Discussion on the Implementation Path of Multi-dimensional Linkage of Sports in Colleges and Universities

1. Introduction

rrespective of whether it concerns physical skill enhancement or mental wellbeing, and regardless of whether it pertains to fostering interest or nurturing abilities, the learning and development of physical education should be accorded utmost priority [1-2]. Cultivating students’ sporting consciousness and laying a solid foundation for their athletic capabilities not only holds paramount significance but also exerts a profound influence on fostering their autonomous, sustainable, and personalized growth trajectories [3-5]. Nonetheless, amidst the prevalence of exam-oriented education, a phenomenon has gradually emerged within the student populace, wherein an affinity for sports activities coexists with an aversion towards the academic pursuit of physical education curricula. To fundamentally address this dichotomy, educators must adeptly harness diverse strategies and methodologies, endeavoring to construct a model of physical activity that aligns with the vision of students’ lived cultural experiences [6-7].

The fundamental distinguishing features of physical education, as contrasted with other forms of sport, reside in its inherently educational and pedagogical essence. Notably, it is under the guidance of teachers, with students occupying the central role in their own learning and physical development process. Through the application of the “integrative approach of internal learning and external application, learning and practice in tandem,” it aims to foster in students a heightened awareness of, and aptitude for, “sunshine sports,” thereby enhancing their overall quality [8-10]. In essence, while other disciplines primarily emphasize cognitive activities to equip students with disciplinary fundamentals, physical education predominantly adopts physical practice as its primary modality, prioritizing health promotion as its core objective. This allows students to engage in comprehensive learning, through which they can experiment, experience, and comprehend, ultimately mastering the subject’s knowledge, techniques, and skills, and achieving holistic development.

The concept of “multiple linkages” primarily emphasizes the diverse methodologies and approaches employed within the actual teaching process. Regardless of whether it pertains to physical fitness and skills training, mental health education, outdoor sports, or indoor instruction, the prolonged reliance on a traditional, monolithic approach, often characterized by a formal, rote manner of instruction in Mandarin, is destined to lead to a state of monotony, passive reception, and ultimately, a predicament marked by low engagement, diminished interest, and stagnated development among students [11-13].In response, teachers must adapt to the evolving needs of the situation, adeptly adopting flexible and dynamic teaching methods tailored to the content being taught. This approach not only caters to students’ innate desires for novelty, variety, and active participation, embodying the new curriculum’s humanistic ideals of education and care, but also implicitly guides them towards a fulfilling path of effective teaching and joyful learning, thereby fostering a lifelong appreciation for physical education and continuously infusing vitality into the educational process [14-15].Literature [16] integrates the hierarchical analysis method with a physical training model to devise a comprehensive physical health evaluation framework specifically tailored for colleges and universities. The findings indicate that integrating physical training into public physical education curricula significantly contributes to enhancing the overall health status of students. Furthermore, incorporating hierarchical analysis into the assessment of college students’ physical health levels augments the comprehensiveness and depth of health analysis.Literature [17], leveraging the design of an artificial intelligence system for intelligent data acquisition and analysis, concludes that the artificial intelligence-based remote multimedia physical education teaching system renders the teaching process flexible, adaptable, and location-independent. This system can dynamically adjust teaching strategies based on individual student needs, thereby facilitating personalized instruction.Literature [18], utilizing the qualitative comparative analysis method to examine two explanatory variables, reveals that diluted teaching objectives and inadequate teaching depth are factors contributing to the suboptimal physical health levels among students. The so-called “three independent” teaching reform in college sports, characterized by a formality that overshadows its substantive content, primarily constitutes a methodological shift rather than a transformation in the teaching content itself.Literature [19] employs a semi-supervised framework to implement a movement input and interactive virtual scene algorithm. Consequently, students’ motion efficiency was enhanced by 30%. Additionally, two-thirds of the participants reported an 80% increase in their interest in sports training, while 90% of college coaches concurred that the integration of virtual reality technology in sports instruction is crucial for enhancing the technical proficiency and training quality of college sports athletes, thereby contributing to China’s competitive sports talent pool.Literature [20] conducted an analysis and evaluation of the computer-based assessment system for sports courses. The developed artificial intelligence-powered computer teaching system elevates the modernization of physical education teaching to unprecedented levels. Experimental outcomes demonstrate its effectiveness in accurately detecting physical activity among college students.Literature [21] explores the methodologies and guidelines for fostering and implementing core values through an investigation into the practices of cultivating national values in developed educational systems. In the context of globalization, it is imperative to initiate by actively constructing a university culture, continually refining system construction, and reinforcing collaborative work platforms. This approach facilitates the integration of first-class and second-class learning, as well as explicit and implicit education, ultimately fulfilling the noble mission of fostering virtue and educating individuals.

Having delved into the external and internal constructs of the overlap effect theory, this paper focuses on two pivotal aspects of linkage goals and content structure to formulate a multifaceted linkage model for college sports. Consequently, a comprehensive evaluation index system was devised, spanning multiple dimensions including linkage subjects, content, implementation, and evaluation. This system aims to holistically appraise the salutary effects of multiple linkages within college sports. To ascertain the precision and impartiality of the evaluation outcomes, the weights of the indicators were derived through a combined assignment approach, integrating subjective weights determined by the Analytic Hierarchy Process (AHP) with objective weights sourced from the Information Entropy method. Furthermore, utilizing the example of multivariate linkage data from X university’s sports programs spanning 2019-2023, the established college sports multivariate linkage evaluation index system was rigorously evaluated employing an enhanced TOPSIS model. This evaluation was then juxtaposed with the conventional TOPSIS model to validate the efficacy of the proposed enhancements. Lastly, commencing from the linkage behavior factors and linkage effect factors inherent to the multiple linkage model, this study delves into the multifarious influencing factors of college sports linkages via correlation and regression analyses.

2. Construction of A Multifaceted Linkage Model for University Sports

The Theory of Overlapping Effects (TOSI) is the most influential theory guiding home-school-society cooperation in the field of education. The linkage efficiency is the direct expression of the effect of multiple linkage implementation, and the evaluation feedback is the key link of the multi-linkage implementation, and also the basic guarantee for the optimization and upgrading of multiple linkage. Based on the idea of “Theory Interpretation – Problem Analysis – Logical Interpretation – Mode Construction – Promotion Path”, this chapter discusses the theoretical and practical issues of the university sports multi-dimensional cooperation by using the literature and text analysis methods. This chapter is based on the idea of “theory interpretation, problem analysis, logical interpretation, mode construction and promotion path”.

A. Theory of overlapping effects

The theory of overlap is primarily employed to guide collaboration among primary and secondary schools, families, and communities in fostering student learning and promoting healthy development. This theory underscores not only the individual contributions of schools, families, and communities to students’ learning behaviors but also the pivotal influence of the “overlap” or intersection between home, school, and community environments on students’ academic performance.

The external (I) and internal (II) structures of the overlap effect theory are illustrated in Figure 1. The external structure comprises three distinct segments external to the school, family, and community, primarily elucidating the influence of the environment shaped by the external mechanisms of these entities on students. It underscores the discrete influences of the school, family, and community, alongside individual factors, on student development. In this framework, the theory categorizes factors impacting students’ learning and development into four types: A (students), B (schools), C (families), and D (communities).The internal structure, on the other hand, encompasses the overlapping segments between the school, family, and community. It primarily explains the influence of shared factors among members of these entities on students’ academic performance and healthy development. These factors include teachers, parents, neighbors, peers, and other individual-level aspects that contribute to the overlapping effects on students.

The internal mechanisms that influence students are mediated through the overlap between the individual levels of teachers, parents, neighbors, and peers. Within the internal structure of the overlap effect theory, from the familial and educational perspectives, the theory elucidates the factors affecting students in terms of the interactions between parents, teachers, and their mutual cooperation. It emphasizes the direct impact of “interpersonal relationships” forged at the individual level, specifically between teachers and parents, on students’ learning and development.Under the framework of the overlapping effects theory, home-school-community cooperation emerges as a vital complement to traditional school-based education. In this collaborative model, the school assumes a leading role, the family fulfills a participatory function, and the community plays a supportive role. The integration of the internal and external structures of the overlapping effects theory offers a unique opportunity to optimize and compensate for the limitations of isolated home, school, and community education.

B. Sports multiple linkage model based on overlapping effect theory

Collegiate sports multiple linkage represents a sports activity that effectively enhances physical activity levels, aiming to fortify individuals, prevent illnesses, and elevate quality of life. It accomplishes these objectives through a range of scientifically grounded, systematic, and standardized social behaviors and lifestyles that involve physical participation. Furthermore, it constitutes a process of intervention implementation, characterized by the involvement of multiple organizations, the coordination of diverse tasks, and the symbiosis of various forms. Figure 2 depicts the sports multiple linkage model grounded in the theory of overlapping effects. This research focuses on the linkage objectives and content structure to construct the collegiate sports multiple linkage model, adhering to the internal and external coherence of the overlapping effects theoretical structure. Additionally, it acknowledges the multiplicity, holistic nature, and permeability of the overlapping effects theoretical practice, employing multiple goal targets, diversified subject participation, varied content design, and multi-channel implementation to foster the synergistic multiple linkage of collegiate sports, which encompasses both internal and external connections.

The multiple linkage of sports in colleges and universities underscores the comprehensive, equitable, and reciprocal engagement of diverse stakeholders from schools, families, communities, and society. Its objective is to foster a supportive social environment that enhances students’ learning capabilities and fosters their holistic physical and mental health development. Consequently, the overarching goal is the establishment of a “Vibrant Sports Ecosystem,” with the creation of vibrant families, schools, and communities as its specific manifestations. The target orientation of a “dynamic sports environment” profoundly encapsulates the intrinsic unity and interconnectivity between the internal and external structures of the linkage participants, grounded in the theory of overlapping effects. These three facets are mutually reinforcing and complementary, collectively forming an integrated and harmonious whole that drives students’ sports participation and health promotion.

The content structure serves as the foundational prerequisite for the effective implementation of college sports and constitutes the pivotal factor in fostering the holistic and healthy development of students. It is primarily devised with a focus on resource integration, aimed at maximizing the utilization of human capital, venues, financial allocations, policy frameworks, and other sports-related resources emanating from schools, families, and communities. Furthermore, it emphasizes harnessing the synergies of diverse resource types to facilitate collaborative sharing and construction. For instance, regarding venue resources, schools boast a plethora of facilities such as playgrounds, gymnasiums, and classrooms, whereas communities offer a variety of activity areas and sports clubs. With regard to policy resources, schools primarily devise and enact multifaceted work plans and programs, integrating multi-dimensional linkages into their strategic agendas. In contrast, communities devise sports activity plans, organize competitions, and actively encourage the participation of families and individuals in physical exercise endeavors.

3. Evaluation Model for the Multifaceted Linkage of University Sports

A. Multi-linkage evaluation indicator system for higher education sports

Guided by the principles of integrating universality with specificity, merging quantitative and qualitative metrics, and harmonizing hierarchical structure with operational aspects, this section employs a multifaceted methodology encompassing literature review, the Delphi method, and logical analysis, among others. Consequently, an index system for the multifaceted linkage of college sports has been devised, as depicted in Table 1. This system encompasses a comprehensive framework comprising four primary indicators and seventeen subordinate secondary indicators, encompassing dimensions such as linkage subjects, content, implementation, and evaluation.

B. Improvement of the TOPSIS model for portfolio weight calculation

The cornerstone of comprehensively evaluating multiple linkages in college sports, utilizing the enhanced TOPSIS model, resides in the rational determination of indicator weights based on the intricate interplay among these indicators. The assignment of weights significantly influences the outcomes of the comprehensive evaluation. Traditional TOPSIS methods predominantly rely on subjective weighting, which can be heavily influenced by human biases. In contrast, objective weighting methods, grounded in index data, effectively capture the genuine disparities in college sports’ multiple linkages. However, these methods, exemplified by the information entropy method, may occasionally yield results that diverge from practical experience.To mitigate the subjectivity inherent in traditional models and ensure that the actual variations in indicator data are accurately reflected, this study enhances the weighting mechanism of the model indicators within the framework of the traditional TOPSIS model. We adopt a hybrid approach that combines the subjective Analytic Hierarchy Process (AHP) with the objective information entropy method, thereby fostering a subjective-objective integration in the empowerment strategy.

1) Calculation of evaluation indicator weights based on the AHP method

1. Constructing pairwise comparison matrix according to the importance of indicators: In the evaluation index system, two indicators are randomly selected to carry out appropriate scaling, which is recorded as \(a_{ij}\), and these indicators can constitute a pairwise comparison matrix \(M\) through principal component importance analysis and expert scoring: \[\label{GrindEQ__1_} M=\left(\begin{array}{ccccc} {a_{11} ,} & {a_{12} ,} & {a_{13} ,} & {\ldots ,} & {a_{1j} } \\ {a_{21} ,} & {a_{22} ,} & {a_{23} ,} & {\ldots ,} & {a_{2j} } \\ {\vdots } & {\vdots } & {\vdots } & {\ddots } & {\vdots } \\ {a_{i1} ,} & {a_{i2} ,} & {a_{i3} ,} & {\ldots ,} & {a_{ij} } \end{array}\right) . \tag{1}\]

2. Calculate the weight vector between evaluation indicators: Before calculating the weight vector, it is first necessary to determine whether the pairwise comparison matrix of the indicators constructed above is a positive and negative matrix, which is a positive and negative matrix and a consistency matrix that needs to meet the following conditions: \[\label{GrindEQ__2_} a_{ij} >0,a_{ji} =\frac{1}{a_{ij} } (i,j=1,2,\ldots ,n) . \tag{2}\] \[\label{GrindEQ__3_} a_{ij} a_{jk} =a_{ik} ,\forall i,j,k=1,2,\ldots ,n . \tag{3}\]

   The weight vector of the evaluation index is \(w=\left(w_{1} ,w_{2} ,\ldots ,w_{n} \right)^{T}\), and the maximum eigenvalue is \(\lambda _{\max }\). The eigenvalue method is used to calculate the weight vector, and the approximate eigenroot and \(\lambda _{\max }\) can be obtained after normalization of \(w\) as: \[\label{GrindEQ__4_} Mw=\lambda _{\max } w . \tag{4}\]

3. Use the consistency test to determine whether the matrix needs to be modified or not: For the \(\lambda _{\max }\) derived in (2), it can be used to determine whether the matrix is a consistent matrix. The indicators of the consistency matrix are denoted by \(CI\), and the smaller the value of \(CI\), the stronger the consistency. The final test coefficient between the multiple linkage evaluation indicators of college sports is recorded as \(CR\), where \(CR,CI\) can be expressed as: \[\label{GrindEQ__5_} CI=\frac{\lambda _{\max } -n}{n-1} . \tag{5}\] \[\label{GrindEQ__6_} CR=\frac{CI}{RI} . \tag{6}\]

   The above formula \(RI\) is related to the order of the judgment matrix, which is an important index to measure whether the matrix has satisfactory consistency or not.The reference value of RI is shown in Table 2. When \(CR<0.10\), the judgment matrix meets the consistency requirements, otherwise the judgment matrix must be adjusted appropriately and made to meet the consistency requirements.

2) Calculation of evaluation index weights based on the information entropy method

The information entropy method is a quantitative approach in physics used to measure the uncertainty or dispersion of information. In the context of determining the weights of evaluation indices for multiple linkages in college sports, we can draw upon the principles of information entropy calculation. Specifically, we treat each linkage to be evaluated as an individual sample, and the evaluation indices within that linkage correspond to the relevant variables under consideration. The primary steps involved in this process are outlined below:

4. Normalize the evaluation index data

1. Normalize the evaluation index data by using the method of deviation normalization to linearly transform the relevant data and obtain the normalization matrix \(M_{ij}\): \[\label{GrindEQ__7_} M_{ij(m\times n)} =\frac{x_{ij} -\bar{x}_{\cdot j} }{\max x_{\cdot j} -\min x_{\cdot j} } , \tag{7}\] where \(x_{ij}\) is the initial data series, \(M_{ij}\) is the normalized data series, and \(M_{ij} \in (0,1)\) the interval, which eliminates the differences in magnitude and scale between the evaluation indicators. \(x_{\cdot j}\) represents the \(j\)th column of \(M_{ij}\).

2. Calculate the weight of the \(j\)th evaluation indicator in the indicator value of the \(i\)th \(p_{ij}\): \[\label{GrindEQ__8_} p_{ij} =\frac{x_{ij} }{\sum _{i=1}^{m}x_{ij} } ,0\le p_{ij} \le 1 . \tag{8}\] As a result, a normalized matrix consisting of the weight of each evaluation indicator under different linkages can be obtained: \[\label{GrindEQ__9_} Y=\left\{p_{ij} \right\}_{m\times n} . \tag{9}\]

3. Calculate the entropy value of the \(j\)st evaluation index \(H_{j}\), at this time the entropy value of the \(j\)rd index \(H_{j}\) can be expressed as: \[\label{GrindEQ__10_} H_{j} =-\frac{1}{\ln m} \sum _{i=1}^{m}\left(p_{ij} \ln p_{ij} \right) ,0\le H_{j} \le 1 . \tag{10}\]

4. Calculate the deviation coefficient of the \(j\)th linkage evaluation index \(g_{j}\): The entropy value of the linkage evaluation index can be obtained in step (3) \(H_{j}\), the larger the entropy value (tends to 1) the higher the degree of confusion, and the smaller the utility of the evaluation index. Therefore, in order to facilitate the calculation of the weights of the subsequent indicators, the deviation coefficient \(g_{j}\) of the evaluation indicators is introduced: \[\label{GrindEQ__11_} g_{j} =1-H_{j} . \tag{11}\]

5. Calculate the objective weight of the \(j\)th linkage evaluation index according to the results of step 4: The key of the information entropy method to calculate the weights of the linkage evaluation indicators is to utilize the utility ratio of each evaluation indicator to reflect the importance of the indicator, and the larger the deviation coefficient of the linkage indicators, the larger the contribution to the linkage evaluation results. Therefore the weight of the \(j\)th indicator can be expressed as: \[\label{GrindEQ__12_} \omega _{j} =\frac{g_{j} }{\sum _{j=1}^{n}g_{j} } . \tag{12}\]

3) Assignment of portfolio optimization based on AHP-information entropy method

From an empowerment perspective, the AHP (Analytic Hierarchy Process) method falls under the category of subjective empowerment methods, whereas the information entropy method constitutes an objective empowerment approach. To address the limitations of solely relying on either method in the traditional TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) model, we propose integrating the indicator weights derived from the information entropy method to adjust those obtained through the AHP method. This approach enables us to calculate the combined subjective-objective weights for the linkage indicators, which are then utilized to enhance the comprehensive evaluation of multiple linkages in college and university sports within the TOPSIS framework.

In the calculation process, the combination assignment can be obtained by using the Lagrange multiplier method: \[\label{GrindEQ__13_} W_{j} =\frac{\sqrt{\alpha _{j} \beta _{j} } }{\sum _{j=1}^{n}\sqrt{\alpha _{j} \beta _{j} } } , \tag{13}\] where \(\alpha _{j}\) is the weight calculated by hierarchical analysis method and \(\beta _{j}\) is the weight calculated by information entropy method.

4) Improvement of the TOPSIS integrated evaluation model

Based on the deficiencies in the traditional TOPSIS model, this paper mainly improves the model in two points:

1. Improved the way of model assignment, using subjective and objective combination assignment to reduce the subjectivity of traditional assignment.

2. Improve the linkage closeness calculation method, using two-dimensional spatial distance, to eliminate the problem of Euclidean linear equidistance. In summary, the steps of improving the TOPSIS model for the multivariate linkage assessment of college sports are as follows:

   • a) Construct the linkage evaluation index \(j\) and the \(i\)nd linkage decision matrix \(p_{ij}\), and carry out normalization and dimensionless processing on the decision matrix \(p_{ij}\) to eliminate the different scales between the indicators.

   • b) Weight the standardized linkage evaluation decision matrix \(p_{ij}\) and obtain the weighted normalized matrix \(z_{ij(m\times n)}\) as: \[\label{GrindEQ__14_} z_{ij(m\times n)} =w_{j} \cdot p_{ij} , \tag{14}\] where \(w_{j}\) denotes the weight value of linkage evaluation indicator \(j\) after using the optimized combination assignment.

   • c) Determine the positive and negative ideal solutions in all linkage indicators. For very large indicators take the best indicator value among all linkages, and for very small indicators take the opposite value. \(Z^{+}\) indicates a positive ideal solution and \(Z^{-}\) indicates a negative ideal solution.

   • Calculate the distance between different linkages \(i\) and the positive and negative ideal solutions according to \(Z^{+}\) and \(Z^{-}\) in the linkage evaluation system, using the formula: \[\label{GrindEQ__15_} D_{i}^{+} =\sqrt{\sum _{j=i}^{n}\left(Z_{ij} -Z_{j}^{+} \right)^{2} } ,\forall i=1,2,\cdots m \tag{15}\] \[\label{GrindEQ__16_} D_{i}^{-} =\sqrt{\sum _{j=i}^{n}\left(Z_{ij} -Z_{j}^{-} \right)^{2} } ,\forall i=1,2,\cdots m, \tag{16}\] where \(D_{i}^{+} ,D_{i}^{-}\) denotes the distance between linkage \(i\) and the ideal solution, respectively.

   • e) Construct the reference point \(X\left(\min \left(D_{i}^{+} \right),\max\right.\) \(\left. \left(D_{i}^{-} \right)\right)\) in the two-dimensional space, and then calculate the closeness \(M_{i} {'}\) of the \(i\)th linkage by adopting the distances of the positive and negative ideal solutions from linkage \(i\) to the spatial reference point \(X\) based on the positive and negative ideal solution distances of the different linkages \(i\) mentioned above, to improve the traditional model Euclidean distances linearly equidistant problem. The traditional closeness \(M_{i}\) and improved closeness \(M_{i} {'}\) are calculated as:

\[\label{GrindEQ__17_} M_{i} =\frac{D_{i}^{-} }{D_{i}^{+} +D_{i}^{-} } ,\forall i=1,2,\cdots ,m , \tag{17}\] \[\label{GrindEQ__18_} M_{i} {'} =\sqrt{\left[\left(D_{i}^{+} -\min \left(D_{i}^{+} \right)\right]^{2} +\left[\left(D_{i}^{-} -\max \left(D_{i}^{-} \right)\right]^{2} \right. \right. } , \tag{18}\] \(\forall i=1,2,\cdots ,m.\)

Among them, the closer the closeness \(M_{i} \in [0,1],M_{i}\) of the traditional model is to 1, the higher the ranking results of the comprehensive evaluation of multiple linkages in college sports are, and the closeness of the improved model is just the opposite, the closer \(M_{i} {'}\) is to 0, the better the evaluation results are, and the better the effectiveness of multiple linkages are, and finally the comprehensive evaluation results of all linkages are obtained.

C. Determination of weights and comprehensive evaluation

In this section, we utilize the improved TOPSIS model’s combined weight method to compute the index weights based on the relevant sports multiple linkage data from X university spanning the years 2019 to 2023, alongside the university sports multiple linkage evaluation index system established previously. These weights are presented in Table 3. The ranking of the action indicator weights for multiple linkages, from highest to lowest, is as follows: linkage subject (0.3199) \(\mathrm{>}\) linkage content (0.2887) \(\mathrm{>}\) linkage implementation (0.2438) \(\mathrm{>}\) linkage evaluation (0.1476). Notably, the linkage subject emerges as a pivotal factor in the context of multiple linkage actions, determined by the positioning and roles of administrative bodies, social organizations, and individuals within the multiple linkage framework, as well as the intricate network of relationships they forge. Social network and social support theories underscore the profound impact of interpersonal and inter-organizational relationships on students’ individual health behaviors and health consciousness. Furthermore, the healthy development of individual students is intertwined with the family, school, community, and the broader network growth environment they inhabit. Consequently, in the realm of multiple linkages, the collaboration of families, schools, communities, and other social organizations forms a complex social network support system, which holds immense potential to foster the advancement of sports in colleges and universities.

In the secondary indicator system, the weight coefficients of four primary indicators rank prominently, specifically, administration (0.5653), comprehensive evaluation (0.3399), action implementation (0.3279), and school-led initiatives (0.2851). Firstly, the successful implementation of the multiple linkage actions pertaining to college sports necessitates not only the extensive involvement of social organizations but also the effective management and regulation by administrative bodies.Only through the synergistic governance involving both the government and the market can the optimal allocation of public resources for social sports be achieved. Secondly, schools ought to assume the pivotal role of the primary arena and principal position in executing multiple linkage actions within college sports. School sports constitutes a fundamental pillar in fulfilling the overarching goal of fostering morality and enhancing students’ comprehensive qualities, and it holds a distinct function in nurturing both the wisdom and the spirit of sports. Subsequently, the focus of multiple linkage ought to lie in the practical execution and implementation of the initiatives, and it is only under the precondition of ensuring the effective enactment of these multiple linkage actions that we can fully harness the educational potential and value inherent in them. Lastly, in evaluating multiple linkage, emphasis should be placed on conducting a comprehensive assessment of the target entities, while monitoring the overall impact of the implementation process. Comprehensive evaluation of students has emerged as a pivotal juncture and starting point for reforming the assessment framework of basic education, which necessitates a shift in the focus of multiple linkage evaluations away from individual student assessments and towards evaluations that encompass diverse groups, such as families and teachers, while also attending to the sustainability and continuity of the societal benefits generated by these linkages.

In order to verify the effectiveness of the improved TOPSIS model proposed in this paper, 10 colleges and universities (A1-A10) with complete statistical data and meeting the requirements were selected as the evaluation object to conduct the comprehensive analysis of multiple linkage in college sports and compared with the traditional TOPSIS model, and the results of the multiple linkage evaluation of college sports are shown in Table 4. It can be seen that the multilinkage evaluation value derived from the improved TOPSIS model is generally lower, and the multilinkage ranking situation has changed accordingly. The top three colleges and universities in terms of multivariate linkage rankings derived from the traditional TOPSIS model were A9, A1, and A10, and the top three colleges and universities in terms of multivariate linkage rankings derived from the improved TOPSIS model were A1, A9, and A10.Comparisons revealed that the multivariate linkage rankings of the improved TOPSIS model were more objective and reasonable. As a result, this section calculates the standard deviation of the evaluation index values of each university to prove the rationality of the improved TOPSIS model. The size of the standard deviation represents the coordination size of the indexes to a certain extent; the smaller the standard deviation is, the smaller the difference between the index values is, and the better the coordination of the indexes is, then the ranking of the multivariate linkage will be relatively higher. For example, the ranking of A1 colleges and universities is changed from No. 2 to No. 1, and the ranking of A9 colleges and universities is changed from No. 1 to No. 2, and the calculation shows that the standard deviation of each evaluation index of A1 and A9 colleges and universities is 0.21 and 0.27, respectively, which shows that the discrepancy of the indexes of A1 colleges and universities is smaller, and the coordination of the indexes is better, therefore, the rank of A1 colleges and universities rises, and the rank of A9 colleges and universities falls. As another example, the ranking of A7 colleges and universities rises from 9th to 6th, while the ranking of A5 colleges and universities, which is ranked 6th, changes to 10th, and the calculation shows that the standard deviation of A7 and A5 colleges and universities is 0.20 and 0.37 respectively, and the standard deviation of the indicators of A7 colleges and universities is smaller, and the coordination of the indicators is better, so the ranking of A7 colleges and universities rises while that of A5 colleges and universities declines. Accordingly, it can be concluded that the improved TOPSIS model has fully considered the coordination characteristics between the indicators, which makes the evaluation results of multiple linkage more reasonable and effective, and more representative of the development level of multiple linkage of sports in colleges and universities.

B. Regression analysis

Regression analysis refers to determining the correlation between the dependent variable and some independent variables through mathematical processing methods, and then establishing regression equations. According to the correlation analysis above, it is known that there is a significant positive influence between the linkage behavior factors of the multiple linkage model and the linkage effect, but the correlation analysis does not reflect whether there is a causal relationship between the variables and the level of multiple linkage in college sports. Therefore, with the help of SPSS statistical software, this study presents the interaction relationship existing between each different influencing factor and the level of multiple linkage of college sports clearly through regression analysis. In this study, physical activity A, being a good parent B, communication C, exercising at home D, sports decision making E, community cooperation F, sports organization cooperation G, behavioral attitudes H, behavioral perceptions I, behavioral habits J, behavioral intentions K, emotional experience L, and behavioral sense of control M were used as independent variables, and regression analysis was conducted by using the level of multiple linkages in college sports as the dependent variable. Table 6 shows the results of regression analysis between the level of multiple linkages of college sports and the influencing factors. 13 variables have a significant P value of 0, and the regression coefficients are positive, indicating that these 13 variables are significantly positively correlated with the level of multiple linkages of college sports, which demonstrates that there is an interrelationship between the systems of “family, school, community, and sports organizations” and other systems. Correlation. The results of ANOVA with statistic F=518.7962 and significance P=0.000 indicate that there is a linear regression relationship between more than one influencing factor and the level of multiple linkages in sports. According to the adjusted R² value of the model, it can be learned that the independent variable set can explain 61.6% of the dependent variable, which can indicate that the linkage behavior factor and the linkage effect factor have an impact on the level of multiple linkages in college sports with 61.6% of the predictive effect. In summary, the level of multiple linkage of college sports = 4.164 + 0.424* physical activity + 0.308* being a good parent + 0.386* communication + 0.308* exercising at home + 0.362* sports decision-making + 0.371* community cooperation + 0.286* sports organization cooperation + 0.176* behavioral attitudes + 0.295* behavioral cognition + 0.342* Behavioral Habits + 0.342* Behavioral Intentions + 0.337* Emotional Experiences + 0.196* Sense of Behavioral Control.

In the implementation of multifaceted sports activities in colleges and universities, collaboration among families, schools, communities, sports organizations, and other relevant systems is imperative to ensure comprehensive and content-rich sports programs. Furthermore, during the participation process, it is crucial to holistically consider the psychological factors of students, aiming to enhance their behavioral cognition, facilitate behavioral control in physical exercise, enrich their emotional experiences, foster a heightened sense of well-being, and amplify successful experiences derived from physical activities. These endeavors will ultimately facilitate the cultivation of positive attitudes, habits, and intentions towards physical exercise among students.

5. Conclusion

The emergence of phenomena, such as the heightened awareness of health and the substantial surge in demand for sports and cultural activities, signifies that the development of college sports has embarked on a novel opportunity and phase. Drawing upon the theory of overlapping effects, this paper establishes a multifaceted linkage model for college sports and assesses its efficacy. In the devised evaluation index system, the order of importance, as assigned weights, for the operational indices of multifaceted linkage is as follows: linkage subject (0.3199), linkage content (0.2887), linkage implementation (0.2438), and linkage evaluation (0.1476). This underscores the pivotal role of the linkage subject in the operation of multifaceted linkage within college sports. Subsequent to a comprehensive evaluation, the validity of the refined TOPSIS model proposed herein is affirmed, enhancing the rationality and efficacy of multifaceted linkage evaluation outcomes. Furthermore, the investigation reveals that 13 independent variables, encompassing physical activity, parental nurturing, communication, home-based exercise, sports decision-making, community collaboration, partnership with sports organizations, behavioral attitudes, behavioral cognition, behavioral habits, behavioral intentions, emotional experiences, and behavioral control, collectively account for 61.6% of the variance in the dependent variable—the degree of multifaceted linkages in college sports. This paper delves into the implementation strategies and influencing factors of multifaceted linkages in college sports, thereby offering a valuable reference for fostering the holistic development of college sports.