# Application of Response Surface Methodology in Dance Health and Training Safety

Authors: Yu Zhang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In order to analyze and optimize the influencing factors of dance health and training safety, response surface methodology is applied to dance health and training safety in this study. Response surface methodology is an optimization method that integrates experimental design and mathematical modeling. It can effectively reduce the number of experiments, provide visual contour maps and three-dimensional diagrams, and investigate the interaction between influencing factors. This paper introduces the basic definition, model construction principle, and characteristics of the response surface method and expounds the test scheme design, model establishment, model inspection, model optimization, and other aspects of the response surface method combined with an example of analysis of influencing factors of dance health and training safety. The results show that the response surface method is an optimization method of comprehensive experiment design and mathematical modeling, which can effectively reduce the number of experiments for the analysis of influencing factors of dance health and training safety and can investigate the interaction between influencing factors.


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## Body

## 1. Introduction

Dance sport is a highly artistic sport in which paired male and female athletes play music on a flat and smooth ground, which is arranged according to a certain level, with the help of human body movements to express emotional skills and show the beauty of the unity of the mind and body [1–5]. Dance is a new sports genre that combines sports and art and has the functions of exercise, competition, and aesthetics. It has a strong sports nature and is prone to accidental injury in the process of sports, which will affect further training and competition [6–11]. Therefore, it is necessary for us to study the dance health and training safety in teaching and competition.Nowadays, dance is developing towards higher competitiveness and performance, and the requirements for dancers’ ability, physical quality, and psychological quality are also higher [12–14]. The occurrence of sports injury increases with the increasing high-intensity training, and the injury is difficult to avoid, which directly affects the performance of dancers [15–17].The nature and characteristics of sports injuries of dancers are summarized and analyzed, including the following points. (1) Male athletes' actions are loud and powerful, with clear movement and stillness; the free and changeable dance posture of female athletes is mainly reflected in the waist, abdomen, and upper limbs. The movement range is large, and the frequency is fast, which makes the sports injury rate of female athletes higher than that of male athletes. (2) Latin dancers bear more intense exercise intensity than modern dancers, and their action frequency and range are also required to be higher. Therefore, the proportion of injuries of Latin dancers is slightly higher than that of modern dancers. The incidence of acute injuries of dancers is higher than that of chronic injuries. (3) Most of Latin dances experience acute injuries, while most of chronic injuries occur in modern dances. (4) Due to the inherent characteristics of dance, the amount of exercise of the lower limbs is much greater than that of the upper limbs, and the requirements for flexibility, strength, speed, and other qualities are higher than those of the upper limbs. Therefore, the incidence of injury in the lower limbs is higher than that in the upper limbs, mainly sports injuries of knee joints, ankles, and ligaments of the lower limbs.In recent years, the progress of computer and information technology [18–23] has also provided a powerful tool for the dance health and training safety analysis of this study. In order to promote the safety of dance teaching and competition, this study adopts response surface methodology [22] to study the accidental injuries and characteristics in dance teaching and competition, analyzes the causes, and puts forward corresponding safety precautions to reduce the incidence of injuries in safety measures. Combined with the causes and nature of accidental injuries caused by inadequate safety precautions in dance teaching and training, this paper puts forward safety precautions suitable for dance sports teaching and competition, which have certain guiding significance for dance athletes to avoid accidental injuries, enhance their awareness of prevention, and prevent injuries in teaching and competition.


## 2. Analysis on the Main Causes of Dance Health and Training Safety

### 2.1. Analysis Object

The analysis object is the accidents and safety precautions in dance teaching and competition. It can be summarized as the following main factors: poor physical quality (strength and flexibility), poor training or precompetition preparation activities, incorrect technical essentials, improper cooperation between dance partners, excessive fatigue, unclear music, and dance rhythm, and the dancers’ lack of attention to sports prevention.


### 2.2. Analysis Methods

The analysis methods mainly include the following five types:(1)
Documentation Method. Through the university libraries and CNKI, we can access the materials related to dance sports injuries and safety precautions.(2)
Expert Interview Method. We interviewed experts who have been engaged in dance teaching in universities for a long time to understand the characteristics of injuries caused by various emergencies in dance sports.(3)
Questionnaire Method. We used the previous research questionnaire, modified the content, prepared the questionnaire, and randomly surveyed the 2012 dance students in the Art Department of the University. A total of 50 questionnaires were distributed, and 48 valid questionnaires (23 men and 25 women) were recovered. The recovery efficiency was 96%.(4)
Mathematical Statistics Method. We conducted statistical routine analysis and processing on the collected data and summarized the frequency of different accidental injury accidents so as to provide data basis for this study.(5)
Logical Analysis Method. Combined with the injuries caused by accidents in dance, we analyze and discuss the awareness of safety precautions in dance and refine it into the conclusion of this study.


### 2.3. Analysis Results

Through the above analysis, we found that the main contributing factors of dance health and training safety are the following aspects.


#### 2.3.1. Poor Physical Fitness

Poor physical fitness is one of the most important factors causing sports injury, ranking first among the injury factors. In fact, strength quality is one of the most important foundations of sports. How to smoothly, stably, and substantially complete the complete set of dance movements with changeable rhythms and diverse movements has put forward higher requirements for the strength quality of dancers. Strengthening strength quality training can reduce the occurrence of sports injuries of dancers, reduce the severity of sports injuries, and achieve self-protection.


#### 2.3.2. Poor Preparation before Training or Competition

The incidence of sports injuries caused by poor preparation of dancers is very high. Inadequate preparation activities will reduce muscle viscosity, inhibit muscle working ability, make it unable to adapt and coordinate with muscle work, and increase the occurrence of muscle sports injury. At the same time, it cannot effectively improve the working state of the heart and internal organs, affecting the role of protecting the heart and internal organs. The amount of preparatory activities is too large. When athletes enter the formal training competition, their physical function has decreased, or they are tired. In this case, they are prone to muscle and ligament strains and sprains.


#### 2.3.3. Incorrect Technical Essentials

Dancers need high accuracy to complete difficult movements. If they are careless, they will deviate from the correct technology, resulting in technical errors and sports injuries. Correct technical movements can not only finish the difficult movements gracefully and lightly but also prevent sports injuries.


#### 2.3.4. Improper Coordination between Partners

Dance is a sport for men and women to cooperate. Male and female dance partners must have a tacit understanding. The strength coordination is inconsistent, and the movement rhythm is not similar. The male leads the female heel and focuses on the lead of the male partner. It is very important to complete each movement and the alternation of dance steps. In the process of dancing, if the male partner does not give the female partner hints and guidance, or the female partner is too active or changes the dance steps at will, it is easy to collide with each other and lead to disconnection, resulting in unnecessary sports injury.


#### 2.3.5. Excessive Fatigue

Dance is a skill-oriented performance, and it takes a long time and a large amount of exercise training to master the relevant technologies. Moreover, the players often participate in multiple rounds of dance competitions and a large number of performances, which makes the dancers consume a lot of physical strength, and it is very easy to cause excessive fatigue or tension in the body and psychology of male and female players, resulting in sports injury.


#### 2.3.6. Lack of Ideological Attention

Dancers do not pay enough attention to sports injuries in ideology and lack knowledge about the generation and prevention of sports injuries, which often leads to unnecessary injuries in the process of sports. The uninjured dancers should learn more and pay more attention to the sports injury and prevention of dance so as to minimize the incidence of sports injury; dance injured people should look for the most timely and effective treatment to reduce the harm caused by the injury in the event of injury. After rehabilitation, we should pay attention to its prevention, understand the most effective means to prevent sports injury, and achieve self-protection.


## 2.1. Analysis Object

The analysis object is the accidents and safety precautions in dance teaching and competition. It can be summarized as the following main factors: poor physical quality (strength and flexibility), poor training or precompetition preparation activities, incorrect technical essentials, improper cooperation between dance partners, excessive fatigue, unclear music, and dance rhythm, and the dancers’ lack of attention to sports prevention.


## 2.2. Analysis Methods

The analysis methods mainly include the following five types:(1)
Documentation Method. Through the university libraries and CNKI, we can access the materials related to dance sports injuries and safety precautions.(2)
Expert Interview Method. We interviewed experts who have been engaged in dance teaching in universities for a long time to understand the characteristics of injuries caused by various emergencies in dance sports.(3)
Questionnaire Method. We used the previous research questionnaire, modified the content, prepared the questionnaire, and randomly surveyed the 2012 dance students in the Art Department of the University. A total of 50 questionnaires were distributed, and 48 valid questionnaires (23 men and 25 women) were recovered. The recovery efficiency was 96%.(4)
Mathematical Statistics Method. We conducted statistical routine analysis and processing on the collected data and summarized the frequency of different accidental injury accidents so as to provide data basis for this study.(5)
Logical Analysis Method. Combined with the injuries caused by accidents in dance, we analyze and discuss the awareness of safety precautions in dance and refine it into the conclusion of this study.


## 2.3. Analysis Results

Through the above analysis, we found that the main contributing factors of dance health and training safety are the following aspects.


### 2.3.1. Poor Physical Fitness

Poor physical fitness is one of the most important factors causing sports injury, ranking first among the injury factors. In fact, strength quality is one of the most important foundations of sports. How to smoothly, stably, and substantially complete the complete set of dance movements with changeable rhythms and diverse movements has put forward higher requirements for the strength quality of dancers. Strengthening strength quality training can reduce the occurrence of sports injuries of dancers, reduce the severity of sports injuries, and achieve self-protection.


### 2.3.2. Poor Preparation before Training or Competition

The incidence of sports injuries caused by poor preparation of dancers is very high. Inadequate preparation activities will reduce muscle viscosity, inhibit muscle working ability, make it unable to adapt and coordinate with muscle work, and increase the occurrence of muscle sports injury. At the same time, it cannot effectively improve the working state of the heart and internal organs, affecting the role of protecting the heart and internal organs. The amount of preparatory activities is too large. When athletes enter the formal training competition, their physical function has decreased, or they are tired. In this case, they are prone to muscle and ligament strains and sprains.


### 2.3.3. Incorrect Technical Essentials

Dancers need high accuracy to complete difficult movements. If they are careless, they will deviate from the correct technology, resulting in technical errors and sports injuries. Correct technical movements can not only finish the difficult movements gracefully and lightly but also prevent sports injuries.


### 2.3.4. Improper Coordination between Partners

Dance is a sport for men and women to cooperate. Male and female dance partners must have a tacit understanding. The strength coordination is inconsistent, and the movement rhythm is not similar. The male leads the female heel and focuses on the lead of the male partner. It is very important to complete each movement and the alternation of dance steps. In the process of dancing, if the male partner does not give the female partner hints and guidance, or the female partner is too active or changes the dance steps at will, it is easy to collide with each other and lead to disconnection, resulting in unnecessary sports injury.


### 2.3.5. Excessive Fatigue

Dance is a skill-oriented performance, and it takes a long time and a large amount of exercise training to master the relevant technologies. Moreover, the players often participate in multiple rounds of dance competitions and a large number of performances, which makes the dancers consume a lot of physical strength, and it is very easy to cause excessive fatigue or tension in the body and psychology of male and female players, resulting in sports injury.


### 2.3.6. Lack of Ideological Attention

Dancers do not pay enough attention to sports injuries in ideology and lack knowledge about the generation and prevention of sports injuries, which often leads to unnecessary injuries in the process of sports. The uninjured dancers should learn more and pay more attention to the sports injury and prevention of dance so as to minimize the incidence of sports injury; dance injured people should look for the most timely and effective treatment to reduce the harm caused by the injury in the event of injury. After rehabilitation, we should pay attention to its prevention, understand the most effective means to prevent sports injury, and achieve self-protection.


## 2.3.1. Poor Physical Fitness

Poor physical fitness is one of the most important factors causing sports injury, ranking first among the injury factors. In fact, strength quality is one of the most important foundations of sports. How to smoothly, stably, and substantially complete the complete set of dance movements with changeable rhythms and diverse movements has put forward higher requirements for the strength quality of dancers. Strengthening strength quality training can reduce the occurrence of sports injuries of dancers, reduce the severity of sports injuries, and achieve self-protection.


## 2.3.2. Poor Preparation before Training or Competition

The incidence of sports injuries caused by poor preparation of dancers is very high. Inadequate preparation activities will reduce muscle viscosity, inhibit muscle working ability, make it unable to adapt and coordinate with muscle work, and increase the occurrence of muscle sports injury. At the same time, it cannot effectively improve the working state of the heart and internal organs, affecting the role of protecting the heart and internal organs. The amount of preparatory activities is too large. When athletes enter the formal training competition, their physical function has decreased, or they are tired. In this case, they are prone to muscle and ligament strains and sprains.


## 2.3.3. Incorrect Technical Essentials

Dancers need high accuracy to complete difficult movements. If they are careless, they will deviate from the correct technology, resulting in technical errors and sports injuries. Correct technical movements can not only finish the difficult movements gracefully and lightly but also prevent sports injuries.


## 2.3.4. Improper Coordination between Partners

Dance is a sport for men and women to cooperate. Male and female dance partners must have a tacit understanding. The strength coordination is inconsistent, and the movement rhythm is not similar. The male leads the female heel and focuses on the lead of the male partner. It is very important to complete each movement and the alternation of dance steps. In the process of dancing, if the male partner does not give the female partner hints and guidance, or the female partner is too active or changes the dance steps at will, it is easy to collide with each other and lead to disconnection, resulting in unnecessary sports injury.


## 2.3.5. Excessive Fatigue

Dance is a skill-oriented performance, and it takes a long time and a large amount of exercise training to master the relevant technologies. Moreover, the players often participate in multiple rounds of dance competitions and a large number of performances, which makes the dancers consume a lot of physical strength, and it is very easy to cause excessive fatigue or tension in the body and psychology of male and female players, resulting in sports injury.


## 2.3.6. Lack of Ideological Attention

Dancers do not pay enough attention to sports injuries in ideology and lack knowledge about the generation and prevention of sports injuries, which often leads to unnecessary injuries in the process of sports. The uninjured dancers should learn more and pay more attention to the sports injury and prevention of dance so as to minimize the incidence of sports injury; dance injured people should look for the most timely and effective treatment to reduce the harm caused by the injury in the event of injury. After rehabilitation, we should pay attention to its prevention, understand the most effective means to prevent sports injury, and achieve self-protection.


## 3. Response Surface Methodology and Its Application in Dance Health and Training Safety

Response surface methodology (RSM) is a test design method proposed by Box et al. It is an optimization method that integrates test design and mathematical modeling [22–27]. By performing tests on representative local points, the functional relationship between factors and results in the global scope is regressed and fitted, and the optimal level value of each element is obtained. Initially used for the fitting of physical experiments, in recent years, it has become a newly developed optimization theory method in the world and is widely used in the fields of the chemical industry, agriculture, pharmacy, environment, and mechanical engineering. Many scholars and researchers at home and abroad have carried out lots of research [27]. Compared with the currently widely used orthogonal test design method, orthogonal experiments cannot obtain a clear functional expression between the test factors and the response target over the specified area, so the optimal combination of design variables and the optimal value of the response target cannot be obtained. Moreover, when the test factor has more numbers of levels, using the orthogonal design method still needs to do many tests, and it is not easy to implement. The response surface method has the advantages of fewer tests, short test periods, high precision, high accuracy of the regression equation, good prediction performance, and the ability to study the interaction between several factors. It has been widely used in many fields.


### 3.1. Principle

The response surface methodology uses a limited test design for a set of sample points within a specified design space. It fits a global approximation of the output variable (system response) in place of the actual response surface. In the engineering optimization design, the application of the response surface method can not only obtain the changing relationship between the response target and the design variables but also obtain the optimization scheme, that is, the optimal combination of the design variables so that the objective function can be optimized.Before building an approximate response surface model, the relationship between the design variables and the analysis objectives should be clearly defined. Choose an appropriate functional form to describe the relationship between the current design variables and the analysis objectives. At present, the methods of constructing the response surface mainly include polynomial, exponential function, logarithmic function fitting, and approximate methods such as a neural network. According to the Weierstrass polynomial best approximation theorem, many types of functions can be approximated by polynomials, and the polynomial approximation model can handle a wide range of nonlinear problems. Therefore, in practical applications, a polynomial approximation model can always be used for analysis regardless of the relationship between the design variables and the objective function.The regression model uses polynomials of different orders to approximate the relationship between the represented target and the design variables. It has the characteristics of a simple mathematical representation, small calculation amount, and fast convergence speed and can be represented explicitly. It is the most in-depth research study at present and the most widely used method.Generally, the system response Y and the design variablex satisfy(1)Y=y˜x+δ.Here,y˜x is the approximate function of the unknown function, and x=x1,x2,⋯,xn is the n-dimensional independent design variable; δ is the total error, including the random error, modeling error, and systematic errors.If a polynomial response surface is used to approximate the relationship between the system input and the response target, then(2)y˜x=β0+∑i=1kβiφix.Here,φix is the basis function; βi is the basis function coefficient; and k is the number of φix.In practical applications, the form of the polynomial response surface model can be determined according to engineering experience. Usually, a low-order polynomial approximation is used within a certain range of design variables, such as fitting with a linear function or a second-order model. The basis functions of the first-order and second-order polynomial approximation models are(3)Y=β0+∑i=1kβixi,Y=β0+∑i=1kβixi+∑i=1kβiixi+∑i=1i<jkβijxixj.Almost all RSM problems can be solved by one or both of the above two models. Of course, a polynomial model cannot be a reasonable approximation of the real functional relationship in the whole space of independent variables, but it usually does well in a relatively small area.The least square method can be used to estimate the parameters of the approximate polynomial and then do response surface analysis on the fitting surface. If the fitting surface is a reasonable approximation of the real response function, the analysis of the fitting surface is approximately equivalent to the analysis of the actual system. If the experimental design can be properly used to collect data, the model parameters can be estimated most effectively.


### 3.2. Test Design

By taking response surface methodology optimization of dance training design as an example, the application of response surface methodology in test design and optimization using Design-Expert software is introduced to improve the level of dance health and training safety.


#### 3.2.1. Selection of Test Factors and Determination of the Optimal Range of Single Factors

There are many methods to determine the design factors and levels of response surface experiments, and the commonly used methods include using the results of the existing literature, single-factor tests, climbing tests, and two-level factorial design tests. Through relevant literature results and single-factor experiments, it was determined that the influencing factors of the improvement of dance health and training safety level mainly depend on the physical fitness, correctness of technical movements, and site facilities. The fuzzy evaluation method [18–21] is used to assign values to them. Their value ranges are 0–1.0, 0–1.0, and 0–1.0, respectively.


#### 3.2.2. Response Surface Experimental Design

There are many response surface test design methods; and the more commonly used are Box–Behnken design (BBD) and central composite design (CCD). The test design divides the experimental points into central, cubic, and axial points, as shown in Figure1.Figure 1
Cubic point, axial point, and center point in test design.BBD is a commonly used test design method for response surface optimization, which is suitable for optimization experiments with 2 to 5 factors. Box–Behnken design with 3 levels for each factor were coded as (−1, 0, 1). The design table is arranged with 0 as the center point, and +1, −1 are the cubic points corresponding high and low values. The distribution of experimental points in the BBD design is shown in Figure2 (taking three factors as an example).Figure 2
Distribution of experimental points in BBD test design.CCD are also sometimes referred to as star point designs. The design table comprises the cubic points of the two-level factorial design plus the axial point and the center point. The CCD method takes 5 levels for each factor and is coded by (0, ±1, ±α), and 0 is the center point, while α is the extreme value corresponding to the axial point α=2k/4kis the number of factors .When the factors are the same, because there is no axial point, the number of tests of the Box–Behnken design is less and more economical. The optimal process-level value solved by optimization will not exceed the highest value range. It is especially suitable for some tests with special needs or safety requirements. According to the determined test factors and levels, the Box–Behnken combinatorial design method in Design-Export software was used to determine the experimental design of the MAP method. Taking the optimal value point of each single test factor as the center, one level value in the upper and lower regions is taken as the design level of the response surface test. The corresponding design factor level values and coding values are shown in Table1.Table 1
Design factor codes and levels.
FactorVariableLevel−10+1Physical fitnessX10.00000.50001.0000Correctness of technical movementsX20.00000.50001.0000Site facilitiesX30.00000.50001.0000Then, we input each factor and its level value into the system, in turn, to generate the test plan table (Table2). We carry out the test according to this scheme, record the test results of each factor combination, and fill in the test results in the corresponding columns. The degree of improvement of dance health and the training safety level (Y/%) of each test is shown in Figure 3.Table 2
Response surface test design and results.
Id of the testX1X2X3Degree of improvement of dance health and the training safety level (Y/%)10.00000.50000.000069.057920.00000.50001.000082.823230.00001.00000.500090.718540.00000.00000.500039.517950.50000.50000.500089.573260.50000.00000.000052.118570.50000.50000.500089.837980.50001.00001.000091.273290.50000.50000.500088.2785100.50000.00001.000054.1379110.50001.00000.000071.8532121.00000.00000.500064.5285131.00001.00000.500062.2379141.00000.50001.000073.5832151.00000.50000.000067.9185Figure 3
The degree of improvement of dance health and the training safety level (Y/%).


#### 3.2.3. Model Building and Testing

We use Design-Expert software to analyze the test results. First, click the Analysis button to test the significance of the linear function, 2FI model, second-order model, and third-order model. By comparing the data of the model significance test, the lack of the fit test, and the correlation test, a suitable model is recommended. The results of this test protocol recommend the use of a second-order model. Then (click the ANOVA button), variance analysis and significance tests are performed according to the selected model. In the analysis of variance, the significance of the constant term, a linear term, the quadratic term (interaction term), and the square term (surface effect) affecting the quadratic equation model will be tested, as shown in Table3.Table 3
The regression equation analysis of variance.
TypeSSDFMSFPr>FModel3659.38879406.5987128.9187<0.0010∗∗X123.9822123.98227.60220.0004∗X21398.611511398.4115443.4015<0.0010∗∗X3208.68731208.687366.16730.0005∗∗X1X2715.29871715.2987226.7987<0.0010∗∗X1X316.4022116.40225.20220.0715Not significanceX2X375.6915175.691524.00150.0045∗∗X12332.43731332.4373105.40730.0002∗∗X22885.78871885.7887280.8587<0.0010∗∗X32151.06221151.062247.90220.0010∗∗Residual15.771553.1515Lack of fit item14.377334.78736.89730.1288Not significancePure error1.388720.6987Note. SS: the sum of squares of variation, DF: degree of freedom, MS: mean square, and Pr > F: the probability of no significant effect; ∗∗represents very significance, and ∗represents significant.The fitted regression equation is(4)Y=88.2315−1.7289X1+13.2156X2+5.1206X3−13.3785X1X2−200312X1X3+4.3526X2X3−10.0231X1X2X3.The error statistics of the fitted regression equation were further analyzed, and Design-Expert software calculated the precision, multivariate correlation coefficient, reliability, and accuracy, as shown in Table4.Table 4
Statistical analysis of regression equation errors.
Statistics projectValueStd. Dev.2.4287Mean73.7722CV/%3.2915PRESS250.4373R-squared0.9928Adj R-squared0.9932Pred R-squared0.9868Adeq precision52.2573Large F values and smallP values represent the significance of the correlation coefficient. The Pr > F value <0.05 of the analysis of the variance model is considered to be significant, the fitting accuracy is good, and the response surface approximate model can be used for subsequent optimization design; lack of fit item Pr > F value >0.05 indicates insignificant; that is, the model fits well in the entire regression area under study; the more significant the multivariate correlation coefficient R2, the better the correlation; Adj R-Squared and Pred R-Squared RAdj2−RPred2<0.2 are high and close to Adj Pred, then the regression model can fully explain the process; if it is not high, it means that the explanation of the process is not sufficient, and it is necessary to consider whether there are other significant influencing factors. CV <10% indicates high reliability and precision of the experiment; Adeq Precision is the ratio of effective signal to noise, and it is considered reasonable if it is greater than 4. It can be seen from Table 4 that the fitted regression equation complies with the above test principles and has good adaptability.


## 3.1. Principle

The response surface methodology uses a limited test design for a set of sample points within a specified design space. It fits a global approximation of the output variable (system response) in place of the actual response surface. In the engineering optimization design, the application of the response surface method can not only obtain the changing relationship between the response target and the design variables but also obtain the optimization scheme, that is, the optimal combination of the design variables so that the objective function can be optimized.Before building an approximate response surface model, the relationship between the design variables and the analysis objectives should be clearly defined. Choose an appropriate functional form to describe the relationship between the current design variables and the analysis objectives. At present, the methods of constructing the response surface mainly include polynomial, exponential function, logarithmic function fitting, and approximate methods such as a neural network. According to the Weierstrass polynomial best approximation theorem, many types of functions can be approximated by polynomials, and the polynomial approximation model can handle a wide range of nonlinear problems. Therefore, in practical applications, a polynomial approximation model can always be used for analysis regardless of the relationship between the design variables and the objective function.The regression model uses polynomials of different orders to approximate the relationship between the represented target and the design variables. It has the characteristics of a simple mathematical representation, small calculation amount, and fast convergence speed and can be represented explicitly. It is the most in-depth research study at present and the most widely used method.Generally, the system response Y and the design variablex satisfy(1)Y=y˜x+δ.Here,y˜x is the approximate function of the unknown function, and x=x1,x2,⋯,xn is the n-dimensional independent design variable; δ is the total error, including the random error, modeling error, and systematic errors.If a polynomial response surface is used to approximate the relationship between the system input and the response target, then(2)y˜x=β0+∑i=1kβiφix.Here,φix is the basis function; βi is the basis function coefficient; and k is the number of φix.In practical applications, the form of the polynomial response surface model can be determined according to engineering experience. Usually, a low-order polynomial approximation is used within a certain range of design variables, such as fitting with a linear function or a second-order model. The basis functions of the first-order and second-order polynomial approximation models are(3)Y=β0+∑i=1kβixi,Y=β0+∑i=1kβixi+∑i=1kβiixi+∑i=1i<jkβijxixj.Almost all RSM problems can be solved by one or both of the above two models. Of course, a polynomial model cannot be a reasonable approximation of the real functional relationship in the whole space of independent variables, but it usually does well in a relatively small area.The least square method can be used to estimate the parameters of the approximate polynomial and then do response surface analysis on the fitting surface. If the fitting surface is a reasonable approximation of the real response function, the analysis of the fitting surface is approximately equivalent to the analysis of the actual system. If the experimental design can be properly used to collect data, the model parameters can be estimated most effectively.


## 3.2. Test Design

By taking response surface methodology optimization of dance training design as an example, the application of response surface methodology in test design and optimization using Design-Expert software is introduced to improve the level of dance health and training safety.


### 3.2.1. Selection of Test Factors and Determination of the Optimal Range of Single Factors

There are many methods to determine the design factors and levels of response surface experiments, and the commonly used methods include using the results of the existing literature, single-factor tests, climbing tests, and two-level factorial design tests. Through relevant literature results and single-factor experiments, it was determined that the influencing factors of the improvement of dance health and training safety level mainly depend on the physical fitness, correctness of technical movements, and site facilities. The fuzzy evaluation method [18–21] is used to assign values to them. Their value ranges are 0–1.0, 0–1.0, and 0–1.0, respectively.


### 3.2.2. Response Surface Experimental Design

There are many response surface test design methods; and the more commonly used are Box–Behnken design (BBD) and central composite design (CCD). The test design divides the experimental points into central, cubic, and axial points, as shown in Figure1.Figure 1
Cubic point, axial point, and center point in test design.BBD is a commonly used test design method for response surface optimization, which is suitable for optimization experiments with 2 to 5 factors. Box–Behnken design with 3 levels for each factor were coded as (−1, 0, 1). The design table is arranged with 0 as the center point, and +1, −1 are the cubic points corresponding high and low values. The distribution of experimental points in the BBD design is shown in Figure2 (taking three factors as an example).Figure 2
Distribution of experimental points in BBD test design.CCD are also sometimes referred to as star point designs. The design table comprises the cubic points of the two-level factorial design plus the axial point and the center point. The CCD method takes 5 levels for each factor and is coded by (0, ±1, ±α), and 0 is the center point, while α is the extreme value corresponding to the axial point α=2k/4kis the number of factors .When the factors are the same, because there is no axial point, the number of tests of the Box–Behnken design is less and more economical. The optimal process-level value solved by optimization will not exceed the highest value range. It is especially suitable for some tests with special needs or safety requirements. According to the determined test factors and levels, the Box–Behnken combinatorial design method in Design-Export software was used to determine the experimental design of the MAP method. Taking the optimal value point of each single test factor as the center, one level value in the upper and lower regions is taken as the design level of the response surface test. The corresponding design factor level values and coding values are shown in Table1.Table 1
Design factor codes and levels.
FactorVariableLevel−10+1Physical fitnessX10.00000.50001.0000Correctness of technical movementsX20.00000.50001.0000Site facilitiesX30.00000.50001.0000Then, we input each factor and its level value into the system, in turn, to generate the test plan table (Table2). We carry out the test according to this scheme, record the test results of each factor combination, and fill in the test results in the corresponding columns. The degree of improvement of dance health and the training safety level (Y/%) of each test is shown in Figure 3.Table 2
Response surface test design and results.
Id of the testX1X2X3Degree of improvement of dance health and the training safety level (Y/%)10.00000.50000.000069.057920.00000.50001.000082.823230.00001.00000.500090.718540.00000.00000.500039.517950.50000.50000.500089.573260.50000.00000.000052.118570.50000.50000.500089.837980.50001.00001.000091.273290.50000.50000.500088.2785100.50000.00001.000054.1379110.50001.00000.000071.8532121.00000.00000.500064.5285131.00001.00000.500062.2379141.00000.50001.000073.5832151.00000.50000.000067.9185Figure 3
The degree of improvement of dance health and the training safety level (Y/%).


### 3.2.3. Model Building and Testing

We use Design-Expert software to analyze the test results. First, click the Analysis button to test the significance of the linear function, 2FI model, second-order model, and third-order model. By comparing the data of the model significance test, the lack of the fit test, and the correlation test, a suitable model is recommended. The results of this test protocol recommend the use of a second-order model. Then (click the ANOVA button), variance analysis and significance tests are performed according to the selected model. In the analysis of variance, the significance of the constant term, a linear term, the quadratic term (interaction term), and the square term (surface effect) affecting the quadratic equation model will be tested, as shown in Table3.Table 3
The regression equation analysis of variance.
TypeSSDFMSFPr>FModel3659.38879406.5987128.9187<0.0010∗∗X123.9822123.98227.60220.0004∗X21398.611511398.4115443.4015<0.0010∗∗X3208.68731208.687366.16730.0005∗∗X1X2715.29871715.2987226.7987<0.0010∗∗X1X316.4022116.40225.20220.0715Not significanceX2X375.6915175.691524.00150.0045∗∗X12332.43731332.4373105.40730.0002∗∗X22885.78871885.7887280.8587<0.0010∗∗X32151.06221151.062247.90220.0010∗∗Residual15.771553.1515Lack of fit item14.377334.78736.89730.1288Not significancePure error1.388720.6987Note. SS: the sum of squares of variation, DF: degree of freedom, MS: mean square, and Pr > F: the probability of no significant effect; ∗∗represents very significance, and ∗represents significant.The fitted regression equation is(4)Y=88.2315−1.7289X1+13.2156X2+5.1206X3−13.3785X1X2−200312X1X3+4.3526X2X3−10.0231X1X2X3.The error statistics of the fitted regression equation were further analyzed, and Design-Expert software calculated the precision, multivariate correlation coefficient, reliability, and accuracy, as shown in Table4.Table 4
Statistical analysis of regression equation errors.
Statistics projectValueStd. Dev.2.4287Mean73.7722CV/%3.2915PRESS250.4373R-squared0.9928Adj R-squared0.9932Pred R-squared0.9868Adeq precision52.2573Large F values and smallP values represent the significance of the correlation coefficient. The Pr > F value <0.05 of the analysis of the variance model is considered to be significant, the fitting accuracy is good, and the response surface approximate model can be used for subsequent optimization design; lack of fit item Pr > F value >0.05 indicates insignificant; that is, the model fits well in the entire regression area under study; the more significant the multivariate correlation coefficient R2, the better the correlation; Adj R-Squared and Pred R-Squared RAdj2−RPred2<0.2 are high and close to Adj Pred, then the regression model can fully explain the process; if it is not high, it means that the explanation of the process is not sufficient, and it is necessary to consider whether there are other significant influencing factors. CV <10% indicates high reliability and precision of the experiment; Adeq Precision is the ratio of effective signal to noise, and it is considered reasonable if it is greater than 4. It can be seen from Table 4 that the fitted regression equation complies with the above test principles and has good adaptability.


## 3.2.1. Selection of Test Factors and Determination of the Optimal Range of Single Factors

There are many methods to determine the design factors and levels of response surface experiments, and the commonly used methods include using the results of the existing literature, single-factor tests, climbing tests, and two-level factorial design tests. Through relevant literature results and single-factor experiments, it was determined that the influencing factors of the improvement of dance health and training safety level mainly depend on the physical fitness, correctness of technical movements, and site facilities. The fuzzy evaluation method [18–21] is used to assign values to them. Their value ranges are 0–1.0, 0–1.0, and 0–1.0, respectively.


## 3.2.2. Response Surface Experimental Design

There are many response surface test design methods; and the more commonly used are Box–Behnken design (BBD) and central composite design (CCD). The test design divides the experimental points into central, cubic, and axial points, as shown in Figure1.Figure 1
Cubic point, axial point, and center point in test design.BBD is a commonly used test design method for response surface optimization, which is suitable for optimization experiments with 2 to 5 factors. Box–Behnken design with 3 levels for each factor were coded as (−1, 0, 1). The design table is arranged with 0 as the center point, and +1, −1 are the cubic points corresponding high and low values. The distribution of experimental points in the BBD design is shown in Figure2 (taking three factors as an example).Figure 2
Distribution of experimental points in BBD test design.CCD are also sometimes referred to as star point designs. The design table comprises the cubic points of the two-level factorial design plus the axial point and the center point. The CCD method takes 5 levels for each factor and is coded by (0, ±1, ±α), and 0 is the center point, while α is the extreme value corresponding to the axial point α=2k/4kis the number of factors .When the factors are the same, because there is no axial point, the number of tests of the Box–Behnken design is less and more economical. The optimal process-level value solved by optimization will not exceed the highest value range. It is especially suitable for some tests with special needs or safety requirements. According to the determined test factors and levels, the Box–Behnken combinatorial design method in Design-Export software was used to determine the experimental design of the MAP method. Taking the optimal value point of each single test factor as the center, one level value in the upper and lower regions is taken as the design level of the response surface test. The corresponding design factor level values and coding values are shown in Table1.Table 1
Design factor codes and levels.
FactorVariableLevel−10+1Physical fitnessX10.00000.50001.0000Correctness of technical movementsX20.00000.50001.0000Site facilitiesX30.00000.50001.0000Then, we input each factor and its level value into the system, in turn, to generate the test plan table (Table2). We carry out the test according to this scheme, record the test results of each factor combination, and fill in the test results in the corresponding columns. The degree of improvement of dance health and the training safety level (Y/%) of each test is shown in Figure 3.Table 2
Response surface test design and results.
Id of the testX1X2X3Degree of improvement of dance health and the training safety level (Y/%)10.00000.50000.000069.057920.00000.50001.000082.823230.00001.00000.500090.718540.00000.00000.500039.517950.50000.50000.500089.573260.50000.00000.000052.118570.50000.50000.500089.837980.50001.00001.000091.273290.50000.50000.500088.2785100.50000.00001.000054.1379110.50001.00000.000071.8532121.00000.00000.500064.5285131.00001.00000.500062.2379141.00000.50001.000073.5832151.00000.50000.000067.9185Figure 3
The degree of improvement of dance health and the training safety level (Y/%).


## 3.2.3. Model Building and Testing

We use Design-Expert software to analyze the test results. First, click the Analysis button to test the significance of the linear function, 2FI model, second-order model, and third-order model. By comparing the data of the model significance test, the lack of the fit test, and the correlation test, a suitable model is recommended. The results of this test protocol recommend the use of a second-order model. Then (click the ANOVA button), variance analysis and significance tests are performed according to the selected model. In the analysis of variance, the significance of the constant term, a linear term, the quadratic term (interaction term), and the square term (surface effect) affecting the quadratic equation model will be tested, as shown in Table3.Table 3
The regression equation analysis of variance.
TypeSSDFMSFPr>FModel3659.38879406.5987128.9187<0.0010∗∗X123.9822123.98227.60220.0004∗X21398.611511398.4115443.4015<0.0010∗∗X3208.68731208.687366.16730.0005∗∗X1X2715.29871715.2987226.7987<0.0010∗∗X1X316.4022116.40225.20220.0715Not significanceX2X375.6915175.691524.00150.0045∗∗X12332.43731332.4373105.40730.0002∗∗X22885.78871885.7887280.8587<0.0010∗∗X32151.06221151.062247.90220.0010∗∗Residual15.771553.1515Lack of fit item14.377334.78736.89730.1288Not significancePure error1.388720.6987Note. SS: the sum of squares of variation, DF: degree of freedom, MS: mean square, and Pr > F: the probability of no significant effect; ∗∗represents very significance, and ∗represents significant.The fitted regression equation is(4)Y=88.2315−1.7289X1+13.2156X2+5.1206X3−13.3785X1X2−200312X1X3+4.3526X2X3−10.0231X1X2X3.The error statistics of the fitted regression equation were further analyzed, and Design-Expert software calculated the precision, multivariate correlation coefficient, reliability, and accuracy, as shown in Table4.Table 4
Statistical analysis of regression equation errors.
Statistics projectValueStd. Dev.2.4287Mean73.7722CV/%3.2915PRESS250.4373R-squared0.9928Adj R-squared0.9932Pred R-squared0.9868Adeq precision52.2573Large F values and smallP values represent the significance of the correlation coefficient. The Pr > F value <0.05 of the analysis of the variance model is considered to be significant, the fitting accuracy is good, and the response surface approximate model can be used for subsequent optimization design; lack of fit item Pr > F value >0.05 indicates insignificant; that is, the model fits well in the entire regression area under study; the more significant the multivariate correlation coefficient R2, the better the correlation; Adj R-Squared and Pred R-Squared RAdj2−RPred2<0.2 are high and close to Adj Pred, then the regression model can fully explain the process; if it is not high, it means that the explanation of the process is not sufficient, and it is necessary to consider whether there are other significant influencing factors. CV <10% indicates high reliability and precision of the experiment; Adeq Precision is the ratio of effective signal to noise, and it is considered reasonable if it is greater than 4. It can be seen from Table 4 that the fitted regression equation complies with the above test principles and has good adaptability.


## 4. Measures to Improve Dance Health and Training Safety

In view of the main influencing factors of dance health and training safety, after the analysis by response surface methodology, we put forward the following countermeasures and preventive measures for the health and training safety problems in dance teaching and competition.


### 4.1. Popularizing the Common Sense of Safety Precautions and Improving the Self-Protection Awareness of Practitioners

Dance requires great intensity and fast pace, and there are many hidden safety hazards in the process of sports. Teachers should explain and pass on the common sense of safety precautions in dance training and competition to the practitioners so that the practitioners can understand the basic situation of accidents prone in dance sports, improve their awareness of active protection, pay attention to the safety precautions in dance teaching and competition ideologically, and reduce the occurrence of accidental injuries.


### 4.2. Strengthening Physical Fitness Training and Paying Attention to Recovery after Training

Teachers should pay attention to the training of the physical quality of the practitioners, focusing on the training of the joints and muscle ligaments that are prone to injury so as to improve the strength of the small joints and the activity of the muscle ligaments, form effective support in sports, and improve the safety in sports. After training, appropriate relaxation and recovery work should be carried out in time to reduce the fatigue caused by training, promote physical recovery, and reduce the accident rate caused by decreased attention due to fatigue.


### 4.3. Reasonably Arranging the Training Load and Carrying Out Adequate Preparation Activities

The training load shall be reasonably arranged to avoid excessive exercise on local areas, such as excessive step training with high intensity, resulting in joint and muscle fatigue. Proper preparation activities can improve muscle excitability and prepare for sports training and competition. Before training and competition, sports dance practitioners should carry out sufficient preparation activities to improve the physical and mental excitement, improve the directivity of attention, and effectively avoid muscle strain caused by insufficient warm-up and inattention caused by low excitement.


### 4.4. Improving Safety Awareness and Eliminating Potential Safety Hazards

Before training and competition, both teachers and students should inspect the training ground to find and eliminate potential safety hazards. In terms of clothing, avoid wearing inappropriate clothes and high dancing shoes and avoid wearing jewelry that is easy to cause injury or articles that are easy to cause injury to yourself or others. Practitioners with physical diseases should carry medicines with them, inform their dance partners or coaches of their diseases before training and competition, and prepare for all problems in advance.


## 4.1. Popularizing the Common Sense of Safety Precautions and Improving the Self-Protection Awareness of Practitioners

Dance requires great intensity and fast pace, and there are many hidden safety hazards in the process of sports. Teachers should explain and pass on the common sense of safety precautions in dance training and competition to the practitioners so that the practitioners can understand the basic situation of accidents prone in dance sports, improve their awareness of active protection, pay attention to the safety precautions in dance teaching and competition ideologically, and reduce the occurrence of accidental injuries.


## 4.2. Strengthening Physical Fitness Training and Paying Attention to Recovery after Training

Teachers should pay attention to the training of the physical quality of the practitioners, focusing on the training of the joints and muscle ligaments that are prone to injury so as to improve the strength of the small joints and the activity of the muscle ligaments, form effective support in sports, and improve the safety in sports. After training, appropriate relaxation and recovery work should be carried out in time to reduce the fatigue caused by training, promote physical recovery, and reduce the accident rate caused by decreased attention due to fatigue.


## 4.3. Reasonably Arranging the Training Load and Carrying Out Adequate Preparation Activities

The training load shall be reasonably arranged to avoid excessive exercise on local areas, such as excessive step training with high intensity, resulting in joint and muscle fatigue. Proper preparation activities can improve muscle excitability and prepare for sports training and competition. Before training and competition, sports dance practitioners should carry out sufficient preparation activities to improve the physical and mental excitement, improve the directivity of attention, and effectively avoid muscle strain caused by insufficient warm-up and inattention caused by low excitement.


## 4.4. Improving Safety Awareness and Eliminating Potential Safety Hazards

Before training and competition, both teachers and students should inspect the training ground to find and eliminate potential safety hazards. In terms of clothing, avoid wearing inappropriate clothes and high dancing shoes and avoid wearing jewelry that is easy to cause injury or articles that are easy to cause injury to yourself or others. Practitioners with physical diseases should carry medicines with them, inform their dance partners or coaches of their diseases before training and competition, and prepare for all problems in advance.


## 5. Conclusions

In order to provide the application of response surface methodology in the experimental design and optimization of the analysis of influencing factors on dance health and training safety, Design-Export software is used for the experimental design and analysis of response surface methodology, providing intuitive contour maps and three-dimensional diagrams, establishing prediction models, and being able to test the model adaptability, the significance of models and coefficients, and the nonfitting items so as to further carry out variance analysis and model diagnosis. By optimizing the model established by response surface methodology, an experimental optimization scheme can be proposed to solve the problems encountered in the practical application of response surface methodology in experimental design and optimization.Response surface methodology not only establishes the prediction model but also tests the adaptability of the model, the significance of the model and coefficient, and the mismatch and further carries out variance analysis and model diagnosis. The response surface methodology can effectively guide the optimization of the design of the test scheme in the analysis of dance health and training safety, which is conducive to improving the benefit of the test. However, it is a complex process to construct the response surface approximation model that can meet the optimal design of practical engineering. It also needs to repeatedly collect the test data, fit the approximate model, and test the accuracy of the response surface.Response surface methodology also has its limitations. The premise of response surface optimization is that the designed experimental points should include the best experimental conditions. If the experimental points are improperly selected, the response surface optimization method cannot get good optimization results. Therefore, before using the response surface optimization method, we should establish reasonable experimental influencing factors and levels.



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*Source: 9818664-2022-08-04.xml*