Final: Texas Higher Education Opportunity Project (THEOP)
Critical Thinking Group 1
Ben Inbar, Cliff Lee, Daria Dubovskaia, David Simbandumwe, Jeff Parks
Abstract
This work examines changes in the racial and ethnic composition of admissions at Texas A&M college following the 1996 Hopwood case, which put a court prohibition on affirmative action. We estimate the extent to which these universities used affirmative action before the ban, and we examine how admissions officers at these universities adjusted the relative weights given to key applicant criteria throughout the suspension. We model the extent to which these new regulations succeeded in preserving minority admission rates at pre-Hopwood levels after examining whether changes in relative weights favored minority applicants. We discovered that most colleges followed the Hopwood rule, so that direct advantages offered to black and Hispanic candidates vanished (and, in some circumstances, became disadvantages). While there is evidence that universities changed the weights they placed on applicant characteristics other than race and ethnicity in ways that aided underrepresented minority applicants, these changes in the admissions process were unable to maintain the share of admitted students held by black and Hispanic applicants. As a result, these alternative admissions procedures have not been a reliable proxy for race and ethnicity.
Introduction
Despite some claims, people have acknowledged a gap in access to higher education still exists, especially for minority students. However, many have objected to affirmative action policies as a solution, and in 1996 in Texas, the Fifth Court Circuit of Appeals outlawed the use of race for college admissions in the state. After a noticeable drop in minority student enrollments, the governor of Texas signed a new bill as a compromise: any high school student finishing in top decile received a guaranteed admission to Texas public university (Texas A&M University). Instead of competing with applicants from the entire state, students only had to compete with their immediate classmates to access higher education.
Literature review
Discuss how other researchers have addressed similar problems, what their achievements are, and what the advantage and drawbacks of each reviewed approach are. Explain how your investigation is similar or different to the state-of-the-art. Please cite the relevant papers where appropriate.
Methodology
Discuss the key aspects of your problem, data set and regression model(s). Given that you are working on real-world data, explain at a high-level your exploratory data analysis, how you prepared the data for regression modeling, your process for building regression models, and your model selection.
Data
We employ administrative records from a Texas A&M University (1992-2022) that include info about admissions selectivity, public/private status, and the ethno-racial composition of their student body for this analysis. Importantly, the time period for the public organization comprises years prior to and following the judicial ban on affirmative action. This is significant because, while the judicial restriction applied to all schools in the 5th Circuit District, the top 10% policy was restricted to public colleges and universities. These records contain a plethora of information about the applicant pool, and have been standardized where necessary, and checked for consistency.
The application dataset contained 163,027 observations of 24 predictor variables and transcripts dataset contained 637,028 observations of 10 predictor variables, where each record represented an individual applicant. These variables describe items typically found on a college admission application such as the year and term an applicant desired to enroll, applicant demographics, applicant academic characteristics, and high school characteristics.
Unfortunately, the data does not often include information regarding a student’s high school academics or application essays. In evaluating the results, we take great note of these data constraints.
Each record in the application dataset included a response variable “admit” (Institution’s admission decision), was a boolean where “1” indicated the person was admitted.
A sample of the data appears in the following table:
While exploring this data, we made the following observations for the applications data:
- 21 variables were categorical, and 3 were numeric.
actR,gradyear,hscentury,hslosvariables had more that 50% of the missing data.
| variable | complete_rate | n_missing | min | max |
|---|---|---|---|---|
| Texas_resident | 1.00 | 0 | NA | NA |
| US_Citizen | 1.00 | 0 | NA | NA |
| actR | 0.46 | 72133 | 1 | 24 |
| admit | 1.00 | 0 | NA | NA |
| admit_prov | 1.00 | 0 | NA | NA |
| decileR | 1.00 | 0 | NA | NA |
| enroll | 1.00 | 0 | NA | NA |
| ethnic | 1.00 | 50 | NA | NA |
| gradyear | 0.35 | 86914 | 2 | 22 |
| hscentury | 0.32 | 91636 | NA | NA |
| hseconstatus | 0.72 | 38094 | NA | NA |
| hsinstate | 1.00 | 103 | NA | NA |
| hslos | 0.41 | 79635 | NA | NA |
| hsprivate | 0.98 | 2375 | NA | NA |
| hstypeR | 0.91 | 12338 | NA | NA |
| major_field | 1.00 | 2 | NA | NA |
| male | 1.00 | 52 | NA | NA |
| quartile | 1.00 | 10 | NA | NA |
| satR | 1.00 | 0 | 3 | 111 |
| studentid | 1.00 | 0 | NA | NA |
| studentid_uniq | 1.00 | 0 | NA | NA |
| termdes | 1.00 | 0 | NA | NA |
| testscoreR | 1.00 | 0 | 3 | 111 |
| yeardes | 1.00 | 0 | 1992 | 2002 |
76% of all the students were admitted to the university.
By checking the frequency of the variables with few unique values
(“male”,“ethnic”, “citizenship”, “restype”,“decileR”,“major_field”), we
checked the frequency of each value. As demonstrated by the graph below,
more than 50% of the data for the citizenship,
restype and ethnic variables was
US Citizen, Texas Resident and
White, Non-Hispanic accordingly:
Transform Data
Students with missing values for ethnic variable are
combined with “White, Non-Hispanic” students, resulting in cautious
estimates of policy effects.
## The original levels Black, Non-Hispanic Hispanic American Indian/Alaskan Native Asian or Pacific Islander White, Non-Hispanic International Other None
## have been replaced by Black, Non-Hispanic Hispanic American Indian/Alaskan Native Asian or Pacific Islander International Other White, Non-Hispanic| variable | complete_rate | n_missing | min | max |
|---|---|---|---|---|
| Texas_resident | 1 | 0 | NA | NA |
| US_Citizen | 1 | 0 | NA | NA |
| admit | 1 | 0 | NA | NA |
| enroll | 1 | 0 | NA | NA |
| ethnic | 1 | 0 | NA | NA |
| hsinstate | 1 | 0 | NA | NA |
| hsprivate | 1 | 0 | NA | NA |
| male | 1 | 0 | NA | NA |
| satR | 1 | 0 | 3 | 111 |
| termdes | 1 | 0 | NA | NA |
| testscoreR | 1 | 0 | 3 | 111 |
| top10 | 1 | 0 | NA | NA |
| yeardes | 1 | 0 | 1992 | 2002 |
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As the first step, we built the generalized linear model based on the dataset with some variables transformed to categorical from numeric for the time before Fall, 1998.
| variable | complete_rate | n_missing | min | max |
|---|---|---|---|---|
| Texas_resident | 1 | 0 | NA | NA |
| US_Citizen | 1 | 0 | NA | NA |
| admit | 1 | 0 | NA | NA |
| enroll | 1 | 0 | NA | NA |
| ethnic | 1 | 0 | NA | NA |
| hsinstate | 1 | 0 | NA | NA |
| hsprivate | 1 | 0 | NA | NA |
| male | 1 | 0 | NA | NA |
| satR | 1 | 0 | 3 | 111 |
| termdes | 1 | 0 | NA | NA |
| testscoreR | 1 | 0 | 3 | 111 |
| top10 | 1 | 0 | NA | NA |
| yeardes | 1 | 0 | 1992 | 1998 |
Since our dependent variable is binary (0 and 1), we used logistic
regression. To do so, the function glm() with
family=binomial was used. At the beginning, variable below
were included.
##
## Call:
## glm(formula = admit ~ . - enroll - yeardes - termdes, family = binomial(link = "probit"),
## data = pre_df)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -3.8936 0.0305 0.1906 0.6786 2.7660
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.0649721 0.0569552 -18.698 < 2e-16 ***
## male1 -0.1443959 0.0131538 -10.978 < 2e-16 ***
## ethnicHispanic -0.0258000 0.0370032 -0.697 0.4857
## ethnicAmerican Indian/Alaskan Native -0.8790031 0.0939285 -9.358 < 2e-16 ***
## ethnicAsian or Pacific Islander -0.9181691 0.0427157 -21.495 < 2e-16 ***
## ethnicInternational -0.4846015 0.0896390 -5.406 6.44e-08 ***
## ethnicOther -0.9894696 0.0689488 -14.351 < 2e-16 ***
## ethnicWhite, Non-Hispanic -0.7983680 0.0325750 -24.509 < 2e-16 ***
## US_Citizen1 0.0766258 0.0420904 1.821 0.0687 .
## Texas_resident1 0.7330951 0.0335813 21.830 < 2e-16 ***
## satR 0.0282780 0.0003536 79.970 < 2e-16 ***
## testscoreR NA NA NA NA
## top10TRUE 1.4734079 0.0196313 75.054 < 2e-16 ***
## hsprivatePrivate -0.1714509 0.0257750 -6.652 2.89e-11 ***
## hsprivateNone -0.2848834 0.0473206 -6.020 1.74e-09 ***
## hsinstateYes 0.0653105 0.0361204 1.808 0.0706 .
## hsinstateNone -0.2056722 0.2039254 -1.009 0.3132
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 73803 on 69018 degrees of freedom
## Residual deviance: 50413 on 69003 degrees of freedom
## AIC: 50445
##
## Number of Fisher Scoring iterations: 6Using StepAic() function helped to improve the model performance.
| Observations | 69019 |
| Dependent variable | admit |
| Type | Generalized linear model |
| Family | binomial |
| Link | probit |
| 𝛘²(15) | 23390.53 |
| Pseudo-R² (Cragg-Uhler) | 0.44 |
| Pseudo-R² (McFadden) | 0.32 |
| AIC | 50444.85 |
| BIC | 50591.12 |
| Est. | S.E. | z val. | p | |
|---|---|---|---|---|
| (Intercept) | -1.06 | 0.06 | -18.70 | 0.00 |
| male1 | -0.14 | 0.01 | -10.98 | 0.00 |
| ethnicHispanic | -0.03 | 0.04 | -0.70 | 0.49 |
| ethnicAmerican Indian/Alaskan Native | -0.88 | 0.09 | -9.36 | 0.00 |
| ethnicAsian or Pacific Islander | -0.92 | 0.04 | -21.49 | 0.00 |
| ethnicInternational | -0.48 | 0.09 | -5.41 | 0.00 |
| ethnicOther | -0.99 | 0.07 | -14.35 | 0.00 |
| ethnicWhite, Non-Hispanic | -0.80 | 0.03 | -24.51 | 0.00 |
| US_Citizen1 | 0.08 | 0.04 | 1.82 | 0.07 |
| Texas_resident1 | 0.73 | 0.03 | 21.83 | 0.00 |
| satR | 0.03 | 0.00 | 79.97 | 0.00 |
| top10TRUE | 1.47 | 0.02 | 75.05 | 0.00 |
| hsprivatePrivate | -0.17 | 0.03 | -6.65 | 0.00 |
| hsprivateNone | -0.28 | 0.05 | -6.02 | 0.00 |
| hsinstateYes | 0.07 | 0.04 | 1.81 | 0.07 |
| hsinstateNone | -0.21 | 0.20 | -1.01 | 0.31 |
| Standard errors: MLE |
Model Results
As it appeared, ethnic variable had the most negative
effect for the university acceptance. While being Texas resident or
being in top 10% could help to be admitted.
The null deviance of 7.380338^{4} defined how well the
target variable could be predicted by a model with only an intercept
term.
The residual deviance of 5.041285^{4} defined how well
the target variable could be predicted by the AIC model that we fit with
the predictor variables listed above. The lower the value, the better
the model’s predictions of the response variable.
The p-value associated with this Chi-Square Statistic
was 0 (less than .05), so the model could be useful.
The Akaike information criterion (AIC) was 5.044485^{4}.
The lower the AIC value, the better the model’s ability to fit the
data.
Checking Model Assumptions
The resulting plots show us similar to model 1 picture: under-fitting at lower predicted values, with the predicted proportions being larger than the observed proportions; over-fitting for a couple of predictor patterns at higher predicted values with the predicted values are much larger than the predicted proportions. The Standardized Pearson Residuals plot shows an approximate Standard Normal distribution if the model fits. The model seems good in the middle range but there are extremes on the right and left sides. Some normality of the residuals for the binomial logistic regression models is just an evidence of a decent fitting model.
By checking the linearity assumptions, we see that only
rm seem like linear relation with the logit results.
## 1 2 3 4 5 6
## "1" "1" "1" "1" "0" "1"The Standardized Residuals plot seems to have a constant variance
though there are some outliers.
The marginal model plots below show reasonable agreement across the two sets of fits indicating that model_2_aic is a valid model.
## Error in names(dat) <- object$term :
## 'names' attribute [1] must be the same length as the vector [0]
## Error in names(dat) <- object$term :
## 'names' attribute [1] must be the same length as the vector [0]
In terms of multicollinearity, all variables have a VIF less than 5. As a result, multicollinearity shouldn’t be a problem for our model.
## GVIF Df GVIF^(1/(2*Df))
## male 1.051668 1 1.025509
## ethnic 1.994261 6 1.059209
## US_Citizen 1.546489 1 1.243579
## Texas_resident 3.086787 1 1.756926
## satR 1.197600 1 1.094349
## top10 1.018228 1 1.009073
## hsprivate 1.383635 2 1.084564
## hsinstate 3.311345 2 1.348966Predict
Once the uniform admission law was fully in force, fall, 1998.
Lasso
## 17 x 1 sparse Matrix of class "dgCMatrix"
## s1
## (Intercept) -1.83381217
## male1 -0.24603449
## ethnicHispanic .
## ethnicAmerican Indian/Alaskan Native -1.42324025
## ethnicAsian or Pacific Islander -1.53683380
## ethnicInternational -0.79574261
## ethnicOther -1.59897292
## ethnicWhite, Non-Hispanic -1.32162247
## US_Citizen1 0.12029934
## Texas_resident1 1.24391271
## satR 0.04823873
## testscoreR .
## top10TRUE 2.73974407
## hsprivatePrivate -0.28104105
## hsprivateNone -0.48453373
## hsinstateYes 0.08437993
## hsinstateNone -0.28174646## 17 x 1 sparse Matrix of class "dgCMatrix"
## s1
## (Intercept) -1.906999e+00
## male1 -2.045992e-01
## ethnicHispanic 1.628669e-01
## ethnicAmerican Indian/Alaskan Native -9.314747e-01
## ethnicAsian or Pacific Islander -1.193380e+00
## ethnicInternational -4.823771e-01
## ethnicOther -1.163898e+00
## ethnicWhite, Non-Hispanic -1.033448e+00
## US_Citizen1 8.443794e-02
## Texas_resident1 1.205775e+00
## satR 4.561657e-02
## testscoreR 8.843864e-17
## top10TRUE 2.627511e+00
## hsprivatePrivate -2.075913e-01
## hsprivateNone -4.242971e-01
## hsinstateYes 6.028163e-02
## hsinstateNone .##
## Call: cv.glmnet(x = X, y = Y, nfolds = 5, family = "binomial", link = "logit", standardize = TRUE, alpha = 1)
##
## Measure: Binomial Deviance
##
## Lambda Index Measure SE Nonzero
## min 0.0004846 64 0.7325 0.001479 14
## 1se 0.0025862 46 0.7338 0.001570 15
## $deviance
## lambda.1se
## 0.7332928
## attr(,"measure")
## [1] "Binomial Deviance"
##
## $class
## lambda.1se
## 0.1826743
## attr(,"measure")
## [1] "Misclassification Error"
##
## $auc
## [1] 0.8654476
## attr(,"measure")
## [1] "AUC"
##
## $mse
## lambda.1se
## 0.2410274
## attr(,"measure")
## [1] "Mean-Squared Error"
##
## $mae
## lambda.1se
## 0.4864324
## attr(,"measure")
## [1] "Mean Absolute Error"## $AICc
## [1] -23258.91
##
## $BIC
## [1] -23130.93## $AICc
## [1] -23162.24
##
## $BIC
## [1] -23025.12| Est | |
|---|---|
| top10TRUE | 2.740 |
| Texas_resident1 | 1.244 |
| US_Citizen1 | 0.120 |
| hsinstateYes | 0.084 |
| satR | 0.048 |
| ethnicHispanic | 0.000 |
| testscoreR | 0.000 |
| male1 | -0.246 |
| hsprivatePrivate | -0.281 |
| hsinstateNone | -0.282 |
| hsprivateNone | -0.485 |
| ethnicInternational | -0.796 |
| ethnicWhite, Non-Hispanic | -1.322 |
| ethnicAmerican Indian/Alaskan Native | -1.423 |
| ethnicAsian or Pacific Islander | -1.537 |
| ethnicOther | -1.599 |
| (Intercept) | -1.834 |
Model Results
The coefficients extracted at the lambda.min value are used to predict the relative crime rate for the testing data set. The confusion matrix highlights an accuracy of 91%.
Analysis
| coef | < 1998 | 1998 | 1999 | 2000 | 2001 | 2002 |
|---|---|---|---|---|---|---|
| (Intercept) | -1.065 | -1.437 | -1.719 | -1.277 | -1.408 | -1.699 |
| male1 | -0.144 | -0.399 | -0.252 | -0.145 | -0.114 | -0.087 |
| ethnicHispanic | -0.026 | 0.117 | -0.116 | -0.100 | 0.074 | 0.173 |
| ethnicAmerican Indian/Alaskan Native | -0.879 | 0.346 | 0.095 | -0.047 | 0.081 | 0.089 |
| ethnicAsian or Pacific Islander | -0.918 | -0.194 | -0.383 | -0.407 | -0.421 | -0.211 |
| ethnicInternational | -0.485 | -0.181 | 0.069 | -0.061 | -0.293 | -0.114 |
| ethnicOther | -0.989 | -0.300 | 0.023 | -0.367 | -0.369 | -0.184 |
| ethnicWhite, Non-Hispanic | -0.798 | 0.204 | 0.070 | -0.246 | 0.058 | 0.181 |
| US_Citizen1 | 0.077 | NA | 0.251 | NA | NA | NA |
| Texas_resident1 | 0.733 | 0.836 | 0.514 | 0.627 | 0.376 | 0.574 |
| satR | 0.028 | 0.031 | 0.019 | 0.021 | 0.021 | 0.021 |
| top10TRUE | 1.473 | 2.001 | 1.916 | 2.591 | 3.052 | 3.276 |
| hsprivatePrivate | -0.171 | -0.162 | -0.157 | NA | -0.123 | -0.242 |
| hsprivateNone | -0.285 | -0.487 | -0.144 | NA | -0.087 | -0.185 |
| hsinstateYes | 0.065 | NA | 0.443 | -0.210 | NA | NA |
| hsinstateNone | -0.206 | NA | 0.388 | 0.326 | NA | NA |
| coef | < 1998 | 1998 | 1999 | 2000 | 2001 | 2002 |
|---|---|---|---|---|---|---|
| top10TRUE | 2.740 | 3.899 | 3.399 | 4.899 | 5.815 | 6.375 |
| Texas_resident1 | 1.244 | 1.225 | 0.791 | 0.820 | 0.510 | 0.968 |
| US_Citizen1 | 0.120 | -0.242 | 0.436 | 0.051 | 0.174 | -0.044 |
| hsinstateYes | 0.084 | 0.312 | 0.581 | -0.245 | 0.117 | -0.064 |
| satR | 0.048 | 0.056 | 0.027 | 0.029 | 0.028 | 0.027 |
| male1 | -0.246 | -0.706 | -0.365 | -0.199 | -0.130 | -0.108 |
| hsprivatePrivate | -0.281 | -0.229 | -0.201 | -0.066 | -0.188 | -0.348 |
| hsinstateNone | -0.282 | NA | NA | 0.431 | -0.340 | 0.527 |
| hsprivateNone | -0.485 | -0.700 | -0.179 | -0.115 | -0.093 | -0.326 |
| ethnicInternational | -0.796 | -0.701 | NA | -0.043 | -0.360 | -0.088 |
| ethnicWhite, Non-Hispanic | -1.322 | 0.341 | 0.100 | -0.349 | 0.012 | 0.266 |
| ethnicAmerican Indian/Alaskan Native | -1.423 | 0.579 | NA | NA | 0.003 | 0.061 |
| ethnicAsian or Pacific Islander | -1.537 | -0.376 | -0.590 | -0.567 | -0.690 | -0.353 |
| ethnicOther | -1.599 | -0.510 | NA | -0.483 | -0.442 | -0.304 |
| (Intercept) | -1.834 | -2.420 | -2.441 | -1.741 | -2.071 | -2.299 |
| ethnicHispanic | NA | 0.149 | -0.101 | NA | 0.111 | 0.332 |
| testscoreR | NA | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Discussion and Conclusions
Conclude your findings, limitations, and suggest areas for future work.