# A Novel Approach for Reducing Attributes and Its Application to Small Enterprise Financing Ability Evaluation

Authors: Baofeng Shi; Bin Meng; Hufeng Yang; Jing Wang; Wenli Shi
Journal: Complexity
 (2018)

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## Abstract

Attribute reduction is viewed as a kind of preprocessing steps for reducing large dimensionality in data mining of all complex systems. A great deal of researchers have proposed various approaches to reduce attributes or select key features in multicriteria decision making evaluation. In practice, the existing approaches for attribute reduction focused on improving the classification accuracy or saving the cost of computational time, without considering the influence of the reduction results on the original data set. To help address this gap, we develop an advanced novel attribute reduction approach combining Pearson correlation analysis withF test significance discrimination for the screening and identification of key characteristics related to the original data set. The proposed model has been verified using the financing ability evaluation data of 713 small enterprises of a city commercial bank in China. And the experimental results show that the proposed reduction model is efficient and effective. Moreover, our experimental findings help to locate the qualified partners and alleviate the difficulties faced by enterprises when applying loan.


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## Body

## 1. Introduction

With the coming of the era of big data, the size of data sets has been increasing sharply, causing the decision makers and management to have difficulty in making decisions based on those data [1]. Then the most important thing for decision makers is to reduce huge attributes or large dimensionality in data sets. Attribute reduction, also called indicators selection or feature screening, ascertains a subset of attributes to reduce the dimensionality of the original data sets. Utilizing reducing attributes, it can select the attributes with the highest information content and save the cost of computational time and memory [2]. Besides, it is also useful to improve the classification accuracy as a result of deleting the information chaos and irrelevant attributes [3]. In practice, attribute reduction has been applied to a great deal of fields such as decision making, pattern recognition, and economic and social system evaluation [4–7].The main attribute reduction approaches can be divided into three categories. One of the most famous methods for attribute reduction is based on rough set theory. Rough set approach proposed by Pawlak provides useful tools for reasoning from data [8]. It is advantageous to other approaches for attribute reduction that typically use multivariate statistics which require specific parametric assumptions [9, 10]. Degang et al. established a model to reduce the attributes of covering decision systems combining traditional rough set. Empirical study indicated that the proposed attribute reduction approach accomplished better classification performance than those of existing rough set methods [11]. In order to improve the classification accuracy containing hybrid type attributes, such as discretizing numerical attributes or categorical attributes, Hu et al. introduced a simple and efficient greedy algorithm for hybrid attribute reduction [12]. When some decision or evaluation systems have some errors, missing data, and missing attributes in observation, neither DRSA (dominance-based rough set approach) [13] nor VC-DRSA (variable-consistency dominance-based rough set approach) [14] can work appropriately. Inuiguchi et al. created a variable-precision dominance-based rough set approach (VP-DRSA) to deal with these problems [15]. Tsang et al. presented an attribute reduction model with covering rough sets based on discernibility matrix to compute all attribute reducts [16]. Furthermore, Wang et al. developed a novel approach for constructing simpler discernibility matrix with covering rough sets, and it improved some characterizations of attribute reduction proposed by Wang et al. [17]. In addition, there are the two most important attribute reduction models, which extend the Pawlak’s rough set, the neighborhood rough set (NRS) model [18] and the fuzzy rough set model [19]. They can tackle continuous numeric data and fuzzy information granulation, and the determination of what objects should be included in a rough set allowed some flexibility [20].The second method for screening key factors is the attribute reduction models based on statistics or econometrics technique. In order to obtain preference information of the decision maker in multiobjective search, Zitzler and Künzli defined an optimization goal in terms of a binary performance measure, to select key information directly utilizing this measure [21]. Polat and Krmac screened the most important attributes using pairwise Fisher score attribute reduction approach (PFSAR) and correlation based attribute reduction [22]. Ju and Sohn developed a technology attribute reduction model that uses logistic regression based on exploratory factor analysis (EFA) of 16 technology-related attributes [23]. Elliott et al. developed a model based on a double hidden Markov model (DHMM), to extract information about the “true” credit qualities of firms [24]. Shi et al. created an indicators extraction model based on Pearson correlation analysis and logistic regression significant discriminant in customers’ classification. The proposed approach ensured the reserved indicators can effectively distinguish default customers from nondefault customers [25].In addition, there are other attributes reduction methods, such as the concept lattice model, the heuristic algorithm, and the colony optimization algorithm. Some researchers developed some new attribute reduction models by using the concept lattice classification theory [26–28]. Wei et al. discussed attribute reduction in information systems by establishing three equivalence relations on the attribute set and its power set [29]. In overwhelming data analysis and machine learning studies, most existing attribute reduction work focused on improving the classification accuracy. However, these studies neglected the problem of how to decrease the test cost. Min et al. proposed a heuristic algorithm to handle this problem in attribute reduction [30]. Chi et al. created an indicators screen model based on correlation analysis and component analysis [31]. Minimal test cost attribute reduction is very important in cost-sensitive machine learning. However, in many cases these heuristic algorithms cannot find the optimal solution. In order to deal with this problem, Xu et al. established an ant colony optimization algorithm for attribute reduction. Experimental results on UCI data sets showed that the proposed method outperforms the information gain-based approach [32]. According to the principle of eliminating redundant information and the principle of the maximum information content, Shi and Chi proposed an attribute reduction model combining R cluster analysis and coefficient of variation [33]. Because people are interested in the maximal rules implicated in attribute reduction, Li et al. developed two new kinds of attribute reduction approaches in the decision formal context based on maximal rules [34].The existing findings can offer important references for reducing attributes. However, there are still some limitations. First of all, in the evaluation of complex systems, the aim of the attribute reduction is to eliminate the factors, which should not have significant effect on the comprehensive evaluation results. However, the existing attribute reduction approaches have not established the comprehensive indexy (i.e., the comprehensive score vector y), which can reflect all of the attributes’ characteristics. This means that the existing attribute reduction approaches have not developed the relationship between attributes and the comprehensive index y (i.e., the comprehensive evaluation result). This results in some reserved attributes, which do not have significant effect on the comprehensive evaluation result. And secondly, most of existing attributes reduction approaches judged the performance of the proposed approach by the standard of saving the cost of computational time. The standard does not analyze the information contribution degree of the reserved attributes to the mass-election attributes. Thirdly, most of existing researches verify the applicability of the proposed attribute reduction methods using numerical simulation, but not utilizing actual data.To solve the shortcomings, this study creates a novel attribute reduction model to screen the key influencing factors. We advance in three aspects. First, this paper establishes an attribute reduction approach by combining Pearson correlation analysis withF test significance discrimination. Pearson correlation analysis is applied to the calculation of the correlation among attributes to delete the similar attributes. F test significance discrimination is used to select the key attributes which have the greatest influence on comprehensive index y. Second, we also define an information contribution ratio to assess this attribute reduction approach from a statistical viewpoint. Third, the proposed attribute reduction approach has been verified by utilizing the financing ability evaluation data of 713 small enterprises of a city commercial bank in China. Empirical evidence presents that the selected attributes reflect 94.7% original information with 27.54% original attributes. Furthermore, this paper also selects 19 key influencing factors for assessing the financing ability of small enterprises.The remainder of this paper is organized as follows. Section2 introduces the design and methodology of this study. Section 3 presents the data and empirical analysis of our attribute reduction model for 713 small enterprises. Section 4 concludes and highlights the future research directions of this paper.


## 2. Design and Methodology of the Study

In this section, we introduce a novel attribute reduction model by combining Pearson correlation analysis withF test significance discrimination approach. First of all, in order to eliminate the influence of the differences of attributes units and dimensions on attribute reduction, the original data should be transformed into real numbers within the interval [0,1]. Secondly, we utilize Pearson correlation analysis to delete the attributes of large correlation from the whole mass-election attributes set, avoiding repeated information. Thirdly, F test significance discrimination approach has been created to select the attributes with the highest information content, which ensures that the selected attribute has the greatest influence on the small enterprise financing performance. A step-by-step instruction is as follows.


### 2.1. Standardization of Attribute Data

In our attribute reduction model, the first step is standardization of attribute data so that the after-calculation processes and parameters use the same standard. According to the features of attributes, the attributes can be divided into two types: quantitative attributes and qualitative attributes. The quantitative attributes include positive attributes, negative attributes, and interval attributes. The positive attributes are attributes showing that the greater their values are, the better the small enterprise financing capacity is. The negative attributes are attributes showing that the less their values are, the better small enterprise financing capacity is. The interval attributes are attributes reasonable only when the original index data are within certain range.The standardization equations of positive attributes, negative attributes, and interval attributes are represented by (1), (2), and (3), respectively, [35]: (1)pij=vij-min1≤j≤nvijmax1≤j≤nvij-min1≤j≤nvij(2)pij=max1≤j≤nvij-vijmax1≤j≤nvij-min1≤j≤nvij(3)pij=1-q1-vijmaxq1-min1≤j≤nvij,max1≤j≤nvij-q2,vij<q1a1-vij-q2maxq1-min1≤j≤nvij,max1≤j≤nvij-q2,vij>q2b1,q1≤vij≤q2cwhere pij is the standardized score of the jth small enterprise on the ith attribute, vij is the attribute original data of the jth small enterprise on the ith attribute, n is the number of small enterprises, q1 is the left boundary of the ideal interval, and q2 is the right boundary of the ideal range.The qualitative attributes refer to these attributes whose attribute values are described by a text, rather than a numerical value. The standard scores of qualitative attributes can be obtained by rational analysis and expert investigation.


### 2.2. Pearson Correlation Coefficients

The Pearson product-momentum correlation coefficient was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s [36]. It is a measure of the linear correlation (dependence) between two random variables. It was also called the PPMCC, PCC, or Pearson’s rxy. Historically, it is the first formal measure of correlation and it is still one of the most widely used measures of relationship.The Pearson correlation coefficient of two attributesx and y is defined as the covariance of the two variables divided by the product of their standard deviations. The Pearson correlation coefficient is commonly represented by the letterr and it can be equivalently defined by [37] (4)rxy=∑i=1nxi-x-yi-y-∑k=1nxk-x-2∑k=1nyk-y-2where x-=(1/n)∑j=1nxj, y-=(1/n)∑j=1nyj are the mean of x and y, respectively. Equation (4) is applied to the calculation of the correlation between two variables x and y. The coefficient rxy ranges from −1 to 1 and it is invariant to linear transformations of either variables. A value of 1 indicates a total positive correlation between x and y, a value of 0 implies no correlation between x and y, and a value of −1 indicates a total negative correlation.Some authors have offered guidelines for the interpretation of the Pearson correlation coefficient [38–41]. If the Pearson correlation coefficient of two attributes is greater than 0.8 [40, 41], we can conclude that these attributes are information redundancy. In this situation, we should remove one of attributes. In the opposite situation, if the Pearson correlation coefficient is smaller than 0.8, it indicates that these attributes are not information redundancy and should keep these two attributes.


### 2.3. Attribute Reduction Model

In our attribute reduction model, the third step is to select the key attribute which has the greatest influence on comprehensive indexy and deleting the uncorrelated attributes. In this part, we first calculate the attribute weightings using entropy weight approach. And then, we can obtain the financing ability evaluation score y (i.e., comprehensive index y) for every small enterprise. Subsequently, the multiple determination coefficient Rm-12 between comprehensive index y and all of these m-1 attributes can be obtained, and the multiple determination coefficient Rm-22 between comprehensive index y and the remaining m-2 attributes after removing an attribute xi can be calculated. By using F test significance discrimination, these key attributes which have the greatest influence on small enterprise financing ability evaluation are selected. At the same time, the reduction idea—that is, the bigger the difference ΔR2 between the multiple repeated determination coefficient Rm-12 and the multiple determination coefficient Rm-22 (ΔR2=Rm-12-Rm-22), the more the significance to comprehensive evaluation results—is reflected. Thus, the right time to make up the existing attribute reduction approaches cannot reflect the influence of attributes on the comprehensive index y, because the attribute reduction process has nothing to do with comprehensive index y.


#### 2.3.1. Weighting Attributes Utilizing Entropy Weight Method

Letfij denote the weight of the ith attribute in the jth small enterprise, let pij denote the standard score of the ith attribute in the jth small enterprise, let n denote the number of small enterprises, and let m denote the number of attributes.The subordinate degree functionfij of the attribute pij is given by (5)fij=pij∑j=1npij.Then, the entropyHi of the ith attribute can be calculated with (6)Hi=-1ln⁡n∑j=1nfijln⁡fij.And then, the entropy weightwi of the ith attribute is [42] (7)wi=1-Him-∑i=1mHi.where ∑i=1mwi=1.


#### 2.3.2. Reducing Attributes Based onF Test Significance Discrimination

After eliminating redundant information in Section2.2, this section will select the key attributes which have the greatest influence on comprehensive index y utilizing F test significance discrimination approach. Now we outline the steps to build an attribute reduction model based on F test significance discrimination.Step 1.
Calculate the comprehensive indexy. Let yj denote the comprehensive index or the comprehensive score for the jth small enterprise financing ability evaluation. We have(8)yj=∑i=1mwipij.
The meanings of the rest of variables in (8) are the same as the variables in (1) and (7).Step 2.
Calculate Pearson correlation coefficientsri(i=1,2,…,m) between attribute xi and the comprehensive index y. We can assume the attributes ranking is x1∗,x2∗,…,xm∗ according to the correlation coefficient absolute value ri in a descending order.Step 3.
Calculate the multiple determination coefficientRm-12 between comprehensive indexy and the remaining m-1 attributes x2∗, x3∗, …, xm∗ after removing the first attribute x1∗ with the biggest correlation coefficient absolute value.
Leta0,a2,a3,…,am denote the estimated parameters, respectively, let x2∗,x3∗,…,xm∗ denote m-1 attributes, and let ε denote the random error term. The regression function is given by (9)y=a0+a2x2∗+a3x3∗+⋯+amxm∗+ε.In (9), the estimated values for parameters a0,a2,a3,…,am can be obtained using the least squares regression estimation method. Furthermore, the estimated value vector y^ of the comprehensive indexy can be calculated. Then, we have [43] (10)R2m-1=∑j=1ny^j-y-2∑j=1nyj-y-2,where y-=(1/n)∑j=1nyj and n denotes the number of small enterprises.
It should be pointed out that the attributex1∗ should be reserved in attribute reduction, because the attribute x1∗ has the maximum pertinency with the comprehensive evaluation results. It also indicates that the attribute x1∗ has the biggest impact on small enterprise financing ability evaluation.Step 4.
Calculate the multiple determination coefficientRm-22 between comprehensive index y and the remaining m-2 attributes x3∗,x4∗,…,xm∗ after removing the first attribute x1∗ with the biggest correlation coefficient absolute value and the second attribute x2∗ with the second biggest correlation coefficient absolute value.
Letb0,b3,b4,…,bm denote the estimated parameters, respectively, let x3∗,x4∗,…,xm∗ denote m-2 attributes, and let η denote the random error term. The regression function is as follows: (11)y=b0+b3x3∗+b4x4∗+⋯+bmxm∗+η.In the same way, we can calculate the estimated value vector y^′ of the comprehensive index y for (11). And the multiple determination coefficient Rm-22 is given by (12)R2m-2=∑j=1ny^j′-y-2∑j=1nyj-y-2.Step 5.
CalculateΔR2. Let ΔR2 denote the difference of the multiple determination coefficient Rm-12 and the multiple determination coefficient Rm-22; namely,(13)ΔR2=Rm-12-Rm-22.
In (13), the difference ΔR2 reflects the influence of the attribute x2∗ on the comprehensive index y. If ΔR2 is not equal to zero significantly, it means that the attribute x2∗ affects the comprehensive evaluation result y significantly, and therefore the attribute x2∗ should be reserved. On the contrary, if ΔR2 is equal to zero significantly, then ΔR2=0, which indicates the attribute x2∗ does not have significant effect on the comprehensive evaluation result y, and the attribute x2∗ should be deleted.Step 6.
Reduce attributes establishingF test significance discrimination.
HypothesisH0: ΔR2≠0; H1: ΔR2=0.
LetFi denote the F test value of the ith attribute xi; we have [44](14)Fi=ΔR2/11-Rm-12/n-m-2=Rm-12-Rm-22n-m-21-Rm-12.
For (14), we can understand its meanings from the following three aspects. Firstly, the bigger the multiple determination coefficient Rm-12 is, the smaller the deviation of the estimated value y^ and the actual comprehensive index y would be. The smaller the multiple determination coefficient Rm-22 is, the bigger the deviation of the estimated value y^′ and the actual comprehensive index y after removing the attribute x2∗ would be. That is to say, when we remove the attribute x2∗, the explanation ability of the m-2 attributes x3∗,x4∗,…,xm∗ to the comprehensive evaluation score y decreases significantly. It also indicates that the attribute x2∗ has significant effect on the comprehensive evaluation result y of small enterprises; thus the attribute x2∗ should be reserved.
Secondly, the bigger the differenceΔR2 of the multiple determination coefficient Rm-12 and the multiple determination coefficient Rm-22 is, the bigger the difference of the explanation ability Rm-12 of the m-1 attributes x2∗,x3∗,…,xm∗ to the comprehensive evaluation score y and the explanation ability Rm-22 of the m-2 attributes x3∗,x4∗,…,xm∗ to the comprehensive evaluation score y would be. It means that the attribute x2∗ affects the comprehensive evaluation result y of small enterprises significantly, and the attribute x2∗ should not be deleted.
Thirdly, the bigger the differenceΔR2 (i.e., the bigger the difference value Rm-12-Rm-22) is, the bigger the F test value Fi would be. In this situation, the F test can be passed easily. And it also expresses the attribute effects on the comprehensive evaluation result y significantly.
Under the condition of the hypothesis ofH0, Fi follows F distribution; that is to say, Fi~F(1,n-(m-2)). Let the confidence level α be equal to 0.05 [45]; the critical value Fα can be checked from F statistics. If Fi≥Fα, accept hypothesis H0:ΔR2≠0. It means that ΔR2 is not equal to zero significantly, and the attribute x2∗ should be reserved. Conversely, if Fi<Fα, reject hypothesis H0:ΔR2≠0, which indicates that ΔR2 is equal to zero significantly, and the attribute x2∗ should be deleted.Step 7.
RepeatStep 3 toStep 6, and select other attributes.For the rest of them-2 attributes x3∗,x4∗,…,xm∗, we can reduce attributes by repeating Step 3 to Step 6. Until you find the first attribute xi∗, the corresponding F test value satisfies the inequation Fi∗(xi∗)≤F0.05(xi∗). At this time, the attribute reduction can be stopped. It suggests that the rest of attributes do not have significant influence on comprehensive evaluation result y.


### 2.4. The Judgment of Reasonability of the Proposed Attribute Reduction Approach

According to the idea that the multiple determination coefficientR2 can be used to describe the explanation ability of the independent variable on the dependent variable, this paper uses an information contribution ratio to assess the performance of attribute reduction model. The information contribution ratio can be defined as the ratio of the explanation ability RReserved2 of the reserved attributes to the comprehensive evaluation score y to the explanation ability RMass-election2 of the mass-election attributes to the comprehensive evaluation score y.LetIn denote an information contribution ratio of the reserved attributes to the mass-election attributes, let RReserved2 denote the multiple determination coefficient of the reserved attributes to the comprehensive evaluation score y, and let RMass-election2 denote the multiple determination coefficient of the mass-election attributes to the comprehensive evaluation score y. The information contribution rate In of the reserved attributes to the mass-election attributes is given by (15)In=RReserved2RMass-election2.Equation (15) is applied to judge the reasonability of the proposed attribute reduction model. The numerator RReserved2 reflects the explanation ability of the reserved attributes to the comprehensive evaluation score y, and the denominator RMass-election2 illustrates the explanation ability of the mass-election attributes to the comprehensive evaluation score y. Equation (15) is the ratio of the explanation ability RReserved2 to the explanation ability RMass-election2. It reveals the information contribution degree of the reserved attributes to the mass-election attributes.As a decision criterion for judging the rationality of the proposed attribute reduction model, the proposed approach is considered reasonable if the reserved attributes are able to contribute more than 90% of the mass-election attributes by using less than 30% of attributes in the mass-election attribute set.


## 2.1. Standardization of Attribute Data

In our attribute reduction model, the first step is standardization of attribute data so that the after-calculation processes and parameters use the same standard. According to the features of attributes, the attributes can be divided into two types: quantitative attributes and qualitative attributes. The quantitative attributes include positive attributes, negative attributes, and interval attributes. The positive attributes are attributes showing that the greater their values are, the better the small enterprise financing capacity is. The negative attributes are attributes showing that the less their values are, the better small enterprise financing capacity is. The interval attributes are attributes reasonable only when the original index data are within certain range.The standardization equations of positive attributes, negative attributes, and interval attributes are represented by (1), (2), and (3), respectively, [35]: (1)pij=vij-min1≤j≤nvijmax1≤j≤nvij-min1≤j≤nvij(2)pij=max1≤j≤nvij-vijmax1≤j≤nvij-min1≤j≤nvij(3)pij=1-q1-vijmaxq1-min1≤j≤nvij,max1≤j≤nvij-q2,vij<q1a1-vij-q2maxq1-min1≤j≤nvij,max1≤j≤nvij-q2,vij>q2b1,q1≤vij≤q2cwhere pij is the standardized score of the jth small enterprise on the ith attribute, vij is the attribute original data of the jth small enterprise on the ith attribute, n is the number of small enterprises, q1 is the left boundary of the ideal interval, and q2 is the right boundary of the ideal range.The qualitative attributes refer to these attributes whose attribute values are described by a text, rather than a numerical value. The standard scores of qualitative attributes can be obtained by rational analysis and expert investigation.


## 2.2. Pearson Correlation Coefficients

The Pearson product-momentum correlation coefficient was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s [36]. It is a measure of the linear correlation (dependence) between two random variables. It was also called the PPMCC, PCC, or Pearson’s rxy. Historically, it is the first formal measure of correlation and it is still one of the most widely used measures of relationship.The Pearson correlation coefficient of two attributesx and y is defined as the covariance of the two variables divided by the product of their standard deviations. The Pearson correlation coefficient is commonly represented by the letterr and it can be equivalently defined by [37] (4)rxy=∑i=1nxi-x-yi-y-∑k=1nxk-x-2∑k=1nyk-y-2where x-=(1/n)∑j=1nxj, y-=(1/n)∑j=1nyj are the mean of x and y, respectively. Equation (4) is applied to the calculation of the correlation between two variables x and y. The coefficient rxy ranges from −1 to 1 and it is invariant to linear transformations of either variables. A value of 1 indicates a total positive correlation between x and y, a value of 0 implies no correlation between x and y, and a value of −1 indicates a total negative correlation.Some authors have offered guidelines for the interpretation of the Pearson correlation coefficient [38–41]. If the Pearson correlation coefficient of two attributes is greater than 0.8 [40, 41], we can conclude that these attributes are information redundancy. In this situation, we should remove one of attributes. In the opposite situation, if the Pearson correlation coefficient is smaller than 0.8, it indicates that these attributes are not information redundancy and should keep these two attributes.


## 2.3. Attribute Reduction Model

In our attribute reduction model, the third step is to select the key attribute which has the greatest influence on comprehensive indexy and deleting the uncorrelated attributes. In this part, we first calculate the attribute weightings using entropy weight approach. And then, we can obtain the financing ability evaluation score y (i.e., comprehensive index y) for every small enterprise. Subsequently, the multiple determination coefficient Rm-12 between comprehensive index y and all of these m-1 attributes can be obtained, and the multiple determination coefficient Rm-22 between comprehensive index y and the remaining m-2 attributes after removing an attribute xi can be calculated. By using F test significance discrimination, these key attributes which have the greatest influence on small enterprise financing ability evaluation are selected. At the same time, the reduction idea—that is, the bigger the difference ΔR2 between the multiple repeated determination coefficient Rm-12 and the multiple determination coefficient Rm-22 (ΔR2=Rm-12-Rm-22), the more the significance to comprehensive evaluation results—is reflected. Thus, the right time to make up the existing attribute reduction approaches cannot reflect the influence of attributes on the comprehensive index y, because the attribute reduction process has nothing to do with comprehensive index y.


### 2.3.1. Weighting Attributes Utilizing Entropy Weight Method

Letfij denote the weight of the ith attribute in the jth small enterprise, let pij denote the standard score of the ith attribute in the jth small enterprise, let n denote the number of small enterprises, and let m denote the number of attributes.The subordinate degree functionfij of the attribute pij is given by (5)fij=pij∑j=1npij.Then, the entropyHi of the ith attribute can be calculated with (6)Hi=-1ln⁡n∑j=1nfijln⁡fij.And then, the entropy weightwi of the ith attribute is [42] (7)wi=1-Him-∑i=1mHi.where ∑i=1mwi=1.


### 2.3.2. Reducing Attributes Based onF Test Significance Discrimination

After eliminating redundant information in Section2.2, this section will select the key attributes which have the greatest influence on comprehensive index y utilizing F test significance discrimination approach. Now we outline the steps to build an attribute reduction model based on F test significance discrimination.Step 1.
Calculate the comprehensive indexy. Let yj denote the comprehensive index or the comprehensive score for the jth small enterprise financing ability evaluation. We have(8)yj=∑i=1mwipij.
The meanings of the rest of variables in (8) are the same as the variables in (1) and (7).Step 2.
Calculate Pearson correlation coefficientsri(i=1,2,…,m) between attribute xi and the comprehensive index y. We can assume the attributes ranking is x1∗,x2∗,…,xm∗ according to the correlation coefficient absolute value ri in a descending order.Step 3.
Calculate the multiple determination coefficientRm-12 between comprehensive indexy and the remaining m-1 attributes x2∗, x3∗, …, xm∗ after removing the first attribute x1∗ with the biggest correlation coefficient absolute value.
Leta0,a2,a3,…,am denote the estimated parameters, respectively, let x2∗,x3∗,…,xm∗ denote m-1 attributes, and let ε denote the random error term. The regression function is given by (9)y=a0+a2x2∗+a3x3∗+⋯+amxm∗+ε.In (9), the estimated values for parameters a0,a2,a3,…,am can be obtained using the least squares regression estimation method. Furthermore, the estimated value vector y^ of the comprehensive indexy can be calculated. Then, we have [43] (10)R2m-1=∑j=1ny^j-y-2∑j=1nyj-y-2,where y-=(1/n)∑j=1nyj and n denotes the number of small enterprises.
It should be pointed out that the attributex1∗ should be reserved in attribute reduction, because the attribute x1∗ has the maximum pertinency with the comprehensive evaluation results. It also indicates that the attribute x1∗ has the biggest impact on small enterprise financing ability evaluation.Step 4.
Calculate the multiple determination coefficientRm-22 between comprehensive index y and the remaining m-2 attributes x3∗,x4∗,…,xm∗ after removing the first attribute x1∗ with the biggest correlation coefficient absolute value and the second attribute x2∗ with the second biggest correlation coefficient absolute value.
Letb0,b3,b4,…,bm denote the estimated parameters, respectively, let x3∗,x4∗,…,xm∗ denote m-2 attributes, and let η denote the random error term. The regression function is as follows: (11)y=b0+b3x3∗+b4x4∗+⋯+bmxm∗+η.In the same way, we can calculate the estimated value vector y^′ of the comprehensive index y for (11). And the multiple determination coefficient Rm-22 is given by (12)R2m-2=∑j=1ny^j′-y-2∑j=1nyj-y-2.Step 5.
CalculateΔR2. Let ΔR2 denote the difference of the multiple determination coefficient Rm-12 and the multiple determination coefficient Rm-22; namely,(13)ΔR2=Rm-12-Rm-22.
In (13), the difference ΔR2 reflects the influence of the attribute x2∗ on the comprehensive index y. If ΔR2 is not equal to zero significantly, it means that the attribute x2∗ affects the comprehensive evaluation result y significantly, and therefore the attribute x2∗ should be reserved. On the contrary, if ΔR2 is equal to zero significantly, then ΔR2=0, which indicates the attribute x2∗ does not have significant effect on the comprehensive evaluation result y, and the attribute x2∗ should be deleted.Step 6.
Reduce attributes establishingF test significance discrimination.
HypothesisH0: ΔR2≠0; H1: ΔR2=0.
LetFi denote the F test value of the ith attribute xi; we have [44](14)Fi=ΔR2/11-Rm-12/n-m-2=Rm-12-Rm-22n-m-21-Rm-12.
For (14), we can understand its meanings from the following three aspects. Firstly, the bigger the multiple determination coefficient Rm-12 is, the smaller the deviation of the estimated value y^ and the actual comprehensive index y would be. The smaller the multiple determination coefficient Rm-22 is, the bigger the deviation of the estimated value y^′ and the actual comprehensive index y after removing the attribute x2∗ would be. That is to say, when we remove the attribute x2∗, the explanation ability of the m-2 attributes x3∗,x4∗,…,xm∗ to the comprehensive evaluation score y decreases significantly. It also indicates that the attribute x2∗ has significant effect on the comprehensive evaluation result y of small enterprises; thus the attribute x2∗ should be reserved.
Secondly, the bigger the differenceΔR2 of the multiple determination coefficient Rm-12 and the multiple determination coefficient Rm-22 is, the bigger the difference of the explanation ability Rm-12 of the m-1 attributes x2∗,x3∗,…,xm∗ to the comprehensive evaluation score y and the explanation ability Rm-22 of the m-2 attributes x3∗,x4∗,…,xm∗ to the comprehensive evaluation score y would be. It means that the attribute x2∗ affects the comprehensive evaluation result y of small enterprises significantly, and the attribute x2∗ should not be deleted.
Thirdly, the bigger the differenceΔR2 (i.e., the bigger the difference value Rm-12-Rm-22) is, the bigger the F test value Fi would be. In this situation, the F test can be passed easily. And it also expresses the attribute effects on the comprehensive evaluation result y significantly.
Under the condition of the hypothesis ofH0, Fi follows F distribution; that is to say, Fi~F(1,n-(m-2)). Let the confidence level α be equal to 0.05 [45]; the critical value Fα can be checked from F statistics. If Fi≥Fα, accept hypothesis H0:ΔR2≠0. It means that ΔR2 is not equal to zero significantly, and the attribute x2∗ should be reserved. Conversely, if Fi<Fα, reject hypothesis H0:ΔR2≠0, which indicates that ΔR2 is equal to zero significantly, and the attribute x2∗ should be deleted.Step 7.
RepeatStep 3 toStep 6, and select other attributes.For the rest of them-2 attributes x3∗,x4∗,…,xm∗, we can reduce attributes by repeating Step 3 to Step 6. Until you find the first attribute xi∗, the corresponding F test value satisfies the inequation Fi∗(xi∗)≤F0.05(xi∗). At this time, the attribute reduction can be stopped. It suggests that the rest of attributes do not have significant influence on comprehensive evaluation result y.


## 2.3.1. Weighting Attributes Utilizing Entropy Weight Method

Letfij denote the weight of the ith attribute in the jth small enterprise, let pij denote the standard score of the ith attribute in the jth small enterprise, let n denote the number of small enterprises, and let m denote the number of attributes.The subordinate degree functionfij of the attribute pij is given by (5)fij=pij∑j=1npij.Then, the entropyHi of the ith attribute can be calculated with (6)Hi=-1ln⁡n∑j=1nfijln⁡fij.And then, the entropy weightwi of the ith attribute is [42] (7)wi=1-Him-∑i=1mHi.where ∑i=1mwi=1.


## 2.3.2. Reducing Attributes Based onF Test Significance Discrimination

After eliminating redundant information in Section2.2, this section will select the key attributes which have the greatest influence on comprehensive index y utilizing F test significance discrimination approach. Now we outline the steps to build an attribute reduction model based on F test significance discrimination.Step 1.
Calculate the comprehensive indexy. Let yj denote the comprehensive index or the comprehensive score for the jth small enterprise financing ability evaluation. We have(8)yj=∑i=1mwipij.
The meanings of the rest of variables in (8) are the same as the variables in (1) and (7).Step 2.
Calculate Pearson correlation coefficientsri(i=1,2,…,m) between attribute xi and the comprehensive index y. We can assume the attributes ranking is x1∗,x2∗,…,xm∗ according to the correlation coefficient absolute value ri in a descending order.Step 3.
Calculate the multiple determination coefficientRm-12 between comprehensive indexy and the remaining m-1 attributes x2∗, x3∗, …, xm∗ after removing the first attribute x1∗ with the biggest correlation coefficient absolute value.
Leta0,a2,a3,…,am denote the estimated parameters, respectively, let x2∗,x3∗,…,xm∗ denote m-1 attributes, and let ε denote the random error term. The regression function is given by (9)y=a0+a2x2∗+a3x3∗+⋯+amxm∗+ε.In (9), the estimated values for parameters a0,a2,a3,…,am can be obtained using the least squares regression estimation method. Furthermore, the estimated value vector y^ of the comprehensive indexy can be calculated. Then, we have [43] (10)R2m-1=∑j=1ny^j-y-2∑j=1nyj-y-2,where y-=(1/n)∑j=1nyj and n denotes the number of small enterprises.
It should be pointed out that the attributex1∗ should be reserved in attribute reduction, because the attribute x1∗ has the maximum pertinency with the comprehensive evaluation results. It also indicates that the attribute x1∗ has the biggest impact on small enterprise financing ability evaluation.Step 4.
Calculate the multiple determination coefficientRm-22 between comprehensive index y and the remaining m-2 attributes x3∗,x4∗,…,xm∗ after removing the first attribute x1∗ with the biggest correlation coefficient absolute value and the second attribute x2∗ with the second biggest correlation coefficient absolute value.
Letb0,b3,b4,…,bm denote the estimated parameters, respectively, let x3∗,x4∗,…,xm∗ denote m-2 attributes, and let η denote the random error term. The regression function is as follows: (11)y=b0+b3x3∗+b4x4∗+⋯+bmxm∗+η.In the same way, we can calculate the estimated value vector y^′ of the comprehensive index y for (11). And the multiple determination coefficient Rm-22 is given by (12)R2m-2=∑j=1ny^j′-y-2∑j=1nyj-y-2.Step 5.
CalculateΔR2. Let ΔR2 denote the difference of the multiple determination coefficient Rm-12 and the multiple determination coefficient Rm-22; namely,(13)ΔR2=Rm-12-Rm-22.
In (13), the difference ΔR2 reflects the influence of the attribute x2∗ on the comprehensive index y. If ΔR2 is not equal to zero significantly, it means that the attribute x2∗ affects the comprehensive evaluation result y significantly, and therefore the attribute x2∗ should be reserved. On the contrary, if ΔR2 is equal to zero significantly, then ΔR2=0, which indicates the attribute x2∗ does not have significant effect on the comprehensive evaluation result y, and the attribute x2∗ should be deleted.Step 6.
Reduce attributes establishingF test significance discrimination.
HypothesisH0: ΔR2≠0; H1: ΔR2=0.
LetFi denote the F test value of the ith attribute xi; we have [44](14)Fi=ΔR2/11-Rm-12/n-m-2=Rm-12-Rm-22n-m-21-Rm-12.
For (14), we can understand its meanings from the following three aspects. Firstly, the bigger the multiple determination coefficient Rm-12 is, the smaller the deviation of the estimated value y^ and the actual comprehensive index y would be. The smaller the multiple determination coefficient Rm-22 is, the bigger the deviation of the estimated value y^′ and the actual comprehensive index y after removing the attribute x2∗ would be. That is to say, when we remove the attribute x2∗, the explanation ability of the m-2 attributes x3∗,x4∗,…,xm∗ to the comprehensive evaluation score y decreases significantly. It also indicates that the attribute x2∗ has significant effect on the comprehensive evaluation result y of small enterprises; thus the attribute x2∗ should be reserved.
Secondly, the bigger the differenceΔR2 of the multiple determination coefficient Rm-12 and the multiple determination coefficient Rm-22 is, the bigger the difference of the explanation ability Rm-12 of the m-1 attributes x2∗,x3∗,…,xm∗ to the comprehensive evaluation score y and the explanation ability Rm-22 of the m-2 attributes x3∗,x4∗,…,xm∗ to the comprehensive evaluation score y would be. It means that the attribute x2∗ affects the comprehensive evaluation result y of small enterprises significantly, and the attribute x2∗ should not be deleted.
Thirdly, the bigger the differenceΔR2 (i.e., the bigger the difference value Rm-12-Rm-22) is, the bigger the F test value Fi would be. In this situation, the F test can be passed easily. And it also expresses the attribute effects on the comprehensive evaluation result y significantly.
Under the condition of the hypothesis ofH0, Fi follows F distribution; that is to say, Fi~F(1,n-(m-2)). Let the confidence level α be equal to 0.05 [45]; the critical value Fα can be checked from F statistics. If Fi≥Fα, accept hypothesis H0:ΔR2≠0. It means that ΔR2 is not equal to zero significantly, and the attribute x2∗ should be reserved. Conversely, if Fi<Fα, reject hypothesis H0:ΔR2≠0, which indicates that ΔR2 is equal to zero significantly, and the attribute x2∗ should be deleted.Step 7.
RepeatStep 3 toStep 6, and select other attributes.For the rest of them-2 attributes x3∗,x4∗,…,xm∗, we can reduce attributes by repeating Step 3 to Step 6. Until you find the first attribute xi∗, the corresponding F test value satisfies the inequation Fi∗(xi∗)≤F0.05(xi∗). At this time, the attribute reduction can be stopped. It suggests that the rest of attributes do not have significant influence on comprehensive evaluation result y.


## 2.4. The Judgment of Reasonability of the Proposed Attribute Reduction Approach

According to the idea that the multiple determination coefficientR2 can be used to describe the explanation ability of the independent variable on the dependent variable, this paper uses an information contribution ratio to assess the performance of attribute reduction model. The information contribution ratio can be defined as the ratio of the explanation ability RReserved2 of the reserved attributes to the comprehensive evaluation score y to the explanation ability RMass-election2 of the mass-election attributes to the comprehensive evaluation score y.LetIn denote an information contribution ratio of the reserved attributes to the mass-election attributes, let RReserved2 denote the multiple determination coefficient of the reserved attributes to the comprehensive evaluation score y, and let RMass-election2 denote the multiple determination coefficient of the mass-election attributes to the comprehensive evaluation score y. The information contribution rate In of the reserved attributes to the mass-election attributes is given by (15)In=RReserved2RMass-election2.Equation (15) is applied to judge the reasonability of the proposed attribute reduction model. The numerator RReserved2 reflects the explanation ability of the reserved attributes to the comprehensive evaluation score y, and the denominator RMass-election2 illustrates the explanation ability of the mass-election attributes to the comprehensive evaluation score y. Equation (15) is the ratio of the explanation ability RReserved2 to the explanation ability RMass-election2. It reveals the information contribution degree of the reserved attributes to the mass-election attributes.As a decision criterion for judging the rationality of the proposed attribute reduction model, the proposed approach is considered reasonable if the reserved attributes are able to contribute more than 90% of the mass-election attributes by using less than 30% of attributes in the mass-election attribute set.


## 3. Empirical Study

### 3.1. Sample Selection and Data Sources

In consideration of research purpose of verifying the applicability of the proposed attribute reduction model, this subsection implements empirical study based on the financing ability data of 713 small enterprises. In order to guarantee the representation of empirical results, this paper collected the data from the headquarter and all of the branches in a city commercial bank of China, including Beijing Branch, Tianjin Branch, Shanghai Branch, Chongqing Branch, Shenyang Branch, Dalian Branch, and Dandong Branch. The data is shown in Column 5 to Column 717 in Table1 [46].Table 1
Financing ability evaluation attribute data of 713 small enterprises.
(
1
) No.
(
2
) Criterion layer
(
3
) Attribute
(
4
) Attribute types
Original datavij
Standard datapij
(
1431
) Reduction results
(
5
) Enterprise 1
⋯
(
717
) Enterprise 713
(
718
) Enterprise 1
⋯
(
1430
) Enterprise 713
1
C
1 enterprise basic situation
X
1 the education background of enterprise legal person
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Reserve
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
11
X
11 the number of contract breaches between enterprises
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Delete by correlation analysis
12
C
2 debt paying ability
X
12 asset liability ratio
Negative
0.501
⋯
0.463
0.474
⋯
0.513
Delete byF test significance discrimination
...
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
32
X
32 the ratio of EBITDA and total debt
Positive
0.084
⋯
−0.115
0.488
⋯
0.485
Delete by correlation analysis
33
C
3 enterprise profitability
X
33 net assets income rate
Positive
0.085
⋯
−0.097
0.495
⋯
0.440
Delete byF test significance discrimination
...
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
44
X
44 net cash flow from operating activities
Positive
0.000
⋯
−2404133.970
0.503
⋯
0.492
Delete byF test significance discrimination
45
C
4 operation ability
X
45 Accounts receivable turnover speed
Positive
4.450
⋯
45.060
0.469
⋯
0.519
Delete byF test significance discrimination
...
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
54
X
54 cash cycle
Positive
3.386
⋯
23.695
0.502
⋯
0.507
Delete by correlation analysis
55
C
5 development potential
X
55 business revenue growth rate
Positive
−0.540
⋯
0.030
0.463
⋯
0.483
Delete byF test significance discrimination
...
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
63
X
63 brand product level
Qualitative
0.000
⋯
0.750
0.000
⋯
0.750
Reserve
64
C
6 enterprise external macro conditions
X
64 industry cycle index
Positive
123.200
⋯
135.300
0.578
⋯
0.700
Reserve
...
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
69
X
69 Engel coefficient
Negative
44.000
⋯
37.000
0.450
⋯
0.755
Delete byF test significance discriminationThe mass-election attribute set for small enterprise financing ability evaluation contains six criterion layers:C1,enterprise basic situation; C2,debt paying ability; C3,enterprise profitability; C4,operation ability; C5,development potential; C6, enterprise external macroconditions, as shown in Column 2 in Table 1. All of the 69 attributes are listed in Column 3 in Table 1. As known from the fourth Column of Table 1, there are 46 positive attributes, 7 negative attributes, 2 interval attributes, and 14 qualitative attributes.


### 3.2. The Attribute Data Standardization

In this paper, we have two interval attributes: “X4the age of enterprise legal person” and “X67consumer price index (CPI).” The ideal range of “X4the age of enterprise legal person” is [31,45] [25]. It means if the age of the business owner is within the interval [31,45], the repayment ability and repayment willingness of the small enterprise are strong. The ideal range of “X67consumer price index (CPI)” is [101,105] [25]. It indicates that there exists neither deflation nor inflation, when the CPI is within the range [101,105].The data standardization for quantitative attributes is as follows: in terms of the attribute type in Column 4 of Table1, substituting the original data of positive attributes vij from Column 5 to 717 of Table 1 into (1), the original data of negative attributes vij into (2), and the original data of interval attributes vij into (3), the standardized data pij of attributes are obtained. The results are shown in Column 718 to 1430 of Table 1.Subsequently, we will compute the standardized score for the qualitative attributes. Learning from a commercial bank nonfinancial attributes scoring standard [46], the scoring standard of qualitative attributes can be obtained by rational analysis, as shown in Table 2. Then, the standardized scores of qualitative attributes are obtained combined with the attribute type in Column 4 of Table 1, as shown in Column 718 to 1430 of Table 1.Table 2
The scoring criteria of the qualitative attributes.
(
1
) No.
(
2
) Criterion layer
(
3
) Attribute
(
4
) Options and its description
(
5
) Score
1
C
1 enterprise basic situation
X
1 the education background of enterprise legal person
1
Bachelor degree or above
1.000
2
2
Junior college graduate
0.900
3
3
Senior high school graduate or education technical secondary school graduate
0.700
4
4
Junior middle school graduate
0.400
5
5
Others or data missing
0.000
⋯
⋯
⋯
⋯
⋯
⋯
63
C
5 development potential
X
63 brand product level
1
The main products of the enterprise are the national brand products
1.000
64
2
The main products of the enterprise are the provincial brand products
0.750
65
3
The main products of the enterprise are the prefecture-level brand products
0.500
67
4
The main products of the enterprise are the district brand products
0.250
68
5
Others or data missing
0.000


### 3.3. Attribute Reduction Utilizing Pearson Correlation Analysis

In practice, due to the presence of related attribute values but independent attributes of meaning, some of the attributes might be mistakenly deleted. This paper calculates attributes’ Pearson correlation coefficients in the same criterion layer. In order to explain the process of Pearson correlation analysis, we take the 10 attributes of the fourth criterion layer “C4 Operation ability” as an example.After substituting the data from Row 45 to 54 and Column 718 to 1430 of Table1 into (4), the correlation coefficients can be obtained for any two attributes, as shown in Table 3. Known from Table 3, the correlation coefficient 0.998 between “X53accounts payable turnover speed” and “X54cash cycle” is greater than the threshold value 0.8, which means that the two attributes reflect information highly repetitively. Because there are other attributes representing cash flow in the attribute set, such as “X17the main business income cash ratio” and “X20all assets cash recovery rate,” we delete the attribute “X54cash cycle.”Table 3
The correlation coefficients between attributes for “C4 operation ability”.
X
45
X
46
X
47
X
48
X
49
X
50
X
51
X
52
X
53
X
54
X
45
1.000
X
46
0.425
1.000
X
47
0.371
0.297
1.000
X
48
0.349
0.297
0.783
1.000
X
49
0.043
0.009
0.064
−0.010
1.000
X
50
0.249
0.088
0.513
0.486
0.070
1.000
X
51
0.016
0.046
0.043
−0.054
−0.045
−0.139
1.000
X
52
0.004
0.005
0.063
0.003
0.006
−0.024
0.030
1.000
X
53
−0.146
−0.185
−0.023
−0.058
−0.032
−0.030
−0.119
0.027
1.000
X
54
−0.098
−0.179
0.002
−0.037
−0.030
−0.020
−0.121
0.029
0.998
1.000Similarly, we can obtain the attributes’ Pearson correlation coefficients for the rest of five criterion layers. Next, we delete the other 15 attributes:X6, X7, X9, X11, X16, X18, X21, X22, X26, X31, X32, X38, X40, X42, and X43. The removed 16 attributes are marked with “delete by correlation analysis” in the last column of Table 1.There are 53 attributes after reducing by Pearson correlation analysis, and the corresponding attributes’ standard data are listed in Column 3 to 55 in Table4.Table 4
53 attributes and the comprehensive scoreyi of 713 small enterprises.
(
1
) No.
(
2
) Enterprise
(
3
)
X
1
⋯
(46)X60
(47)
X
61
(48)X62
(49)
X
63
⋯
(55)X69
(56) Comprehensive scoreyi
1
Enterprise 1
1.000
⋯
0.000
0.000
1.000
0.000
⋯
0.450
0.155
2
Enterprise 2
0.000
⋯
0.000
0.000
1.000
0.000
⋯
0.790
0.177
3
Enterprise 3
1.000
⋯
0.250
0.000
1.000
0.000
⋯
0.755
0.176
4
Enterprise 4
0.000
⋯
0.000
0.000
1.000
0.000
⋯
0.790
0.177
5
Enterprise 5
1.000
⋯
0.250
0.000
1.000
0.000
⋯
0.755
0.176
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
711
Enterprise 711
1.000
⋯
0.250
0.500
0.500
0.250
⋯
0.751
0.314
712
Enterprise 712
1.000
⋯
0.250
0.000
0.500
0.000
⋯
0.751
0.263
713
Enterprise 713
1.000
⋯
0.750
0.500
0.500
0.750
⋯
0.755
0.544


### 3.4. Attribute Reduction UsingF Test Significance Discrimination

Taking the data of Table4 into (5) to (7), the entropy weights of 53 attributes can be obtained: (16)w=0.0178,0.0102,0.0030,…,0.000353×1T.Substituting the data of the first row in Table4 and the entropy weights of 53 attributes into (8), then the comprehensive score y1=0.155 of enterprise 1 can be calculated. Similarly, we can calculate the rest of 712 enterprises’ comprehensive scores yi, as shown in the last column of Table 4.After taking the data from Table4 into (4), 53 Pearson correlation coefficients between the 53 attributes Xi and the comprehensive score y can be obtained. According to the correlation coefficient absolute value ri in a descending order, the attributes’ ranking results are listed in Table 5. Obviously, the attribute X61 has the maximum correlation coefficient with the comprehensive score y; therefore the attribute X61 should be reserved.Table 5
Attribute reduction process based onF test significance discrimination.
(
1
)No.
(
2
)
X
i
(
3
) Correlation coefficient between attribute Xi and y
(
4
) Relevance ranking between attribute Xi and y
(
5
)
Δ
R
2
(
6
)
F
(
X
i
)
(
7
)
F
0.05
1
,
n
-
m
-
2
(8) Reduction results
1
X
61
0.782
1
0.114
1217.2
3.8556
Reserve
2
X
63
0.721
2
0.351
1322.2
3.8555
Reserve
3
X
60
0.717
3
0.033
41.579
3.8555
Reserve
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
19
X
3
0.136
19
0.01
7.9439
3.8552
Reserve
20
X
12
0.126
20
0.003
2.3591
3.8552
Delete byF test significance discrimination
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
53
X
14
0.000
53
—
—
—
Delete byF test significance discriminationIn terms of Step3 in Section 2.3.2, substituting the 52 attributes’ data after removing the attribute X61 and the comprehensive score y into (9) and (10), then the multiple determination coefficient R522=0.938 between the comprehensive index y and the remaining 52 attributes can be got. In a similar way, we can calculate the multiple determination coefficient R512=0.824 between the comprehensive index y and the remaining 51 attributes after removing the two attributes X61 and X63. Then ΔR2=R522-R512=0.938-0.824=0.114. Take ΔR2=0.114, Rm-12=R522=0.938, and n-(m-2)=713-51=662 into (14); F(X63)=1217.226 can be obtained.Let the confidence levelα be equal to 0.05; the critical value Fα(1,n-(m-2))=F0.05(1,662)=3.8556 can be checked from F statistics. Because F(X63)=1217.226>F0.05(1,662)=3.8556, we should accept the hypothesis H0:ΔR2≠0. It means that ΔR2 is not equal to zero significantly, and the attribute X63 affects small enterprise financing ability evaluation significantly, and therefore it should be reserved.Similarly, after repeating this process as mentioned above, we can select other 17 attributes, as shown in Row 3 to Row 19 of Table5. Until the 20th attribute X12, the F test value F(X12) equals 2.3591. And the corresponding F0.05(1,681) equals 3.8552. Obviously, F(X12)=2.3591<F0.05(1,681)=3.8552; thus we should accept the hypothesis H1:ΔR2=0. It indicates that the attribute X12 does not have significant influence on the comprehensive evaluation resulty, and it should be deleted. At this time, the attribute reduction can be stopped according to Step 7 in Section 2.3.2.The selected 19 attributes are marked with“reserve” in the last column of Table 5. And the deleted 34 attributes are marked with“delete byFtest significance discrimination” in the last column of Table 5. The selected 19 key influencing factors of small enterprise financing ability are shown in Table 6. And the detailed attribute reduction process of financing ability evaluation for 713 small enterprises is shown in Table 7.Table 6
The key influencing factors of small enterprise financing ability.
(
1
) No.
(
2
) Criterion layer
(
3
) Influencing factors
1
C
1 enterprise basic situation
X
1 the education background of enterprise legal person
2
X
2 local residence permission of enterprise legal person
3
X
3 the gender of enterprise legal person
4
X
5 years for the job of enterprise legal person
5
X
8 enterprise tax records
6
X
10 enterprise law-abiding business conditions
7
C
2 debt paying ability
X
15 quick ratio
8
X
17 the main business income cash ratio
9
X
19 full capitalization rate
10
X
30 noncurrent liabilities business activities net cash flow ratio
11
C
3 enterprise profitability
X
36 total assets return rate
12
X
39 gross profit rate
13
X
41 EBITDA
14
C
4 operation ability
X
52 rate of return on investment
15
C
5 development potential
X
60 new product identification level
16
X
61 patent status
17
X
62 product sales scope
18
X
63 brand product level
19
C
6 enterprise external macroconditions
X
64 industry cycle indexTable 7
The attribute reduction process of financing ability evaluation for 713 small enterprises.
(1) No.
(
2
) Criterion layer
(
3
) Attribute
(
4
) Attribute types
Original data
Standard data
(1431) Reduction results
(5)Enterprise 1
⋯
(717) Enterprise 713
(718)Enterprise 1
⋯
(1430)Enterprise 713
1
C
1 enterprise basic situation
X
1 the education background of enterprise legal person
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Reserve
2
X
2 local residence permission of enterprise legal person
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Reserve
3
X
3 the gender of enterprise legal person
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Reserve
4
X
4 the age of enterprise legal person
Interval
36.000
⋯
40.000
1.000
⋯
1.000
Delete byF test significance discrimination
5
X
5 years for the job of enterprise legal person
Qualitative
0.400
⋯
1.000
0.400
⋯
1.000
Reserve
6
X
6 enterprise registered capital category
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Delete by correlation analysis
7
X
7 enterprise credit, nearly three years
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Delete by correlation analysis
8
X
8 enterprise tax records
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Reserve
9
X
9 enterprise legal disputes
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Delete by correlation analysis
10
X
10 enterprise law-abiding business conditions
Qualitative
0.500
⋯
1.000
0.500
⋯
1.000
Reserve
11
X
11 the number of contract breaches between enterprises
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Delete by correlation analysis
12
C
2 debt paying ability
X
12 asset liability ratio
Negative
0.501
⋯
0.463
0.474
⋯
0.513
Delete byF test significance discrimination
13
X
13 current liabilities business activities net cash flow ratio
Positive
0.000
⋯
−0.024
0.492
⋯
0.492
Delete byF test significance discrimination
14
X
14 interest safeguard multiple
Positive
0.000
⋯
0.000
0.440
⋯
0.440
Delete byF test significance discrimination
15
X
15 quick ratio
Positive
0.200
⋯
0.320
0.457
⋯
0.459
Reserve
16
X
16 current ratio
Positive
0.830
⋯
0.600
0.471
⋯
0.469
Delete by correlation analysis
17
X
17 the main business income cash ratio
Positive
1.240
⋯
1.000
0.694
⋯
0.625
Reserve
18
X
18 profit current liabilities ration before interest and tax
Positive
0.084
⋯
−0.115
0.489
⋯
0.486
Delete by correlation analysis
19
X
19 full capitalization rate
Negative
0.000
⋯
0.000
1.000
⋯
1.000
Reserve
20
X
20 all assets cash recovery rate
Positive
0.000
⋯
−0.011
0.475
⋯
0.466
Delete byF test significance discrimination
21
X
21 shareholders’ equity ratio
Positive
0.499
⋯
0.537
0.474
⋯
0.513
Delete by correlation analysis
22
X
22 superquick ratio
Positive
0.097
⋯
0.288
0.476
⋯
0.479
Delete by correlation analysis
23
X
23 net profit cash content
Positive
0.000
⋯
0.210
0.496
⋯
0.496
Delete byF test significance discrimination
24
X
24 the ratio of net assets and year-end loan balance
Positive
1.970
⋯
2.350
0.493
⋯
0.494
Delete byF test significance discrimination
25
C
2 debt paying ability
X
25 capital fixed ratio
Negative
1.170
⋯
1.345
0.485
⋯
0.479
Delete byF test significance discrimination
26
X
26 Cash ratio
Positive
0.010
⋯
0.240
0.468
⋯
0.474
Delete by correlation analysis
27
X
27 long-term assets appropriate rate
Positive
0.850
⋯
0.740
0.467
⋯
0.467
Delete byF test significance discrimination
28
X
28 total outstanding loans accounted for the proportion of net assets
Negative
0.000
⋯
0.000
0.530
⋯
0.497
Delete byF test significance discrimination
29
X
29 total outstanding loans accounted for the proportion of general assets
Negative
0.000
⋯
0.000
1.000
⋯
0.950
Delete byF test significance discrimination
30
X
30 noncurrent liabilities business activities net cash flow ratio
Positive
−155.323
⋯
−155.323
0.000
⋯
0.000
Reserve
31
X
31 total liabilities business activities net cash flow ratio
Positive
0.000
⋯
−0.024
0.492
⋯
0.492
Delete by correlation analysis
32
X
32 the ratio of EBITDA and total debt
Positive
0.084
⋯
−0.115
0.488
⋯
0.485
Delete by correlation analysis
33
C
3 enterprise profitability
X
33 net assets income rate
Positive
0.085
⋯
−0.097
0.495
⋯
0.440
Delete byF test significance discrimination
34
X
34 sales net rate
Positive
0.000
⋯
−0.012
0.506
⋯
0.505
Delete byF test significance discrimination
35
X
35 sales net profit rate
Positive
0.098
⋯
−0.059
0.716
⋯
0.693
Delete byF test significance discrimination
36
X
36 total assets return rate
Positive
0.070
⋯
−0.030
0.497
⋯
0.450
Reserve
37
X
37 operating profit rate
Positive
0.100
⋯
−0.060
0.563
⋯
0.539
Delete byF test significance discrimination
38
X
38 cost net interest rate
Positive
0.109
⋯
−0.056
0.497
⋯
0.484
Delete by correlation analysis
39
X
39 gross profit rate
Positive
0.320
⋯
0.010
0.822
⋯
0.741
Reserve
40
X
40 cost-profit ratio
Positive
0.109
⋯
−0.057
0.495
⋯
0.482
Delete by correlation analysis
41
X
41 EBITDA
Positive
1016249.670
⋯
−11686752.290
0.658
⋯
0.599
Reserve
42
X
42 EBITDA profit rate
Positive
0.098
⋯
−0.060
0.509
⋯
0.492
Delete by correlation analysis
43
X
43 net profit
Positive
1016249.670
⋯
−11451533.390
0.661
⋯
0.604
Delete by correlation analysis
44
X
44 net cash flow from operating activities
Positive
0.000
⋯
−2404133.970
0.503
⋯
0.492
Delete byF test significance discrimination
45
C
4 operation ability
X
45 accounts receivable turnover speed
Positive
4.450
⋯
45.060
0.469
⋯
0.519
Delete byF test significance discrimination
46
X
46 inventory turnover speed
Positive
1.150
⋯
6.880
0.478
⋯
0.482
Delete byF test significance discrimination
47
X
47 total assets turnover speed
Positive
0.480
⋯
0.890
0.350
⋯
0.400
Delete byF test significance discrimination
48
X
48 current assets turnover speed
Positive
1.038
⋯
3.207
0.428
⋯
0.648
Delete byF test significance discrimination
49
X
49 fixed assets turnover speed
Positive
0.797
⋯
1.308
0.484
⋯
0.484
Delete byF test significance discrimination
50
X
50 shareholders’ equity turnover speed
Positive
0.866
⋯
1.656
0.443
⋯
0.455
Delete byF test significance discrimination
51
X
51 working capital allocation ratio
Positive
−0.204
⋯
−0.668
0.977
⋯
0.969
Delete byF test significance discrimination
52
X
52 rate of return on investment
Positive
0.000
⋯
0.000
0.490
⋯
0.490
Reserve
53
X
53 accounts payable turnover speed
Negative
2.214
⋯
28.245
0.791
⋯
0.780
Delete byF test significance discrimination
54
X
54 cash cycle
Positive
3.386
⋯
23.695
0.502
⋯
0.507
Delete by correlation analysis
55
C
5 development potential
X
55 business revenue growth rate
Positive
−0.540
⋯
0.030
0.463
⋯
0.483
Delete byF test significance discrimination
56
X
56 profit growth rate
Positive
−0.620
⋯
0.160
0.495
⋯
0.496
Delete byF test significance discrimination
57
X
57 total assets growth rate
Positive
0.024
⋯
0.020
0.471
⋯
0.471
Delete byF test significance discrimination
58
X
58 capital accumulation rate
Positive
3.350
⋯
−1.780
0.494
⋯
0.494
Delete byF test significance discrimination
59
X
59 retained earnings growth rate
Positive
0.008
⋯
5.781
0.494
⋯
0.498
Delete byF test significance discrimination
60
X
60 new product identification level
Qualitative
0.000
⋯
0.750
0.000
⋯
0.750
Reserve
61
X
61 patent status
Qualitative
0.000
⋯
0.500
0.000
⋯
0.500
Reserve
62
X
62 product sales scope
Qualitative
1.000
⋯
0.500
1.000
⋯
0.500
Reserve
63
X
63 brand product level
Qualitative
0.000
⋯
0.750
0.000
⋯
0.750
Reserve
64
C
6 enterprise external  macroconditions
X
64 industry cycle index
Positive
123.200
⋯
135.300
0.578
⋯
0.700
Reserve
65
X
65 urban residents per capita savings yearend balance
Positive
31965.990
⋯
44133.430
0.343
⋯
0.491
Delete byF test significance discrimination
66
X
66 the GDP growth rate
Positive
21.900
⋯
15.000
0.697
⋯
0.346
Delete byF test significance discrimination
67
X
67 consumer price index (CPI)
Interval
105.100
⋯
103.000
0.994
⋯
1.000
Delete byF test significance discrimination
68
X
68 urban residents per capita disposable income
Positive
17123.380
⋯
19920.000
0.424
⋯
0.508
Delete byF test significance discrimination
69
X
69 Engel coefficient
Negative
44.000
⋯
37.000
0.450
⋯
0.755
Delete byF test significance discrimination


### 3.5. The Reasonability Judgment for the Proposed Attribute Reduction Model

In Table4, taking the data of 19 reserved attributes and the comprehensive evaluation scorey into (9) and (10), the multiple determination coefficient RReserved2 = 0.947 can be obtained. In Table 1, taking the data of 69 original attributes and the comprehensive evaluation score y into (9) and (10), the multiple determination coefficient RMass-election2 = 1.000 can be obtained. Thus In=RReserved2/RMass-election2=94.7%. It illustrates that the selected attributes reflect 94.7% original information with 27.54% attributes (27.54%=19/69) by using the proposed attribute reduction model. And the experimental results show that the proposed reduction model is efficient and effective.


### 3.6. Some Notes about the Proposed Model

In Section2.3.1, this paper takes entropy weight method as an example for the purpose of illustrating the feasibility and rationality of the proposed attribute reduction idea. As a matter of fact, the weight methods can be substituted in terms of the needs of decision makers. They can select other weight methods, such as AHP, G1, G2, and interval numbers weight approaches [47].In Section2.3.2, the paper takes linear regression model as an example so as to explain the feasibility of the proposed model. In reality, decision makers can select other nonlinear regression models [48].


## 3.1. Sample Selection and Data Sources

In consideration of research purpose of verifying the applicability of the proposed attribute reduction model, this subsection implements empirical study based on the financing ability data of 713 small enterprises. In order to guarantee the representation of empirical results, this paper collected the data from the headquarter and all of the branches in a city commercial bank of China, including Beijing Branch, Tianjin Branch, Shanghai Branch, Chongqing Branch, Shenyang Branch, Dalian Branch, and Dandong Branch. The data is shown in Column 5 to Column 717 in Table1 [46].Table 1
Financing ability evaluation attribute data of 713 small enterprises.
(
1
) No.
(
2
) Criterion layer
(
3
) Attribute
(
4
) Attribute types
Original datavij
Standard datapij
(
1431
) Reduction results
(
5
) Enterprise 1
⋯
(
717
) Enterprise 713
(
718
) Enterprise 1
⋯
(
1430
) Enterprise 713
1
C
1 enterprise basic situation
X
1 the education background of enterprise legal person
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Reserve
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
11
X
11 the number of contract breaches between enterprises
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Delete by correlation analysis
12
C
2 debt paying ability
X
12 asset liability ratio
Negative
0.501
⋯
0.463
0.474
⋯
0.513
Delete byF test significance discrimination
...
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
32
X
32 the ratio of EBITDA and total debt
Positive
0.084
⋯
−0.115
0.488
⋯
0.485
Delete by correlation analysis
33
C
3 enterprise profitability
X
33 net assets income rate
Positive
0.085
⋯
−0.097
0.495
⋯
0.440
Delete byF test significance discrimination
...
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
44
X
44 net cash flow from operating activities
Positive
0.000
⋯
−2404133.970
0.503
⋯
0.492
Delete byF test significance discrimination
45
C
4 operation ability
X
45 Accounts receivable turnover speed
Positive
4.450
⋯
45.060
0.469
⋯
0.519
Delete byF test significance discrimination
...
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
54
X
54 cash cycle
Positive
3.386
⋯
23.695
0.502
⋯
0.507
Delete by correlation analysis
55
C
5 development potential
X
55 business revenue growth rate
Positive
−0.540
⋯
0.030
0.463
⋯
0.483
Delete byF test significance discrimination
...
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
63
X
63 brand product level
Qualitative
0.000
⋯
0.750
0.000
⋯
0.750
Reserve
64
C
6 enterprise external macro conditions
X
64 industry cycle index
Positive
123.200
⋯
135.300
0.578
⋯
0.700
Reserve
...
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
69
X
69 Engel coefficient
Negative
44.000
⋯
37.000
0.450
⋯
0.755
Delete byF test significance discriminationThe mass-election attribute set for small enterprise financing ability evaluation contains six criterion layers:C1,enterprise basic situation; C2,debt paying ability; C3,enterprise profitability; C4,operation ability; C5,development potential; C6, enterprise external macroconditions, as shown in Column 2 in Table 1. All of the 69 attributes are listed in Column 3 in Table 1. As known from the fourth Column of Table 1, there are 46 positive attributes, 7 negative attributes, 2 interval attributes, and 14 qualitative attributes.


## 3.2. The Attribute Data Standardization

In this paper, we have two interval attributes: “X4the age of enterprise legal person” and “X67consumer price index (CPI).” The ideal range of “X4the age of enterprise legal person” is [31,45] [25]. It means if the age of the business owner is within the interval [31,45], the repayment ability and repayment willingness of the small enterprise are strong. The ideal range of “X67consumer price index (CPI)” is [101,105] [25]. It indicates that there exists neither deflation nor inflation, when the CPI is within the range [101,105].The data standardization for quantitative attributes is as follows: in terms of the attribute type in Column 4 of Table1, substituting the original data of positive attributes vij from Column 5 to 717 of Table 1 into (1), the original data of negative attributes vij into (2), and the original data of interval attributes vij into (3), the standardized data pij of attributes are obtained. The results are shown in Column 718 to 1430 of Table 1.Subsequently, we will compute the standardized score for the qualitative attributes. Learning from a commercial bank nonfinancial attributes scoring standard [46], the scoring standard of qualitative attributes can be obtained by rational analysis, as shown in Table 2. Then, the standardized scores of qualitative attributes are obtained combined with the attribute type in Column 4 of Table 1, as shown in Column 718 to 1430 of Table 1.Table 2
The scoring criteria of the qualitative attributes.
(
1
) No.
(
2
) Criterion layer
(
3
) Attribute
(
4
) Options and its description
(
5
) Score
1
C
1 enterprise basic situation
X
1 the education background of enterprise legal person
1
Bachelor degree or above
1.000
2
2
Junior college graduate
0.900
3
3
Senior high school graduate or education technical secondary school graduate
0.700
4
4
Junior middle school graduate
0.400
5
5
Others or data missing
0.000
⋯
⋯
⋯
⋯
⋯
⋯
63
C
5 development potential
X
63 brand product level
1
The main products of the enterprise are the national brand products
1.000
64
2
The main products of the enterprise are the provincial brand products
0.750
65
3
The main products of the enterprise are the prefecture-level brand products
0.500
67
4
The main products of the enterprise are the district brand products
0.250
68
5
Others or data missing
0.000


## 3.3. Attribute Reduction Utilizing Pearson Correlation Analysis

In practice, due to the presence of related attribute values but independent attributes of meaning, some of the attributes might be mistakenly deleted. This paper calculates attributes’ Pearson correlation coefficients in the same criterion layer. In order to explain the process of Pearson correlation analysis, we take the 10 attributes of the fourth criterion layer “C4 Operation ability” as an example.After substituting the data from Row 45 to 54 and Column 718 to 1430 of Table1 into (4), the correlation coefficients can be obtained for any two attributes, as shown in Table 3. Known from Table 3, the correlation coefficient 0.998 between “X53accounts payable turnover speed” and “X54cash cycle” is greater than the threshold value 0.8, which means that the two attributes reflect information highly repetitively. Because there are other attributes representing cash flow in the attribute set, such as “X17the main business income cash ratio” and “X20all assets cash recovery rate,” we delete the attribute “X54cash cycle.”Table 3
The correlation coefficients between attributes for “C4 operation ability”.
X
45
X
46
X
47
X
48
X
49
X
50
X
51
X
52
X
53
X
54
X
45
1.000
X
46
0.425
1.000
X
47
0.371
0.297
1.000
X
48
0.349
0.297
0.783
1.000
X
49
0.043
0.009
0.064
−0.010
1.000
X
50
0.249
0.088
0.513
0.486
0.070
1.000
X
51
0.016
0.046
0.043
−0.054
−0.045
−0.139
1.000
X
52
0.004
0.005
0.063
0.003
0.006
−0.024
0.030
1.000
X
53
−0.146
−0.185
−0.023
−0.058
−0.032
−0.030
−0.119
0.027
1.000
X
54
−0.098
−0.179
0.002
−0.037
−0.030
−0.020
−0.121
0.029
0.998
1.000Similarly, we can obtain the attributes’ Pearson correlation coefficients for the rest of five criterion layers. Next, we delete the other 15 attributes:X6, X7, X9, X11, X16, X18, X21, X22, X26, X31, X32, X38, X40, X42, and X43. The removed 16 attributes are marked with “delete by correlation analysis” in the last column of Table 1.There are 53 attributes after reducing by Pearson correlation analysis, and the corresponding attributes’ standard data are listed in Column 3 to 55 in Table4.Table 4
53 attributes and the comprehensive scoreyi of 713 small enterprises.
(
1
) No.
(
2
) Enterprise
(
3
)
X
1
⋯
(46)X60
(47)
X
61
(48)X62
(49)
X
63
⋯
(55)X69
(56) Comprehensive scoreyi
1
Enterprise 1
1.000
⋯
0.000
0.000
1.000
0.000
⋯
0.450
0.155
2
Enterprise 2
0.000
⋯
0.000
0.000
1.000
0.000
⋯
0.790
0.177
3
Enterprise 3
1.000
⋯
0.250
0.000
1.000
0.000
⋯
0.755
0.176
4
Enterprise 4
0.000
⋯
0.000
0.000
1.000
0.000
⋯
0.790
0.177
5
Enterprise 5
1.000
⋯
0.250
0.000
1.000
0.000
⋯
0.755
0.176
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
711
Enterprise 711
1.000
⋯
0.250
0.500
0.500
0.250
⋯
0.751
0.314
712
Enterprise 712
1.000
⋯
0.250
0.000
0.500
0.000
⋯
0.751
0.263
713
Enterprise 713
1.000
⋯
0.750
0.500
0.500
0.750
⋯
0.755
0.544


## 3.4. Attribute Reduction UsingF Test Significance Discrimination

Taking the data of Table4 into (5) to (7), the entropy weights of 53 attributes can be obtained: (16)w=0.0178,0.0102,0.0030,…,0.000353×1T.Substituting the data of the first row in Table4 and the entropy weights of 53 attributes into (8), then the comprehensive score y1=0.155 of enterprise 1 can be calculated. Similarly, we can calculate the rest of 712 enterprises’ comprehensive scores yi, as shown in the last column of Table 4.After taking the data from Table4 into (4), 53 Pearson correlation coefficients between the 53 attributes Xi and the comprehensive score y can be obtained. According to the correlation coefficient absolute value ri in a descending order, the attributes’ ranking results are listed in Table 5. Obviously, the attribute X61 has the maximum correlation coefficient with the comprehensive score y; therefore the attribute X61 should be reserved.Table 5
Attribute reduction process based onF test significance discrimination.
(
1
)No.
(
2
)
X
i
(
3
) Correlation coefficient between attribute Xi and y
(
4
) Relevance ranking between attribute Xi and y
(
5
)
Δ
R
2
(
6
)
F
(
X
i
)
(
7
)
F
0.05
1
,
n
-
m
-
2
(8) Reduction results
1
X
61
0.782
1
0.114
1217.2
3.8556
Reserve
2
X
63
0.721
2
0.351
1322.2
3.8555
Reserve
3
X
60
0.717
3
0.033
41.579
3.8555
Reserve
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
19
X
3
0.136
19
0.01
7.9439
3.8552
Reserve
20
X
12
0.126
20
0.003
2.3591
3.8552
Delete byF test significance discrimination
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
53
X
14
0.000
53
—
—
—
Delete byF test significance discriminationIn terms of Step3 in Section 2.3.2, substituting the 52 attributes’ data after removing the attribute X61 and the comprehensive score y into (9) and (10), then the multiple determination coefficient R522=0.938 between the comprehensive index y and the remaining 52 attributes can be got. In a similar way, we can calculate the multiple determination coefficient R512=0.824 between the comprehensive index y and the remaining 51 attributes after removing the two attributes X61 and X63. Then ΔR2=R522-R512=0.938-0.824=0.114. Take ΔR2=0.114, Rm-12=R522=0.938, and n-(m-2)=713-51=662 into (14); F(X63)=1217.226 can be obtained.Let the confidence levelα be equal to 0.05; the critical value Fα(1,n-(m-2))=F0.05(1,662)=3.8556 can be checked from F statistics. Because F(X63)=1217.226>F0.05(1,662)=3.8556, we should accept the hypothesis H0:ΔR2≠0. It means that ΔR2 is not equal to zero significantly, and the attribute X63 affects small enterprise financing ability evaluation significantly, and therefore it should be reserved.Similarly, after repeating this process as mentioned above, we can select other 17 attributes, as shown in Row 3 to Row 19 of Table5. Until the 20th attribute X12, the F test value F(X12) equals 2.3591. And the corresponding F0.05(1,681) equals 3.8552. Obviously, F(X12)=2.3591<F0.05(1,681)=3.8552; thus we should accept the hypothesis H1:ΔR2=0. It indicates that the attribute X12 does not have significant influence on the comprehensive evaluation resulty, and it should be deleted. At this time, the attribute reduction can be stopped according to Step 7 in Section 2.3.2.The selected 19 attributes are marked with“reserve” in the last column of Table 5. And the deleted 34 attributes are marked with“delete byFtest significance discrimination” in the last column of Table 5. The selected 19 key influencing factors of small enterprise financing ability are shown in Table 6. And the detailed attribute reduction process of financing ability evaluation for 713 small enterprises is shown in Table 7.Table 6
The key influencing factors of small enterprise financing ability.
(
1
) No.
(
2
) Criterion layer
(
3
) Influencing factors
1
C
1 enterprise basic situation
X
1 the education background of enterprise legal person
2
X
2 local residence permission of enterprise legal person
3
X
3 the gender of enterprise legal person
4
X
5 years for the job of enterprise legal person
5
X
8 enterprise tax records
6
X
10 enterprise law-abiding business conditions
7
C
2 debt paying ability
X
15 quick ratio
8
X
17 the main business income cash ratio
9
X
19 full capitalization rate
10
X
30 noncurrent liabilities business activities net cash flow ratio
11
C
3 enterprise profitability
X
36 total assets return rate
12
X
39 gross profit rate
13
X
41 EBITDA
14
C
4 operation ability
X
52 rate of return on investment
15
C
5 development potential
X
60 new product identification level
16
X
61 patent status
17
X
62 product sales scope
18
X
63 brand product level
19
C
6 enterprise external macroconditions
X
64 industry cycle indexTable 7
The attribute reduction process of financing ability evaluation for 713 small enterprises.
(1) No.
(
2
) Criterion layer
(
3
) Attribute
(
4
) Attribute types
Original data
Standard data
(1431) Reduction results
(5)Enterprise 1
⋯
(717) Enterprise 713
(718)Enterprise 1
⋯
(1430)Enterprise 713
1
C
1 enterprise basic situation
X
1 the education background of enterprise legal person
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Reserve
2
X
2 local residence permission of enterprise legal person
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Reserve
3
X
3 the gender of enterprise legal person
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Reserve
4
X
4 the age of enterprise legal person
Interval
36.000
⋯
40.000
1.000
⋯
1.000
Delete byF test significance discrimination
5
X
5 years for the job of enterprise legal person
Qualitative
0.400
⋯
1.000
0.400
⋯
1.000
Reserve
6
X
6 enterprise registered capital category
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Delete by correlation analysis
7
X
7 enterprise credit, nearly three years
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Delete by correlation analysis
8
X
8 enterprise tax records
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Reserve
9
X
9 enterprise legal disputes
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Delete by correlation analysis
10
X
10 enterprise law-abiding business conditions
Qualitative
0.500
⋯
1.000
0.500
⋯
1.000
Reserve
11
X
11 the number of contract breaches between enterprises
Qualitative
1.000
⋯
1.000
1.000
⋯
1.000
Delete by correlation analysis
12
C
2 debt paying ability
X
12 asset liability ratio
Negative
0.501
⋯
0.463
0.474
⋯
0.513
Delete byF test significance discrimination
13
X
13 current liabilities business activities net cash flow ratio
Positive
0.000
⋯
−0.024
0.492
⋯
0.492
Delete byF test significance discrimination
14
X
14 interest safeguard multiple
Positive
0.000
⋯
0.000
0.440
⋯
0.440
Delete byF test significance discrimination
15
X
15 quick ratio
Positive
0.200
⋯
0.320
0.457
⋯
0.459
Reserve
16
X
16 current ratio
Positive
0.830
⋯
0.600
0.471
⋯
0.469
Delete by correlation analysis
17
X
17 the main business income cash ratio
Positive
1.240
⋯
1.000
0.694
⋯
0.625
Reserve
18
X
18 profit current liabilities ration before interest and tax
Positive
0.084
⋯
−0.115
0.489
⋯
0.486
Delete by correlation analysis
19
X
19 full capitalization rate
Negative
0.000
⋯
0.000
1.000
⋯
1.000
Reserve
20
X
20 all assets cash recovery rate
Positive
0.000
⋯
−0.011
0.475
⋯
0.466
Delete byF test significance discrimination
21
X
21 shareholders’ equity ratio
Positive
0.499
⋯
0.537
0.474
⋯
0.513
Delete by correlation analysis
22
X
22 superquick ratio
Positive
0.097
⋯
0.288
0.476
⋯
0.479
Delete by correlation analysis
23
X
23 net profit cash content
Positive
0.000
⋯
0.210
0.496
⋯
0.496
Delete byF test significance discrimination
24
X
24 the ratio of net assets and year-end loan balance
Positive
1.970
⋯
2.350
0.493
⋯
0.494
Delete byF test significance discrimination
25
C
2 debt paying ability
X
25 capital fixed ratio
Negative
1.170
⋯
1.345
0.485
⋯
0.479
Delete byF test significance discrimination
26
X
26 Cash ratio
Positive
0.010
⋯
0.240
0.468
⋯
0.474
Delete by correlation analysis
27
X
27 long-term assets appropriate rate
Positive
0.850
⋯
0.740
0.467
⋯
0.467
Delete byF test significance discrimination
28
X
28 total outstanding loans accounted for the proportion of net assets
Negative
0.000
⋯
0.000
0.530
⋯
0.497
Delete byF test significance discrimination
29
X
29 total outstanding loans accounted for the proportion of general assets
Negative
0.000
⋯
0.000
1.000
⋯
0.950
Delete byF test significance discrimination
30
X
30 noncurrent liabilities business activities net cash flow ratio
Positive
−155.323
⋯
−155.323
0.000
⋯
0.000
Reserve
31
X
31 total liabilities business activities net cash flow ratio
Positive
0.000
⋯
−0.024
0.492
⋯
0.492
Delete by correlation analysis
32
X
32 the ratio of EBITDA and total debt
Positive
0.084
⋯
−0.115
0.488
⋯
0.485
Delete by correlation analysis
33
C
3 enterprise profitability
X
33 net assets income rate
Positive
0.085
⋯
−0.097
0.495
⋯
0.440
Delete byF test significance discrimination
34
X
34 sales net rate
Positive
0.000
⋯
−0.012
0.506
⋯
0.505
Delete byF test significance discrimination
35
X
35 sales net profit rate
Positive
0.098
⋯
−0.059
0.716
⋯
0.693
Delete byF test significance discrimination
36
X
36 total assets return rate
Positive
0.070
⋯
−0.030
0.497
⋯
0.450
Reserve
37
X
37 operating profit rate
Positive
0.100
⋯
−0.060
0.563
⋯
0.539
Delete byF test significance discrimination
38
X
38 cost net interest rate
Positive
0.109
⋯
−0.056
0.497
⋯
0.484
Delete by correlation analysis
39
X
39 gross profit rate
Positive
0.320
⋯
0.010
0.822
⋯
0.741
Reserve
40
X
40 cost-profit ratio
Positive
0.109
⋯
−0.057
0.495
⋯
0.482
Delete by correlation analysis
41
X
41 EBITDA
Positive
1016249.670
⋯
−11686752.290
0.658
⋯
0.599
Reserve
42
X
42 EBITDA profit rate
Positive
0.098
⋯
−0.060
0.509
⋯
0.492
Delete by correlation analysis
43
X
43 net profit
Positive
1016249.670
⋯
−11451533.390
0.661
⋯
0.604
Delete by correlation analysis
44
X
44 net cash flow from operating activities
Positive
0.000
⋯
−2404133.970
0.503
⋯
0.492
Delete byF test significance discrimination
45
C
4 operation ability
X
45 accounts receivable turnover speed
Positive
4.450
⋯
45.060
0.469
⋯
0.519
Delete byF test significance discrimination
46
X
46 inventory turnover speed
Positive
1.150
⋯
6.880
0.478
⋯
0.482
Delete byF test significance discrimination
47
X
47 total assets turnover speed
Positive
0.480
⋯
0.890
0.350
⋯
0.400
Delete byF test significance discrimination
48
X
48 current assets turnover speed
Positive
1.038
⋯
3.207
0.428
⋯
0.648
Delete byF test significance discrimination
49
X
49 fixed assets turnover speed
Positive
0.797
⋯
1.308
0.484
⋯
0.484
Delete byF test significance discrimination
50
X
50 shareholders’ equity turnover speed
Positive
0.866
⋯
1.656
0.443
⋯
0.455
Delete byF test significance discrimination
51
X
51 working capital allocation ratio
Positive
−0.204
⋯
−0.668
0.977
⋯
0.969
Delete byF test significance discrimination
52
X
52 rate of return on investment
Positive
0.000
⋯
0.000
0.490
⋯
0.490
Reserve
53
X
53 accounts payable turnover speed
Negative
2.214
⋯
28.245
0.791
⋯
0.780
Delete byF test significance discrimination
54
X
54 cash cycle
Positive
3.386
⋯
23.695
0.502
⋯
0.507
Delete by correlation analysis
55
C
5 development potential
X
55 business revenue growth rate
Positive
−0.540
⋯
0.030
0.463
⋯
0.483
Delete byF test significance discrimination
56
X
56 profit growth rate
Positive
−0.620
⋯
0.160
0.495
⋯
0.496
Delete byF test significance discrimination
57
X
57 total assets growth rate
Positive
0.024
⋯
0.020
0.471
⋯
0.471
Delete byF test significance discrimination
58
X
58 capital accumulation rate
Positive
3.350
⋯
−1.780
0.494
⋯
0.494
Delete byF test significance discrimination
59
X
59 retained earnings growth rate
Positive
0.008
⋯
5.781
0.494
⋯
0.498
Delete byF test significance discrimination
60
X
60 new product identification level
Qualitative
0.000
⋯
0.750
0.000
⋯
0.750
Reserve
61
X
61 patent status
Qualitative
0.000
⋯
0.500
0.000
⋯
0.500
Reserve
62
X
62 product sales scope
Qualitative
1.000
⋯
0.500
1.000
⋯
0.500
Reserve
63
X
63 brand product level
Qualitative
0.000
⋯
0.750
0.000
⋯
0.750
Reserve
64
C
6 enterprise external  macroconditions
X
64 industry cycle index
Positive
123.200
⋯
135.300
0.578
⋯
0.700
Reserve
65
X
65 urban residents per capita savings yearend balance
Positive
31965.990
⋯
44133.430
0.343
⋯
0.491
Delete byF test significance discrimination
66
X
66 the GDP growth rate
Positive
21.900
⋯
15.000
0.697
⋯
0.346
Delete byF test significance discrimination
67
X
67 consumer price index (CPI)
Interval
105.100
⋯
103.000
0.994
⋯
1.000
Delete byF test significance discrimination
68
X
68 urban residents per capita disposable income
Positive
17123.380
⋯
19920.000
0.424
⋯
0.508
Delete byF test significance discrimination
69
X
69 Engel coefficient
Negative
44.000
⋯
37.000
0.450
⋯
0.755
Delete byF test significance discrimination


## 3.5. The Reasonability Judgment for the Proposed Attribute Reduction Model

In Table4, taking the data of 19 reserved attributes and the comprehensive evaluation scorey into (9) and (10), the multiple determination coefficient RReserved2 = 0.947 can be obtained. In Table 1, taking the data of 69 original attributes and the comprehensive evaluation score y into (9) and (10), the multiple determination coefficient RMass-election2 = 1.000 can be obtained. Thus In=RReserved2/RMass-election2=94.7%. It illustrates that the selected attributes reflect 94.7% original information with 27.54% attributes (27.54%=19/69) by using the proposed attribute reduction model. And the experimental results show that the proposed reduction model is efficient and effective.


## 3.6. Some Notes about the Proposed Model

In Section2.3.1, this paper takes entropy weight method as an example for the purpose of illustrating the feasibility and rationality of the proposed attribute reduction idea. As a matter of fact, the weight methods can be substituted in terms of the needs of decision makers. They can select other weight methods, such as AHP, G1, G2, and interval numbers weight approaches [47].In Section2.3.2, the paper takes linear regression model as an example so as to explain the feasibility of the proposed model. In reality, decision makers can select other nonlinear regression models [48].


## 4. Conclusions and Future Work

In order to reduce large dimensionality in complex data sets, we create an attribute reduction approach based on Pearson correlation analysis andF test significance discrimination. First of all, we delete redundancy attributes using Pearson correlation coefficient, avoiding information chaos of the original attribute data sets. Secondly, developing attribute reduction methodology utilizing F test significance discrimination can find the key attributes that have the greatest influence on the evaluation results. Thirdly, the paper also defines an information contribution ratio to assess the performance of attribute reduction model from a statistical viewpoint.The proposed attribute reduction model has been verified utilizing the financing ability evaluation data of 713 small enterprises of a city commercial bank in China. The empirical evidence shows the accuracy and applicability of the proposed model. Moreover, we also establish an evaluation indicator system for small enterprise financing ability. It will help the downstream organizations of supply chain to choose more qualified partners and alleviate the difficulties faced by enterprises when applying loan. Furthermore, applications of the proposed model to real world data are expected in future.It is well known that the problems of attribute reduction are ubiquitous in data mining activities. The empirical study in this paper is only an example in order to verify the accuracy of the proposed model. A topic of future research can be the application of the proposed approach to data sets in other attribute reduction areas. Researchers can easily conduct attribute reduction through cases and empirical studies.



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*Source: 1032643-2018-01-15.xml*