5.1. Facility Accessibility
- (1)
Empirical analysis
To clarify the spillover effect of facility scale Si and facility distance Di on the accessibility index A, we conducted an empirical analysis based on the benchmark model. Firstly, under the distance attenuation coefficient α = 0.015, the regression results of model (1) were established. In this model, the explanatory variables only include facility scale Si, not facility distance Di. Through analysis, the benchmark regression results of model (1) are obtained, which are used to evaluate the impact of facility scale on the accessibility index.
Model (2) adds facility scale Si and facility distance Di to the benchmark model for regression analysis. By comparing the results of these two models, it can be found that the role of facility scale Si is very important in both models. Specifically, the regression coefficient of Si is 0.981, with a very high significance level (p < 0.01, denoted by ***). The corresponding t-values are 180.856 and 181.52, further confirming the robustness of the results and indicating a significant positive correlation between facility scale and the accessibility index. This means that as the facility scale increases, the contribution of the facility to the accessibility index will enhance, and this effect is statistically significant and reliable.
The high goodness of fit (R2 = 0.997) of the two models further confirms the significant role of facility scale in explaining the facility accessibility index. This indicates that the regression model has a very strong ability to explain, and the remaining unexplained variance is extremely small.
However, when facility distance
Di is included in model (2), it is found that its impact on the accessibility index is not significant. The regression coefficient of
Di is −1.66 × 10
4, and the t-value is −1.573, which does not pass the significance test (
p > 0.1). This indicates that facility distance has no significant impact on the facility accessibility index. Therefore, the research results show that within the framework of this study, the facility accessibility index
A is affected by facility scale
Si, while the role of facility distance is relatively minor. (
Table 1)
The influence of the facility distance studied in this paper is not significant. There are three reasons for this, as follows.
Influence of case studies: Different cities have distinct characteristics. Xi’an has a vast territory, and the influence of urban density on facility accessibility is limited. Therefore, the facility distance may not be the main influencing factor. In other cities, such as those with well-developed transportation or relatively low population density, the influence of facility distance on the accessibility index is more crucial.
Influence of comprehensive factors: There are many factors affecting accessibility, such as traffic conditions and population density. These factors lead to the statistically insignificant influence of the facility distance. For example, even though the facility is far away, if the public transportation is well developed, good accessibility can still be achieved.
Limitations of data collection: There may be limitations in the data collection, which will have a certain impact on the research results and may lead to a decrease in accuracy.
- (2)
Residual analysis
Since the goodness of fit in the above experiment was 0.997, which may lead to overfitting, a residual test was added.
We first conducted a regression analysis on the Di variable to obtain the residuals of the model, then drew a residual plot for analysis. The following are the results obtained.
From the results of the residual analysis, it can be seen that the mean value of the residuals should ideally be close to 0, and this item meets the requirement. There should be no obvious heteroscedasticity in the residuals, that is, the variance should be a constant. The graph is a straight line. The residuals should exhibit a normal distribution or an approximately normal distribution. The histogram basically conforms to the normal distribution, indicating that there is no situation of overfitting. (
Table 2,
Figure 1,
Figure 2,
Figure 3 and
Figure 4)
Overall, the research results show that in areas with higher facility accessibility, residents can more easily reach various facilities (such as schools, hospitals, parks, etc.), significantly improving the convenience of life, thereby promoting the improvement of urban spatial quality. This verifies the correctness of Hypothesis 1.
5.2. Facility Correlation
To examine the spillover effects of facility scale and distance, we conducted an empirical analysis based on the benchmark model. Under the assumption that Cij = 1 and = 0.15, by incorporating the interaction term Si∙Sj into Model (1), the benchmark regression results without considering the facility distance Dij were obtained. In contrast, Model (2) simultaneously includes the interaction term Si∙Sj and the facility distance Dij in the regression.
A comparative analysis of the regression results of the two models reveals that the interaction term Si∙Sj is statistically significant in both cases, with coefficients of 0.584 and 0.581, respectively, at a significance level of p < 0.01 (indicated by ***). These results suggest that the interaction of facility scales has a robust and positive impact on the dependent variable G, and the coefficients of the two models are very similar. This reinforces the view that the product of facility scales consistently influences the facility connection index in a significant manner.
Furthermore, the R2 values of both models are 0.998, indicating that the models have high explanatory power, explaining 99.8% of the variance. This highlights the models’ advantage in fitting the data and emphasizes the role of the facility scale interaction term in explaining the connection index.
In sharp contrast, the facility distance
Dij is only included in Model (2), with a regression coefficient of −1.66e04 and a t-value of −1.573, which did not pass the significance test (i.e.,
p > 0.10). This indicates that facility distance has no significant impact on the dependent variable
G. Therefore, the research results suggest that the facility connection index is mainly influenced by the product of scales, while the influence of facility distance is relatively weak (
Table 3).
In areas with strong facility correlation, various facilities form a good complementary and synergistic relationship, providing residents with diverse services. This diverse range of services not only meets the different needs of residents but also enhances the overall quality of the urban space. This supports Hypothesis 2.
5.3. Resident Satisfaction
To verify whether the questionnaire results truly reflected the situation and whether the questionnaire design was reasonable, reliability and validity tests, factor analysis, correlation tests, and Student’s t-test were conducted for verification.
- (1)
Reliability Test
Reliability refers to the consistency of results when the same method is used to measure the same variable multiple times. The higher the reliability, the more reasonable the questionnaire design and the higher the credibility. In this test, a total of 16 items were included in the reliability analysis. The average covariance between the items was 0.4947366. A high value indicates strong correlations among the items, which is the basis for high reliability. According to the general standard of Cronbach’s α coefficient: above 0.9: very high (excellent); 0.8–0.9: high; 0.7–0.8: acceptable; 0.6–0.7: low; below 0.6: unacceptable. In this case, the α coefficient was 0.8931, indicating that the scale has high consensus in measuring the same latent variable. (
Table 4)
- (2)
Validity Test
Validity refers to the degree to which a measurement method can effectively and accurately measure the variable to be measured. The closer the measurement result is to the required variable, the higher the validity. Conversely, the further it is, the lower the validity. Factor analysis was conducted using Stata software. Its determination coefficient is Det = 0.000, which is of the correlation matrix. A value close to 0 indicates that the correlation matrix may have multicollinearity or a strong linear relationship between variables. In factor analysis, the closer the determination coefficient of the correlation matrix is to 0, the more it supports the use of factor analysis, as this indicates a certain correlation between variables. For Bartlett’s test of sphericity, chi-squared = 1824.225, degrees of freedom = 120,
p-value = 0.0000. The null hypothesis (Ho) of this test is that “the correlation matrix of the variables is an identity matrix”, that is, there is no significant correlation between the variables.
p = 0.000 indicates rejection of the null hypothesis, indicating that there is a significant correlation between the variables and is suitable for factor analysis. In its KMO measure of sampling adequacy, KMO = 0.958. The KMO index is used to measure whether the correlation between variables is suitable for factor analysis, with a value range of 0 to 1: KMO > 0.9: excellent (very suitable for factor analysis); 0.8–0.9: very good; 0.7–0.8: moderate; <0.5: not suitable for factor analysis. In this case, KMO = 0.958, indicating that the data were very suitable for analysis. (
Table 5)
- (3)
Factor Analysis
Factor analysis is a commonly used dimensionality reduction method that is employed to explore the latent structure between variables and explain the relationships between variables through fewer factors. This analysis was based on 204 samples, using the principal factor method to extract factors, and ultimately retaining eight factors. This paper reports the results of factor analysis in detail, including characteristic values, factor loadings, and the uniqueness of variables.
Data overview and method of this analysis: sample size: 204; extraction method: principal factors; factor rotation: none; number of retained factors: 8; free parameters: 100. The fit of the factor model was evaluated through the chi-squared test. The chi-squared statistic χ
2(120) = 1833.49/chi^2(120) = 1833.49 χ
2(120) = 1833.49, with a significant
p-value (
p = 0.0000), indicating that the factor model has a good explanatory ability for the data. From the table, it can be seen that the eigenvalue of Factor 1 is much higher than that of other factors (7.47037), and its variance contribution rate is as high as 95.75%, indicating that Factor 1 is the dominant factor in this analysis. (
Table 6)
The factor loading matrix shows the loading values of each variable on each factor. The absolute value of a variable’s loading on a factor (usually >0.4) indicates the greater contribution of that variable to the factor. In the table, it can be seen that variables
X5,
X7,
X8 and
X10 have higher loading values on Factor 1, suggesting that they are mainly influenced by Factor 1. Some variables (such as
X1 and
X3) have a wider distribution across factors. The uniqueness of the variables is shown in the table below. From the results, it can be seen that the uniqueness of
X5,
X12 and
X13 is relatively low, indicating that the factor model can explain the variance of these variable well s. The higher uniqueness of
X1 and
X3 indicates that the variance of these variables is largely unexplained by the factors. (
Table 7)
- (4)
Correlation Analysis
Correlation refers to the degree of association between two variables and has three forms: positive correlation, negative correlation, and no correlation. The correlation coefficient ranges from 0 to 1, with a higher value indicating a stronger correlation and a value closer to 0 indicating a weaker correlation. The Pearson correlation coefficients among multiple variables are shown in the correlation coefficient matrix. The values on the diagonal of the matrix are always 1.000, indicating that each variable is perfectly correlated with itself. In the matrix, the following pairs of variables have a strong positive correlation. For X12 and X8, the correlation coefficient is 0.709 (p < 0.01), indicating a strong positive correlation between the two. For X10 and X8, the correlation coefficient is 0.679 (p < 0.01), indicating a strong positive correlation. For X15 and X12, the correlation coefficient is 0.668 (p < 0.01), further confirming the close positive correlation between the two.
Some variables have a weak correlation, but still have statistical significance. For example,
X4 and
X2 is 0.322 (
p < 0.01), indicating a weak positive correlation. Many pairs of variables have a low correlation and are not statistically significant. For example, the correlation coefficient between
X6 and
X2 is 0.078 (
p = 0.278, not significant). The correlation coefficients between
X1 and other variables are relatively low, with most
p-values greater than 0.1 and no statistical significance. (
Table 8)
- (5)
T-test
The following are the results of multiple one-sample t-tests. For each variable (X1, X2, ..., X16), the hypothesis test compares the sample mean of the variable with the hypothesized value, which is the average score of the variable. Null hypothesis (Ho): The sample mean is equal to the given value. Alternative hypothesis (Ha): The sample mean is not equal to the given value. t-value: A measure of the difference between the sample mean and the hypothesized value. The larger the t-value, the greater the difference between the sample mean and the hypothesized value.
p-value: A measure of the fit between the observed data and the null hypothesis. If the
p-value is less than a certain significance level (usually 0.5), the null hypothesis is rejected. Only the results of
X1,
X3, and
X4 are significant, while the others (
X2,
X5,
X6, ...,
X16) do not significantly deviate from the hypothesized value. Based on the
p-value results, only those with
p-values close to zero (such as
X1,
X3, and
X4) can reject the hypothesis, indicating that their sample means are significantly different from the hypothesized value. The
p-values of other variables are close to 1, indicating that their means are not significantly different from the hypothesized value. (
Table 9,
Table 10,
Table 11,
Table 12,
Table 13,
Table 14,
Table 15,
Table 16,
Table 17,
Table 18,
Table 19,
Table 20,
Table 21,
Table 22,
Table 23 and
Table 24)
Areas with higher resident satisfaction often imply that urban residents have higher recognition and satisfaction with the spatial quality of their area. The spatial quality of these areas is usually relatively high, reflecting a positive correlation between resident satisfaction and urban spatial quality. This verifies the validity of Hypothesis 3.