The Effect of Choosing Key Versus Ordinary Schools on Student’s Mathematical Achievement in China

Abstract

The current study examined the impacts on students’ cognitive performance of the key versus ordinary school system in China, using an analytic approach that combines hierarchical linear modeling with propensity score stratification. The results show that students from key schools score significantly higher on a mathematical achievement test than their counterparts in ordinary schools, after controlling for student characteristics and their family background. The specific magnitude of the school effect varies substantially across the geographic locations of the school. The advantages of key schools over ordinary schools are found to be generally greater among urban schools compared with suburban schools. The results are noteworthy as both key and ordinary schools are state-funded and the system was formed directly by policy initiatives and differential resource allocation. As such, they bear important policy implications on systematic-level school management in general.

Appendices

Appendix 1: Propensity Score Estimation and Stratification

Using all 20 covariates in Table 2, an iterative approach was taken to estimate each student’s propensity score via logistic models. Within each iteration, students were divided into five strata according to their propensity scores. The balance of each covariate across school sectors was examined within each stratum by Cochran–Mantel–Haenszel (CMH) χ ² test and bar chart examination. Higher-order term of a covariate and/or interaction terms with other important covariates would be added into the initial model, if a large value of the CMH statistic was obtained after propensity score stratification. This process continues until all covariates are balanced across school sectors.

It took three iterations before the final propensity scores were obtained. The final model contained 32 terms, including four interaction terms, six quadratic terms, one cubic term, and one quartic term. Figure 2 displays the distributions of the final estimated propensity scores for key and ordinary school students on a logit scale. The two distributions overlap substantially, which implies that for most students in key schools, there is a comparable student in ordinary schools in the sense of having similar probability of getting into key schools. However, a few students in key schools have propensity scores higher than any of the ordinary school students. Similarly, a few ordinary school students have propensity scores lower than any of the key school students. For these students, there are no students in the other sector with comparable propensity scores. To take these students into account, students were classified into seven strata based on their estimated propensity scores, in which the first (and the seventh) stratum contains exclusively these students with highest (and lowest) propensity scores, and the other five strata contain key or ordinary school students with comparable propensity scores.

Fig. 2

Distributions of estimated propensities of getting into key schools for students in different school types

The last column of Table 2 shows the results of balance check for each covariate after the stratification. As expected, there is considerably greater balance on the observed covariates after propensity score stratification. None of the 20 covariates now exhibit significant differences across school sectors after stratification. To show this is indeed the case, Fig. 3 depicts the balance within each stratum for covariate Mother’s educational level, the only variable that needs to add more higher-order terms after the second iteration. Compare to across-sector differences before stratification, greater balances are achieved within each stratum of the propensity scores.

Fig. 3

Balance check before and after propensity score stratification for Mother’s Educational Level (MEL)

Similar process was taken to obtain refined propensity score after adding students’ problem solving scores as a covariate. Figure 4 displays the balance check after stratifying students on the re-estimated propensity scores. As expected, great balances are achieved for almost all strata.

Fig. 4

Balance check for problem solving scores

Appendix 2: Hierarchical Linear Models Used for School Effect Estimation

After stratifying students based on their propensity scores, within-stratum school effect between key and ordinary schools were estimated using the following model

$$Y_{ij} = \gamma_{00} + \gamma_{01} {\text{Sector}}_{j} + u_{j} + e_{ij} ;\,u_{j} \sim \left( {0,\tau } \right);\,e_{ij} \sim \left( {0,\sigma^{2} } \right),$$

(1)

where $Y_{ij}$ is the mathematics score for student $i$ in school $j$, ${\text{Sector}}_{j}$ is the school type (0 = ordinary school, 1 = key school), and $\gamma_{01}$ indicates school effect on $Y_{ij}$. In the random part of the model, $u_{j}$ and $e_{ij}$ are residuals at school and student level, respectively. Both follow normal distributions with mean 0 and variances $\tau$ and $\sigma^{2}$, respectively.

The combined school effect across strata was estimated using the following model

$$Y_{ij} = \gamma_{00} + \gamma_{01} {\text{Sector}}_{j} + \sum\limits_{h = 1}^{4} {\gamma_{0(1 + h)} {\text{Strata}}_{hij} } + u_{j} + e_{ij} ;\,u_{j} \sim \left( {0,\tau } \right);\,\,e_{ij} \sim \left( {0,\sigma^{2} } \right),$$

(2)

where ${\text{Strata}}_{hij}$ are a series of dummy variables created to indicate four of the five propensity score strata and were added to model 1 to control for selection bias.

Before stratification, school effect estimation with regression adjustment for propensity score was estimated using the following model

$$Y_{ij} = \gamma_{00} + \gamma_{01} {\text{Sector}}_{j} + \gamma_{02} {\text{Logit}}P_{ij} + e_{ij} ;\,e_{ij} \sim \left( {0,\sigma^{2} } \right),$$

(3)

where ${\text{Logit}}P_{ij}$ is the logit of propensity score for student $i$ in school $j$.

After stratification, Eq. 3 was used to estimate within-stratum school effects with regression adjustment for propensity score. For the combined school effect across strata, estimate with regression adjustment for propensity score was obtained by adding ${\text{Logit}}P_{ij}$ to model 2.