Concluding Comments
Appendix 4: Models Used in This Report
Chapter 2 Models
For our analyses, we used two-level hierarchical linear modeling, with students at level- and schools at level-. Since all of our analyses used models with binary outcomes, our HLM analyses use a binomial sampling model with a logit link.
To provide a baseline for later analyses, our base model contained only demographic information and eighth-grade ITBS scores. The base model, which is used for the HLM analyses in Chapter , is shown below in the “Pre-high school achievement model.” It models the probability of attending college, controlling for the other variables; however, identical models were analyzed for the probability of attending a four-year versus two-year college, given that a student attended college, and the probability of attending a selective or very selective college, given that a student attended a four-year college. Each of the variables in the base model was grand mean centered in order to allow the intercept to represent the value for an “average” CPS graduate. Subsequent variables were either grand mean centered or uncentered.
The equations below show both the Level- and Level- models. In the Level- model, “Distance from HS” is a measure of how far, in miles, a student lives from the high school the student attends (whether a student is likely to have made the decision to attend a high school outside of his or her neighborhood); “Freshman Age” refers to a student’s age at the beginning of the freshman year of high school; “male,” “white,” “Latino,” and “Asian” are dummy variables indicat- ing a student’s gender and race/ethnicity (African-American is the omitted category); and “Neighborhood Poverty” is a measure of the concentration of poverty in the students’ census block group; “Neighborhood SES” is the mean social status, without income, of students’ census block group; “Average ITBS” is the average of a student’s latent reading and
math eighth-grade scores on the Iowa Tests of Basic Skills; “Average ITBS”is the square of the Average ITBS variable
to account for non-linearity; “No ITBS” is an indicator for students for whom eighth-grade ITBS scores were not avail- able (to avoid significantly reducing sample size, and possibility biasing the results, students without ITBS scores were assigned the sample average); and “No Freshman Age” is an indicator for students for whom freshman ages were not available.
In the Level- model, “Average School SES” is the average of the students’ Neighborhood SES values (not the average SES of the area in which the school is located); “Average School ITBS” is the average of the students’ eighth-grade latent ITBS scores; “Primarily Latino School” refers to a school in which 70 percent or more students are Latino; “Integrated School” refers to a school in which 0 percent or more of students are white or Asian; and “Mixed School” refers to a school in which less than 0 percent or the students are white or Asian and the percentages of African-American and Latino students are less than 70 percent.
Level-1
Level-2
Building on the base model, in Chapter we included students’ ACT scores, unweighted GPAs, and a measure of the rigor of the coursework they completed while in high school into Level- of the model in order to determine these variables’ effects on the probability of attending college, enrolling in a four-year college (given that the student attended college), and enrolling in a selective or very selective college (given the student attended a four-year college). The ACT score included in the analyses is the student’s ACT composite score. Most students in CPS take the ACT as a part of the Illinois state achievement test, the Prairie State Achievement Examination (PSAE). Thus, students’ ACT test results are not self-reported, and because they are not obtained from ACT, may not reflect a student’s highest ACT score if a student opted to retake the ACT.
In some analyses, ACT scores were left in their original metric. In others, ACT scores were broken down by deciles, based on the scores of the CPS students in our sample, or they were broken down into the following categories: less than or equal to 4, 5-7, 8-0, -, 4-6, and 7 and above. Students for whom ACT scores were not available were assigned the mean ACT score for CPS students, 7., in order to prevent a sizable reduction in sample size (ACT scores were not available for approximately 0 percent of students) and possible bias. When ACT scores were included in a model, a “No ACT” dummy variable was included.
The unweighted GPAs were computed based on students’ grades in core courses: mathematics, English, science, social science, and foreign language. Students for whom GPAs were not available were not included in the analyses with GPA in the model. Like ACT scores, GPAs were included in the models in their original metric, as deciles, based on the GPAs of the CPS students in our sample, or as categories: less than or equal to .0, .-.5, .6-.0, .-.5, and .6-4.0. The final models also included indicators of the rigor of students’ coursework. Based on the number of AP and honors courses students took, we placed students into the following categories: standard curriculum, limited honors/AP, honors track, honors with one AP, and advanced track. Students designated as taking the standard curriculum had not taken honors or AP classes at any point during high school. Students categorized as being in the honors track took four or more honors classes while in high school. Students placed into the limited honors/AP category took fewer than four honors
€ Prob College
(
=)
= ϕij € Log[
ϕij/ − ϕ(
ij)]
= ηij € ηij= β0 j+ βj(
Distance from HS)
ij + β j
(
Freshman Age)
ij + β j( )
Male ij+ β4 j(
White)
ij+ β5 j(
Latino)
ij+β6 j
(
Asian)
ij + β7(
Neighborhood Poverty)
ij+ β8 j(
Neighborhood SES)
ij + β9 j(
Average ITBS)
ij+β0 j
(
Average ITBS)
ij + βj
(
No ITBS)
ij + β j(
No Freshman Age)
ij
€
β0 j= γ00+ γ0
(
Average School SES)
j+ γ0(
Average School ITBS)
j+ γ04(
Primarily Latino School)
j+classes and not more than one AP class throughout high school. Students in the honors with one AP category took four or more honors class and not more than one AP class. Finally, students designated as being in the advanced track took six or more honors courses and at least two AP courses while in high school. These variables were included in the analyses as dummy variables.