Literature review, questionnaire survey, interview survey and statistical analysis were used in this study. This study used stratified sampling and simple random sampling to select samples. Questionnaires on the demand of public sports services for rural older adults were distributed, targeting rural older adults of Fuding City (Fujian Province, eastern China), Shaoyang County (Hunan Province, central China) and Guanghan City (Sichuan Province, western China). When distributing questionnaires, investigators conducted door-to-door surveys and assisted respondents in filling out the forms through interviews. A total of 1,800 questionnaires were distributed in our study. Excluding 133 invalid questionnaires, 1,667 valid questionnaires remained, with an effective recovery rate of 92.6% (invalid responses are determined based on the following criteria: ① missing key questions (matching criteria/stratification variables), ② duplicate records for the same respondent or address, ③ age < 50 or failure to meet screening criteria, ④ logical conflicts (e.g., contradictory responses to mutually exclusive questions), ⑤ abnormal responses (extremely short completion time/straight-line answering patterns). The sample showed that in terms of gender, there were 906 males (54.35%), which was slightly higher than the number of females. In terms of age, the sample consisted mainly of young older adults (under 70 years old), which accounted for 70.12% of the total sample. In terms of education, more than 70% of older adults in the sample had an education level of “primary school and below.” In terms of income, the largest group is those with “average” financial status, numbering 1,047 people (62.81%). The group with “more difficult” economic situations is relatively small, and the group with “affluent and above” families is the smallest group. The group belonging to “well-off and above” families is the smallest. In terms of marital status, the proportion of older persons with a spouse (74.09%) is much higher than that of older persons without a spouse. In terms of regional structure, the sample was relatively evenly distributed, with 437 (26.24%) in the eastern region, 507 (30.44%) in the central region and 723 (43.32%) in the western region. The sample of older adults and village cadre was merged with equal weighting at a 50/50 ratio to balance the two groups and prevent bias from sample imbalance. Robustness checks were performed on population weights (stratified by “East/Central/West × village economic tier,” using iterative quota/raking, extreme weights truncated at 1%-99%). The “achieved matching” rate and 95% confidence interval (CI) were recalculated under these weights. Results were nearly identical to the equal-weight approach (differences within ± 0.8 percentage points). It is important to note that the demographic characteristics of the demand-side sample—such as the slight male majority (54.35%), the high proportion with primary school education or below (>70%), and the predominance of younger older adults (under 70 years old, 70.12%)—are highly reflective of the current socio-demographic reality of rural China. This is largely due to long-term patterns of out-migration where more young and middle-aged women leave rural areas for urban employment, and the historical context of limited educational access for this older cohort. Therefore, while the sample may appear skewed, it accurately represents the target population for public sports services in these regions. Overall, the demographic structure of the sample is relatively consistent with the actual living status of rural older adults. The sample sizes in the eastern, central, and western regions are appropriate and even, making it highly representative.

At the same time, in order to understand the matching of supply and demand of public sports services for rural older adults in China, a questionnaire on the current situation of the supply of public sports services for rural older adults was designed, which included the matching of supply and demand of public sports services for rural older adults in China. Fuding City of Fujian in the eastern China, Shaoyang County of Hunan in the central China, and Guanghan City of Sichuan in the western China were selected, according to the degree of economic development, one town (township) each with high, medium, and low levels was selected from each city (county), and the questionnaire on the current situation of the supply of public sports services for rural older adults was distributed to all villages (communities) in the nine townships. In addition, some villages (communities) were selected from other townships in the three cities (counties) for investigation. In order to fully understand the national situation, we took the opportunity of the national “village chiefs” forum held in Baoshan village (Pengzhou, Sichuan province) to distribute the questionnaires to the cadres in charge of culture and sports work in administrative villages. A total of 587 questionnaires were collected, 25 invalid questionnaires were excluded, leaving 563 valid questionnaires, with an effective recovery rate of 95.9%. In the sample, there were 91 samples in ethnic minorities and 472 samples in non-ethnic minorities. In the ethnic minority regions, the sample sizes in the east, central, and west accounted for 25.3%, 4.4%, and 70.3%, respectively. In the non-ethnic minority regions, the sample sizes in the east, central, and west accounted for 6.4%, 8.5%, and 82.6%, respectively. As a whole, the proportions in the east, central, and west were 9.4%, 8.0%, and 82.6%, respectively.

The questionnaire items in this study all employed a 5-point Likert-type scale. In the confirmatory factor analysis (CFA), the WLSMV estimator was primarily used. This method is based on polychoric correlation matrices and employs mean- and variance-adjusted Chi-square statistics. For comparison, robust maximum-likelihood estimation (MLR, Satorra-Bentler scaled) was also conducted. Regarding the evaluation of model fit, the reported indices include χ²(df), Comparative Fit Index (CFI), Tucker-Lewis Index (TLI), Root Mean Square Error of Approximation (RMSEA) (with 90% CI), and Standardized Root Mean Square Residual (SRMR). For all questionnaire items, standardized factor loadings with their corresponding 95% CIs are presented. Additionally, the composite reliability (CR), average variance extracted (AVE), and McDonald's ω for the latent constructs were calculated to comprehensively reflect the internal consistency and convergent validity of the measurement instrument.

Contingency table analysis (hypotheses, effect sizes, and multiplicity). For comparisons between groups involving categorical variables, Pearson's χ² test was used, consistently reporting the degrees of freedom (df), p-value, and the effect size Cramér's V with its 95% CI (calculated using the non-central Chi-square method). For each contingency table, the minimum expected frequency was first checked: if the proportion of cells with expected count < 5 was ≥20% or if any zero counts were present, Fisher's exact test (using the Monte Carlo method with 10,000 replications) was employed instead, or adjacent categories were merged. The Benjamini-Hochberg (BH) procedure was applied to control the false discovery rate (FDR) at 5%. For significant results, the q-value is also reported. Unless otherwise specified, conclusions with p < 0.001 remained significant after FDR control, whereas results with p close to 0.05 were mostly non-significant after FDR adjustment.

Factor loadings reflect the relationship between an item and its underlying latent variable, not the degree of supply-demand “match.” Therefore, we avoid directly using the magnitude of factor loadings to rank the “degree of matching.” If comparison is necessary, the proportion of respondents indicating a “match” for each item in the sample, along with its 95% CI, serves as a more appropriate basis, as it aligns better with the meaning of the observed data. Furthermore, to ensure the comparability of results across different groups, we tested for measurement invariance across (eastern, central, and western) regions and village economic levels (underdeveloped, average, and developed). The testing sequence followed these steps sequentially: configural invariance → metric invariance → scalar invariance. The criteria for judging invariance were ΔCFI ≤ 0.010 and ΔRMSEA ≤ 0.015. If full scalar invariance was not achieved, partial scalar invariance was established by releasing some intercept parameter constraints before proceeding with cross-group comparisons of latent means.

In this study, the Chi-square test was used to analyze the differences in the degree of match between different ethnic groups, regions, and villages with different financial status in terms of the supply and demand of public sports services for rural older adults (including investment in sports, construction of venues and facilities, organizational construction, training of backbones, organization of sports activities, fitness guidance, provision of health knowledge, publicity and mobilization, system construction and physical fitness monitoring service). Whether there is a statistically significant difference between the variables in each group was determined by calculating the degree of deviation between the observed frequencies and the theoretical frequencies. The formula for calculating the Chi-square statistic is as follows:

where, O_ij denotes the observed frequency in row i, column j, and E_ij is the theoretical frequency of the cell, which is calculated as follows:

where, R_i is the marginal total of row i, C_j is the marginal total of column j, and N is the total number of samples. If p < 0.05, H₀ will be rejected. If p > 0.05, H₀ will be accepted.

Canonical correlation analysis is a statistical analysis method to study the interrelationship between two groups of variables. This study analyzes the correlation between the supply and demand of public sports services for rural older adults and their various related factors with the help of this method. The method adopts the principle of principal component analysis to extract components, and each of the two groups of variables is linearly combined into canonical variables. The original correlation between the two groups of variables is transformed into the study of canonical correlation between a few canonical variables proposed from each group, thus reducing the number of variables in the study. In practical application, several pairs of canonical variables are retained based on the significance test of the canonical correlation coefficient and the amount of information contained in the canonical variables. The mathematical principle of the CCA can be expressed as follows:

Let the random vector X = (x1, x2, …, xp), Y = (y1, y2, …, yq), the variance matrix of X, Y is as follows:

where ∑11 is the covariance matrix of the first group of variables, ∑12 and ∑21 are the covariance matrices of the first and second group of variables, and ∑22 is the covariance matrix of the second group of variables. In order to study the canonical correlation between two groups of variables, X and Y, a linear combination between them is made as follows:

Given x, y, and ∑ that is to say, a, b, to maximize the correlation coefficients between U and V:

This study addressed measurement quality assessment and the examination of variable relationships separately. First, measurement model analysis was conducted using AMOS 26.0. The model was estimated with the Weighted Least Squares Mean and Variance adjusted (WLSMV)/MLR estimators. Model fit was reported using the following fit indices: χ²(df), CFI, TLI, RMSEA (with 90% CI), and SRMR. For all questionnaire items, standardized factor loadings with their corresponding 95% CIs are provided. Additionally, CR, AVE, and McDonald's ω were calculated for the latent constructs. To enable cross-group comparisons, measurement invariance across regions (east/central/west) and village economic level (underdeveloped/average/developed) was tested sequentially for configural, metric, and scalar invariance. The criteria for judging invariance were ΔCFI ≤ 0.010 and ΔRMSEA ≤ 0.015. Partial measurement invariance was established if full invariance was not met. Subsequently, CCA was performed using SPSS/R. Supply-side variables were designated as set X, and demand/matching variables as set Y. The analysis was based on the (polychoric) correlation matrix, proceeding only after assumptions of linearity, multicollinearity, and the absence of significant outliers were checked. For each canonical function, the following were reported: the canonical correlation (Rc), Wilks' Lambda (Λ), degrees of freedom (df), Rao's F, and p-value. Additionally, standardized canonical coefficients, structure coefficients (canonical loadings), cross-loadings, and redundancy indices (for both X → Y and Y → X) were examined. To prevent misinterpretation, the magnitude of factor loadings or canonical loadings was not used to rank the “degree of matching.” Where comparisons were necessary, only the observed proportion of “match achieved” for each item, along with its 95% CI, was provided. This analytical approach elucidated the primary correlational pathways and structural relationships between the supply and demand of public sports services for rural older adults and their various influencing factors. It thereby provides statistical support and a theoretical foundation for subsequent analysis of the allocation efficiency and intervention pathways for sports service resources for this demographic.

As shown in Table 1:

As shown in Table 2:

As shown in Table 3:

As shown in Table 4:

As shown in Table 5:

As shown in Table 6:

As shown in Table 7:

As shown in Table 8:

As shown in Table 9:

As shown in Table 10:

As shown in Figure 2 and Table 11:

In this canonical correlation, among the variables in group X, four variables, including “educational level,” “residential status,” “family economic status,” and “sedentary behavior,” have absolute values of coefficients of the first canonical variable (ε₁) that are all ≥0.30, which are 0.554, 0.833, 0.763, and −0.704, respectively. In the variables in group Y, five variables, including “funding input,” “facilities,” “backbone training,” “activity organization,” and “activity guidance,” have absolute values of coefficients in the first canonical variable (η₁) that are all ≥0.30, which are 0.603, 0.794, 0.605, 0.801, and 0.741, respectively. This indicates that the four variables (“educational level,” “living status,” “family economy,” and “sedentary behavior”) of older adults can establish a significant correlation with the five variables (“funding input,” “facilities,” “backbone training,” “activity organization,” and “activity guidance”) reflecting the equilibrium degree of supply-demand matching in public sports services for rural older adults through the first canonical correlation (ε₁ and η₁).

In this canonical correlation, among the variables in group X, three variables, including “physical activity,” “chronic disease status,” and “sleep behavior,” have absolute values of the coefficients of the second canonical variable (ε₂) that are ≥0.30, which are 0.552, 0.817, and −0.613, respectively. In the variables in group Y, four variables, including “facilities,” “knowledge provision,” “publicity and mobilization,” and “monitoring service,” have absolute values of coefficients in the second canonical variable (η₂) that are all ≥0.30, which are −0.754, −0.804, −0.569, and −0.508, respectively. This indicates that the three variables (“physical activity,” “chronic disease status,” and “sleep behavior”) of older adults can establish a significant correlation with the four variables (“facilities,” “knowledge provision,” “publicity and mobilization,” and “monitoring service”) reflecting the equilibrium degree of supply-demand matching in public sports services for rural older adults through the second canonical correlation (ε₂ and η₂).

From the absolute values and positive/negative signs of the variable coefficients contributing to the second canonical correlation (ε₂ and η₂), the equilibrium degree of matching for public sports services in terms of “facilities,” “knowledge provision,” “publicity and mobilization,” and “monitoring service” shows an inverse relationship with older adults' “physical activities” and “chronic disease status,” while showing a positive relationship with their “sleep behaviors.” This finding is quite strange. This counterintuitive finding does not imply that poor services improve health. Instead, it likely reflects the complex reality of rural life: in villages with scarce public sports facilities and services, older adults' primary form of “physical activity” often consists of strenuous, non-leisure agricultural or domestic labor. This type of physically demanding lifestyle is associated with a high incidence of chronic diseases, poor sleep quality, and a lack of formal sports services, thus explaining the negative statistical correlation. This highlights a critical misallocation: service supply is lowest where health needs are greatest.

In addition, the percentage of overlapping variance showed that the overlapping variances under the two canonical variables ε₁ and ε₂ in group X were 12.27 and 4.24% (sum 16.51%), and the overlapping variances under the two canonical variables η₁ and η₂ in group Y were 16.82 and 4.81% (sum 21.63%), which indicates that the canonical variables ε₁ and ε₂ of group X can jointly explain 21.63% of the variance of the 10 variables in group Y, while η₁ and η₂ in group Y can only jointly account for 16.51% of the variance of the seven variables in group X. Evidently, from these two groups of correlation relationships, it is not difficult to observe that the variables in group X essentially pertain to the issue of the demands of older adults. The overlapping variance indicates that the current match of supply and demand of public sports services for rural older adults cannot yet meet the needs of older adults in terms of health status and lifestyle, and it also highlights the inadequacy in the supply-demand matching degree of public sports services for rural older adults.

As Figure 3 shows, the combined Table 12 can be found:

In this canonical correlation, among the variables in group X, two variables, including “physical fitness enhancement and disease prevention” and “mental health and achievement interaction,” have absolute values of coefficients of the first canonical variable (ε₁) that are all ≥0.30, which are 0.924 and 0.595, respectively. In the variables in group Y, five variables, including “funding input,” “facilities,” “organizational construction,” “activity organization,” and “activity guidance,” have absolute values of coefficients in the first canonical variable (η₁) that are all ≥0.30, which are 0.671, 0.745, 0.582, 0.739, and 0.648, respectively. It shows that to improve older adults' awareness that physical exercise is beneficial to enhance physical fitness and prevent diseases as well as mental health and achievement interaction, a significant positive correlation can be generated with the equilibrium degree of supply-demand matching in public sports services such as “funding input,” “facilities,” “organizational construction,” “activity organization,” and “activity guidance” through the first canonical correlation (ε₁ and η₁).

In this canonical correlation, among the variables in group X, two variables, including “physical fitness enhancement and disease prevention” and “mental health and achievement interaction,” have absolute values of the coefficients of the second canonical variable (ε₂) that are ≥0.30, which are 0.501 and 0.821, respectively. In the variables in group Y, six variables, including “organizational construction,” “backbone training,” “knowledge provision,” “publicity and mobilization,” and “monitoring service,” have absolute values of coefficients in the second canonical variable (η₂) that are all ≥0.30, which are 0.914, 0.851, 0.610, 0.534, and 0.685, respectively. It shows that to improve older adults' awareness that physical exercise is beneficial to enhance physical fitness and prevent diseases as well as mental health and achievement interaction, a significant positive correlation can be generated with the equilibrium degree of supply-demand matching in public sports services such as “organizational construction,” “backbone training,” “knowledge provision,” “publicity and mobilization,” and “monitoring service” through the second canonical correlation (ε₂ and η₂).

From the second canonical correlation (ε₂ and η₂, ρ = 0.442, p < 0.001), the variant that makes an important contribution to ε₂ is “mental health and achievement interaction” (0.821), and the variant that makes an important contribution to η₂ are “backbone training” (0.914) and “knowledge provision” (0.851). The standardized coefficients of the variables contributing to both ε₂ and η₂ are also in the same direction. This indicates that the higher the supply-demand matching degree in many aspects, such as “training of key members” and “knowledge provision” in public sports services for older adults, the more beneficial it is to improve older adults' cognitive ability regarding the role of physical exercise in “mental health and achievement interaction.”

In addition, the percentage of overlapping variance shows that the canonical variables ε₁ and ε₂ in group X can explain 31.07% (23.83% + 7.24%) of the 10 variables in group Y, while the canonical variables η₁ and η₂ in group Y can explain 28.65% (22.15% + 6.50%) of the two variables in group X. Obviously, the difference between the two percentages of overlapping variance is not significant, which indicates that the current supply-demand matching of rural public sports services has basically met the needs of older adults in terms of their cognition of physical exercise.

Figure 4, given in Table 13, can be found:

In this canonical correlation, among the variables in group X, two variables, including “organization and service” and “safety and convenience,” have absolute values of coefficients of the first canonical variable (ε₁) that are all ≥0.30, which are 0.515 and 0.968, respectively. In the variables in group Y, eight variables including “funding input,” “facilities,” “organizational construction,” “backbone training,” “knowledge provision,” “publicity and mobilization,” “system construction,” and “monitoring service” have absolute values of coefficients in the first canonical variable (η₁) that are all ≥0.30, which are −0.755, −0.798, 0.614, 0.514, 0.806, 0.504, −0.705, and −0.573, respectively. It shows that the main barriers for older adults to participate in physical exercise is “safety and convenience,” can have a significant negative correlation with “funding input,” “facilities,” “system construction,” and “monitoring service” in public sports services through the first canonical correlation (ε₁ and η₁), and a significant positive correlation with “organizational construction,” “backbone training,” “knowledge provision,” and “publicity and mobilization.”

In this canonical correlation, among the variables in group X, the absolute value of the coefficient of “social support” in ε₂ is ≥0.30 (R = 0.976). In the variables in group Y, five variables, including “activity organization,” “activity guidance,” “knowledge provision,” “publicity and mobilization,” and “system construction,” have absolute values of coefficients in the second canonical variable (η₂) that are all ≥0.30, which are 0.526, 0.503, 0.744, 0.694, and 0.601, respectively. It shows that older adults need more “social support” for physical exercise, and such “support” can be addressed by strengthening sports services in aspects, such as “activity organization,” “activity guidance,” “knowledge provision,” “publicity and mobilization,” and “system construction.”

In this canonical correlation, among the variables in group X, the absolute value of the coefficient of “organization and service” in ε₃ is ≥0.30 (R = 0.875). In the variables in group Y, four variables, including “organizational construction,” “backbone training,” “activity organization,” and “activity guidance,” have absolute values of coefficients in the third canonical variable (η₃) that are all ≥0.30, which are 0.801, 0.664, 0.701, and 0.760, respectively. It shows that “organization and service,” as an obstacle to older adults' physical exercise, can be addressed by strengthening “activity organization,” “activity guidance,” “knowledge provision,” “publicity and mobilization,” and “system construction” in public sports services.

In the second canonical correlation (ε₂ and η₂), the variable that makes an important contribution to ε₂ is “social support,” which mainly refers to the need for support from family members and friends for older adults' participation in physical exercise. The variables that make important contributions to η₂ are “knowledge provision” (0.744^**) and “publicity and mobilization” (0.694^**). The standardized coefficients of the variables contributing to both ε₂ and η₂ are in the same direction. This indicates that overcoming the obstacle of “social support” in older adults' physical exercise can be achieved through measures such as strengthening “knowledge provision,” enhancing the level of “publicity and mobilization,” improving “activity guidance,” and perfecting “system construction” in public sports services.

In the third canonical correlation (ε₃ and η₃), the variable that makes an important contribution to ε₃ is “organization and service.” The obstacle of “organization and service” in older adults‘ physical exercise refers to the fact that their exercise needs guidance, planned and organized activities, especially the availability of relevant services on demand. The variables that make important contributions to η₃ are “organizational construction” (0.801^**), “activity guidance” (0.760^**), and “activity organization” (0.701^**), etc. The standardized coefficients of the variables contributing to both ε₃ and η₃ are in the same direction. This indicates that overcoming the obstacle of “organization and service” in older adults' physical exercise can be achieved by strengthening “organizational construction,” “activity guidance,” and “activity organization” in public sports services.

In addition, the percentage of overlapping variance shows that the canonical variables ε₁, ε₂, and ε₃ in group X can explain 20.2% (11.79% + 5.43% + 2.98%) of the variance of the 10 variables in group Y, while the canonical variables η₁, η₂, and η₃ in group Y can explain 17.95% (10.48% + 5.06% + 2.41%) of the three variables in group X. Obviously, the difference between the two percentages of overlapping variance is not significant, which indicates that the current obstacles encountered by rural older adults in physical exercise can basically be solved through the current supply-demand matching of public sports services.

Figure 5, given in Table 14, can be found:

In this canonical correlation, among the variables in group X, two variables, including “physical activity participation level” and “satisfaction with sports services,” have absolute values of coefficients of the first canonical variable (ε₁) that are all ≥0.30, which are −0.569 and 0.698, respectively. In the variables in group Y, six variables including “facilities,” “backbone training,” “activity guidance,” “knowledge provision,” “publicity and mobilization,” and “monitoring service” have absolute values of coefficients in the first canonical variable (η₁) that are all ≥0.30, which are 0.656, −0.556, −0.903, −0.558, 0.622, and 0.548, respectively. This shows that improving older adults' “satisfaction” and “participation level” in physical exercise is related to strengthening six aspects of public sports services, namely “facilities,” “backbone training,” “activity guidance,” “knowledge provision,” “publicity and mobilization,” and “monitoring service.”

In this canonical correlation, among the variables in group X, two variables, including “intention to participate in physical activity” and “satisfaction with sports services,” have absolute values of coefficients of the second canonical variable (ε₂) that are all ≥0.30, which are −0.527 and −0.870, respectively. In the variables in group Y, four variables, including “funding input,” “organizational construction,” “backbone training,” and “activity organization,” have absolute values of coefficients in the second canonical variable (η₂) that are all ≥0.30, which are −0.918, 0.704, −0.761, 0.748, and 0.727, respectively.

In the second canonical correlation (ε₂ and η₂), the variable that makes an important contribution to ε₂ is “intention to participate in physical exercise” (−0.870), followed by “physical activity participation level” (−0.527). They show an inverse correlation with the important contributing variables “organizational development” (0.704), “activity organization” (0.748), and “system construction” (0.727) in η₂. This indicates that in the current rural public sports services for older adults, the supply-demand matching of “organizational development,” “activity organization,” and “system construction” needs to be improved to ensure that they can truly enhance older adults' intention and intensity of participation in physical exercise.

In addition, the percentage of overlapping variance shows that the canonical variables ε₁ and ε₂ in group X can explain 15.47% (12.39% + 3.08%) of the 10 variables in group Y. The canonical variables η₁ and η₂ in group Y can explain 11.6% (8.72% + 2.88%) of the four variables in group X. Obviously, there is a significant difference between the two percentages of overlapping variance, which indicates that the current supply-demand matching of rural public sports services is still far from meeting the needs of older adults in terms of satisfaction, participation intention, and participation level in physical exercise.