The post analyzes predictions from two regression
models of the relationship between probability of completing school in six years
or less and school average SAT scores. Models were estimate over separate samples
of private and public universities inside the state of California. All data was obtained from the College Scorecard.
Question: Regression models where ontime graduation rates
were explained by an average college wide SAT score were estimated for 31
public universities and 20 private
universities in the state of California.
The dependent variable in
this model was log of the odds of the likelihood a person in the college
graduates within 6 years of entering the college.
The explanatory variable in
the model was the average of four SAT numbers  the 25^{th} percentiles
SAT verbal and math scores and the 75^{th} percentile SAT verbal and
math scores.
The results from two regression
models are presented below.
On
Time Graduation Results


Public
Universities


Coeff

t


SATAVG

0.0122

13.83

_cons

6.008

12.95

n

31


F

191.2


Private
universities


Coef.

t


SATAVG

0.0098

10.17

_cons

4.714

8.32

n

20


F

103.4

Discuss the implications of
the regression results?
Calculate the predicted ontime
graduation probability of a public and private college with SAT averages of 550,
600, and 650 consistent with this model.
Are the distribution of SATAVG
values the same for public and private institutions? Compare predicted ontime graduation rates
for the median public SATAVG school and the median private SATAVG school.
What other factors might influence
ontime graduation rates?
Discussion of Regression Estimates:
Note the constant term for both
regression estimates is negative and significantly different from zero. This is nonsensical because the estimated log
odds can’t be negative. However, note the
constant term is the ontime graduation rate when SATAVG is
0 and this can never occur because the worse possible SAT score is 200.
The coefficient of the SATAVG variable in both regressions is
positive and significantly different from zero. The coefficient for public schools is over 24 %
higher than the coefficient for private schools indicating SAT scores have a
larger impact on ontime graduation at public schools than at private schools.
Method for obtaining predicted ontime graduation
rates:
The predicted value of the dependent variable, the
log of odds, must be converted to the ontime probability.
First step is to calculate the
predicted log of odds which is
P=_cons+b*SATAVG
Second step is to set P equal
to the log of odds. Denote PROB as ontime
graduation prob.
log(PROB/(1PROB))=P
Third step is to solve the
above equation for PROB. Take EXP of
both sides and rearrange to get.
PROB= EXP(P)/(1+EXP(P))
Ontime Graduation Probability Calculations:
Relationship
between ontime graduation and SAT Average for Public and Private
Universities


Public
Universities


_cons

6.008

6.008

6.008

satavg

0.0122

0.0122

0.0122

SATAVG

550

600

650

P

0.727

1.340

1.952

EXP(P)

2.069

3.817

7.041

PROB

0.674

0.792

0.876

Private
Universities


_cons

4.714

4.714

4.714

satavg

0.0098

0.0098

0.0098

SATAVG

550

600

650

P

0.676

1.166

1.656

EXP(P)

1.966

3.209

5.238

PROB

0.663

0.762

0.840

Observations:
Over the SATAVG domain 550 to
650 Public universities have a higher ontime graduation rate for identical SATAVG
levels.
Difference in SATAVG for Private vs
Public Universities
SAT
Average Distribution


Public
University

Private University


1%

425

467.5

5%

440

471.25

10%

452.5

482.5

25%

462.5

550.375

50%

507.5

580

75%

547.5

612.5

90%

620

671.25

95%

650

713.75

99%

677.5

737.5

Implications:
Private school SAT scores are
consistently higher than public school SAT scores. It may be more relevant to compare the
ontime graduation rates at the same SATAVG percentile for both public and
private school.
Ontime Graduation Comparisons Public vs
Private
at Median SATAVG
Below is a comparison of ontime
graduation rates at the 50^{th} SATAVGs percentile.
Relationship
between ontime graduation and
SAT
Average for Public and Private Universities


Public
Universities

Private
Universities


_cons
Ceoeff.

6.008

4.714

Satavg
Coeff.

0.0122

0.0098

Median
SATAVG
Variable

507.5

580

P

0.207

0.970

EXP(P)

1.230

2.638

PROB

0.552

0.725

Implications:
The probability a person in
the median SATAVG private university graduates within six years of starting
education is 31 percent higher than a person starting college in the median SATAVG
public school.
Caveats:
It would be wrong to assert that
the observed differences are entirely related to SAT scores. Family wealth and income allow people to stay in school and these variables should also
be considered.
The ontime graduation rate involves
people who graduate within six years, the measure published on the College
Score cared. Actual ontime graduation from
a four year program could arguably be set at four years. It would be highly useful to estimate models
with the fouryear graduation rate as the dependent variable.
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