# Modeling preferential admissions at elite liberal arts colleges

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6 admits versus ED admits, and overall deposits versus ED deposits that are consistent with actual data. Each student in the ED sub-pool then chooses a single school to which he or she sends an application. To model the assumption that students with higher SAT scores will generally apply to higher-ranked schools, the ED sub-pool is divided into 50 blocks by SAT score; all students in the same block use the same process for choosing a school. More precisely, each ED student in the (1 + t)-th block (where t begins at 0, representing the highest scoring students) randomly generates a number from a chi-squared distribution with mean ν = t, then multiplies by 4 and rounds down to an integer value; this number is the rank of the single school to which the student applies. The intended effect is to have students in the t-th block apply to schools with ranks that are roughly normally distributed with mean t, with a variance larger than that of the usual chi-squared distribution. Each school now has a pool of ED applicants. All ED applicants whose SAT scores are strictly less than the school s cut-off score are rejected and thrown back into the general student pool. The number m ED of ED applicants that a school admits is a fixed percentage of its qualified ED applicant pool; this percentage is modeled as a linear function of the school rank r, with the top school admitting 33% and the bottom school admitting 82%, resulting in the formula ED acceptance rate (r) r. This formula was chosen to reflect IC 2010 data for a subset of these schools reported by Newsweek (Cohen, 2011); the regression line fitted to this data is ED acceptance rate (r) r. Each school determines which m ED students in its qualified ED applicant pool to admit using the process described earlier, based on the common ED round preferential rate; 75% are chosen from the top SAT scores, and 25% are chosen randomly from the remaining scores. Taking a 20-run average, the number of ED applicants admitted to all 50 schools was 12, 240 (with standard deviation 20), roughly 53% of the ED sub-pool. This means that the regular round pool consists of 15, 760 students (56% of the original student pool). Regular Round The regular round pool is divided into 50 blocks by SAT score, with all students in the same block using the same process for choosing a set of schools to which to apply. More precisely, each student in the (1 + t)-th block (where t begins at 0) randomly generates 20 numbers from a chi-squared distribution with ν = t, then multiplies each by 7.5 and rounds down to an integer value; if the resulting number is greater than 49, it is rejected and a new number is generated. Each number is the rank of a school to which the student applies. The intended effect is to have students in the t-th block apply to schools that are roughly normally distributed with mean t, with a variance larger than that of the usual chi-squared distribution. Note that because of rounding, these 20 schools will rarely all be different; in fact, the average number of schools to which a regular round student applies is 14. Students with high SAT and students with low SAT scores may in fact apply to even fewer schools, because it is more likely that two or more values of their school choice random variable will round to the same integer. This reasonably models the assumptions that (a) better students are more confident of getting into one of the schools they apply to, and thus send in fewer applications, and (b) weaker students may also be submitting applications to schools outside the top 50 institutions. Table 5 (Appendix) shows the set of application schools for 10 sample students obtained in one run of the model; more precisely, the table shows the set of schools for one student from the t-th block, Modeling preferential admissions, page 7

9 r 0 r 8, Y 0,0 (r) (r 7) 8 r 18, 0.12, 18 r 50. Figure 7 shows how this assumed regular round yield function compares to the simulated regular round yield (20-run average). The total number of deposits at the top 35 schools is 19, 899, with an average of and standard deviation of 24.5; by the stated criteria, these are comparable to the status quo numbers. With the new generally lower assumed regular round yield, the number of admission offers per student rises to 6 (from 5 in the status quo model). Table 9 (Appendix) shows the academic profile numbers generated by the model, using Y 0,0. In the case where all admissions decisions are preferential ( P ED P REG 100%), reasonable class sizes are obtained by changing the assumed regular round yield function to r 0 r 15, Y 1, (r 15) 15 r 50. Figure 8 (Appendix) shows that this closely resembles the actual regular round yield produced by the model (20-run average). Using Y 1,1, the model gives 19, 950 total deposits, giving an average of per school, with standard deviation of 39.4; these are comparable to status quo numbers. With higher assumed yields, the average number of admission offers a regular round student gets has dropped from 5 to 4. The academic profile numbers produced by the model are shown in Table 10 (Appendix). When P ED P REG 100%, each school is accepting a randomly chosen subset of its qualified applicant pool. (In fact, the effect of each school s cut-off SAT score is negligible; if the model is run with no cut-off scores, the regression line changes from mean SAT score, where r is the school rank, to mean SAT score (r) r.) Thus, in this case the distribution of mean SAT scores among the incoming classes is just a reflection of the distribution of mean SAT scores among the applicant pools. What Table 10 shows is that a great deal of the variation in the academic profile of the top 35 schools is due to the self-selection of student applicants, not to the admissions decisions each of these schools makes. Running the model using extreme rates P ED P REG 0% and P ED P REG 100% does give a sense of the range of possibilities. However, more likely scenarios have preferential rates are half the status quo rates ( P ED 12.5%, P REG 15%) and double the status quo rate ( P ED 50%, P REG 60%). The corresponding adjusted assumed regular round yield functions used were r 0 r 7, Y.125,.15 (r) (r 7) 7 r 13, r 50, and Y.50,.60 (r) r. As usual, the model ran 20 simulations and then averages were computed. Figure 9 (Appendix) illustrates that the numbers of total deposits under the different preferential rates with corresponding adjusted regular round yield functions are comparable. Figures 10 and 11 (Appendix) show the effect on overall acceptance rates and overall yields; the IC 2010 data are included to demonstrate that the curves are all the right basic shape. However, note that reducing preferential rates increases acceptance rates and decreases yield across all institutions in our group of selective liberal arts colleges. Given that acceptance rate and yield are among the measures US Modeling preferential admissions, page 10

12 APPENDIX Table 1 IC 2010 at top 35 liberal arts institutions Institution Name Appl ns Admits Deposits Mean SAT Accept. Rate Yield USN&WR 1 Amherst College Rate Ra 2 nk 2 Swarthmore College Williams College Bowdoin College Carleton College Wesleyan University Haverford College Middlebury College Washington & Lee University Vassar College Hamilton College Bard College Oberlin College Colgate University Macalester College Barnard College Davidson College Grinnell College Kenyon College Colby College Bates College Connecticut College College Bucknell University University Gettysburg College Franklin & Marshall College Trinity College University of Richmond College of the Holy Cross Union College Dickinson College Lafayette College Furman University Rhodes College Centre College Skidmore College Modeling preferential admissions, page 13

13 Table 2 IC 2010 data vs. Simulation (20-run average): Sizes Applications Admits Deposits IC 2010 Total Mean per School Model Total (20-run average) Mean per School Table 3 Simulation Results (20-run average, status quo rates) Applications Admits Deposits Mean SAT Acceptance Rate Yield Modeling preferential admissions, page 14

14 Table 4 Simulated Differences in SAT scores (20-run average, status quo rates) Overall (Admits-Applicants) ED (Admits-Applicants) (Overall-ED) Admits (Overall-ED) Deposits Table 5 Sample Sets of Application Schools t Set of Application Schools 4 {0, 1, 2, 3, 4, 6, 7, 8, 16, 30} 9 {0, 1, 2, 3, 4, 7, 8, 11, 14, 16, 23, 24, 27} 14 {2, 3, 4, 5, 8, 9, 11, 17, 18, 20, 25, 30, 31, 32, 44} 19 {1, 4, 5, 6, 8, 10, 13, 16, 17, 20, 21, 24, 32, 36, 47} 24 {0, 2, 3, 5, 6, 7, 8, 10, 11, 13, 16, 19, 21, 23, 29, 30, 38} 29 {1, 2, 12, 15, 16, 18, 19, 21,22, 24, 25, 26, 31, 32, 35, 36, 40, 41} 34 {4, 8, 9, 11, 16, 17, 18, 20, 22, 23, 27, 28, 29, 30, 37, 38, 42, 46} 39 {10, 13, 15, 20, 21, 23, 24, 26, 27, 33, 42, 43, 46, 48, 49} 44 {5, 6, 8, 9, 12, 14, 18, 22, 23, 24, 28, 29, 31, 34, 36} 49 {10, 12, 13, 15, 16, 17, 20, 25, 27, 32, 40, 41, 42, 43, 46, 48} Modeling preferential admissions, page 15

15 Table 6 Simulated Yields (20-run average, status quo rates) Assumed Regular Round Yield Simulated Regular Round Yield Simulated Overall Yield IC 2010 Overall Yield Modeling preferential admissions, page 16

16 Table 7 Sample Admission Offers t Set of admission offers 4 {2, 3, 4, 6, 7, 8, 16, 30} 9 {1, 11, 14, 16, 23, 24, 27} 14 {9, 20, 25, 30, 31, 32, 44} 19 {6, 17, 32, 36, 47} 24 {38} 29 {35, 36, 40, 41} 34 {17, 20, 28, 37, 38, 42, 46} 39 {42, 43, 46, 48, 49} 44 {29} 49 {41, 42, 43, 46, 48} Modeling preferential admissions, page 17

17 Table 8 Extreme Preferential Rates vs. Status Quo Rates P ED P REG 0% P ED 25%, P REG 50% P ED P REG 100% Deposits Mean SAT Deposits Mean SAT Deposits Mean SAT Modeling preferential admissions, page 18

18 Table 9 Simulation Results (20-run average); P ED = P REG = 0% Applications Admits Deposits Mean SAT Acceptance Rate Yield Table 10 Simulation Results (20-run average): P ED = P REG = 100% Modeling preferential admissions, page 19

19 Applications Admits Deposits Mean SAT Accept. Rate Yield Modeling preferential admissions, page 20

20 Overall Acceptance Rate Mean SAT Research in Higher Education Journal IC 2010 simulated Figure 1. IC 2010 vs. Simulation (Status Quo Rates, 20-run average): Mean SAT IC 2010 simulated Figure 2. IC 2010 vs. Simulation (Status Quo Rates, 20-run average): Overall Acceptance Rate Modeling preferential admissions, page 21

21 Yield Overall Yield Research in Higher Education Journal IC 2010 simulated Figure 3. IC 2010 vs. Simulation (Status Quo Rates, 20-run average): Overall Yield assumed reg. yield simulated reg. yield simulated overall yield IC 2010 overall yield Figure 4. Yields with Status Quo Rates P ED = 25%, P REG = 30% Modeling preferential admissions, page 22

22 Total Deposits Mean SAT Research in Higher Education Journal %,0% 25%, 30% (status quo) 100%, 100% Figure 5. Extreme Preferential Rates vs. Status Quo Rates: Mean SAT Scores %, 0% 25%, 30% (status quo) 100%, 100% Figure 6. Extreme Preferential Rates vs. Status Quo Rates: Total Deposits Modeling preferential admissions, page 23

23 Regular Round Yield Regular Round Yield Research in Higher Education Journal simulated reg. yield assumed reg. yield Figure 7. Regular Round Yields with Y 0,0, P ED = P REG = 0% simulated reg. yield assumed reg. yield Figure 8. Regular Round Yields with Y 1,1, P ED = P REG = 100% Modeling preferential admissions, page 24

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