Transcript MSC 317 - Michigan State University

AGENDA
MULTIPLE REGRESSION REVIEW

Overall Model Test (F Test for Regression)

Test of Model Parameters

Test of βi = βi*

Coefficient of Multiple Determination (R2)
Formula

Confidence Interval
CORRELATION BASICS
VI.
Hypothesis Test on Correlation
Multiple Regression Basics
Y=b0 + b1X1 + b2X2 +…bkXk
 Where Y is the predicted value of Y,
the value lying on the estimated
regression surface. The terms b0,…,k
are the least squares estimates of the
population regression parameters ßi
I. ANOVA Table for Regression Analysis
Source of
Variation
Regression
Residual
Total
Degrees of Sums of
Freedom Squares
Mean Squares
F
MSR/ MSE
k
SSR
MSR = SSR / k
n-k-1
SSE
MSE=SSE/(n-k-1)
n-1
SST
II. Test of Model Parameters
H0:
H1:
β1= 0
β1 ≠ 0
t-calc =
No Relationship
Relationship
bi   i
S bi
n = sample size
t-critical:
 t / 2,nk 1
III. Test of βi = βi*
H0:
H1:
t-calc =
n=
t-critical:
β1= βi*
β1≠ βi*
bi   i
S bi
sample size
 t / 2,nk 1
IV. Coefficient of Multiple Determination (R2)
Formula
2
R
=
SSR
SST
SSE
1
SST
or
Adjusted R2 =
 n  1 
2 
R  1  
 1 R 
 n  k  1 

2


V. Confidence Interval
bi  t 2,n  k 1  Sbi
Range of numbers believed to include
an unknown population parameter.
Multiple Regression Review
Great rebounding is going to offer your team more opportunities to
score, and give the opposing team less opportunities to score. Think
about it: just one rebound could add a 6 point swing to your team’s
score! Good rebounding is going to give your team more possessions,
which means more scoring.
-powerbasketball.com
Players play — tough players win," was the motto made famous
Michigan State University men's basketball coach Tom Izzo — who built
rebuilt the MSU program during the mid-1990s around toughness and
rebounding, taking the Spartans to five Finals 4s in the last 15 years.
Much of the success Izzo's Spartans have attained is attributed to their
brutal practices and the now signature "war drill" that places a special
emphasis on rebounding, toughness, getting after loose balls and
accountability to your teammates.
-newburyportnews.com
Determinants of Points Scored
(X1) = Field Goal Percentage
(X2) = Number of Assists
(X3) = Number of Total Rebounds
N=
20 Games
Output from Computer
Yˆ  24.267  0.639 X 1  1.120 X 2  0.059 X 3
sb1  0.240 sb2  0.508 sb3  0.444
ANOVAb
Model
1
Sum of Squares
df
Mean Square
F
Sig.
Regression
1263.222
13
421.074
7.593
.002a
Residual
887.328
16
55.458
Total
2150.550
19
a. Predictors: (Constant), REB, FG, AST
b. Dependent Variable: PTS
Multiple Regression Example
Conduct the following tests:
•What is the R2? the adjusted R2?
•Overall Model F test
•Test whether β1 = 0
•Test whether one more assist leads to 2 more points
•Construct a 95% confidence interval for β3
Correlation Review
 Measures the strength of the
linear relationship between two
variables
 Ranges from -1 to 1
 Positive = direct relationship
r
S xy
SxS y
 1 to  1
 Negative = inverse relationship
 Near 0 = no strong linear
relationship
 Does NOT imply causality
Sxy  covariance of X and Y
Sx  standard deviation of X
Sy  standard deviation of Y
Illustrations of correlation
Y
r=-1
Y
X
Y
r=-.8
X
Y
r=0
r=1
X
Y
X
Y
r=0
X
r=.8
X
VI. Hypothesis Test on Correlation
 To test the significance of the linear
relationship between two random variables:
H0:  = 0 no linear relationship
H1:   0 linear relationship
 This is a t-test with (n-2) degrees of freedom:
t / 2, n  2 
r
1 r2
n2
VI. Hypothesis Test on Correlation (cont.)
 Is the number rebounds related to the
number of points scored
Sxy = 8.958
Sx = 4.160
Sy = 10.639
r = .670 (0.001)
r = .635 (0.003)
r = .202 (0.392)