Class 28
Get Ready….
Height and Weight
• Is CM or Inches the better predictor of KG?
– Whichever has the lower standard error
• Will also have a variety of better stats
– NOT whichever has the bigger coefficient
• A multiple regression lets you test
– H0: all b’s = 0 (nothing in the model matters)
– H0: b1=0 given all the other b’s
• When using both CM and INCHES
– We reject H0 b1=b2=0
– We fail to reject H0 b1=0 given b2
– We fail to reject H0 b2=0 given b1
• You need either CM or INCHES but not both
– Because they are highly correlated
• Regressions ALWAYS go thru the sample averages
Things I expect you will know
• How to interpret a regression using p-1 dummy
variables
– The p possible forecasts will equal the sample average
Y for each of the p groups
– The intercept is the average of the left-out group
– The coefficients are differences in group averages.
– The p-value/significance F will match that from
ANOVA single factor
Things I expect you will know
• How to interpret a residual (error)
– It is Y - 𝑌
– It is the distance each Y is from the line.
– Positive means above the line.
– They measure the difference between actual Y
and expected Y (based on the X’s)
– The most over-weight girl (for her height) is the
girl with the largest positive residual.
• Check the box to get residuals.
Things I expect you will know
• How to interpret a coefficient in a multiple
regression.
– It measures the change in expected Y for a unit
change in that X keeping all other Xs constant.
• If I keep miles and stops constant and change from williams
to spencer, expect 0.97 hours less.
• If I change from Williams to Spencer, expect 0.33 hours
more.
– It is the easy way to answer some questions.
• If the previous rating goes from 17.5 to 20, how will the
expected ratings change? (by 0.18571 per point)
Things I expect you will know
• How to use a regression model to calculate a
point forecast.
– Plug and chug.
• I use SUMPRODUCT
• You must know what Xs to plug in.
• It is a package deal….you must know and plug in ALL
the Xs.
Things I expect you will know
• How to use a regression model to calculate a
probability.
– The question gives you the Y.
– You Plug and chug to get the 𝑌.
– You calculate t = (Y - 𝑌 )/ standard error
– Use t.dist.rt( t , dof)
• Dof is n – total number of regression terms.
– Requires the FOUR assumptions.
Things I expect you will know
• If the coefficient of X1 changes when X2 is
included in the model…..
– You know X1 and X2 are correlated.
– You can use the two regression results to tell whether
X1 and X2 are positively or negatively correlated.
•
•
•
•
•
Ds was positively correlated with Miles
Fact was negatively correlated with Stars
Nobel was positively correlated with Yanks
Speed was positively correlated with Dcorporate
Exam 1 was negatively correlated with Exam 2.
UNDERSTANDING
Coefficient
Regression Table
Constant
Fact
13.24615
1.40107
Coefficient
Regression Table
Constant
Fact
Stars
Oh…Fact Movies
had fewer Stars!
12.568
1.799
1.259
Secret Formula
Coefficient
Regression Table
Constant
Fact
13.24615
1.40107
Coefficient
Regress
Y on X1
Regression Table
Constant
Fact
Stars
12.568
1.799
1.259
𝑏 − 𝑏1
Regress Y on
X1 and X2
𝑏2Movies
Oh…Fact
had fewer Stars!
Regress Y on
X1 and X2
𝑐=
Regress X2 on
X1
Secret Formula
Coefficient
Coefficient
Regression Table
Regression Table
Constant
Fact
13.24615
1.40107
Regress
Y on X1
Constant
Fact
Stars
1.40 − 1.80
𝑐=
1.26
𝑐 = −0.32
Regress X2 on
X1
12.568
1.799
1.259
Regress Y on
X1 and X2
Regress Y on
X1 and X2
UNDDERSTANDING
Coefficient
Regression Table
Constant
Fact
13.24615
1.40107
Coefficient
Regression Table
Constant
Fact
Stars
Oh…Fact Movies
had fewer Stars!
12.568
1.799
1.259
UNDERSTANDING
Secret Formula
Coefficient
Regression Table
Constant
Fact
13.24615
1.40107
Coefficient
Regression Table
Constant
Fact
Stars
Fact Movies
averaged 0.32
fewer Stars!
12.568
1.799
1.259
Regression is the line through a cloud
of points
• Scatter-plot the cloud
• It is up to YOU to interpret the results.
• Don’t assume X causes Y
– Y might be causing X
– Both might be caused by Z
• Don’t assume better fitting lines are better at
forecasting
– They usually are not…..too good a fit means too
complicated a model…..means poorer performance.
Class 28 Assignment
Variable
School
% of Classes
Graduation Under 20
Rate
Description The name Percentage Percentage of
of the
of enrollees Classes offered
Universit who
with <= 20
y
graduate
students.
Student/Faculty
Ratio
Alumni Giving Rate
Number of
students enrolled
divided by total
number of faculty
Percentage of living
alumni who gave to
the University in
2000
Mean
Median
Mode
Standard
Deviation
Skewness
Minimum
Maximum
Count
83.042
83.5
92
8.607
55.729
59.5
65
13.194
11.542
10.5
13
4.851
29.271
29
13
13.441
-0.282
66
97
48
-0.501
29
77
48
0.582
3
23
48
0.370
7
67
48
1. Test the hypothesis that graduation rate and alumni giving rate are
(linearly) independent. We expect universities with higher graduation
rates to have higher mean giving rates. [15 points]
• Regress Giving Rate on Grad Rate
• Check if coeff is positive
• Divide reported p-value (found in two places)
by 2.
• Reject if less than 0.05.
Intercept
Graduation Rate
Coefficients
Standard
Error
t Stat
P-value
-68.76
12.58
-5.46
1.82E-06
1.18
0.15
7.83
5.24E-10
2. If the graduation rate of school A is 5 percentage
points higher than that of school B, how much higher
do we expect school A’s giving rate to be? [10 points]
• Using the above regression (graduation rate
is all we know), the expected giving rate will
be 1.18*5 = 5.9 percentage points higher for
school A.
3. If you learn that A and B above have identical student to faculty ratios,
what is your revised answer to question 2? Be certain to explain why it went
up (if it went up) or why it went down (if it went down) or why it stayed the
same. Direct your response to a university administrator. [15 points]
Intercept
Graduation Rate
Student/Faculty
Ratio
•
•
•
•
•
•
Coefficients
-19.10631
0.75574
Standard Error
15.55006
0.16023
t Stat
-1.22870
4.71669
P-value
0.22557
0.00002
-1.24595
0.28430
-4.38250
0.00007
IF we keep SFR constant, expected Giving Rate goes up 0.76 points per point of
graduation rate.
If we don’t keep SFR constant, expected Giving Rates went up 1.18 points per point.
Schools with higher grad rates had LOWER SFR (that makes sense)
If we don’t hold SFR constant, increases in grad rate mean decreases in SFR and the
combined effect of the two is 1.18.
So….if grad rate is higher (but SFR is not), expected 0.76 increase.
If grad rate is higher (and SFR is lower as in the data), expect 1.18 increase.
4. Provide a point forecast of alumni giving rate for a university
with graduation rate of 80, 65 percent of its classes with 20 or
fewer students, and a student/faculty ratio of 20. [25 points]
Intercept
Graduation Rate
% of Classes Under 20
Student/Faculty Ratio
Intercept
Graduation Rate
Student/Faculty
Ratio
Intercept
Graduation Rate
Student/Faculty
Ratio
POINT FORECAST
Coefficients
Standard Error
-20.7201
17.5214
0.7482
0.1660
0.0290
0.1393
-1.1920
0.3867
Coefficients
-19.10631
0.75574
t Stat
-1.1826
4.5082
0.2084
-3.0823
P-value
0.2433
0.0000
0.8358
0.0035
Don’t Use this
variable.
Use this
model.
-1.24595
1
80
20
16.43
Plug and
Chug.
• The best model includes Grad Rate and
SFR (% classes <20 not needed)
5. Of the 48 universities in the data set, which one has the most
surprisingly low alumni giving rate? [10 points]
• The university with the most negative
residual.
• Use the best model, ask for residuals, find
the minimum.
• MICHIGAN!
6. Bo notices that some of the 48 have “university” in their names, some have “college” and the
rest have “institute”. Bo wonders whether these names are predictive of student/faculty ratio?
(Formulate and test a relevant hypothesis.) [25 points]
• Three groups (p=3)
• ANOVA or Regression of SFR on 2
dummies.
SUMMARY OUTPUT
ANOVA
df
Regression
Residual
Total
Intercept
Dcollege
Dinstitute
SS
103.7348
1002.1818
1105.9167
MS
51.8674
22.2707
F
2.3290
Coefficients Standard Error
11.8636
0.7114
-0.3636
3.4120
-7.3636
3.4120
t Stat
16.6754
-0.1066
-2.1582
P-value
0.0000
0.9156
0.0363
2
45
47
Significance
F
0.1090
Get Ready…..
• More practice problems (answers) on
website.
• I’ll host Sunday night Office Hours.
• I am available Monday and Tuesday until
2pm.
– Email pfeiferp@virginia.edu
– Check the website to see where I am…you
are welcome to join us.