AGENDAMULTIPLE 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

Degrees of

Freedom

Sums of

Squares

Mean Squares

F

Regression

k SSR MSR = SSR / k MSR/ MSE

Residual n-k-1 SSE MSE=SSE/(n-k-1)

Total n-1 SST

H0: β1= 0 No Relationship

H1: β1 ≠ 0 Relationship

t-calc =

n = sample size

t-critical:

bi

ii

S

b

II. Test of Model Parameters

1,2/ knt

III. Test of βi = βi* H0: β1= βi*

H1: β1≠ βi*

t-calc =

n = sample sizet-critical:

bi

ii

S

b

1,2/ knt

R2 = orSST

SSR

IV. Coefficient of Multiple Determination (R2) Formula

SST

SSE1

22 11

11 R

kn

nR

Adjusted R2 =

V. Confidence Interval

Range of numbers believed to include an unknown population parameter.

bikni Stb 1,2

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

ANOVAb

ModelSum of Squares df Mean Square F Sig.

1 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

444.0508.0240.0

059.0120.1639.0267.24ˆ

321

321

sbsbsb

XXXY

Multiple Regression ExampleConduct 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

Negative = inverse relationship

Near 0 = no strong linear relationship

Does NOT imply causality

11 toSS

Sr

yx

xy

Y ofdeviation standard

X ofdeviation standard

Y and X of covariance

Sy

Sx

Sxy

Illustrations of correlation

Y

X

r=0

Y

X

r=-.8 Y

X

r=.8Y

X

r=0

Y

X

r=-1 Y

X

r=1

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:

2

1,2/ 2 2

nr

rn

t

VI. Hypothesis Test on Correlation (cont.)

Is the number rebounds related to the number of points scored

Sxy = 8.958Sx = 4.160Sy = 10.639

r = .670 (0.001)

r = .635 (0.003)

r = .202 (0.392)