Regressions for 36-315 Project
2024-04-17
Introduction
Dataset Description
This data is on 139 universities in the United
States, collected by Opportunity Insights. It contains 81 variables
detailing application rates, admissions, tests score brackets, student
income, and university prestige. This dataset provides a great way to
view how admissions, applications, scores, and income vary between
different tiers of universities and public versus private institutions.
The rows of this dataset represent the student matriculation and
admission outcomes from one of the 14 income brackets at a specific
university. There are 1946 rows in total. The 14 income brackets are
based on the student’s parents income percentile in relation to the US
national income distribution. Of the expanse of variables in this
dataset, this report focuses on income level, relative application and
attendance rates, university tier, whether the school is public, whether
it is a flagship university. Relative attendance is the proportion of
students attending a given college among all test-takers within a
specific parent income bin. This variable is represented by the
proportion of mean attendance rate for all income bins for the given
university. Relative application is the relative proportion of all the
standardized test takers who sent their scores to that college.
University tiers in this dataset go from 1 to 6, representing Ivy-Plus,
Other elite college, Highly selective public college, Highly selective
private college, Selective public college, and Selective private
college, respectively. These tiers were based on Barron’s selectivity
groupings. For some exploratory data analysis, tiers were combined to
represent both private and public schools in the same Barron’s
groupings. Additionally, for some plots, a dummy variable that indicated
whether a school was among the top “tier” grouping, or not, was created,
taking the value of 1 if it was and 0 if not.
Research Questions
This report focuses on the following three research questions:
1.) Does income level affect the application or attendance rate to a given college, or to more/less prestigious colleges as a group? Does this change relative to SAT scores?
2.) How does university tier (variable rearranged to be based on prestige) vary based on whether the school is public, private, and or a flagship school?
3.) What does the relationship between relative attendance and relative application look like? How does income affect this relationship?
To answer these questions, we will first
create several plots for exploratory data analysis before diving in to
graphical analysis, as well as testing some of our theories using a few
regressions.
Exploratory Data Analysis
Figure 1: MDS Plot, Colored by Tier
The Multidimensional Scaling (MDS) plot
depicted above shows a representation of the relationships between
universities based on their respective prestige tiers. MDS is a
dimensionality reduction technique that takes high-dimensional data and
transforms it to a lower-dimensional space, preserving pairwise
distances between the data. Universities are represented as points with
similarity of their attributes determining their position. The colors
assigned to each point indicate the prestige tier to which the
university belongs:
- Green: Ivy Plus institutions.
- Red: Highly Selective Private universities.
- Orange: Highly Selective Public universities.
- Blue: Selective Private universities.
- Purple: Selective Public universities.
- Cyan: Other Elite Schools, encompassing both public and private institutions.
The clusters of points that are close together suggest a similarity in prestige tier among the corresponding universities while points that are more distant from one another indicate differences in prestige levels or unique characteristics. Taking a look at the graph above, we observe two primary groups both with a similar distribution of colored points with a majority being other elite schools with the other tiers in them as well. It is important to note that there is also a smaller group of other elite schools between the two clusters, which shows that there is a little more variation in other elite schools compared to other schools. This overall shows that there are lots of similarities between more selective schools, but there are still some elite schools that have differences between the other schools in this dataset of higher tier schools.
Figure 2: Bar Plot of University Prestige, Facetted by School Type and Colored by Flagship Status(Public)
This graph was created as part of the
exploratory data analysis on the universities in this dataset in order
to see the distribution of public and private schools within each
university tier as well as the distribution of flagship schools, which
is related to the second research question. This relates to other
research questions involving the income of student’s parents as public
schools tend to have a lower tuition. This graph shows that most of the
schools in the dataset are private schools. The Ivy League schools are
all private but those in the highly selective tier are about 40% public
and those in the selective tier are majority public. The other elite
schools are mostly private schools. The highest count of flagship
schools are in the selective tier which makes sense given this is the
highest concentration of public schools.
Figure 3: Heatmap of Relative Application and Relative Attendance
The heat map above was created to view the
different counts for the relative application and relative attendance
variable. Above it can be seen that there appears to be a greater amount
of data where relative application and relative attendance is lower.
There are not many points where relative application is low but relative
attend is high meaning there are not many cases where the application
rate is low but the attendance rate is high for a college. From the
information gathered from this map, it seems that the colleges in the
data set are medium sized since most students tend to apply and attend
larger colleges. By understanding this information, we can further
investigate the third research question regarding the relationship
between relative application and relative attendance through the lens of
parental income.
Figure 4: Histogram of Application Rates, Facetted by Income
Here, it appears clear that there are more
instances of higher application rates among higher parental income
groups, though this does not obviously seem to change based on prestige.
These variables were included primarily to shed light on the first
research question, though more analyses is obviously needed.
Graphical Analysis
Figure 5: Relationship between Relative Attendance Rate over Relative Application
The plot above is a linear regression plot by
school selectivity tiers investigating the relationship between relative
application rate and attendance rate. This plot is informative for the
dataset as it allows for discovery of the effect of school selectivity
tiers in the ratio of attendance over application rates. This plot
clearly shows the positive correlation between relative application rate
and relative attendance rate across any school, however, it is
interesting to see the slopes of the linear regression lines for each
tier. Interestingly, the slopes show that private schools and more
competitive school tiers tend to have higher relative attendance over
application ratios. Although highly selective public schools are more
competitive than selective private, they seem to be lower which raises
more questions about the dataset. These questions include the effect of
private and public on attendance rates, why is there a higher
application and attendance rate for private schools, as their regression
lines are longer, and why do people tend to attend private schools even
if they are less selective?
Figure 6: Correlation between Relative Application and Relative Attendance by Income
The scatterplot above was created to attempt to
answer the third research question regarding the relationship between
relative attendance and relative application. The scatterplot shows that
there appears to be a positive correlation between relative attendance
and relative application since as one variable increases, the other does
as well. Additionally, the points were colored by parental income, where
the higher the income the bluer the points and the lower the income the
blacker the points. Above it can be seen that the bluer points appear to
have a greater spread which means that those of a higher income tend to
have a larger spread across relative application and relative
attendance. The darker points which represent those of lower income
appear to have a cluster where relative application is 1, and relative
attendance is less than 2.5 which shows a much smaller spread. In
regards to the research question, it appears that there could be a
relationship between relative attendance and relative application, but
at minimum there appears to be a positive correlation. In regards to how
income effects this, since the relationship between relative attendance
and relative application does not seem to be impacted by income, thus
there is not a relationship between relative attendance/application and
parental income.
Figure 7: Dendrogram Displaying Relationship Between Relative Attendance and Application, Colored by Tier
The above dendrogram was created to observe the
similarity between relative attendance/application levels and assist
with the answering of research question 1. The blue labels here indicate
the “most prestigious” colleges–as seen above, they are incredibly
clustered together among one area of the dendrogram. This dendrogram was
created using the averages for each college such that general trends
could be analyzed and the labels could be properly read. This seems to
indicate a high degree of similarity between prestigious colleges when
it comes to relative attendance and relative application rates, while
less prestigious colleges have more variation. This would seem to
indicate that there is a difference between more and less presitigous
colleges when it comes to relative attendance and application
rates.
Hypothesis Testing and Analysis of Variance
Model Fitting and Summaries
Four models were fit to assist in answering our
research questions–primarily the first. Two predict relative attendance
rate, while 2 predict relative application rate. Both use only parental
income and the dummy indicating college prestige as predictors. However,
the two regressions predicting either of the dependence variables differ
in that 1 includes only the fixed effects of both variables, while the
other includes an interaction between the two. This was done in order to
see if the effect of one depends on the classification of another for
each income group, for example if attendance rate is lower for the
99.5th percentile level for prestigious colleges as compared to non
prestigious colleges.
##
## Relative Application Rates, with an Interation with College Prestige
## ========================================================================================================
## Dependent variable:
## ---------------------------------------------------
## rel_apply
## (1) (2)
## --------------------------------------------------------------------------------------------------------
## factor(par_income_bin)30 0.035 (0.056) 0.035 (0.059)
## factor(par_income_bin)50 -0.030 (0.056) -0.027 (0.059)
## factor(par_income_bin)65 -0.064 (0.056) -0.056 (0.059)
## factor(par_income_bin)75 -0.053 (0.056) -0.043 (0.059)
## factor(par_income_bin)85 0.036 (0.056) 0.052 (0.059)
## factor(par_income_bin)92.5 0.198*** (0.056) 0.223*** (0.059)
## factor(par_income_bin)95.5 0.326*** (0.056) 0.359*** (0.059)
## factor(par_income_bin)96.5 0.406*** (0.056) 0.442*** (0.059)
## factor(par_income_bin)97.5 0.483*** (0.056) 0.522*** (0.059)
## factor(par_income_bin)98.5 0.586*** (0.056) 0.633*** (0.059)
## factor(par_income_bin)99.400002 0.730*** (0.056) 0.784*** (0.059)
## factor(par_income_bin)99.5 0.732*** (0.056) 0.786*** (0.059)
## factor(par_income_bin)100 0.765*** (0.056) 0.822*** (0.059)
## factor(prestigious)1 -0.138*** (0.038) 0.179 (0.141)
## factor(par_income_bin)30:factor(prestigious)1 0.001 (0.199)
## factor(par_income_bin)50:factor(prestigious)1 -0.039 (0.199)
## factor(par_income_bin)65:factor(prestigious)1 -0.098 (0.199)
## factor(par_income_bin)75:factor(prestigious)1 -0.123 (0.199)
## factor(par_income_bin)85:factor(prestigious)1 -0.180 (0.199)
## factor(par_income_bin)92.5:factor(prestigious)1 -0.292 (0.199)
## factor(par_income_bin)95.5:factor(prestigious)1 -0.377* (0.199)
## factor(par_income_bin)96.5:factor(prestigious)1 -0.416** (0.199)
## factor(par_income_bin)97.5:factor(prestigious)1 -0.457** (0.199)
## factor(par_income_bin)98.5:factor(prestigious)1 -0.543*** (0.199)
## factor(par_income_bin)99.400002:factor(prestigious)1 -0.623*** (0.199)
## factor(par_income_bin)99.5:factor(prestigious)1 -0.626*** (0.199)
## factor(par_income_bin)100:factor(prestigious)1 -0.658*** (0.199)
## Constant 0.888*** (0.040) 0.861*** (0.041)
## --------------------------------------------------------------------------------------------------------
## Observations 1,946 1,946
## R2 0.305 0.319
## Adjusted R2 0.300 0.309
## Residual Std. Error 0.470 (df = 1931) 0.466 (df = 1918)
## F Statistic 60.506*** (df = 14; 1931) 33.239*** (df = 27; 1918)
## ========================================================================================================
## Note: *p<0.1; **p<0.05; ***p<0.01##
## Relative Attendance Rates, with an Interation with College Prestige
## ========================================================================================================
## Dependent variable:
## ---------------------------------------------------
## rel_attend
## (1) (2)
## --------------------------------------------------------------------------------------------------------
## factor(par_income_bin)30 0.014 (0.094) 0.020 (0.098)
## factor(par_income_bin)50 0.004 (0.094) 0.015 (0.098)
## factor(par_income_bin)65 -0.004 (0.094) 0.015 (0.098)
## factor(par_income_bin)75 0.013 (0.094) 0.034 (0.098)
## factor(par_income_bin)85 0.075 (0.094) 0.097 (0.098)
## factor(par_income_bin)92.5 0.232** (0.094) 0.266*** (0.098)
## factor(par_income_bin)95.5 0.380*** (0.094) 0.419*** (0.098)
## factor(par_income_bin)96.5 0.482*** (0.094) 0.523*** (0.098)
## factor(par_income_bin)97.5 0.645*** (0.094) 0.688*** (0.098)
## factor(par_income_bin)98.5 0.859*** (0.094) 0.910*** (0.098)
## factor(par_income_bin)99.400002 1.163*** (0.094) 1.207*** (0.098)
## factor(par_income_bin)99.5 1.186*** (0.094) 1.221*** (0.098)
## factor(par_income_bin)100 1.448*** (0.094) 1.439*** (0.099)
## factor(prestigious)1 -0.106* (0.063) 0.190 (0.237)
## factor(par_income_bin)30:factor(prestigious)1 -0.072 (0.335)
## factor(par_income_bin)50:factor(prestigious)1 -0.120 (0.335)
## factor(par_income_bin)65:factor(prestigious)1 -0.215 (0.335)
## factor(par_income_bin)75:factor(prestigious)1 -0.241 (0.335)
## factor(par_income_bin)85:factor(prestigious)1 -0.263 (0.335)
## factor(par_income_bin)92.5:factor(prestigious)1 -0.400 (0.335)
## factor(par_income_bin)95.5:factor(prestigious)1 -0.453 (0.335)
## factor(par_income_bin)96.5:factor(prestigious)1 -0.475 (0.335)
## factor(par_income_bin)97.5:factor(prestigious)1 -0.496 (0.335)
## factor(par_income_bin)98.5:factor(prestigious)1 -0.596* (0.335)
## factor(par_income_bin)99.400002:factor(prestigious)1 -0.503 (0.335)
## factor(par_income_bin)99.5:factor(prestigious)1 -0.410 (0.335)
## factor(par_income_bin)100:factor(prestigious)1 0.096 (0.335)
## Constant 0.819*** (0.067) 0.794*** (0.070)
## --------------------------------------------------------------------------------------------------------
## Observations 1,944 1,944
## R2 0.287 0.291
## Adjusted R2 0.281 0.281
## Residual Std. Error 0.784 (df = 1929) 0.785 (df = 1916)
## F Statistic 55.366*** (df = 14; 1929) 29.058*** (df = 27; 1916)
## ========================================================================================================
## Note: *p<0.1; **p<0.05; ***p<0.01Anova Results
| Regressions for Relative Application | Regressions for Relative Attendance | |
|---|---|---|
| F-stat | 2.9974000 | 0.8043 |
| P-value | 0.0002244 | 0.6562 |
The model using an interaction between parental
income and college prestige appears to better explain the the data and
predict relative application rates, as compared to a model using just
the fixed effects of both without an interaction. This can be seen by
the F-test conducted by the ANOVA function–the p-value of comparison
between the two models was 0.0002244, much less than the typical .05
significance level. On the other hand, the model using the same
interaction to predict relative attendance rates was proven to be less
effective than the model without the interactions–the f-statistic when
comparing the nested models was 0.8043, resulting in a p-value of
0.6562, much greater than the typical significance level of .05. Thus,
for the first comparison, we must reject the null hypothesis that the
coefficients added by the interaction of prestige and income is 0, while
for the second we cannot reject this null hypothesis. This would
indicate that a school being prestigious does not change the effect
parental income on relative attendance rates, but it does change the
effect of parental income on relative application rates. However, given
all the interaction coefficients in both regressions are observed to be
negative but the majority of the parental income coefficients were
positive, particularly the higher income bins, we must assume that the
positive effect of of parental income on relative application rates is
somewhat negated by the school being prestigious. That is to say,
application rates to prestigious colleges do not increase as much due to
parental income being higher.
Unanswered Questions and Future Work
Overall, in regards to the first research
question, we have observed that there is certainly a relationship
between parental income, college prestige, and relative application and
attendance rates. Higher parental income groups tended to increase
relative application rates and attendance rates, while college prestige
tended to decrease both. However, we did not do the work to include SAT
scores within our analysis, which could have led to different results.
In regards to the second question, we observed that the vast majority of
more prestigious schools were private schools, while many of the
flagship schools(which are all public) were not among the most selective
schools. In regards to the third question, we observed that the
relationship between relative application rates and relative attendance
rates does appear to vary based on parental income and college
prestige–highly prestigious schools like Ivy league schools and highly
selective private schools had attendance rates that increased faster
relative to application rates than other tiers of schools.