3.2.3. Control Variables

Based on the survey data and existing studies, this study selected the basic personal information and social security status of the elderly as control variables. The control variables included gender, age, marital status, residential area, co-residence mode, number of surviving children, whether or not he or she has retirement pension/public old-age insurance/private or commercial old-age insurance, and whether or not he or she has medical insurance [8,9,12–14]. In this study, we classified marital status into without spouse and with spouse, and residential area into urban and rural areas. The co-residence mode included living alone, living with family members, and living in an institution, which are generated into two dummy variables: whether or not living with family members and whether or not living in an institution.

#### 3.2.4. Mediating Variables

The mediating variables were the overall living status, leisure and recreation status, and health-care status of the elderly. In the current study, two dichotomous variables, "is all of the financial support sufficient to pay for daily expenses" and "self-reported quality of life ("very bad", "bad", and "general" were merged into "bad", and "good" and "very good" were merged into "good")," were selected to construct the overall living status. If the financial support is sufficient and the quality of life is good, the overall living status is "good"; otherwise, it is "bad". The two dichotomous variables, "whether he or she exercises or not now" and "whether he or she has traveled in the past 2 years", were selected to construct the leisure and recreation status. If they exercise regularly and have traveled, the leisure and recreation status is good; otherwise, it is "bad". The two dichotomous variables, "whether he or she can get adequate medical service at present" and "whether he or she has regular physical examination once a year", were selected to construct the health-care status. If they can get adequate medical service at present and have regular physical examination once a year, the health-care status is good; otherwise, it is "bad". Table 1 shows the variables and data statistics.


**Table 1.** Variables and data statistics.

#### *3.3. Models*

3.3.1. Multiple Linear Regression Model

In the current study, we used the frailty index of the elderly as the explained variable and the family socioeconomic status index as the explanatory variable and added various control variables to establish a multiple linear regression model to analyze the influence of family socioeconomic status on the health of the elderly.

$$Y\_i = \alpha + \beta\_0 X\_i + \sum \beta\_j Z\_{ij} \tag{1}$$

In Equation (1), *Yi* is the frailty index of the *i*th elderly person, and *α* is a constant term. *Xi* is the explanatory variable, which indicates the family socioeconomic status index of the *i*th elderly person, and *β*<sup>0</sup> is its coefficient. *Zij* is the *j*th control variable of the *i*th elderly person, and *β<sup>j</sup>* is the coefficient of each control variable.

### 3.3.2. Quantile Regression Model

Because of the high heterogeneity of the health status of older adults, the same family socioeconomic status may have different effects on older adults with different frailty status. Therefore, we also used the quantile regression model to analyze the effects of family socioeconomic status on frailty indices at different quartiles to verify whether the findings of the multiple linear regression model are still supported.

$$Q\_{i\emptyset}(Y\_i) = \alpha + \beta\_{0\emptyset} X\_i + \sum \beta\_{j\emptyset} Z\_{ij} \tag{2}$$

In Equation (2), *Qi<sup>θ</sup>* (*Yi*) denotes the conditional quantile of the frailty index for a given distribution of explanatory and control variables, where *θ* denotes the quantiles, and 10%, 25%, 50%, 75%, and 90% are selected in turn. The remaining variables and parameters are explained as in the multiple linear regression model above.

#### 3.3.3. Mediating Effect Model

The mediating variables are the overall living status, leisure and recreation status, and health-care status of the elderly, all of which are dichotomous variables. When the mediating variable is a categorical variable, the mediation effect analysis needs to be conducted by calculating a confidence interval through a two-step regression method. The procedure for testing the mediating effect is as follows:

$$M = aX + \varepsilon \varepsilon \tag{3}$$

$$Y = c'X + bM + \varepsilon\_3 \tag{4}$$

where *Y* denotes the explained variable, *X* denotes the explanatory variable, and *M* denotes the mediating variable. Equation (3) represents the regression of the mediating variable on the explanatory variable, and logistic regression is used. Equation (4) represents the regression of the explained variable on both the explanatory and mediating variables, and linear regression is used. In the present study, we first used the Stata 15.0 software to obtain the estimated values of regression coefficients and robust standard errors of a and b. Then, we used the Medci command in the package of RMediation (downloaded from https://cloud.r-project.org/bin/windows/contrib/3.5/RMediation\_1.1.4.zip (accessed on 5 November 2021)) of R 3.5.1 software (R Foundation for Statistical Computing, Vienna, Austria) to conduct the coefficient product distribution test to obtain the confidence interval of the mediating effect [28]. Moreover, if this confidence interval does not contain 0, it indicates the existence of the mediating effect [29].

#### **4. Results**

### *4.1. The Effect of Family Socioeconomic Status on Elderly Health*

A multiple linear regression model was established as a benchmark model to analyze the effect of family socioeconomic status on elderly health. From Model 1 in Table 2, it can be seen that the family socioeconomic status of the elderly has a significantly negative effect on the frailty index at the 1% level, and for every 1 unit increase in the family socioeconomic status, the frailty index decreases by 0.050 units. We performed a multicollinearity test which indicated that the problem of multicollinearity was excluded in the multiple linear regression model.


**Table 2.** Regression results of the impact of family socioeconomic status on elderly health.

The robust standard errors are in parentheses in Model 1, and the statistic for measuring goodness-of-fit is R2. The bootstrap standard errors are in parentheses in Models 2–6, with a sample size of 100, and the statistic for measuring goodness-of-fit is Pseudo R2. \* *p* < 0.1, \*\* *p* < 0.05, \*\*\* *p* < 0.01.

Because of the high heterogeneity of the health status of older adults, the same family socioeconomic status may have different effects on older adults with different frailty status. Then we also developed quantile regression models to analyze the effects of family socioeconomic status on the elderly health in different quantiles. Models 2–6 in Table 2 show that the effects of family socioeconomic status of older adults on the frailty index remain significantly negative at the 1% level, and the coefficients of the effects are, respectively, −0.041, −0.058, −0.058, −0.055, −0.067.

The results of the multivariate linear regression model and quantile regression models suggest that improving family socioeconomic status can reduce the frailty index and promote the health of the elderly.

#### *4.2. Robustness Test*

In the current study, the economic status compared with local people, the average years of education, and the average occupational level before retirement of elderly couples are integrated into the replaced family socioeconomic status index using the entropy weight method to conduct a robustness test. Models 7–12 in Table 3 show that the effects of replaced family socioeconomic status of older adults on the frailty index remain significantly negative at the 1% level, and the coefficients of the effects are, respectively, −0.070, −0.050, −0.071, −0.075, −0.081, −0.100. These results also demonstrate that the increase of family socioeconomic status can decrease the frailty index and promote the elderly health, indicating that the empirical results obtained above are reliable.


**Table 3.** Regression results after replacing the explanatory variable.

The robust standard errors are in parentheses in Model 7, and the statistic for measuring goodness-of-fit is R2. The bootstrap standard errors are in parentheses in Models 8–12, with a sample size of 100, and the statistic for measuring goodness-of-fit is Pseudo R2. \*\*\* *p* < 0.01. The control variables in each model have been controlled.

#### *4.3. Heterogeneity Analysis*

The above analysis found that both residential area and age have significant influence on the health of the elderly. Therefore, we used the multiple linear regression model to continue to analyze the different effects of family socioeconomic status on the health of the elderly in different residential areas and at different ages.

Table 4 shows that the impacts of the family socioeconomic status of the urban and rural elderly on the frailty index are −0.043 and −0.088, both significant at the 1% level. However, the impact in urban areas is lower than rural areas.

**Table 4.** Results of multiple linear regression (with explanatory variable) by residential area.


\*\*\* *p* < 0.01. The control variables in each model have been controlled.

Table 5 shows that the effects of family socioeconomic status on the frailty index for the elderly aged 60–69 and 70–79 years (lower and middle age) are −0.043 and −0.088, both significant at the 1% level, whereas the effect of family socioeconomic status on the frailty index is no longer significant for the elderly aged 80 years and above (higher age).

**Table 5.** Results of multiple linear regression (with explanatory variable) by age.


\*\* *p* < 0.05, \*\*\* *p* < 0.01. The control variables in each model have been controlled.

#### *4.4. Mediating Effect Analysis*

The level of family socioeconomic status generally directly affects the overall living status, leisure and recreation status, and health-care status of the elderly. Therefore, those were selected as mediating variables to analyze their mediation effects in the influence of family socioeconomic status on elderly health. Table 6 shows the results after adding each mediating variable to the baseline linear regression model. From the comparison with Model 1, it can be seen that the absolute values of the impact coefficients of family socioeconomic status in Models 18–20 are becoming smaller and still significant at the 1% level, and that the impact coefficients of each mediating variable are significantly negative at the 1% level, which initially indicates the existence of mediation effects for each of the above mediating variables.

**Table 6.** Results of multiple linear regression adding mediating variables.


\*\*\* *p* < 0.01. The control variables in each model have been controlled.

We further tested the mediation effect by calculating the confidence interval through a two-step regression method. The results in Table 7 show that the 95% confidence intervals of the estimated mediation effects for the overall living status and leisure and recreation status are, respectively, [−0.116, −0.074] and [−0.184, −0.127]; they do not contain 0, indicating the existence of mediating effects in the impact of family socioeconomic status on elderly health. Moreover, the 95% confidence interval of the estimated mediation effect for the health-care status is [−0.014, 0.004] and contains 0, indicating the absence of the mediation effect.

**Table 7.** Estimated results of mediating effects.


We set rho as 0, alpha as 0.1, and type as "mc" in the R software.

#### **5. Discussion**

In the current study, based on the CLHLS in 2018, the total family income, the comprehensive years of education, and the comprehensive occupational rank before retirement of the elderly couples were synthesized into a family socioeconomic status index that was used as the explanatory variable using the entropy weight method, and the frailty index was used as a measurement of the comprehensive health status of the elderly. First, we established the multiple linear regression model and quantile regression models to analyze the effects of family socioeconomic status on the health status of the elderly and conducted a robustness test using the replaced explanatory variable. Then, the heterogeneity of the

effect of family socioeconomic status on the health status of the elderly among different residential areas and at different ages was analyzed. Finally, the overall living status, leisure and recreation status, and health-care status of older adults were used as mediating variables to analyze their mediation effects in the influence of family socioeconomic status on elderly health.

Family socioeconomic status has a positive impact on the health status of the elderly. This result is same to those of other researchers [18,20,21,30]. Family socioeconomic status reflects the individual's ability to obtain material and social resources [31]. Higher family socioeconomic status usually means higher total family income, education, and occupational rank [18]. Older adults with higher total family income tend to have better living conditions, participate in more leisure and entertainment activities to meet higher-lever needs, and purchase better health-care services to increase investment in health. Elderly families with higher years of education have acquired higher health awareness and literacy during their continuous learning and developed healthier living habits; they are more aware of various health risk factors and therefore more aware of their prevention, and are able to respond more quickly and effectively when they encounter diseases. Older households with higher occupational rank tend to have a higher proportion of pensions and have higher pensions; in addition, higher occupational rank tends to be accompanied by more available access to health-care services. Thus, higher family income, education, and occupational rank generally result in better health outcomes for older adults. Family socioeconomic status is a comprehensive reflection of the individual socioeconomic status of elderly couples, and it is the social class or status of elderly couples based on the family cooperation model. Therefore, the increase of family socioeconomic status can decrease the frailty index and promote the elderly health.

Due to the typical dual economic structure of urban and rural areas in China, the family socioeconomic status of the urban elderly is relatively higher (the average family socioeconomic status indices of the urban and rural elderly in the present study are, respectively, 0.237 and 0.126). Additionally, public health and medical resources in urban areas are more abundant and the allocation is more reasonable, while these conditions in rural areas are relatively poor. Under the influence of the law of diminishing marginal utility of the health production function [4,18], the family socioeconomic status of the urban elderly has lower influence on the frailty index. Some scholars also hold the analogous view [13,18]. In other words, when the family socioeconomic status changes by an equal amount, it has a higher impact on the health of the rural elderly.

Family socioeconomic status has a significantly positive influence on the health status of middle and lower age elderly, but not on higher age elderly, which is similar to the conclusion of other related studies [32,33]. When older adults reach the higher age, their physical functions continue to decline, and their health status becomes increasingly dependent on the individual and less influenced by other factors, including the family socioeconomic status, whereupon the effect of family socioeconomic status on the health of the higher age elderly is no longer significant.

A review study by Huang showed that there are four mediating pathways between socioeconomic status and health: material factors, lifestyle factors, psychosocial factors, and neighborhood [34]. Unlike that, we think that overall living status and leisure and recreation status have mediation effects in the influence of family socioeconomic status on the health status of the elderly, whereas health-care status has no mediation effect, which is different from the conclusion of other studies as well [12–14,35,36]. When older adults have a higher family socioeconomic status, on the one hand, their sources of living are often more abundant and their quality of life is usually higher, and thus their overall living status is better. They will pay more attention to direct investment in health. On the other hand, their health awareness tends to be higher, and it is more likely to increase the physical resistance through exercise and to relax by participating in various leisure and recreational activities. Therefore, overall living status and leisure and recreation status have mediation effects in the influence of family socioeconomic status on elderly health. With the expansion

of medical insurance coverage and regular physical examination in China, not subject to the family socioeconomic status, more and more elderly people are able to be hospitalized in time when they fall ill and participate in annual routine medical checkups. As the basic public health services become more equalized, health-care status has no mediation effect in the effect of family socioeconomic status on elderly health.

There were several limitations to this research. First, the study only used the 2018 crosssectional data, so we did not reveal the dynamic impact of family socioeconomic status on elderly health. Second, limited by the variables in CLHLS data, only 32 indicators were used to construct the frailty index. If more indicators can be obtained, the measurement of frailty index will be more accurate. Third, total family income and primary occupation rank before retirement of the elderly might be related to their health status, so there might be a reverse causality between family socioeconomic status and frailty index to some extent.
