Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education

Structural equation modeling (SEM) is also often referred to as structural equation modeling. SEM is a statistical method to analyze the relationship between variables based on the covariance matrix of the variables. It can process latent variables and their measurement indicators at the same time. Simply put, the structural equation model can be divided into two parts: measurement model and structural model. The former describes the relationship between latent variables and measurement indicators, and the latter describes the relationship between latent variables [3]. The following measurement equation usually expresses the relationship between measurement indicators and latent variables: (1) x=Λxξ+δ \begin{equation}x = \Lambda_x \xi + \delta\end{equation} (2) y=Λyη+ε \begin{equation}y = \Lambda_y \eta + \varepsilon\end{equation}

Among them x, y are the exogenous measurement index and the endogenous measurement index, respectively. ξ, η are exogenous latent variables and endogenous latent variables, respectively [4]. ∧_x is the factor loading matrix of the exogenous measurement index on the exogenous latent variable, which represents the relationship between the exogenous measurement index and the exogenous latent variable. ∧_y is the factor loading matrix of the endogenous measurement index on the endogenous latent variable, representing the relationship between the endogenous measurement index and the endogenous latent variable. δ, ɛ are the error terms of the exogenous measurement index x and the endogenous measurement index y, respectively. The following structural equation usually represents the relationship between latent variables: (3) η=Bη+Γξ+ξ \begin{equation}\eta = B\eta + \Gamma\xi + \xi\end{equation}

B represents the relationship between the endogenous latent variable η. Γ describes the effect of an exogenous latent variable ξ on the endogenous latent variable η. ξ is the residual term in the structural equation, representing the information that η cannot be explained in the equation.

High-order confirmatory factor analysis (HCFA) is also known as the full Y model. The core idea is to coordinate the primary factors through high-level factors, thereby simplifying the correlation between the primary factors [5]. An HCFA model only needs to be determined by the following two equations: (4) y=Λyη+ε \begin{equation}y = \Lambda_y \eta + \varepsilon\end{equation} (5) η=Bη+Γξ+ξ \begin{equation}\eta = B\eta + \Gamma\xi + \xi\end{equation}

When there is a consistent high correlation between the first-order factors, high-order confirmatory factor analysis can often achieve better results.

The item combination method is a process of reintegrating two or more question items in the same scale and using the composite score (total score or average) as the score of the new index to analyze. For example, a scale originally had 9 question items. We can use each 3 question item to calculate its composite score to obtain 3 new indicators and then use these 3 new indicators for subsequent analysis.

Some scholars have shown that the project portfolio method can correct the non-normality of the data to a certain extent. When we reduce the number of parameters to be estimated, we can effectively avoid problems such as multicollinearity between measurement indicators, excessive error correlation, and cross-loading of measurement indicators from adversely affecting the model. Furthermore, this method can improve the fitting effect of higher-order factor models. Some scholars have proposed a two-stage simplification method that combines project portfolio and high-order factor analysis [6]. This method can significantly simplify the model and obtain the perfect goodness of fit. It is especially suitable for the situation with many model parameters.

This article adopts the project portfolio method to form higher-order factors of learning emotion. This article has six potential first-order factors (emotion, cognitive ability, value, difficulty, interest, effort), as shown in Table 1. We take the arithmetic average of each factor's measured variables as the value of the element. The original six first-order factors become “explicit variables” (AFFPC, COGPC, VALPC, DIFPC, INTPC, and EFFPC) and jointly measure the “attitude “High-order factor (ATTITUDE). The “attitude” high-level factor will further affect the “expected mastery” factor (EXPMSTRY), and it will be affected by the “sports ability” factor (MATHCOMP). At the same time, “sports ability” will also directly affect the “expected mastery” (as shown in Figure 1).

Fig. 1

In theory, the more active the learning mood of university physical education courses, the easier it is to expect that one has a good grasp of statistical knowledge. The stronger the physical ability, the more confident they are in their physical fitness [7]. This also often leads to the better the expected mastery of the physical education curriculum. In addition, the stronger the physical ability, the easier it is to get a sense of accomplishment from similar courses. The easier it is for students to take a positive attitude towards university physical education courses. Therefore, this article proposes the following hypotheses:

H1: “Attitude” positively affects “expected mastery.”

We perform confirmatory factor analysis on the “attitude” factor measurement model, and the initial fitting results are given in the “initial results” row of Table 3. The standardized chi-square and each appropriate index have not reached the ideal outcome. Therefore, we need to modify the initial model: add error-related terms according to the adjustment index. The “adjusted result” row of Table 3 shows the fit of the adjusted model. The standardized chi-square value has been dramatically improved, and each fitting index meets the requirements of empirical criteria. Sports ability and expected mastery have to sound appropriate effects.

Table 4, the “Standardized Load Factor” column, gives the estimated results of the adjusted measurement model. In terms of the positive and negative coefficients, except for the difficulty factor, the other five first-order factors have positive loading coefficients on the “attitude” high-order factors. This result indicates that the more positive learning emotions mean, the more positive emotions are, the more willing to work hard to study university physical education courses [10]. The negative loading of the difficulty factor means a more positive attitude towards university physical education courses. The more students who often accompany this result agree with the difficulty of college physical education courses.

From the significance of the coefficients, all loads are significant. Except for the first-order factor of difficulty, the absolute values of the load coefficients of the other six first-order factors on the second-order factors of attitude are more significant than 0.50. Among them, philosophy has the most muscular interpretation of cognitive ability, while understanding value, emotion, interest, and effort weakened in turn.

Table 5 shows the reliability and validity test of the questionnaire data. Attitude and sports ability belong to high reliability. The Cronbach alpha coefficient value of the expected mastery level is slightly lower but still acceptable. We then test the convergent validity and discriminative validity of the three factors. First, according to the study in Table 4, it is found that the load coefficients of each measurement index and its corresponding element are all significant, indicating that the questionnaire data has good convergence validity [11]. Secondly, from the average variation extraction amount (AVE) in Table 5, the sports ability factor is more significant than 0.5, and the mastery degree factor is expected to be close to 0.5. Thus, although the attitude factor is slightly lower, the overall convergence validity of the model is better. Finally, the correlation coefficient between attitude and expected mastery is somewhat higher than the AVE \sqrt{AVE} value of the two. This result shows that the discriminative validity is slightly worse, but it is still within the acceptable range. In summary, we can consider the estimated impacts of the model to be of reference significance.