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 on-time 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 25th percentiles
SAT verbal and math scores and the 75th 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 on-time
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 on-time graduation rates
for the median public SATAVG school and the median private SATAVG school.
What other factors might influence
on-time 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 on-time 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 on-time graduation at public schools than at private schools.
Method for obtaining predicted on-time graduation
rates:
The predicted value of the dependent variable, the
log of odds, must be converted to the on-time 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 on-time
graduation prob.
log(PROB/(1-PROB))=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))
On-time Graduation Probability Calculations:
Relationship
between on-time 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 on-time 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
on-time graduation rates at the same SATAVG percentile for both public and
private school.
On-time Graduation Comparisons Public vs
Private
at Median SATAVG
Below is a comparison of on-time
graduation rates at the 50th SATAVGs percentile.
Relationship
between on-time 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 on-time graduation rate involves
people who graduate within six years, the measure published on the College
Score cared. Actual on-time graduation from
a four year program could arguably be set at four years. It would be highly useful to estimate models
with the four-year graduation rate as the dependent variable.
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