Mathematical and Statistical Strategies for Evaluating the Efficiency of Educational Resource Allocation in Tertiary Education

China has always advocated the concept of “fairness in education, which ultimately benefits the whole society”. In recent years, China has been promoting higher education, which has gradually been classified as mass education, which has broken the original egalitarian principle of distribution --- the gap between the rich and the poor is getting bigger and bigger [1-3]. At the same time, the infinite expansion of education demand, rising tuition fees, shortage of education resources, shortage of funds for higher education institutions, etc. make improving the efficiency of higher education institutions to become a hot topic that needs urgent attention [4-5]. “Efficiency” has always been the core issue of higher education resource allocation must be concerned about, so how to fully, reasonably and equitably allocate educational resources this issue must cause us to think [6-7].

Educational resources of higher education refers to the human resources, financial resources, material resources and other resources (such as system, reputation, etc.) provided for the normal and orderly educational activities of colleges and universities [8-9]. It is the foundation of the development of higher education business, and the rational allocation and use of these resources are closely related to the healthy and orderly development of higher education [10]. However, at present in China, people’s demand for higher education is constantly expanding, although the Chinese government has increased its investment in education, but the higher education resources are still relatively insufficient, which makes more and more colleges and universities are constrained by the healthy and sustainable development [11-13]. Due to the rapid development of China’s economy, science and technology, the demand for higher education by Chinese society as a whole as well as individuals is also expanding rapidly. And in this situation of very shortage of educational resources in higher education, there still exist phenomena such as irrational allocation of educational resources and inefficient utilization of resources [14-16]. For example, in human resources, there is an over-allocation of school administrative and logistic management personnel, while the allocation of full-time teachers and so on is too small, and in terms of material resources, there is the phenomenon of misallocation such as blind investment in teaching instruments, equipments, laboratories and library materials, and so on, and the phenomenon of some resources being idle while some resources being in short supply [17-19]. According to relevant statistics, more than 20% of the teaching equipment in China’s institutions of higher education is idle, the utilization rate of expensive large-scale scientific research facilities is less than 15%, and there is the phenomenon of repeated purchases [20].

Therefore, in the above Chinese national conditions, in order to meet people’s demand for higher education, it is necessary to optimize the allocation of higher education resources, in order to place all kinds of educational resources in the most appropriate position, so as to play its maximum effect [21]. In order to optimize the allocation of higher education resources, it is necessary to find out the influencing factors that hinder and limit the improvement of the efficiency of the allocation of higher education resources, and to explore how these influencing factors act on the allocation of educational resources and the mechanism within the influencing factors, in order to provide references for the optimization strategy of educational resources [22-23].

This paper uses factor analysis to extract input and output factors. Then according to the systematic principle and feasibility principle that the evaluation index system should follow, the index system for assessing the efficiency of educational resource allocation in colleges and universities is established. The factor scores are inputted into the classical DEA model to complete the evaluation of the allocation efficiency of college resources. Province A was selected to carry out the example analysis on the assessment of resource allocation efficiency of colleges and universities, and the calculated pure technical efficiency was used as the explanatory variable, and the Tobit model was established to carry out the regression analysis on the influencing factors of resource allocation efficiency of colleges and universities in Province A. It provides a direction for the education department of province A to allocate education resources more scientifically and reasonably.

2

Factor analysis

2.1

Factor analysis structure

Factor analysis is a statistical method that reduces a large number of complex relationships to a small number of composite factors by datamining the interdependencies within the original variables.

When confronted with vectors or variables that are intrinsically related to each other, factor analysis reduces the amount of data to an acceptable range by using a small number of factor combinations to describe the conceptual structure inherent in this multitude of variables.

The factor analysis model is described below:

1) X = (x₁,x₂,…,x_p)^T is the observable random vector with mean vector E(X) = 0 and covariance matrix Cov(X) = ∑, here the covariance matrix ∑ is equal to the correlation matrix R, this is achieved by normalizing the variables.

2) F = (F₁,F2,…,F_m)^T (m < p) is the common factor, an unobservable random vector with mean vector E(F) = 0, covariance matrix Cov(F) = 1, and the vectors are independent of each other.

3) ε = (e₁,e₂,…,e_p)^T and F are independent of each other, and the covariance matrix ∑ of E(e) = 0, e is diagonal, so each e is also independent of each other, with the model as follows: $\begin{matrix} x_{1} = a_{11} F_{1} + a_{12} F_{2} + \dots + a_{1 m} F_{m} + e_{1} \\ x_{2} = a_{21} F_{1} + a_{22} F_{2} + \dots + a_{2 m} F_{m} + e_{2} \\ \dots \dots \dots \dots \\ x_{p} = a_{p l} F_{1} + a_{p 2} F_{2} + \dots + a_{p m} F_{m} + e_{p} \end{matrix}$

It is called a factor analysis model which is orthogonal and can be referred to as a type R orthogonal factor model, which has a combinatorial basis of combinations of vectors. Its matrix form is: (1) $X = A F + ε$

Where A = (a_ij)_p*m is the initial required matrix called factor loading matrix and a_ij responds to the loadings of the i rd variable on the j th F factor.

Each random variable X_i linearly depends on a small number of common factors i.e. F₁,F₂,…F_m and and a representative error e_i which is the basic assumption of the factor model.

The factor analysis method performs the analysis of the logistic between the variables through the relationship of the standardized variance-covariance matrix transformed responses, and it makes use of the common factors for the interpretation of this relationship. In the factor model, the variables have the following mathematical relationships: (2) $\begin{matrix} V a r (X_{i}) = a_{i l}^{2} + \dots + a_{i m}^{2} + e \\ C o v (X_{i}, X_{j}) = a_{i 1} a_{j 1} + \dots + a_{i m} a_{j m} \\ C o v (X_{i}, F_{j}) = a_{i j} \end{matrix}$

Namely:

1) The variance of variable X_i in the initial matrix is jointly represented by AF and E, reflecting the loading of that variable on the common factor, and the loss of the special factor;

2) The covariance between variables X_i and X_j of the initial matrix can be obtained by a linear combination of the loadings corresponding to A, i.e., the common factor F;

3) A i.e. the covariance between X and F, which can also be called factor loadings.

The model is known by itself: (3) $X_{i} = a_{i 1} F_{1} + a_{i 2} F_{2} + \dots + a_{i m} F_{m} + ε_{i}$

(Factor models are not general and are standardized across variables)

Multiply both ends right by F_j to get: (4) $X_{i} F_{j} = a_{i 1} F_{1} F_{j} + a_{i 2} F_{2} F_{j} + \dots + a_{i m} F_{m} F_{j} + ε_{i} F_{j}$

And so: (5) $E (X_{i} F_{j}) = a_{i 1} E (F_{1} F_{j}) + a_{i 2} E (F_{2} F_{j}) + \dots + a_{i m} E (F_{m} F_{j}) + E (ε_{i} F_{j})$

As there is under standardization: (6) $E (F) = 0, E (ε) = 0, V a r (ε_{i}) = 1, E (X_{i}) = 0, V a r (X_{i}) = 1$

So E(X_iF_j) = r_{X_iF_j}, E(F_iF_j) = r_{F_iF_j}, E(ε_iF_j) = r_{ε_iF_j}.

So the above equation can be written as: (7) $\begin{matrix} r_{X_{i} F_{j}} = a_{i_{1}} r_{F_{i} F_{j}} + a_{i_{2}} r_{F_{2} F_{j}} + \dots + r_{ε_{i} F_{j}} \\ = a_{i j} (\begin{matrix} The factor combinations are \\ uncorrelated with each other i .e . \\ uncorrelated i .e . correlation \\ coefficient is equal to 0 \end{matrix}) \end{matrix}$ $$\matrix{ {{r_{{X_i}{F_j}}} = {a_{{i_1}}}{r_{{F_i}{F_j}}} + {a_{{i_2}}}{r_{{F_2}{F_j}}} + \ldots + {r_{{\varepsilon _i}{F_j}}}} \cr { = {a_{ij}}\left( {\matrix{ {{\rm{The\;factor\;combinations\;are\;}}} \cr {{\rm{uncorrelated\;with\;each\;other\;i}}{\rm{.e}}{\rm{.}}} \cr {{\rm{uncorrelated\;i}}{\rm{.e}}{\rm{.\;correlation}}} \cr {{\rm{coefficient\;is\;equal\;to\;0}}} \cr } } \right)} \cr } $$

The explanatory significance of the factor loadings is described by obtaining the correlation coefficients between the i st variable and the j nd common factor, the response X_i by more or less the combination of F_j can be described.

The common degree of variable X_i is obtained by summing the squares of the elements of row i of the factor loading matrix: (8) $h_{i}^{2} = \sum_{j = 1}^{m} a_{i j}^{2} i = 1, \dots, p$

Solve for the variance on both sides of the following equation, i.e: (9) $\begin{matrix} X_{i} = a_{i 1} F_{1} + a_{i 2} F_{2} + \dots + a_{i m} F_{m} + ε_{i} \\ V a r (X_{i}) = a_{i 1}^{2} V a r (F 1) + a_{i 2}^{2} V a r (F_{2}) + \dots + a_{i m}^{2} V a r (F_{m}) + V a r (ε_{i}) \\ = a_{i 1}^{2} + a_{i 2}^{2} + \dots + a_{i m}^{2} + σ_{i}^{2} \\ = h i^{2} + σ_{i}^{2} \end{matrix}$

Since X_i has been standardized, there is: (10) $l = h_{i}^{2} + σ_{i}^{2}$

In the formula, l represents the variance of the complete X matrix, which consists of two parts: the first part is the common degree $h_{i}^{2}$ which measures the percentage of the total variance accounted for by the selected common factor to X the total variance, also known as the degree of contribution, $h_{i}^{2}$ the larger, the closer to 1, the more the intrinsic information of this X is retained in the common factor, i.e., the better the transformation of this factor is, the more the amount of information of the original is retained, and subsequent data processing will also have relatively better results.

2.2

Factor analysis process

To finally factor analyze the actual problem and construct the factor model, it is crucial to construct the factor loading array based on the original information matrix. This section mainly introduces the following two factor extraction algorithms. Among them, the principal component method [24] is a commonly used method for the construction of factor analysis loading matrix, and the subsequent experiments will be based on this method to obtain the loading matrix.

The steps of principal component algorithm are as follows:

1) Calculate its correlation matrix R or covariance matrix by the initial matrix X: $\bar{X} = \frac{1}{n} \sum_{t = 1}^{n} X_{(t)} = {(\bar{X_{1}}, \dots, \bar{X_{p}})}^{'}$ (11) $S = \frac{1}{n - 1} \sum_{t = 1}^{n} (X_{(t)} - \bar{X}) {(X_{(t)} - \bar{X})}^{'} \overset{d e f}{=} {(s_{i j})}_{p \neq p}$

$s_{i j} = \frac{1}{n - 1} \sum_{t = 1}^{n} (x_{t i} - {\bar{x}}_{l}) {(x_{t j} - {\bar{x}}_{j})}^{'}$ of them.

(12)

R = {(r_{i j})}_{p * p}

where $r_{i j} = \frac{s_{i j}}{\sqrt{s_{i i} s_{j j}}} (i, j = 1, 2, \dots, p)$ .

Where S is the covariance matrix and R is the correlation matrix.

2) Find the eigenvalues and eigenvectors of R or S.

The eigenvalue is λ₁ ≧ λ₂ ≧…≧ λ_p > 0 and its corresponding unit orthogonal eigenvector is l₁,l₂,l₃…,l_p.

3) Find the loading matrix of the factor model.

(1) Determine the number of common factors m according to the user’s choice.

(2) Let $a_{i j} = \sqrt{λ} l_{i} (i = 1, 2, \dots, m)$ , then A = (a₁,a₂,…,a_m) is the factor loading matrix.

(3) Find the special factor variance $σ_{i}^{2} = 1 - h_{i}^{2}$ , the common degree: (13) $h_{i}^{2} = \sum_{j = 1}^{m} a_{i j}^{2} i = 1, \dots, p$

Maximum likelihood estimation is a commonly used method of parameter estimation that uses the combination of observations with the greatest probability of occurring as parameter data. This method was first used by the statistician Sir Ronald Fisher between 1912 and 1922.

Assuming that the common factor F and the special factor ε obey the normal distribution, can get the factor loading matrix and the special variance of the great likelihood estimation, the sample likelihood function for the function of μ, ∑ to yield, L = (μ,∑).

Take the production $μ = \bar{X}$ , set ∑ = A′A+D, then the likelihood function L(μ,A′A+D) of the logarithm of the function of A, D, notated as φ(A,D) so that φ reaches the maximum. By the proof can be known to make φ to reach the maximum solution A and D to meet the following system of equations: (14) ${\begin{matrix} S D^{- 1} A = A (I + A D^{- 1} A) \\ D = d i a g (S - A A) \end{matrix}$

where $S = \frac{1}{n} \sum_{t = 1}^{n} (X_{(t)} - \bar{X}) {(X_{(t)} - \bar{X})}^{'}$ , in order to guarantee a unique solution to the above system of equations, a uniqueness condition is attached: A′D⁻¹A = Diagonal matrix.

The algorithm is as follows: give a random initial value of D₀, then solve for A₀. Bring in A0 to get D_i, and so on.

D_i to A_i, is obtained by solving $1 / \sqrt{D_{i}} S 1 / \sqrt{λ D_{i}}$ for the eigenvalue λ₁ ≧ λ₂ ≧…≧ λ_m, whose corresponding unit-orthogonal eigenvector is l₁,l₂,l₃…,l_p. Then several D_i, A_i satisfy the condition. It is known that D_i, A_i converge to the desired solution.

The method of transforming the factor loading matrix is called factor rotation, and there are various ways of rotation, such as orthogonal rotation, oblique rotation, and so on. The Promax oblique rotation method is described below [25].

In factor analysis due to the variance of the maximum orthogonal rotation changes, the factor loading matrix still can not explain the actual significance of the common factor well. It is necessary to transform the factor loading matrix after variance maximum orthogonal rotation into a more suitable simple structure by Promax rotation method.

The Promax oblique rotation algorithm is as follows:

1) Factor loading matrix obtained after variance-maximizing orthogonal rotation A_{var i max} = (a_ij)_p*m

2) Matrix $P_{i j} = {| \frac{a_{i j}}{\sum_{j = 1}^{m} a_{i j}^{2}} |}^{k + 1} (\sum_{j = 1}^{m} a_{i j}^{2}) / a_{i j}$ , where k(k > 1) is the weight of the Promax rotation

3) Matrix $L = {({A^{'}}_{var i \max} A_{var i \max})}^{- 1} {A^{'}}_{var i \max} P$

4) Q = LD

where D = (diag(L′L))^−1/2, C = (diag(Q′Q)⁻¹)^−1/2

5) Apro = A_{var i max}QC⁻¹

Promax rotation makes the factor loadings bifurcated, thus reflecting more clearly the correlation of the original variables to the factors. It is also therefore more suitable for the processing of data categorization after dimensionality reduction.

By the factor matrix: (15) $X = A F + ε - > F = {(A^{'} A)}^{- 1} A^{'} X - E$

Ignored when factor E is less than a certain threshold, there is F: (16) ${\begin{matrix} f_{1} = a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 m} x_{m} \\ f_{2} = a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 m} x_{m} \\ \dots \\ f_{p} = a_{p 1} x_{1} + a_{p 2} x_{2} + \dots + a_{p m} x_{m} \end{matrix}$

This formula is used to calculate the main factor scores.

Factor scores are obtained by estimating the values of each observed variable using the factor scores, i.e., the common factors, which can be used to replace the original variables for subsequent data processing. Eliminating a great deal of redundancy, eliminating correlation, and reducing scale, this method is better able to distinguish the main ideas in the various categories of text and facilitates categorization.

3

Model for assessing the efficiency of educational resource allocation

In order to ensure the scientific truthfulness and validity, this study selected 60 colleges and universities in Province C as the research object, and the data mainly came from the China Education Statistics Yearbook, China Education Expenditure Statistics Yearbook, and the Annual Report on the Quality of Education in Colleges and Universities (2022), Annual Report on the Quality of Graduates’ Employment (2022) released by colleges and universities in Province C, and the data resources of official representative data, and some indicators of the missing data were filled in through the corresponding official website of the university school to make up for it.

3.1

Construction of the index system for assessing the efficiency of educational resource allocation

In practice, the selection of indicators for the input and output of universities should be scientific and effective, but also reflect the university’s own characteristics, and the difficulty of obtaining data should also be taken into account. Therefore, the following principles should be considered when constructing the efficiency evaluation index system:

The principle of systematicity is the need to carry out an all-round and multi-angle evaluation of colleges and universities, and the selected indicators should be able to fully reflect the operational characteristics of the input and output of the college and university education system, and the evaluation indicator system should be as comprehensive as possible, taking into account various factors. In addition, the existence of strong correlation between the indicators will cause some bias to the calculation results, so the indicators should be independent of each other, differentiate from each other, reduce the degree of correlation, and try to use fewer indicators to reflect more information.

The principle of feasibility is that when organizing and selecting the input and output indicators of colleges and universities, they should be combined with the actual situation, fully consider the possibility of obtaining data, and ensure the collection, collation and analysis of relevant data. At the same time, attention should also be paid to the consistency and comparability of the statistical scope, standards, caliber and other aspects of the relevant data, which is conducive to effective comparison between the various evaluation objects.

According to the theory of input and output of college education resources, the input of college resources can be analyzed from the perspective of human, material, and financial resources to choose the relevant indicators.Colleges and universities have three main functions, such as talent training, scientific research, social services, and so on.According to the functions undertaken by colleges and universities to classify their outputs and seek relevant indicators to indicate the performance of colleges and universities.With comprehensive consideration, the selected indicators of input and output of educational resources in colleges and universities are shown in Table 1.

Table 1.

Education resource allocation efficiency evaluation index system

Input output	Symbol	Concrete meaning
University resource input	A1	Special teacher number (person)
	A2	Number of teachers in senior professional technical positions
	A3	Number of courses in the teaching program
	A4	The school occupies the land product (10,000 square meters)
	A5	Library library
	A6	Education cost income (10,000 yuan)
	A7	Teaching research instrument equipment value (10,000 yuan)
Functional output of universities	A8	Number of graduates
	A9	Monthly income (yuan)
	A10	Professional relevance(%)
	A11	Technical turnover (10,000 yuan)
	A12	Vertical scientific funding (10,000 yuan)
	A13	Horizontal technical service to amount (10,000 yuan)
	A14	Non-education training program number (item)
	A15	Non-education training fund to amount (10,000 yuan)

3.2

Construction of the efficiency evaluation model

The first model of the DEA method [26], the CCR model, is based on the principle of using the evaluated units as decision units. In this study, 60 colleges and universities are used as 60 decision-making units, and these 60 decision-making units are comparable, while each college and university has m type of input and s types of output, where x_ij represents the input of the jth college and university to the ith type of input, x_ij > 0; y_rj represents the output of the jth college and university decision-making unit to the rth type of output, y_rj > 0; v_i represents a metric (or weight) for type i inputs; u_r represents a metric (or weight) for type r outputs.

Based on the above description, the formula can be written as follows: (17) ${\begin{matrix} X_{i j} = {(x_{1 j}, x_{2 j}, \dots, x_{i j})}^{T}, j = 1, 2, \dots, n \\ Y_{i j} = {(y_{1 j}, y_{2 j}, \dots, y_{r j})}^{T}, j = 1, 2, \dots, n \\ v = {(v_{1}, v_{2}, \dots, v_{i})}^{T} \\ u = {(u_{1}, u_{2}, \dots, u_{r})}^{T} \end{matrix}$

X_j and Y_j denote the input and output vectors of HEIs, respectively, and j denotes the number of HEIs, which can be obtained from data or statistics. v and u are the weight vectors corresponding to m inputs and s outputs respectively. Therefore, the corresponding evaluation index of each university can be obtained: (18) $h_{j} = \frac{u^{T} Y_{j}}{v^{T} X_{j}}, j = 1, 2, \dots, n$

Appropriate weighting coefficients v and u are selected so that the evaluation index satisfies h_j ≤ 1(j = 1,2⋯,n). The following CCR model is composed with the weighting coefficients as variables for the efficiency evaluation of each university, the efficiency index of the jth university as the reference objective, and the efficiency index of all the universities h_j ≤ 1(j = 1,2⋯,n as the constraint: (19) $(C C R) {\begin{matrix} \max \frac{u^{T} Y_{0}}{v^{T} X_{0}} = V_{P}^{I} \\ s . t . \frac{u^{T} Y_{j}}{v^{T} X_{j}} \leq 1, j = 1, 2 \dots, n \\ u \geq 0, v \geq 0 \end{matrix}$

Then Chames-Cooper transformation is applied to this model. According to the linear programming dyadic theory it is known that the dyadic programming model of CCR is: (20) $(C C R) {\begin{matrix} \min [θ - ε (e^{T} S^{-} + e^{T} S^{+})] \\ s . t . \sum_{j = 1}^{n} X_{j} λ_{j} + S^{-} = θ X_{0} \\ s . t . \sum_{j = 1}^{n} Y_{j} λ_{j} - S^{+} = Y_{0} \\ λ \geq 0, j = 1, 2 \dots, n \\ s^{+} \geq 0, s^{-} \geq 0 \end{matrix}$

After obtaining the CCR pairwise planning model, the efficiency of educational resource allocation in the university is reflected by the optimal solution of the model. If the optimal solution is θ = 1, s⁺ = 0, s = 0, it means that the university effectively utilizes educational resources; if the optimal solution is θ ≠ 1, s⁺ ≠ 0 or s ≠ 0, it means that the university has a non-efficient utilization of resources, and it needs to be further determined whether there is a problem with technology or scale. In the modeling results, the efficiency value will be decomposed into “technical efficiency” and “scale efficiency”, and the product of the two is the overall efficiency of the evaluated unit. “Technical efficiency” refers to the output results brought about by technical changes, reflecting the optimal allocation of inputs and outputs. “Scale efficiency”, on the other hand, indicates that efficiency is affected by the scale of production, and that the appropriate scale of production needs to be maintained to prevent the emergence of a diminishing margin effect.

According to the evaluation results of the CCR model, it is only possible to judge whether the efficiency of decision unit allocation reaches the DEA efficiency, but it is not possible to further compare the effective decision units. In order to improve this problem, a super-efficiency model is proposed on the basis of the CCR model, which can further measure the efficiency value of effective decision-making units and compare the efficiency of multiple decision-making units more intuitively. The principle application is that when the measurement results are DEA effective colleges and universities, their production frontier surface will move backward. The production frontier remains unchanged when the DEA non-effective colleges are measured. In the calculation results, the efficiency value of DEA effective colleges under the super-efficiency model is greater than the traditional CCR model, while the efficiency value of DEA non-effective colleges is equal to the traditional CCR model. Therefore, the mathematical form of the super-efficiency model is constructed as follows: (21) $(S E - C C R) {\begin{cases} \min [θ - ε (e^{T} S^{-} + e^{T} S^{+})] \\ s . t . \sum_{j = 1, j \neq k}^{n} X_{j} λ_{j} + S^{-} \leq θ X_{0} \\ s . t . \sum_{j = 1, j \neq k}^{n} Y_{j} λ_{j} - S^{+} \geq Y_{0} \\ λ \geq 0, j = 1, 2 \dots, n \\ s^{+} \geq 0, s^{-} \geq 0 \end{cases}$

Unlike the CCR model, which evaluates the efficiency of all colleges and universities, the super-efficiency model, when calculating the efficiency value of decision-making units, first judges the efficiency value and excludes the colleges and universities whose efficiency value is less than 1, i.e., the DEA non-effective colleges and universities. When calculating the DEA effective colleges, the constraints of the CCR model are removed, so as to accurately obtain the efficiency value of the effective colleges.

4

Analysis of the empirical results of the assessment of the efficiency of the allocation of educational resources

In order to reduce the data dimensionality and eliminate the correlation between the indicators, the selected indicators were factor analyzed separately from the university resources and functions, i.e., inputs and outputs. According to the KMO test and Bartlett’s test of sphericity to conclude that it is suitable for factor analysis, according to the principal component extraction factors, as well as the maximum variance method of rotation to obtain three factors (taking into account the three aspects of the resource inputs, the number of extracted factors is directly specified as 3), the relevant information is organized, and the university input factor component matrix and the rotated component matrix are shown in Table 2.

Table 2.

College input factor matrix and rotating composition matrix

Factor	Component matrix			Rotational composition matrix
Factor	I-F1	I-F2	I-F3	I-F1	I-F2	I-F3
Eigenvalue	4.028	1.548	0.885	3.048	1.897	1.516
Contribution (%)	57.53	22.06	12.72	43.58	27.04	21.69
Cumulative contribution (%)	57.53	79.59	92.31	43.58	70.62	92.31
A1	0.747	-0.212	0.459	0.243	0.321	0.842
A2	0.346	0.517	0.759	0.222	0.364	0.877
A3	0.843	-0.418	-0.189	0.896	0.35	0.017
A4	0.882	-0.33	0.121	0.881	0.188	0.306
A5	0.922	-0.092	0.175	0.75	0.289	0.481
A6	0.904	0.265	-0.124	0.49	0.761	0.369
A7	0.802	0.405	-0.402	0.301	0.888	0.305

According to Table 2, it can be seen that the three principal component factors bring together 92.36% of the information from the seven original variables, with a high degree of generalization.According to the rotated component matrix, factor I-F1 mainly gathers information from A3, A4, and A5 indicators and is defined as the physical factor.Factor I-F2 mainly gathers information from indicators A6 and A7 and is defined as a financial factor. Factor I-F3 mainly gathers information from indicators A1 and A2, defined as the human factor. The three factors summarize the three aspects of human, material, and financial inputs of educational resources in colleges and universities respectively.Based on the raw indicator data and the coefficient matrix, the factor scores of the three aspects can be calculated for each of the 60 colleges and universities as input data for data envelopment analysis.

Similarly, factor analysis of the eight indicators of university functions that is output, KMO test and Bartlett’s test of sphericity, can also be factor analysis, according to the principal components to extract the factors and the maximum variance method of rotation to get three factors (eigenvalue greater than 1 of the three indicators), the relevant information collated to get the table 3. Table 3 for the output of the university factor component matrix and the rotated matrix of the components, the three main components the factors bring together 93.61% of the information of the eight original variables, the ability to summarize the original information is very high, and it is appropriate to analyze the factors on behalf of the original indicators. According to the rotated component matrix, factor O-F1 mainly collects the information of variables A8, A9 and A10, and is defined as the talent development factor. Factor O-F2 mainly gathers the information of indicators A11, A12 and A13, and is defined as the scientific research factor. Factor O-F3 mainly gathers information on indicators A14 and A15, and is defined as the social service factor.The three factors fully summarize the three aspects of the educational functions of universities and their resource performance output. Based on the raw indicator data and the coefficient matrix, the factor scores of the three aspects of the 60 colleges and universities can be calculated separately as data envelopment analysis output data.

Table 3.

University output factor matrix and rotating composition matrix

Factor	Component matrix			Rotational composition matrix
Factor	O-F1	O-F2	O-F3	O-F1	O-F2	O-F3
Eigenvalue	4.635	1.795	1.062	3.017	2.506	1.967
Contribution (%)	57.92	22.45	13.24	37.72	31.32	24.57
Cumulative contribution (%)	57.92	80.37	93.61	37.72	69.04	93.61
A8	0.871	0.292	-0.385	0.963	0.255	0.062
A9	0.874	0.276	-0.376	0.952	0.261	0.079
A10	0.837	-0.468	-0.169	0.991	0.165	0.07
A11	0.776	0.005	0.422	0.314	0.82	0.105
A12	0.761	-0.092	0.581	0.176	0.936	0.153
A13	0.866	0.04	0.336	0.441	0.811	0.117
A14	0.366	0.897	-0.204	0.052	0.096	0.983
A15	0.223	-0.269	0.874	0.022	0.369	0.877

Using DEAP2.1 software, the higher education resource allocation efficiency of 60 colleges and universities in urban areas of province A is analyzed from 2018 to 2023, and the average values of comprehensive efficiency, pure technical efficiency, and scale efficiency are 0.954, 0.975, and 0.978, respectively, among which 17 decision-making units have the value of 1 in the comprehensive efficiency, 33 decision-making units have reached the value of 1 in the pure technical efficiency, 20 decision-making units are in the state of constant returns to scale, 7 decision-making units are in the state of decreasing returns to scale, and 33 decision-making units are in the state of increasing returns to scale, and the specific empirical results are shown in Table 4.

Table 4.

Evaluation of resource allocation efficiency of higher education

Decision unit(DMU)	Integrated efficiency(TE)	Pure technical efficiency(PTE)	Scale efficiency(SE)	Scale compensation
1	0.999	1.000	0.999	irs
2	0.934	0.944	0.989	irs
3	1.000	1.000	1.000	-
4	0.901	0.901	1.000	-
5	0.852	0.868	0.982	irs
6	0.945	0.955	0.990	irs
7	0.968	0.977	0.991	irs
8	0.874	0.891	0.981	irs
9	1.000	1.000	1.000	-
10	0.999	0.999	1.000	-
11	1.000	1.000	1.000	-
12	0.933	0.951	0.981	irs
13	0.887	0.902	0.983	irs
14	1.000	1.000	1.000	-
15	0.941	1.000	0.941	irs
16	1.000	1.000	1.000	-
17	0.983	1.000	0.983	drs
18	0.978	0.98	0.998	drs
19	1.000	1.000	1.000	-
20	1.000	1.000	1.000	-
21	0.872	0.999	0.873	irs
22	0.941	0.951	0.989	drs
23	1.000	1.000	1.000	-
24	0.893	0.913	0.978	irs
25	0.932	0.958	0.973	irs
26	0.889	1.000	0.889	drs
27	0.911	0.913	0.998	irs
28	0.965	0.976	0.989	irs
29	0.891	1.000	0.891	irs
30	0.931	1.000	0.931	drs
31	0.981	0.991	0.990	irs
32	0.999	1.000	0.999	irs
33	0.902	0.924	0.976	drs
34	0.959	1.000	0.959	irs
35	0.999	1.000	0.999	irs
36	0.836	0.859	0.973	irs
37	0.999	1.000	0.999	drs
38	1.000	1.000	1.000	-
39	0.881	0.901	0.978	irs
40	0.999	1.000	1.000	-
41	1.000	0.999	1.000	-
42	1.000	1.000	1.000	-
43	1.000	1.000	1.000	-
44	0.992	1.000	0.992	irs
45	0.928	0.945	0.982	irs
46	0.88	1.000	0.880	irs
47	0.999	1.000	0.999	irs
48	0.999	1.000	0.999	irs
49	1.000	1.000	1.000	-
50	1.000	1.000	1.000	-
51	1.000	1.000	1.000	-
52	0.829	1.000	0.829	irs
53	0.981	0.998	0.983	irs
54	1.000	1.000	1.000	-
55	0.932	0.935	0.997	irs
56	0.957	0.961	0.996	irs
57	0.858	1.000	0.858	irs
58	0.983	0.991	0.992	irs
59	0.924	0.931	0.992	irs
60	1.000	1.000	1.000	-
Mean	0.954	0.975	0.978

Comprehensive efficiency, which refers to the maximum output capacity of higher education organizations, takes a value between 0-1, the closer it is to 1, indicating that the more appropriate the ratio of inputs and outputs of higher education resources, the more the resources have been effectively utilized, and the greater the benefits. The average value of the comprehensive efficiency of higher education resource allocation is 0.954, indicating that the utilization rate of resources has reached 95.4%, but there are still 4.6% of higher education resources not effectively utilized. Among them, there are 17 decision-making units with comprehensive efficiency value equal to 1, which indicates that these 17 decision-making units are in DEA effective state, that is, the input-output ratio of decision-making units is appropriate, that is, the input-output of higher education resources in 17 colleges and universities has reached the optimal allocation.

Pure technical efficiency, pure technical efficiency is the technical efficiency when considering the scale benefit, taking the value between 0-1, the closer to 1 indicates that the higher the efficiency. The average value of pure technical efficiency is 0.975, which is in the non-effective range, i.e., there is the problem of inefficient utilization of resources in higher education. Pure technical efficiency value of 1 decision-making unit reflects that these schools in higher education in the operation of the maximum output effect, the management level is high, the resources have reached the effective use. 27 decision-making units did not reach the pure technical efficiency state, there is a waste of resources, resource allocation management still has room for improvement, the efficiency of resource utilization needs to be further improved and optimized.

Scale efficiency, scale efficiency is the output efficiency affected by scale factors, that is, whether the input and output between the optimal state, take the value between 0-1, the closer to 1, indicating that the scale is more effective. Scale efficiency value = comprehensive technical efficiency value / pure technical efficiency value. When the value of scale efficiency is 1, it indicates that higher education in the province is in a state of fixed scale remuneration, and the amount of output will show an equal proportional increase with the increase in the amount of inputs. When the value of scale efficiency is less than 1, it indicates that higher education in the province is inefficient due to the change in scale remuneration. The average scale efficiency value is 0.978, which is in the non-effective range. 20 decision-making units with unchanged returns to scale can either expand the scale or keep it unchanged. 7 decision-making units are in the situation of decreasing returns to scale, which indicates that there are too many inputs in the process of higher education resource allocation, and we should pay attention to the streamlining of inputs and qualitative improvement in the process of improvement, so as to avoid the problem of wastefulness and inefficient use of resources caused by excessive input. 33 units are in the situation of inefficient use of resources, and 33 units are in the state of inefficient use of resources. Thirty-three decision-making units are in the stage of increasing returns to scale, indicating that it is necessary to increase the input of resources in the process of higher education development and expand the scale in order to achieve the optimal scale state.

5

Analysis of the factors affecting the efficiency of allocation of educational resources in colleges and universities

5.1

Selection of indicators

In this paper, we first studied many factors that affect the efficiency of resource allocation for higher education by reading the literature and dividing them into independent and dependent variables. Among them, the independent variables include factors such as the level of economic development, the total population, the school area, the total value of fixed assets, the percentage of full-time teachers with senior titles and above, and the percentage of full-time teachers with bachelor’s degree or above, etc., and the analytical efficiency value VTE (i.e., pure technical efficiency) derived from the DEA model is used as a dependent variable.

5.2

Tobit modeling

The efficiency value calculated by the data envelopment model (DEA) is discrete, and its value is distributed from 0 to 1. Generally speaking, in the process of coefficient regression analysis, the very common method is the ordinary least squares method, although the operation of this method is relatively simple, but its defects are also obvious, i.e., the use of this method, the value of the dependent variable has the value of the parameter with a discrete type of regression, the parameter estimate is easy to have a certain bias. Some bias. In order to solve this problem and make the results more scientific, the most important feature of the Tobit model is to use the maximum likelihood method instead of the ordinary least squares method. This thesis determines the Tobit regression model to study the analysis of factors affecting the efficiency of higher education resource allocation in province A based on the indicators described in the previous section, and the meaning of $y_{i}^{*}$ is the value of pure technical efficiency of education resource allocation in region i measured in the first stage by applying the DEA model.

(22)

y_{i}^{*} = α_{1} F_{1} + α_{2} F_{2} + α_{3} F_{3} + C

(23)

y_{i} = {\begin{array}{l} y_{i}^{*}, y_{i}^{*} < 1 \\ 1, y_{i}^{*} \geq 1 \end{array}

The principle of the test is that in real research, due to the limitation of subjective and objective conditions, it is often difficult to comprehensively investigate all variables, and it is easy to miss and miss variables, in this case, it is easy to have a certain correlation between some independent variables and distractors, resulting in the lack of accuracy of parameter estimates.In order to make the research results more scientific and accurate, we need to test the model, and test whether the omission of variables is more difficult, in general, the test of the model is to test whether there is a correlation between the variables and interfering items. The author used Eviews 6.0 software to conduct the Hausman test, the results show that the output value of its statistics is 83.534446, and its corresponding concomitant probability is 0.022, which obviously does not reach 0.1, indicating that the test results rejected the original assumptions of the random effects model, and a fixed effects model should be established.

6

Analysis of Tobit model regression results

Eviews8.0 software is used to do Tobit regression analysis on the data of independent variables, so as to make a deeper study on the influencing factors of the efficiency of educational resource allocation in province A. The Tobit regression results are shown in Table 5.

Table 5.

Tobit regression

Variable	Coefficient	Standard deviation	Z-statistic	P value
Economic development level	-3.97E-06	3.37E-06	-1.174725	0.2406
Total population	0.000312	6.25E-05	4.979456	0.0000
Compulsory school occupies land product	5.31E-06	7.36E-06	0.720106	0.4714
Fixed assets	0.000218	8.91E-05	2.423568	0.0156
Senior professional title and above special teachers	0.001709	0.003298	0.518021	0.6052
Bachelor degree or above	-0.440256	0.151689	-2.901954	0.0038
Constant term	-1.223569	0.526481	-2.332891	0.0198

Next, according to the above Tobit regression results, and combined with the actual situation, this paper explains the influencing factors of educational resource allocation in Province A one by one.

1) Although there is a negative relationship between the level of economic development and total technical efficiency in Province A, the P-value shows that the regression results are not significant. This indicates that although Province A has experienced rapid economic development in recent years, even though it has invested more in education, perhaps the regions are not perfect enough in the education management system, resulting in a certain redundancy in the government’s expenditure on the scale of education resources, resulting in an unreasonable waste of resource allocation, so it is necessary to further optimize the investment in education financial resources.

2) The utilization rate of educational resources in Province A is positively affected by the total population of the province. It shows that the increase of the total local population helps to fully utilize the resources invested in education.

3) The regression results of the two variables of school area and total value of educational fixed assets, which are used to reflect the scale and conditions of schools in province A, show that the total technical efficiency of the allocation of educational resources in province A is not significant, although it varies according to the area of schools. The total value of educational fixed assets is positively related to total technical efficiency at the 5% significance level. Both of them are difficult conditions that can help visualize the allocation of educational resources. From the above, it can be concluded that in view of some problems existing in Province A at this stage, it should increase its investment in fixed assets, expand the scale of education schools, improve the quality of education schools and optimize the conditions of education schools, so as to achieve the goal of improving the efficiency of the allocation of education resources in Province A.

4) The ratio of full-time teachers with senior titles and above and the ratio of full-time teachers with bachelor’s degree or above represent the level of teachers’ strength, and the ratio of full-time teachers with senior titles and above has no significant effect on the efficiency of allocation of educational resources in Province A, while the ratio of full-time teachers with bachelor’s degree or above obviously has a negative effect. It also shows that there is an imbalance in the development of teachers in province A. Investing more senior teachers or teachers with high education level in the education stage can not bring obvious effects in a short time, and it will also lead to the waste of resources of teachers with high education level. Ordinary full-time teachers can provide as much counseling as they can to students in primary and secondary schools, and the proportion of teachers with senior titles or advanced degrees only needs to meet the relevant standards and requirements.

7

Conclusion

1) This paper uses factor analysis to extract the input and output factors related to the allocation of educational resources in higher education. The three principal component factors of the input and output factors pool 92.36% and 93.61% of the information of the original variables respectively, indicating that the principal component factors have a better ability to summarize the original information, and it is appropriate to analyze the factors on behalf of the original indicators.

2) Applying DEAP.2.1 software to measure the efficiency of higher education resource allocation based on Province A. According to the results calculated by the DEA model, the average values of comprehensive efficiency, pure technical efficiency, and scale efficiency of higher education resource allocation in Province A are 0.954, 0.975, and 0.978, respectively. 33 decision-making units in Province A are in the stage of increasing returns to scale, accounting for 55%, which means that the province’s overall efficiency of educational resource allocation has the problems of underutilization of resources and insufficient total input. It is necessary to increase the input of educational resources and expand the scale to reach the optimal scale state.

3) The Tobit regression model analyzes the factors affecting the allocation efficiency of education resources in Province A. The analysis results show that the level of economic development, the area of schools and the proportion of full-time teachers with senior titles and above do not have a significant impact on the allocation efficiency of education resources in Province A. The total population, the fixed assets used for education, and the total amount of resources used for education are not significant, and the total amount of resources used for education is not significant. The total population and the total value of fixed assets used for education have a significant and positive influence on the allocation efficiency of education resources in Province A. The factor that has a significant and negative effect on the efficiency of allocating educational resources in Province A is the percentage of full-time teachers with a bachelor’s degree or higher.Therefore, the province can improve the teaching level of schools by improving the allocation of education funds, improving the allocation of education talents, and improving the allocation of education facilities.

Język:: Angielski

Częstotliwość wydawania:: 1 razy w roku
Dziedziny czasopisma:: Nauki biologiczne, Nauki biologiczne, inne, Matematyka, Matematyka stosowana, Matematyka ogólna, Fizyka, Fizyka, inne

Kanał RSS czasopisma

Mathematical and Statistical Strategies for Evaluating the Efficiency of Educational Resource Allocation in Tertiary Education

Danqiong Wang

Data publikacji: 21 mar 2025

Otrzymano: 16 lis 2024

Przyjęty: 24 lut 2025

DOI: https://doi.org/10.2478/amns-2025-0617

Słowa kluczoweEducational resource allocation, Factor analysis method, DEA model, Tobit model

© 2025 Danqiong Wang, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Słowa kluczowe
Educational resource allocation, Factor analysis method, DEA model, Tobit model