5.1 Empirical results of three-stage DEA

5.1.1 First stage: Input-oriented VRS DEA model results.

To begin with, without accounting for environmental and random factors, a DEA model was constructed based on the selected input and output indicators, using a global benchmark technology to establish the production frontier. This model enabled an initial analysis of the input-output efficiency of higher education across 12 provinces in western China from 2010 to 2022. The model was solved using MaxDEA software, and the results are presented in Table 3.

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Table 3. Initial Technical Efficiency of Higher Education in Western China (2010-2022).

https://doi.org/10.1371/journal.pone.0325901.t003

Technical efficiency reflects the overall efficiency level of each DMU given the established resource inputs. Based on the average technical efficiency scores of the 12 provinces in western China from 2010 to 2022, Chongqing and Shaanxi rank first, each with an average technical efficiency score of 1, signifying DEA-efficient units. Additionally, both Chongqing and Shaanxi show zero slack variables, indicating no redundant inputs or insufficient outputs, which suggests that higher education input-output in these two provinces has achieved strong DEA efficiency, maximizing output under current input conditions. Following them are Yunnan (0.9867) and Sichuan (0.9832), both approaching DEA-efficient levels. Provinces with lower technical efficiency include Guizhou (0.8550), Ningxia (0.8858), and Tibet (0.9005), indicating certain deficiencies in the allocation and management of higher education resources in these regions.

Overall, the input-output technical efficiency of higher education in western China is relatively favorable, with high technical efficiency observed in Shaanxi, Chongqing, Yunnan, and Sichuan. However, there exists an imbalance in input-output technical efficiency across provinces, with Guizhou, Ningxia, and Tibet exhibiting lower levels of efficiency. Notwithstanding this disparity, the difference between the provinces with highest and lowest average technical efficiency scores is not particularly pronounced, with the maximum difference being 0.145 points.

5.1.2 Second stage: SFA regression results.

The slack variables of the seven input variables selected in the first stage of measurement were used as dependent variables, while five environmental variables—per capita GRP, the proportion of government fiscal expenditure on education, urbanization rate, density of highways and railways, and the proportion of tertiary industry in GRP—served as independent variables. Regression analysis was conducted using Frontier 4.1 software, with results shown in Table 4.

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Table 4. SFA Regression Results.

https://doi.org/10.1371/journal.pone.0325901.t004

The one-sided generalized likelihood ratio (LR) statistic has a critical value of 14.325 at the 1% significance level. As presented in the Table 4, the LR values for all seven models exceed 14.325, indicating that each of the seven slack variables passed the 1% significance test, thereby rejecting the null hypothesis of no managerial inefficiency. This suggests that managerial inefficiency significantly impacts each of the seven input variables, confirming the presence of efficiency losses in the input-output process of higher education in western China. Therefore, it is both reasonable and essential to further examine the relationship between external environmental factors and higher education input-output efficiency by constructing a SFA model.

In the regression results, the γ values are all greater than 0.9, indicating that managerial inefficiency plays a dominant role in input-output efficiency. The input slack variables represent the amount of input that could be reduced by improving management levels. In the regression model, a negative regression coefficient implies that an increase in the value of an environmental variable helps reduce input slack, thereby decreasing input waste and enhancing resource utilization efficiency. On the other hand, a positive regression coefficient indicates that an increase in the environmental variable raises input slack, leading to input waste or reduced output, thus impeding effective resource utilization. The subsequent section presents an analysis of the impact of each environmental variable on input slack.

Per capita GRP has a significant negative impact on the slack in general public budget education expenditure of HEIs, the number of HEIs, and campus area, indicating that an increase in per capita GRP helps reduce redundancies in these input resources. However, it exerts a significant positive impact on the slack in intramural expenditures on R&D in higher education and the total fixed assets value of HEIs, implying that an increase in per capita GRP leads to waste in these types of inputs. Thus, while growth in per capita GRP can optimize the allocation of certain educational resources, it may also exacerbate redundancy in others.

The proportion of government fiscal expenditure on education, the urbanization rate, and the proportion of tertiary industry in GRP all have a significant positive effect on the slack of all input variables. This indicates that increased government expenditure on education, a higher level of urbanization, and the development of the tertiary industry exacerbate the waste of higher education resources, leading to relatively reduced educational output.

The increase in highway and railway density has a significant impact on the allocation of higher education resources, particularly by enhancing the accessibility of the transportation network and reducing the distance between remote or underdeveloped regions and other areas. As shown in Table 4, improvements in transportation infrastructure have a significant negative effect on several higher education input slacks, notably reducing redundancy in general public budget education expenditure, intramural expenditures on R&D, full-time equivalent of R&D personnel, campus area, and total fixed assets value of HEIs. This suggests that enhanced transportation accessibility improves resource mobility and personnel flow, optimizing the allocation of higher education resources and reducing unnecessary input waste. However, an increase in highway and railway density has a significant positive impact on the slack in the number of HEIs. Although improved transportation fosters regional economic development and population mobility, potentially leading to the establishment of more institutions within the region, it does not necessarily guarantee the effective utilization of these new institutions. Population mobility and uneven distribution in western China may result in a mismatch between the increase in the number of HEIs and actual educational demand, thereby exacerbating redundancy in institutional numbers.

The above analysis indicates that various environmental variables have significant impacts on input slack in higher education, though their mechanisms of influence differ, resulting in varying degrees of redundancy across different types of input variables. This variation can affect the scientific validity of input-output efficiency results. This result aligns with the findings of Wu et al. [29], whose study on the efficiency of HEIs across 31 provinces in China similarly concluded that environmental variables and stochastic noise exerted a significant influence on the performance of DMUs. Therefore, when measuring the input-output efficiency of higher education, it is imperative to adjust input variables by accounting for the impact of external environmental and stochastic factors, ensuring that all provinces in western China face the same environmental and random conditions. Recalculating higher education input-output efficiency under these standardized conditions helps to ensure the fairness and accuracy of the efficiency measurements.

5.1.3 Third stage: Efficiency adjusted for environmental influences and random noise.

According to the regression results from the second phase, the original input variables were scientifically and systematically adjusted, eliminating the impacts of external environmental factors and random noise. The adjusted input variables, along with the original output variables were then defined as the input and output variables for the third phase. These variables were re-evaluated using the DEA model from the first phase to obtain the adjusted efficiency values for higher education input-output in western China. This adjusted efficiency value more accurately reflects the true input-output efficiency of higher education in this region.

1. (1). Analysis of provincial average technical efficiency.

Table 5 presents the technical efficiency values of each province in western China from 2010 to 2022.

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Table 5. Adjusted Technical Efficiency of Higher Education in Western China (2010-2022).

https://doi.org/10.1371/journal.pone.0325901.t005

In terms of average technical efficiency, provinces such as Shaanxi (1), Chongqing (0.9946), Sichuan (0.9928), and Gansu (0.9772) exhibited relatively high average efficiency values. This indicates that these provinces effectively convert existing resources and technology into outputs, demonstrating high levels of resource utilization and management. It suggests that higher education in these provinces is near the production frontier, with output levels approaching the optimal values for the current inputs, thus leaving limited room for further efficiency improvements. Notably, Shaanxi, with an average technical efficiency score of 1, showed no efficiency loss in its higher education sector, with no redundancy in inputs or outputs, indicating strong DEA efficiency. In contrast, provinces such as Tibet (0.3828), Qinghai (0.5648), Ningxia (0.7286), and Guizhou (0.8416) had lower scores, implying that higher education in these regions have not effectively transformed inputs into outputs under the current resource and technological conditions. These regions have yet to reach the production frontier, revealing issues of resource wastage and insufficient management efficiency that urgently require improvement.

1. (2). Decomposition analysis of technical efficiency.

To further analyze the specific components of input-output efficiency in higher education within western China and to objectively assess whether the allocation of inputs and outputs is near an optimal configuration, the study applies the decomposition formula “TE = PTE × SE” to examine the average values of technical efficiency, pure technical efficiency, and scale efficiency for western China from 2010 to 2022. In addition, the study compares the results from the first and third phases. The decomposition results are presented in Table 6, S4 Table and S5 Table.

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Table 6. Decomposition of Technical Efficiency in First and Third Stages.

https://doi.org/10.1371/journal.pone.0325901.t006

Overall, the pure technical efficiency (average of 0.9761) of higher education in western China during the third stage demonstrated a favorable performance. However, both technical efficiency (average of 0.8510) and scale efficiency (average of 0.8707) remained relatively low, indicating issues of suboptimal resource allocation and inadequate management within higher education, with scale inefficiency further constraining potential improvements in technical efficiency. Following adjustments to the input variables, DMUs with initially lower efficiency scores saw further declines, suggesting that the disparity in input-output efficiency across higher education in western China widened after controlling for external environmental effects.

Analyzing the adjusted technical efficiency values, Tibet, Qinghai, and Ningxia notably fell below the mean, reflecting substantial inefficiencies related to management or scale limitations in their higher education sectors. The adjusted pure technical efficiency indicated that most provinces approached optimal technical management within their existing scale, demonstrating high technical effectiveness. Notably, Shaanxi and Sichuan had pure technical efficiency scores of 1, signifying that higher education in these provinces has achieved optimal levels in technical management and resource utilization. In contrast, Guizhou exhibited the lowest pure technical efficiency score, while Guangxi, Tibet, Inner Mongolia, Qinghai, and Xinjiang also scored below the average, indicating that these provinces have not fully leveraged resources under their current scale, resulting in technical and managerial inefficiencies. The adjusted scale efficiency revealed that scale inefficiency persisted across most provinces. Shaanxi, Chongqing, and Sichuan had scale efficiency values equal to or near 1, suggesting that higher education in these provinces has largely attained optimal scale, avoiding significant resource wastage or scale misalignment. Conversely, Tibet, Qinghai, and Ningxia reported the lowest scale efficiency scores at 0.3943, 0.5778, and 0.7380, respectively, indicating severe scale inefficiency. Further analysis of the third-stage DEA results revealed that these provinces operated under increasing returns to scale, implying that enhancing input-output efficiency in higher education could be achieved through scale expansion and optimized resource allocation.

(3) Comparative analysis of efficiency between the first and third stages.

Table 6, in conjunction with S1-S3 Figs, presents a comparative analysis of the initial DEA efficiency values from the first stage and the adjusted efficiency values from the third stage. The average technical efficiency in the first stage was 0.9512, but following the three-stage DEA adjustment, this average decreased to 0.8510. The decline in technical efficiency for most provinces after adjustment indicates that initial measurements overestimated the actual technical efficiency. Tibet exhibited the largest reduction, with technical efficiency dropping from 0.9005 to 0.3828, followed by Qinghai and Ningxia, suggesting that the technical efficiency in these three provinces heavily depends on favorable external conditions. After excluding these factors, the actual input-output efficiency of higher education in these western provinces proves lower than initially estimated.

The average pure technical efficiency in the first stage was 0.9704, which slightly increased to 0.9761 after the third-stage adjustment. This suggests that, after controlling for external environmental factors, the pure technical efficiency in higher education across most provinces remains relatively stable, indicating that the technical management level is not significantly influenced by external conditions. Notably, Shaanxi and Sichuan maintained a pure technical efficiency of 1, further demonstrating that their optimal level of technical management was unaffected by external environmental factors.

The initial average scale efficiency was 0.98, nearing optimal levels. Nonetheless, following the third-stage adjustment, the average scale efficiency declined to 0.8707, highlighting the role that favorable external conditions initially played in supporting scale efficiency. After adjustment, scale efficiency declined to varying degrees in most provinces (with the exception of Sichuan and Shaanxi), particularly in Tibet, Qinghai, and Ningxia, where scale efficiency fell from 0.9014, 0.9463, and 0.9569 to 0.3943, 0.5778, and 0.7380, respectively. This indicates that the scale of higher education in these regions is far from optimal, with pronounced scale inefficiency. Factors such as sparse population, weak economic foundations, and underdeveloped infrastructure contribute to a mismatch between resource inputs and outputs. Once favorable external conditions were excluded, the scale inefficiency issues in regions like Tibet, Qinghai, and Ningxia became more pronounced, revealing an urgent need for optimized adjustments in scale.

1. (4). Overall trend analysis.

The trends in technical efficiency, pure technical efficiency, and scale efficiency of higher education in western China from 2010 to 2022 are illustrated in Fig 1.

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Fig 1. Trends in TE, PTE, and SE in Western China (2010-2022).

https://doi.org/10.1371/journal.pone.0325901.g001

Overall, from 2010 to 2022, the technical management capabilities of higher education in western China have shown a steady improvement, although scale inefficiency remains a key constraint on technical efficiency. Technical efficiency has exhibited a fluctuating upward trend since 2010, reaching its peak of 0.9057 in 2022. Pure technical efficiency has remained consistently high throughout the study period, with an average of 0.9761, and notably sustained levels above the average during 2015–2017 and 2020–2022, reflecting relatively mature management and technical operations within the region’s higher education sector. In contrast, scale efficiency has been comparatively lower over these 13 years, with an average of 0.8707. Although scale efficiency has also shown a fluctuating upward trend, its fluctuations are less pronounced than those of technical efficiency, with a noticeable improvement beginning in 2018, indicating a gradual resolution of scale inefficiency issues in recent years. However, despite the upward trend, from a time-series perspective, scale inefficiency remains the primary bottleneck restricting overall technical efficiency gains. This suggests that in certain provinces within western China, economies of scale in higher education have not yet been fully realized, underscoring an urgent need to optimize scale in order to enhance resource utilization efficiency and thereby promote improvements in overall technical efficiency.

5.2 Empirical results of global Malmquist productivity index

Building on the efficiency analysis of DEA using the global benchmark technology presented earlier, this section employs the GMI to decompose changes in TFP, thereby assessing temporal variations in higher education productivity in western China and capturing the causes for these changes. Specifically, the GMI is expressed as GMI = TC × EC = TC × (PEC × SEC). The annual TFP and its decomposed indices for western China are presented in Table 7 and S4 Fig.

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Table 7. Annual GMI and Decomposition Indices for Western China.

https://doi.org/10.1371/journal.pone.0325901.t007

As shown in Table 7, the mean annual GMI for higher education in western China was 1.0105, indicating an overall upward trend in input-output performance for the period 2010–2022, with an average annual growth of 1.05%. During the period, GMI experienced three notable fluctuations, with the most significant occurring between 2016 and 2020. The lowest GMI value of 0.9613 was observed from 2017 to 2018, reflecting the sharpest productivity decline in this interval. The other two low values appeared in 2012–2013 (0.9656) and 2020–2021 (0.9785). Conversely, GMI reached a peak of 1.0443 in 2019–2020. This peak indicates that the higher education system exhibited its highest level of productivity during this period, with a marked enhancement in educational output capacity.

The technical efficiency change (EC) measures variations in the efficiency of resource utilization within HEIs at a given technological level. An EC value exceeding 1 suggests an enhancement in relative technical efficiency, while an EC value less than 1 demonstrates a deterioration in relative technical efficiency. Over the study period, the average EC was 1.0061, suggesting a general upward trend in technical efficiency change, with an annual growth of approximately 0.61%. The years 2016–2017 (1.0619) and 2018–2019 (1.0580) marked the most significant improvements in technical efficiency, with annual increases of 6.19% and 5.80%, respectively, reflecting notable advances in resource utilization and management during these periods. Except for the years 2013–2014, 2016–2017, 2018–2019, and 2021–2022, EC values in other years were below 1, indicating a decline in relative technical efficiency. As a whole, the mean of EC surpasses that of TC, suggesting that EC is the principal driver of TFP improvement. This finding is further substantiated by recent studies on productivity shifts within both Europe’s and China’s higher education system, which collectively conclude that efficiency change, rather than technological improvement, represents the dominant force for productivity growth in this sector [15,16,87], in contrast to evidence derived from British and Australian experiences, which suggests a predominant role for technological change in driving overall TFP growth in HEIs [63,164,165].

Technological change (TC) represents the shift in the production frontier, representing either advancement or regression in the technological frontier, primarily driven by technological innovation. A TC value above 1 signifies technological progress, while a TC value below 1 denotes technological regression. Over the study period, the average TC value was 1.0044, suggesting an annual increase in technological level of approximately 0.44%. Notably, during 2019–2020 (1.0480), 2014–2015 (1.0430), and 2011–2012 (1.0428), the TC index was relatively high, indicating significant technological progress in these years. Conversely, in years such as 2012–2013 and 2017–2018, the TC index fell below 1, indicating some degree of technological regression. During 2013–2014 and 2021–2022, both EC and TC exceeded 1, with TC surpassing EC, resulting in a GMI greater than 1. This demonstrates that technological progress contributed more to TFP during these periods than did improvements in technical efficiency, emerging as the main driver of productivity growth. In 2011–2012, 2014–2015, and 2019–2020, although EC fell below 1, reflecting a decline in technical efficiency, TC remained above 1, maintaining GMI at a level above 1. This demonstrates that technological progress not only offset the decline in technical efficiency but also acted as a critical factor driving TFP growth in these years, underscoring the essential role of technological advancement in enhancing productivity within higher education.

Pure technical efficiency change (PEC) measures shifts in the efficiency of resource allocation and management within HEIs under a given level of technology. It reflects whether, within existing technological constraints, HEIs have improved output efficiency through more effective resource allocation or management optimization. Over the study period, the mean PEC value was 1, indicating that, generally, resource allocation efficiency and management systems in western HEIs remained stable, with no significant changes. Specifically, in 2014–2015, 2017–2018, and 2021–2022, PEC values exceeded 1, suggesting that improvements in management practices or resource allocation occurred in these years within western HEIs. Conversely, in 2012–2013, 2013–2014, 2015–2016, and 2020–2021, PEC values were below 1, indicating a decline in resource allocation or management efficiency during these years, revealing issues of mismanagement or suboptimal resource allocation.

Scale Efficiency Change (SEC) measures whether a production unit has reached the optimal production scale, reflecting the influence of economies of scale on the production frontier. An SEC value exceeding 1 indicates an enhancement in scale efficiency, suggesting that the DMU operates closer to the optimal production scale; an SEC value below 1 denotes a decline in scale efficiency, signaling that the DMU’s production scale deviates from the optimal level. The mean SEC value of 1.0061 suggests that, on average, the scale efficiency of HEIs in western China increased by approximately 0.61% per year over the study period. This implies that SEC was a critical factor in driving TFP growth. Specifically, in 2013–2014, 2015–2016, 2016–2017, 2018–2019, and 2021–2022, SEC values exceeded 1, indicating that the production scale of western HEIs approached an optimal state during these years. Particularly in 2016–2017 (1.0616) and 2018–2019 (1.0580), SEC values were at their highest, reflecting the most significant improvements in scale efficiency. However, in other years, SEC values fell below 1, indicating a decline in scale efficiency and suggesting that HEI scales in western China failed to reach optimal configurations. In western China, multiple factors—such as relatively low economic development levels, low population density, fewer students of school age, insufficient public resources, and limited financial support—collectively constrain the expansion of university scale, preventing the achievement of economies of scale and thereby impacting scale efficiency.

In summary, SEC has been pivotal in improving TFP. From 2010 to 2022, TFP and SEC in western HEIs have shown a modest upward trend. However, in certain years, the SEC index declined, particularly after 2019, with a relatively smaller increase in scale efficiency, indicating that western HEIs still need to optimize resource allocation and adjust their scale structures to achieve optimal production scales. Simultaneously, the contribution of TC to TFP should not be overlooked. Although TC’s contribution to TFP has been slightly lower than that of EC overall, in some years, technological progress surpassed improvements in technical efficiency, becoming the primary driver of TFP growth. Therefore, in promoting TFP growth in western HEIs, it is essential to not only focus on the enhancement of scale efficiency within technical efficiency but also to emphasize the long-term role of technological progress to ensure the sustainable development of higher education in western China.

Similarly, the GMI is further employed to evaluate the dynamic efficiency of higher education input-output across provinces in western China, yielding the TFP index and its decomposed indices for each province, as presented in Table 8 and S5 Fig.

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Table 8. GMI and Decomposition Indices for Each Province in Western China.

https://doi.org/10.1371/journal.pone.0325901.t008

The annual average GMI values for Qinghai (1.0785), Inner Mongolia (1.0252), Tibet (1.0217), Xinjiang (1.0201), and Gansu (1.0015) all exceeded 1, suggesting a significant improvement in the productivity of higher education in these provinces over the course of the study period. Notably, Qinghai exhibited the most substantial increase, achieving an average annual growth rate of 7.85%. Conversely, the GMI values for Guizhou (0.9862) and Ningxia (0.9960) fell below 1, indicating a decline in higher education productivity in these provinces, with Guizhou experiencing an average annual decrease of 1.38% and Ningxia 0.4%. The GMI values for other provinces remained at 1, indicating stable higher education productivity with no significant changes observed throughout the study period.

Regarding EC, most provinces exhibited an EC value of 1, indicating stable technical efficiency throughout the study period with no significant improvements or declines. Consequently, the contribution of technical efficiency to TFP in these provinces is relatively limited. This suggests that in much of the western region, HEIs have not experienced notable improvements or deteriorations in resource utilization efficiency, management, or operational capability at the current technological level. Furthermore, changes in resource supply, educational demand, or policy adjustments have not exerted sufficient influence to alter technical efficiency within the higher education sector. Notably, Qinghai (1.0705) and Xinjiang (1.0132) demonstrated improvements in technical efficiency, with Qinghai showing the most significant gains, suggesting progress in resource utilization efficiency or positive impacts from relevant policy adjustments. Conversely, the EC values for Ningxia (0.9980) and Tibet (0.9940) were slightly below 1, suggesting a decline in resource allocation and management efficiency in these areas, highlighting a need for further optimization of management practices.

Concerning TC, Tibet (1.0279), Inner Mongolia (1.0252), and Gansu (1.0015) demonstrated noticeable technological progress, with Tibet and Inner Mongolia particularly distinguished in technological innovation and advancements in production techniques. In contrast, Guizhou (0.9862) and Ningxia (0.9980) had TC values below 1, indicating technological regression, suggesting the need to strengthen research investment and innovation capacity to address their lag in technological progress.

All provinces exhibited a PEC value of 1, indicating stable resource allocation efficiency under existing technological conditions during the study period, with no significant changes observed. This finding suggests that while western HEIs have made no substantial advances in resource allocation at the current technology level, there has also been no decline in resource utilization efficiency.

With respect to SEC, Qinghai (1.0705) and Xinjiang (1.0132) showed the most significant scale efficiency improvements, suggesting that the scale of higher education in these provinces is accelerating toward optimization. Qinghai, in particular, achieved an average annual increase in scale efficiency of 7.05%. Conversely, the SEC values for Tibet (0.9940) and Ningxia (0.9980) were below 1, indicating that higher education in these provinces had not yet reached optimal scale efficiency, necessitating further optimization of resource allocation or adjustments in institutional scale.

In summary, Qinghai, Inner Mongolia, Tibet, Xinjiang and Gansu exhibited significant TFP improvement during the study period, with Qinghai standing out in particular. However, Guizhou and Ningxia showed a downward trend in TFP, suggesting the need for enhanced management and technological innovation. In Tibet, Inner Mongolia, and Gansu, technological progress is the primary driver of TFP growth, whereas in Qinghai and Xinjiang, TFP growth is primarily attributable to improvements in scale efficiency. The decline in TFP in Guizhou is primarily attributed to technological regression, while Ningxia’s decline is affected by both technological regression and decreased scale efficiency. Additionally, there remains room for further optimization in scale efficiency in Tibet.

5.3 Empirical results of Tobit regression

5.3.1 Analysis of Tobit regression results.

In the Tobit model, the dependent variable is the adjusted technical efficiency measured in the third-stage DEA, while the independent variables include the proportion of regional fiscal revenue to gross regional product (RFR/GRP), the proportion of full-time faculty with senior and associate senior academic titles (SAS/FTF), the proportion of employees with college education or above (PECE), per-student education expenditure (PSEE), and the proportion of higher education operating expenses to total education expenditure (HEOE/TEE). Using Stata 18 software, a Tobit regression analysis was conducted to explore the factors influencing higher education input-output efficiency in western China, with the results displayed in Table 9.

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Table 9. Tobit Regression Results.

https://doi.org/10.1371/journal.pone.0325901.t009

According to the regression results in Table 9, the regression coefficients, standard errors, t-values, p-values, and 95% confidence intervals for the key explanatory variables clearly illustrate the impact of each variable on the dependent variable and their respective significance levels.

Firstly, the regression coefficient of RFR/GRP is −0.0028, indicating a negative relationship between the proportion of regional fiscal revenue to GRP and the technical efficiency of higher education input-output. However, with a p-value of 0.675, this variable does not reach a level of statistical significance, suggesting that in this model, the influence of RFR/GRP on the technical efficiency of higher education input-output is not statistically significant, and its explanatory power within the study sample is relatively weak.

Secondly, the regression coefficient for SAS/FTF is −0.0162, with a significance level of p < 0.01, indicating that an increase in the proportion of full-time faculty with senior academic titles (full and associate professorships) has a significantly negative impact on the technical efficiency of higher education input-output. Although senior and associate senior faculty members typically possess strong research and teaching capabilities, in the resource-constrained western China, a higher proportion of senior faculty may lead to imbalanced resource allocation. This finding contrasts with the conclusions of [97,103,119], which assert that the proportion of full-time faculty holding senior academic titles exerts a significantly positive impact on the technical efficiency of HEIs. Three main reasons underpin our finding. Primarily, although the Chinese government has introduced the “Western Development” strategy, with central and local governments actively promoting the expansion of tertiary education, encouraging talent return, and strengthening local high-level talent cultivation, HEIs in western China have not effectively allocated these human resources in the recruitment and cultivation of senior faculty. Resources tend to concentrate in economically more developed cities, key institutions, and certain fields, leading to further centralization of educational resources within the region, ultimately failing to benefit all HEIs and diminishing overall educational resource utilization efficiency. Additionally, the relatively higher salaries and benefits associated with senior faculty positions further exacerbate resource allocation imbalances. While a higher proportion of senior faculty can enhance certain teaching and research outcomes, if not managed effectively—particularly in the resource- and funding-limited western China—an excessively high proportion of senior faculty may generate increased management costs, thereby hindering improvements in higher education input-output efficiency. Moreover, senior faculty members often undertake managerial roles or academic leadership responsibilities in addition to their teaching and research duties. The excessive involvement in administrative tasks can detract from their direct contributions to teaching and research, resulting in a decline in technical efficiency. The negative correlation between the proportion of senior faculty and technical efficiency does not imply inefficiency on the part of senior faculty themselves but rather reflects issues stemming from traditional management structures and suboptimal task allocation.

Thirdly, the regression coefficient for PECE is 0.0179, with a significance level of p < 0.01, indicating that the proportion of employees with college education or above has a significant positive impact on the improvement of technical efficiency in higher education in western China. Attributable to factors such as geographic location and economic foundations, western China has historically faced relatively unequal distribution of educational resources, with underdeveloped education levels and limited talent development serving as bottlenecks to economic and social progress. However, with the increased governmental investment in education in western China, especially in higher education, the quality of higher education has enhanced significantly, greatly facilitating human capital accumulation. From the perspective of human capital theory, education is a critical investment in enhancing the productivity and innovation capacity of the workforce. By increasing investment in higher education, the western region not only develops a labor force with advanced professional skills and innovation abilities but also enhances the adaptability of workers to new technologies and industries—essential factors for driving economic transformation and improving overall productivity. Workers with higher education attainment demonstrate greater efficiency in economic activities, fostering modernization within organizations and industries and strengthening regional competitiveness. Therefore, the significant positive coefficient of PECE reflects that, with the growing accessibility of higher education in western China, the proportion of well-educated labor has risen, accelerating human capital accumulation. This process directly contributes to the improvement of technical efficiency in higher education in the region. In other words, the enhancement of overall regional human resource levels has played a crucial role in advancing higher education development.

Fourthly, the regression coefficient for PSEE is −7.53E-06, with a high level of significance (p < 0.01), indicating a negative relationship between per-student education expenditure and higher education input-output technical efficiency. This result suggests a phenomenon of diminishing marginal returns in the increase of per-student educational funding in western China. Although the government, through the “Double First-Class” initiative and other policy-based funding support, aims to enhance the teaching quality and research level of HEIs in western China, the effective utilization of these funds may face challenges. In many cases, the substantial input in per-student education expenditure has not directly translated into tangible educational outcomes but has been diverted to infrastructure projects or unnecessary expansions, resulting in low efficiency in fund utilization. Simultaneously, given the underdeveloped management systems and oversight mechanisms for fund allocation, the increased per-student education expenditure may lead to resource wastage or overexpansion of certain projects that exceed the actual educational needs of western China, thus inflating management costs and suppressing overall technical efficiency. This phenomenon also highlights that, while financial support for education in western China continues to increase, significant room for improvement remains in the rational allocation and utilization of these funds to maximize input-output efficiency. For HEIs in western China, which benefit from favorable national policies and financial support, the effective application of funds to improve teaching quality, facilitate research outcomes, and cultivate talent should be a primary focus of future policy and management efforts. This finding is reinforced by a recent study, which indicates that current governmental financial support for universities may be sufficient, and that indiscriminate increases in financial investment are likely to yield limited improvements in performance [166].

Fifthly, the regression coefficient for HEOE/TEE is 0.0126, with a significance level of p < 0.01, indicating that the proportion of higher education operating expenses to total education expenditure has a significant positive impact on improving the technical efficiency of higher education in western China. When the share of higher education operating expenses within total educational spending increases, resources are allocated and utilized more effectively, thereby improving the overall operational efficiency of the educational system. Higher education operating expenses are typically allocated to infrastructure development, equipment procurement, and faculty training, all of which lay a foundation for improved educational quality and long-term benefits. A higher proportion of operating expenses ensures that educational resources are efficiently distributed and utilized, thereby fostering efficiency improvements.

Overall, SAS/FTF, PECE, PSEE, and HEOE/TEE are statistically significant at the 1% level, reflecting the substantial impact of human capital structure and educational funding allocation on the input-output efficiency of higher education in western China. The positive significance of HEOE/TEE and PECE underscores the importance of higher education operating expenses and external human capital, while the negative significance of SAS/FTF and PSEE suggests potential issues of diminishing marginal returns in faculty utility and resource misallocation.

5.3.2 Robustness test.

To verify the robustness of the regression results, this study employs Tobit regression with robust standard errors and Probit regression as comparative analysis, with the results shown in Table 10.

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Table 10. Regression results: Tobit model with robust SE and probit Model.

https://doi.org/10.1371/journal.pone.0325901.t010

Comparing Tables 10 and 9 reveals a high degree of consistency between the Tobit regression with robust standard errors and the standard Tobit regression results. The coefficient direction, magnitude, and significance levels of the main explanatory variables (SAS/FTF, PECE, PSEE, HEOE/TEE) remain largely unchanged. The Probit regression results are also broadly consistent with those of the Tobit regression. The significance level of the variable HEOE/TEE differs slightly between the Probit and Tobit models, showing higher significance in the Tobit model (p < 0.01) compared to the Probit model (p < 0.05). Nonetheless, the effect direction of HEOE/TEE is consistent across both models and remains significant, indicating a robust positive effect of HEOE/TEE on the dependent variable in both models. Additionally, there is a minor difference in the significance of the constant term between the Probit and Tobit models: it is insignificant in the Probit model, while the explanatory power of the overall model and the impact of the variables remain stable. Through the comparison with the Probit model and the introduction of robust standard errors in the Tobit model, the study achieves highly consistent regression results, particularly regarding the significance and direction of the main explanatory variables. Therefore, it can be concluded that the regression results in this study exhibit robustness and reliability.