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Project STAR covers kindergarten to third-grade pupils in 80 Tennessee schools launched in the 1985/86 school year. Boyd-Zaharias (1999) emphasizes that all schools in Tennessee, which were able to provide a minimum of one of each of the three different treatment classes, more specifically small (13-17 pupils), regular (22-25 pupils) and regular with a teaching aide (22-25 pupils), were asked to take part in the project to guarantee an unmatched sample. As well as to ensure that students from different backgrounds are included through different types of schools, e.g. rural and inner-city institutions. Participating schools randomly allocated their students and teachers to one of the class types. To compare and analyze the outcome of each Project STAR participant, standardized tests were used, and which results are presented in percentile ranks.
This paper utilizes parts of the data set used by Krueger (1999), more precisely an identifier, a student and teacher background dataset, which were merged into one big dataset. As the project ran over four years and observed the same participating students over this period, the data is presented in a panel format. However, due to the dropouts of pupils and entries of new students during the observation period, it is an unbalanced panel. Fluctuations in the student body of the participating schools create certain limitations and challenges regarding the process of random distribution which led to a large number of missing values as some students were randomized in kindergarten, while others in first grade. Nevertheless, a drop of the missing values would limit the overall effect, so they are included in the following estimations. Krueger (1999) underscores that it should be remembered that kindergarten attendance was optional in Tennessee, which results in a rising number of new STAR participants in first grade. In total, the dataset contains 11,598 observations, i.e. pupils and originally 34 variables. Table 1 shows the means of pupils’ characteristics by treatment status, i.e. assignment to a small, regular or regular aide class in kindergarten and first grade for a student’s first year of participation in the project. An observation of only new STAR participants accounts for the disproportional addition of new students into the experiment over time and thus, will be used in some of the following estimations to control for randomness following Krueger (1999). The means are consistent with Krueger’s (1999) results. In the case of free lunch, the White/Asian, attrition rate and actual class size are almost identical. Here, receiving free lunch is a dummy indicator for the socio-economic status of the pupil’s family, thus their parent’s income, while White/Asian refers to the racial mix in the classes. Angrist and Pischke (2009) point out that the attrition rate relates to the share of pupils leaving the project at one point before finishing third grade. It should generally be noted that kindergarten outcomes are more credible than first-grade outcomes as they do not include the attrition rate problem and are mainly unaffected by the randomization problem. The characteristics of a pupil’s age in 1985 and their average test score in the underlying grade show some differences in comparison to Krueger (1999). This can be explained by the incomplete data used in this paper and the different composition of variables. While Krueger (1999) calculates the age of the students by using values quarterly, this paper calculates them on a yearly basis. Further, the average achieved test scores show slight discrepancies because Krueger (1999) uses the average of three subjects of the “Stanford Achievement Test “(SAT), namely maths, reading and word recognition, while this paper takes the average of the maths and reading SAT only.
Figure 1 illustrates the distribution of the pupils’ average percentile scores for kindergarten and first grade in a kernel density graph for the treatment statuses small and regular without a teaching aide. It is striking that students in smaller-sized classes achieve on average higher test scores in both grades. Beyond that, a comparison of the two grades shows that pupils who joined project STAR in kindergarten generally perform better than the ones joining in the following grade. As a reason for this, Krueger (1999) proposes that there is a high chance of better-performing pupils going to kindergarten in contrast to weaker students traced back to the non-compulsory kindergarten attendance in Texas at the time, similar to weaker-performing students unproportionally added to the sample at a later point.
Figure 1: Density Graph – Distribution of the average math and reading percentile test score
In addition, Table 1 provides the first insights into the successfulness of the desired randomized class assignment and distinctions in class sizes. A full understanding of the presence of an effective randomization process with balanced treatment groups is, according to Angrist and Pischke (2009), usually done by comparing pre-treatment results with treatment results. As project STAR does not include such pre-treatment outcomes, Krueger (1999) examines the question by comparing variables of the three class size groups in means and p-values of F-Tests of equality. Firstly, small classes had an average of 15.4 students in kindergarten and 15.88 students in first grade compared to over 22 in regular-sized classes. This shows that Project STAR was able to create the wanted differences in treatment status. Secondly, the similar values of the means across treatment status in both grades, e.g. White/Asian with a mean of 0.68 in small and regular and a mean of 0.66 in regular aide classes in kindergarten suggest that the random allocation of students turned out to be successful. The joint p-values in the last column of Table 1 support this statement partly as they indicate that there are no significant differences in means of the pupils’ background characteristics in kindergarten, for example, a joint p-value of 0.45 for age. Even though the means of the student background in the first-grade show only slight variances, some of them are statistically significant, e.g. the racial mix dummy variable White/Asian with a joint p-value of 0.00. This might be the result of a random assignment within schools only and not over the whole sample. To investigate random assignment within schools, Table 2 shows differences among treatment groups through the p-values of students’ background and school variables conditional on the school of attendance. The p-values of the students’ background variables with values between 0.28 and 0.45 in kindergarten, respectively 0.12-0.33 in first grade allow rejection of a statistically significant link between those values and the assigned class size at the 10 percent level. This leads to the conclusion that the method of randomly assigning students within schools led on average to balanced treatment groups. THE MODEL AND ITS IMPLEMENTATION
The subsequent model (1) following Krueger (1999) and Schanzenbach (2007) shows the superiority of randomized experiments on the effect of class size on student performance:
Y_ij=〖aS〗_(ij )+ 〖bF〗_ij+ ε_ij (1)
where Y_ij represents the average score of student i in school j, while S_(ij )and F_ijabsorb school, respectively family background effects and ε_ij is a well-behaved error term. Schanzenbach (2007) explains that S_(ij )and F_ij cover information about a student’s life that might be hidden to the researcher, but which possibly have an influence on the variables in the model. Thereby, Schanzenbach (2007) demonstrates that a randomized trial prevents a relation between treatment status and omitted characteristics. Krueger (1999) concludes that this allows an unbiased estimation. To estimate the empirical effect of class size assignment on student performances in the underlying year the subsequent model (2) following Krueger (1999) was used:
Y_ics=β_0+β_1 〖Small〗_cs+β_2 〖REGAIDE〗_cs+β_3 X_ics+a_s+ε_ics (2)
where Y_ics denotes the average test score of pupil i in class c at school s. 〖Small〗_cs and 〖REGAIDE〗_cs are dummy variables reporting small and regular aide class assignments, while X_ics is a vector including student characteristics. a_s captures school effects to control for treatment effects within schools, along with the error term ε_ics which is again assumed to be independent and identically distributed. Firstly, a simple “least-squares”-estimation (OLS) was conducted based on treatment status only, followed by regressions including school-fixed effects and students’ characteristics.
Estimation and analysis
Table 3 displays the regression results. In all estimations across grades and methods, a small class assignment has a greater effect on average test score performance than a regular aide class assignment and reveals statistical significance. Column 1 displays the simple OLS regression without controlling for school effects. The estimation output shows that assignment to a small-sized class improves average test scores ceteris paribus by almost 5 percentage points compared to students in regular-sized classes with an extra teaching aide in kindergarten, respectively 8.5 percentile points in first grade. Additionally, it is noteworthy that assignment to a regular-sized class with an aide shows a negative coefficient in the regression for kindergarten but is statistically insignificant with a p-value of 0.84. An integration of school fixed effects (column 2), only changes the coefficient of the treatment groups to a small extent, more precisely decreases the effect. Krueger (1999) suggests that this is a good indicator for random assignment. Angrist and Pischke (2009) add that this being the case, it is possible to say that the problem of selection on observables dissolves, as the treatment groups and the dependent variable are independent but highlight the presence of causal inference since observed and unobserved factors are well-balanced. The model’s explanatory capacity increases even further through the integration of the background variables. Column 3 shows the estimation results for the regression including student background characteristics and school-fixed effects. Again, the effect of a small class is greater than the regular size treatment with an improvement of percentile test scores of 4.84 percentage points in kindergarten and 6.81 percentage points in first grade. The assignment to a regular aide class is in both grades statistically insignificant. In the two observation periods, the additional covariates decrease the coefficients of the treatment group small but increase the coefficient of regular with an aide. Receiving free lunch lowers average test scores in the observed grades c.p. by more than 10 percentile points compared to those not applicable for full or partly free lunch. Furthermore, free lunch may not only indicate the income of a student’s parents but reflect how much parents foster their child’s educational success. The predicted average score in first grade for White and Asian pupils is c.p. 7 percentile points higher than for Black and Hispanic students, while it is 6.1 percentile points higher in kindergarten. The gender dummy points out that girls perform c.p. 5.63 percentage points higher than boys in kindergarten, whereas the gap between girls and boys in first grade is c.p. 4.45 percentage points. All in all, the regressions demonstrate a greater influence of small class assignments on the average test performance of students than of regular aide assignments, which is consistent with Krueger’s (1999) results. Moreover, the effect of assignment to a regular-sized class with a teaching aide seems to be statistically insignificant. In the regressions above the usage of the OLS and Fixed Effects Model was justified with the random assignment within schools. However, there might be a correlation within the different classes due to a common environment of the students, more precisely getting taught by the same teacher or rather mutual background characteristics like age and being former classmates. Teaching methods and teaching experience of the teacher, for example, have an influence on the learning of students and thus their educational success. This would no longer allow the assumption of a well-behaved error term within the classes, but just across clusters, i.e. schools. Angrist and Pischke (2009) propose to cluster the standard errors to account for the absence of an independently and identically distributed error term. This is of great importance as standard errors specify the precision of estimation. Table 4 shows the same estimations as Table 3 with average test score as the dependent variable and class size as explanatory variables, but with clustered standard errors on class ID for both grades. As expected, the clustered standard errors for small and regular with aide dummies are in both grades higher in comparison to Table 3, for instance, the standard error of a small class for kindergarten without clusters is 0.319 while it is 2.257 taking the correlation within classes into account. Since clustering tolerates correlation within classes, the confidence intervals widen up and thus result in a higher standard error. An additional complication to the above-presented estimations constitutes in the change of treatment group after kindergarten and non-random movements within the observation period. Under these circumstances, it is of interest to see if the initial assignment to a small class, is a good forecast for an actual assignment to a small class in first grade. The estimation output in Table 5 shows that if a pupil was assigned to a small class in kindergarten, he or she was reassigned to a small class in first grade on average by 0.84 percentage points. A high R-squared of 0.68 and a low standard error of 0.009 indicate a high explanatory power of the initial assignment for future assignments. That is to say, that assignment to a small class in kindergarten is a good indicator for future small class size treatment. This outcome allows us to carry out of a “Two-Stages-Least-Squares”-estimation, not only including instrumental variables but also clustered standard errors. Thereby, the initial small class assignment in kindergarten is used as an instrument to explain small class treatment in first grade because it cannot be ruled out that there is no correlation between the error term and the latter, as well as to account for the non-random movements. Krueger (1999) and Schanzenbach (2007) refer this to an “Intent-to-treat setup”. Table 6 presents the output of the estimation on average percentile test scores in first grade for students only who joined Project STAR in kindergarten. Column (1) illustrates the Instrumental Variables-2SLS regression with an only a small class assignment as an additional explanatory variable, while column (2) adds student characteristics. Consistent with the foregone results, students assigned to small classes in first grade perform c.p. over 6 percentile points higher than the ones not receiving a small class treatment. Furthermore, the student background variables are corresponding to the ones presented in Table 3. Additionally, in line with Table 4, the standard errors are higher than in Table 3. Therefore, it is possible to say that non-random transitions did not limit the project outcome.
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