Regression Analysis:
Explaining Graduation Rates at Public and Private Colleges
Question: A database on my statistical resource blog
has information on the graduation rate and SAT score for 31 public universities
and 20 private universities in the state of California. Use the information in this database to
determine the impact of SAT scores and private versus public university status
on the ontime graduation rate for these universities.
In particular consider:
After holding constant SAT scores do private universities
have a better ontime graduation rate than public universities?
Does the impact of SAT score on ontime graduation rate
differ for private and public universities?
Go to this blog on the statistical resource page to get the
raw data:
Analysis: The variable of interest in this study is a
proportion, specifically the proportion of students who start at a university
who graduate within six years. Proportions are bounded between 0 and 1 so
the error term of a regression where the proportion is the dependent variable
is not likely to be normally distributed.
This affects the standard error of the regression coefficients. Also, predictions from such a model could be
below 0 or above 1. To avoid these
problems we transform the graduation ontime proportion into the log of the
odds where
LODDS=Log(p/(1p))
where p is the
proportion of students who graduate within six years.
SAT scores vary from 200 to 800. We transform the SAT measure in the database
by subtracting 200. This transformation
affects the estimation and interpretation of the constant term of the
regression. By transforming the SAT
measure in this manner we can interpret the constant term as the likely
graduation rate when SAT measure is 200.
An aside to teachers
of statistics: Put the formulas for
regression coefficients on the blackboard and have your students explain why
the subtraction of 200 from the SAT measure impacts the constant term in the
regression but does not impact the slope of the SAT measure variable.
Back to the analysis:
Analysis of the impact of private school status on ontime
graduation rates holding constant SAT scores:
Dependent variable is log of the odds of graduation on time.
Explanatory variables are SATMEAS from database minus 200
and a dummy variable set to 1 if the school is private and set to 0 otherwise.
Graduation Rate as a
function of SAT Measure
and Private School Dummy


Coeff

tstat


Constant Term

3.2

14.5

SATMEAS

0.011

16.7

DUMPRIV

0.04

0.4

Adjusted R^{2}

0.868

Observations:
SATMEAS is a highly significant explainer of graduation
rate.
Private school status is not significantly impact graduation
rate in a model which includes the SAT variable.
Questions for
students:
The coefficient of the SATMEAS gives the impact of the SAT
score on the log of the odds. Can you
figure out how to get the impact of SATMEAS on the ontime graduation
proportion?
How might you improve this model?
Discuss the meaning of the constant term. Are you troubled that this value is negative?
Analysis of differences in the impact of SAT scores on graduation
rates in private and public schools:
In order to evaluate differences in the impact of SAT scores
on public and private graduation rates I estimate two separate regression
models one for private schools and the other for public schools. Results are presented below.
Graduation Rate Models for
Public and Private Universities in California


Public Schools

Private Schools


Coeff.

tstat

Coeff

tstat


Constant Term

3.5

12.3

2.7

7.3

SATMEAS

0.0122

13.8

0.0098

10.2

Adjusted R^{2}

0.8638

0.8435

Observations:
Graduation rate is a significant explainer in ontime
graduation rate for both public and private colleges in California.
However, the coefficient of the SATMEAS is 25% higher for
public universities.
Concluding Question: This post ends with a question. Why is the impact of SAT scores on
graduation rates so much higher for the public schools than for the private
schools? I suspect that other factors
perhaps student debt, socioeconomic characteristics of the student population,
or the percent of students attending part time play a role. I may collect more data and analyze these
questions further.
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