*2.2. Measures*

Student data were measured using objective test scores. Students' scores on the first simulated test for the college entrance examination and the senior high school entrance examination were continuous variables. To compare the data across years, this study standardized the scores of each exam in the entire Haidian District based on students' graduation year, liberal arts and sciences, and types of exams.

PSTP was measured using a scale consisting of 10 questions covering multiple aspects of PSTP, including subject matter knowledge, teaching competencies, pedagogy, and student management. The question on the PSTP scale in the teacher questionnaire is, "Does the specialized course you have taken include the following? If so, do you think you are well prepared when you graduate?" The specific scale is shown in Table 1, which collects information on how teachers who are already in service feel about the relevant pre-service training before their employment. All questions are measured using a 5-point scale. In the empirical analysis, "Not included" is assigned a value of 1, "Inclusion; no preparation" 2, "Inclusion; preparation to some degree" 3, "Inclusion; well prepared" 4, and "Inclusion; very well prepared" 5. Thus, each question is transformed into a fixed interval variable.


**Table 1.** Pre-Service Teacher Preparation Scale.

#### *2.3. Models*

Based on the data structure of student data nested in teacher data, a duo-tier teacherstudent model can be developed to estimate the PSTP effects on student performance in Chinese, mathematics, and chemistry. The measurement model is shown below.

$$\text{Level I}: \ Q\_{\vec{\imath}\mathbf{j}} = \beta\_{0\mathbf{j}} + \beta\_{1\mathbf{j}}Q\_{\vec{\imath}\mathbf{j}-1} + \beta\_{2\mathbf{j}}X\_{\vec{\imath}\mathbf{j}\text{var}} + \beta\_{3\mathbf{j}}X\_{\vec{\imath}\mathbf{j}\text{trace}} + \gamma\_{\vec{\imath}\mathbf{j}}$$

$$\text{Level II}: \ \beta\_{0\dot{j}} = \gamma\_{00} + \gamma\_{01} \\ M\_{\dot{j}} + \gamma\_{02} \\ P\_{\dot{j}} + \mu\_{0j}, \beta\_{1\dot{j}} = \gamma\_{10}, \beta\_{2\dot{j}} = \gamma\_{20}, \beta\_{3\dot{j}} = \gamma\_{30}$$

where tier *I* is an estimate of students *Qij*, *Qij* is the exit score of student *i* taught by teacher *j*, *Qij*−<sup>1</sup> is the student's baseline score, *Xijyear* is the student's graduation year, *Xijtrack* is the liberal arts or sciences the student studied, *γij* is the residual, and *β*0*<sup>j</sup>* denotes that a random intercept is used at the teacher level. Tier II is an estimation of *β*0*j*, *Mj* denotes the pre-service preparation of the jth teacher, *Pj* denotes the personal characteristic variable of the jth teacher, *γ*<sup>00</sup> is a constant term, and *μ*0*<sup>j</sup>* is a residual term.

In the estimation of the measurement model, the teacher tier was used with the methods of a random intercept and fixed slope. To exclude the interference factors at the school level, the school-fixed effect was considered in the estimation of the model. PSTP variables were replaced by the standardized values of the same subject in the multi-tier linear regression.

## **3. Results**
