Appendix D
METHODS AND RESULTS FROM THE LOGISTIC ANALYSES

• THE DATA AND SAMPLE
• A LOGISTIC MODEL OF VOCATIONAL CONCENTRATION
• A LOGISTIC MODEL OF PARTICIPATION IN COLLEGE-PREP MATH
• A LOGISTIC MODEL OF PARTICIPATION IN COLLEGE-PREP ENGLISH

This appendix describes the data and empirical methods used in the logistic analyses discussed in Secs. IV and V. In Sec. IV, we presented results based on a logistic analysis that predicted the probability that a student would be a vocational concentrator. Section V contained results based on logistic analyses that predicted the probability that a student would be in the college-prep math track and the college-prep English track in the 11th grade. The predicted probabilities in those two sections were based on the results of the logistic regressions that are presented in Tables D.3 to D.5.

Logistic models are used to analyze dependent variables that have a binary outcome, i.e., the outcome that we want to explain takes on only two values. In the case of the coursetaking behavior that we are modeling, we observe that the student participates or does not participate in the particular coursetaking behavior we are interested in. The logistic model is specified as:

where p_i is the probability that student itakes the course we are modeling (e.g., college-prep math) and X_i is a vector of individual characteristics that affect the probability of taking the course. The logistic model thus allows us to determine which independent or explanatory variables, the Xs, predict the probability that a student does or does not participate in the coursetaking behavior we are modeling. The coefficient estimates, the ßs, show the effect of the variable X on the logarithm of the odds or "log odds" (the logarithm of the ratio of probability that the outcome is 1 to the probability that the outcome is 0). In a multivariate logistic model, the coefficient estimate on any one independent variable measures the effect of a change in that variable on the log odds, holding all other variables constant.

To control for differences by student race/ethnicity, a series of indicator variables was defined for the four primary race/ethnic groups at the three schools: WHITE, BLACK, ASIAN, and LATINO.[76]

We also included an indicator variable in the regressions for McKinley and the pooled sample when race/ethnicity was missing (MISS_RACE).[77] The logistic results in the tables show the coefficient estimates for WHITE, BLACK, and ASIAN. Latino students, the only group common across the three schools, is the omitted group. Consequently, the coefficient estimates on the three race/ethnicity variables measure a difference in the probability of participating in the coursetaking behavior for a student in the particular race/ethnicity group and a Latino student. For example, a positive coefficient on WHITE is interpreted to mean that a white student has a higher probability than a Latino student of participating in the coursetaking behavior that is being modeled.

The effect of student ability on coursetaking behavior is measured by 10th grade reading and math achievement scores, READ and MATH. We had information on 8th grade math and reading achievement scores only for Washington and Coolidge. The regression estimates for those two schools were similar when 8th and 10th grade scores were used to measure ability. To make the results comparable across schools and for estimating the pooled model, we present only the results for models using the 10th grade scores. In addition, the achievement scores were missing for 7 to 15 percent of the sample, depending on the school. Students with missing math or reading scores were assigned the school mean (based on the sample of known data). In addition, an indicator variable, equal to one when the test score was missing, was included in the regressions (MISS_MATH and MISS_READ).

In the model predicting the probability of being a vocational concentrator, an additional independent variable was included. For students who had been at their respective schools for four years, an indicator variable was set equal to one (FOUR_YEAR). Four-year students may be more likely than three-year students to take a large number of vocational courses, since they have had a longer period of "exposure." This variable was included to control for any differences between the two types of students.

Two additional independent variables were available for only a subset of the schools and consequently were included in the models estimated for those schools only. For Coolidge, we had a measure of student SES using a three-point scale. We created three indicator variables, LOWSES, MIDSES, and TOPSES, for students with family incomes of less than $20,000, between $20,000 and $50,000, and greater than $50,000, respectively. A fourth category was created for those students with missing SES information, MISS_SES (about 10 percent of the sample). In the regression results presented in the tables, HIGHSES is the omitted category. Thus, the coefficient estimates for LOWSES, MIDSES, and MISS_SES capture any differences between those three groups and the TOPSES group.

Finally, we had information on country of birth for students at Washington and McKinley. This information was used to create an indicator variable, FOREIGN, for students who were born outside of the United States. In addition, an indicator variable was created for McKinley students when the country of birth was not known, MISS_FOR.

The sample means for the variables used in the logistic analyses are presented separately for each school and for the pooled sample in Table D.2. The percentage of vocational concentrators and the participation rates in college-prep math and college-prep English are the same as those presented in Tables 4.5, 5.4, and 5.5.

The data on student demographics and test scores are the same as those presented in Tables 2.1, 2.2, and 2.3. The sample sizes for each of the three schools and the pooled sample are shown at the bottom of Table D.2.

The logistic models were estimated separately for each school and pooled across schools. The analyses by school allow for the effect of each independent variable to vary across schools, and for the inclusion of independent variables that are available for only a subset of the schools. The pooled model restricts the coefficients on each of the independent variables to be the same across schools.[78] To allow for between-school differences, the pooled model was estimated with dummy variables for each school, MCKINLEY and WASHINGTON (where COOLIDGE was the omitted category). The school dummy variables capture any difference, holding all other independent variables constant, between students at McKinley and students at Coolidge, and students at Washington and students at Coolidge.

In addition, the pooled model was estimated using two different specifications. In the first, test scores were measured in absolute terms. In the second case, test scores were measured as deviations from the respective school means. The first model assumes that it is a student's absolute ability that predicts coursetaking behavior. Using this model, students with the same test score, e.g., with the percentile score equal to 30, 50, or 80, can be compared. The second model assumes that it is a student's ability relative to the cohort of students at that school that affects coursetaking behavior. In this model, students at the same point in the test score distribution can be compared, e.g., in the 25th, 50th, or 75th percentile of the test score distribution. The distinction between these two models is important, since there are significant differences in the test score distributions between the three schools. The two models differ, however, only in the estimate for the intercept and school dummy variables; all other coefficients remain unchanged. Consequently, Table D.3, which presents the regression results for the pooled model, shows two sets of coefficients for the intercept and school dummy variables depending upon how the test scores are measured.

A LOGISTIC MODEL OF VOCATIONAL CONCENTRATION

Table D.3 presents the results for the logistic model, estimated separately by school, predicting the probability that a student is a vocational concentrator. The table shows the results for Washington with and without the adjustment for the practical arts requirement. The first column of estimates for each school shows the results when the common set of independent variables is included in the model. The estimates in the second column for Washington and McKinley include FOREIGN in the model, whereas the estimates in the second column for Coolidge include the SES variables in the model.

The estimates show that in all cases, four-year students are more likely than three-year students to be vocational concentrators, although the effect is significant at conventional levels (p < 0.10) only for students at Coolidge and Washington (when there is no adjustment for the practical arts requirement). The difference between boys and girls is significant only at Washington (without the adjustment) and indicates that girls are less likely than boys to take six or more vocational courses.

The coefficients on the race/ethnic group variables show that there are significant differences between some of the race/ethnic groups, even after controlling for differences in student achievement. At Washington, white students have a significantly higher probability than Latino students of being vocational concentrators, whereas Asian students at Coolidge are significantly less likely than their Latino counterparts to be concentrators. At McKinley, the estimates show no differences between black and Latino students. When the model is estimated with other race/ethnic groups as the omitted category (not shown), the results show that there is always a significantly lower probability that an Asian student and a significantly higher probability that a white student will be a vocational concentrator than the other race/ethnic groups at Coolidge, whereas there are no significant differences between blacks and Latinos. At Washington, with and without the adjustment for the practical arts requirement, white students always stand out with a significant and higher probability of being concentrators, whereas Latino and Asian students are not significantly different from one another.[79]

The relationship between achievement scores and the probability of being a vocational concentrator is similar for all three schools. The negative and significant coefficient on the math achievement scores indicates that the likelihood of being a vocational concentrator declines as a student's math achievement score increases. The magnitude of the negative relationship between test scores and vocational coursetaking is not as large for reading scores, although the relationship is significant at Coolidge and McKinley. The positive and significant coefficient on MISS_MATH, the indicator that the math score was missing, indicates that this information was not randomly missing. Instead, students with missing math achievement score data at Washington are more likely than the average student to be concentrators.

The inclusion of the variable FOREIGN in the regressions for Washington and McKinley show that students born outside of the United States are not significantly different from native students. The second set of estimates for Coolidge include the SES variables and show that students with low SES and students with missing SES data are more likely than high-SES students to be vocational concentrators, and that the difference is significant. When the model is estimated with other SES groups as the omitted category (not shown), the results show that low-SES students are also more likely than middle-SES students to be concentrators, and that there is no significant difference between middle- and high-SES students.

The results from the pooled model are shown in the first two columns of Table D.6, below. The two columns show the results without and with the adjustment for the practical arts requirement at Washington. In general, the results for the independent variables are similar to those when the model is estimated separately by school. The interesting results from the pooled regressions are the signs of the school dummy variables, WASHINGTON and MCKINLEY. The first set of intercept terms (constant and school dummy variables) shows the differences between the schools when the model is estimated with test scores measured in absolute terms; the second set of intercept terms shows the differences between the schools when a student's test scores are measured relative to their respective school means.

First, consider the results when there is no adjustment for the practical arts requirement at Washington (the first column of Table D.6). These results show that students at Washington and McKinley, holding constant the level of test scores and all other characteristics, have a significantly higher probability of being vocational concentrators than students at Coolidge (the omitted category). When the model is estimated with Washington as the omitted category, the resulting estimates (not shown) show that students at McKinley are also more likely to be vocational concentrators than students at Washington (the difference is significant at the .10 level).

However, when the test scores are measured relative to the school mean, the coefficient measuring the differences between Washington and Coolidge is reduced and is no longer significant, whereas the coefficient measuring the difference between McKinley and Coolidge is more positive and still significant. Similarly, the coefficient measuring the difference between McKinley and Washington (not shown) also becomes more positive and very significant. These results indicate that, when comparing students with the same absolute test scores at Washington and Coolidge, there is a higher probability that the Washington student will be a concentrator. However, this finding is due to the higher overall mean test scores at Washington. Once students with the same relative standing at the two schools are compared, there is no difference.

The second set of estimates shows the differences between the schools when an adjustment is made for the practical arts requirement at Washington. This adjustment reverses the ranking of Washington and Coolidge, as there is now a significantly lower probability that a Washington student will be a concentrator compared to a Coolidge student. This ranking holds when test scores are measured in absolute terms and is even larger when test scores are measured relative to the school mean. As before, the probability is higher that a McKinley student will be a concentrator compared to a Coolidge student or a Washington student.

A LOGISTIC MODEL OF PARTICIPATION IN COLLEGE-PREP MATH

The results for the logistic model of participation in college-prep math are shown in Table D.4 for the models estimated separately by school and in Table D.6 (third column) for the pooled model. The school-specific models are estimated both with the same set of independent variables and with the addition of the SES variables for Coolidge, and FOREIGN for Washington and McKinley.

At all three schools, holding constant other characteristics and achievement scores, there is no significant difference between boys and girls in the probability of taking college-prep math. At Washington, white students are not significantly different from Latino students, whereas Asian students are significantly more likely to take college-prep math than Latino students (at the 10 percent level of significance). A comparison of white and Asian students (not shown) shows that Asian students are also significantly more likely than white students to take college-prep math (at the 1 percent level of significance). The coefficient estimates on the race/ethnicity variables for Coolidge also show significant differences across the groups. The estimates in Table D.4 show the comparison for each group relative to Latino students, and the coefficients are always positive and significant. When the model is estimated with other race/ethnicity groups as the omitted categories, pairwise comparisons show that Asian students are more likely than every other race/ethnic group to take college-prep math. As was the case for vocational concentration, there is no difference between black and Latino students at McKinley.

The estimates show that there is a strong positive relationship between students' test scores and the probability of taking college-prep math at all three schools (except for the reading score at Washington, which is not significant). As might be expected, the magnitude of the coefficients is always greater for the math score than the reading score.

The estimates for Washington and McKinley when FOREIGN is included in the model show that Washington students born outside the United States are not significantly different from those born in the United States. In contrast, foreign-born students at McKinley are more likely to take college-prep math. A student's SES is significant at Coolidge, with low- and middle-SES students less likely than high-SES students to take college-prep math. Pairwise comparisons (not shown) indicate that low- and middle-SES students are not significantly different from one another, and that students with missing SES information are also significantly less likely than high-SES students to take college-prep math.

The estimates from the pooled model, shown in column 3 of Table D.6, allow comparisons across the three schools. Again, the results for the independent variables are similar to the findings when the models are estimated separately by school. A comparison of the school dummy variables shows that, compared to a Coolidge student with the same absolute test score, a Washington student is less likely and a McKinley student is more likely to take college-prep math. When the model is estimated with McKinley as the omitted category (not shown), the results show that McKinley students, for a given absolute test score, are also more likely to take college-prep math than Washington students. Again, these findings are due to the difference in the distributions of the test scores at the three schools. When the model is estimated using a student's relative test scores to predict the probability of taking college-prep math, there is no difference between McKinley, Washington, and Coolidge students. Furthermore, for students with the same relative test score, McKinley students are now less likely than Washington students to take college-prep math.

A LOGISTIC MODEL OF PARTICIPATION IN COLLEGE-PREP ENGLISH

Table D.5 presents the results by school for the logistic model predicting the probability of taking college-prep English. For all three schools, girls are significantly more likely than boys to take college-prep English. A student's race/ethnicity is important only at Coolidge, where white and Asian students are significantly more likely than Latino students to take college-prep English. Pairwise comparisons (not shown) indicate that there are no significant differences between the other groups.

Again, at all three schools there is a significant and positive relationship between a student's test scores and his or her probability of taking college-prep English. It is interesting to note that, unlike the model for college-prep math, the coefficients on the math and reading scores are similar in magnitude at Washington and Coolidge.

The inclusion of the SES variables in the model for Coolidge shows no significant difference in the probability of taking college-prep English for students with different SES levels. Foreign-born students at Washington are less likely to take college-prep English than those born in the United States, and there is no difference between foreign- and U.S.-born students at McKinley.

The probability of taking college-prep English can be compared for students at the three schools based on the pooled model shown in column 4 of Table D.6. A student at Coolidge with a given absolute test score is more likely to take college-prep English than a student at Washington, but less likely than a student at McKinley. A student at McKinley is more likely to take college-prep English than a student at Washington. Again, these pairwise comparisons change when students with the same relative standing are compared. For students at the same point in the test score distribution, Washington students are no different from students at Coolidge. However, McKinley students are still significantly more likely than Coolidge students to take college-prep English.

[76]Coolidge is the only school with sufficient representation of the four race/ethnic groups to allow estimation of separate effects for each group. At Washington, there is only one black student in the sample, and there are no white students and only three Asian students in the McKinley sample. Consequently, these students are included in the omitted group, BLACK is omitted from the regressions for Washington, and WHITE and ASIAN are omitted in the regressions for McKinley. In addition, there are three students at Coolidge and one student at McKinley who are classified as "other" race/ethnicity. They are also included in the omitted group in the regressions.

[77]Information about a student's race/ethnicity was missing for about 3 percent of the McKinley sample.

[78]The pooled model was also estimated allowing some of the coefficients, such as those on the race/ethnicity variables, to vary by school.

[79]By estimating the models for Washington and Coolidge with different race/ethnic groups as the omitted category, pairwise comparisons of the significance of the differences between all of the different race/ethnic groups can be made. This is not an issue for McKinley, which has only two groups.