12 introduction to multiple regression model

© 2002 Prentice-Hall, Inc. Chap 14-1
Introduction to
Multiple Regression Model

© 2002 Prentice-Hall, Inc.
Chap 14-2
Chapter Topics
 Multiple linear regression (MLR) model
 Residual analysis
 Influence analysis
 Testing for the significance of the regression
model
 Inferences on the population regression
coefficients
 Testing portions of the multiple regression
model

Chap 14-3
0 1 1 2 2i i i k ki iY b b X b X b X e= + + + + +L
Population
Y-intercept
Population slopes
Random Error
Multiple Linear Regression
Model
A relationship between one dependent and two or
more independent variables is a linear function
Dependent (Response)
variable for sample
Independent (Explanatory)
variables for sample model
1 2i i i k ki iY X X Xβ β β β ε0 1 2= + + + + +L
Residual

Chap 14-4
Population Multiple
Regression Model
X 2
Y
X 1
µ Y X = β 0 + β 1 X 1 i + β 2 X 2 i
β 0
Y i = β 0 + β 1 X 1 i + β 2 X 2 i + ε i
R e s p o n s e
P la n e
( X 1 i,X 2 i)
( O b s e r v e d Y )
ε i
Bivariate model

Chap 14-5
Sample Multiple
Regression Model
X 2
Y
X 1
b 0
Y i
= b 0
+ b 1
X 1 i
+ b 2
X 2 i
+ e i
R e s p o n s e
P la n e
( X 1 i, X 2 i)
( O b s e r v e d Y )
^
e i
Y i = b 0 + b 1 X 1 i + b 2 X 2 i
Bivariate model
Sample Regression PlaneSample Regression Plane

Chap 14-6
Simple and Multiple Linear
Regression Compared: Example
 Two simple regressions:


 Multiple regression:

0 1
0 1
Oil Temp
Oil Insulation
β β
β β
= +
= +
0 1 2Oil Temp Insulationβ β β= + +

Chap 14-7
Multiple Linear
Regression Equation
Too
complicated
by hand! Ouch!

Chap 14-8
Interpretation of
Estimated Coefficients
 Slope (bi)
 Estimated that the average value of Y changes by
bi for each one unit increase in Xi holding all other
variables constant (ceterus paribus)
 Example: if b1 = -2, then fuel oil usage (Y) is
expected to decrease by an estimated two gallons
for each one degree increase in temperature (X1)
given the inches of insulation (X2)
 Y-intercept (b0)
 The estimated average value of Y when all Xi = 0

Chap 14-9
Multiple Regression Model:
Example
Oil (Gal) Temp Insulation
275.30 40 3
363.80 27 3
164.30 40 10
40.80 73 6
94.30 64 6
230.90 34 6
366.70 9 6
300.60 8 10
237.80 23 10
121.40 63 3
31.40 65 10
203.50 41 6
441.10 21 3
323.00 38 3
52.50 58 10
(0
F)
Develop a model for estimating
heating oil used for a single
family home in the month of
January based on average
temperature and amount of
insulation in inches.

Chap 14-10
1 2
ˆ 562.151 5.437 20.012i i iY X X= − −
Sample Multiple Regression
Equation: Example
Coefficients
Intercept 562.1510092
X Variable 1 -5.436580588
X Variable 2 -20.01232067
Excel Output
For each degree increase in
temperature, the estimated average
amount of heating oil used is
decreased by 5.437 gallons,
holding insulation constant.
For each increase in one inch
of insulation, the estimated
average use of heating oil is
decreased by 20.012 gallons,
holding temperature constant.
0 1 1 2 2
ˆ
i i i k kiY b b X b X b X= + + + +L

Chap 14-11
Explanatory Power of
Regression
Oil
Temp
Variations in oil
explained by temp
or variations in
temp used in
explaining variation
in oil
Variations in
oil explained
by the error
term
Variations in
temp not used
in explaining
variation in Oil
( )SSE
( )SSR

Chap 14-12
Regression
Oil
Temp
2
r
SSR
SSR SSE
=
=
+
(continued)

Chap 14-13
Regression
Oil
Temp
Insulation
OverlappingOverlapping
variation in
both Temp and
Insulation are
used in
explaining the
variationvariation in Oil
but NOTNOT in the
estimationestimation of
nor
1β
2β
Variation NOTNOT
explained by
Temp nor
Insulation
( )SSE

Chap 14-14
Coefficient of
Multiple Determination
 Proportion of total variation in Y explained by
all X variables taken together

 Never decreases when a new X variable is
added to model
 Disadvantage when comparing models
2
12
Explained Variation
Total Variation
Y k
SSR
r
SST
• = =L

Chap 14-15
Regression
Oil
Temp
Insulation
2
12Yr
SSR
SSR SSE
• =
=
+

Chap 14-16
Adjusted Coefficient of Multiple
Determination
 Proportion of variation in Y explained by all X
variables adjusted for the number of X
variables used

 Penalize excessive use of independent variables
 Smaller than
 Useful in comparing among models
( )2 2
12
1
1 1
1
adj Y k
n
r r
n k
•
− 
= − − − − 
L
2
12Y kr • L

Chap 14-17
Coefficient of Multiple
Determination
Regression Statistics
Multiple R 0.982654757
R Square 0.965610371
Adjusted R Square 0.959878766
Standard Error 26.01378323
Observations 15
Excel Output
SST
SSR
r ,Y =2
12
Adjusted r2
 reflects the number
of explanatory
variables and sample
size
 is smaller than r2

Chap 14-18
Interpretation of Coefficient of
Multiple Determination

 96.56% of the total variation in heating oil can be
explained by difference in temperature and amount
of insulation

 95.99% of the total fluctuation in heating oil can be
explained by difference in temperature and amount
of insulation after adjusting for the number of
explanatory variables and sample size
2
,12 .9656Y
SSR
r
SST
= =
2
adj .9599r =

Chap 14-19
Using The Model to Make
Predictions
Predict the amount of heating oil used for a
home if the average temperature is 300
and the
insulation is six inches.
The predicted heating oil
used is 278.97 gallons
( ) ( )
1 2
ˆ 562.151 5.437 20.012
562.151 5.437 30 20.012 6
278.969
i i iY X X= − −
= − −
=

Chap 14-20
Residual Plots
 Residuals vs.
 May need to transform Y variable
 Residuals vs.
 May need to transform variable
 Residuals vs.
 May need to transform variable
 Residuals vs. time
 May have autocorrelation
ˆY
1X
2X
1X
2X

Chap 14-21
Residual Plots: Example
Insulation Residual Plot
0 2 4 6 8 10 12
No discernable pattern
Temperature Residual Plot
-60
-40
-20
0
20
40
60
0 20 40 60 80
Residuals
May be some non-
linear relationship

Chap 14-22
Influence Analysis
 To determine observations that have
influential effect on the fitted model
 Potentially influential points become
candidates for removal from the model
 Criteria used are
 The hat matrix elements hi
 The Studentized deleted residuals ti
*
 Cook’s distance statistic Di
 All three criteria are complementary
 Only when all three criteria provide consistent
results should an observation be removed

Chap 14-23
The Hat Matrix Element hi

 If , Xi is an Influential Point
 Xi may be considered a candidate for removal
from the model
( )
( )
2
2
1
1 i
i n
i
i
X X
h
n
X X
=
−
= +
−∑
( )2 1 /ih k n> +

Chap 14-24
The Hat Matrix Element hi :
Heating Oil Example
Oil (Gal) Temp Insulation h i
275.30 40 3 0.1567
363.80 27 3 0.1852
164.30 40 10 0.1757
40.80 73 6 0.2467
94.30 64 6 0.1618
230.90 34 6 0.0741
366.70 9 6 0.2306
300.60 8 10 0.3521
237.80 23 10 0.2268
121.40 63 3 0.2446
31.40 65 10 0.2759
203.50 41 6 0.0676
441.10 21 3 0.2174
323.00 38 3 0.1574
52.50 58 10 0.2268
 No hi > 0.4
 No observation
appears to be a
candidate for removal
from the model
( )
15 2
2 1 / 0.4
n k
k n
= =
+ =

Chap 14-25
The Studentized Deleted
Residuals ti
*

 : difference between the observed and
predicted based on a model that includes all
observations except observation i
 : standard error of the estimate for a model that
includes all observations except observation i
 An observation is considered influential if

is the critical value of a two-tail test at a alpha
level of significance
( )
( )
*
1
i
i
ii
e
t
S h
=
−
iY
ˆ
iY
( )i
S
( )i
e
*
2i n kt t − −>
2n kt − −

Chap 14-26
The Studentized Deleted
Residuals ti
* :Example
Oil (Gal) Temp Insulation ti
*
275.30 40 3 -0.3772
363.80 27 3 0.3474
164.30 40 10 0.8243
40.80 73 6 -0.1871
94.30 64 6 0.0066
230.90 34 6 -1.0571
366.70 9 6 -1.1776
300.60 8 10 -0.8464
237.80 23 10 0.0341
121.40 63 3 -1.8536
31.40 65 10 1.0304
203.50 41 6 -0.6075
441.10 21 3 2.9674
323.00 38 3 1.1681
52.50 58 10 0.2432
2 11
15 2
1.7957n k
n k
t t− −
= =
= =
 t10
* and t13
* are
influential points for
potential removal from
the model
*
10t
*
13t

Chap 14-27
Cook’s Distance Statistic Di

 is the Studentized residual
 If , an observation is considered
influential

is the critical value of the F distribution at a
50% level of significance
( )
2
2 1
i i
i
i
SR h
D
h
=
−
1
i
i
YX i
e
SR
S h
=
−
1, 1i k n kD F + − −>
1, 1k n kF + − −

Chap 14-28
Cook’s Distance Statistic Di :
Heating Oil Example
Oil (Gal) Temp Insulation Di
275.30 40 3 0.0094
363.80 27 3 0.0098
164.30 40 10 0.0496
40.80 73 6 0.0041
94.30 64 6 0.0001
230.90 34 6 0.0295
366.70 9 6 0.1342
300.60 8 10 0.1328
237.80 23 10 0.0001
121.40 63 3 0.3083
31.40 65 10 0.1342
203.50 41 6 0.0094
441.10 21 3 0.4941
323.00 38 3 0.0824
52.50 58 10 0.0062
 No Di > 0.835
 No observation appears to
be candidate for removal
from the model
Using the three criteria,
there is insufficient evidence
for the removal of any
observation from the model
1, 1 3,12
15 2
0.835k n k
n k
F F+ − −
= =
= =

Chap 14-29
Testing for Overall Significance
 Show if there is a linear relationship between all
of the X variables together and Y
 Use F test statistic
 Hypotheses:
 H0: β1 = β2 = … = βk = 0 (no linear relationship)
 H1: at least one βi ≠ 0 ( at least one independent
variable affects Y )
 The null hypothesis is a very strong statement
 Almost always reject the null hypothesis

Chap 14-30
Testing for Overall Significance
Test statistic:
where F has p numerator and (n-p-1)
denominator degrees of freedom
(continued)
( )
( )
all /
all
SSR pMSR
F
MSE MSE
= =

Chap 14-31
Test for Overall Significance
Excel Output: Example
ANOVA
df SS MS F Significance F
Regression 2 228014.6 114007.3 168.4712 1.65411E-09
Residual 12 8120.603 676.7169
Total 14 236135.2
p = 2, the number of
explanatory variables n - 1
p value
Test Statistic
MSR
F
MSE
=

Chap 14-32
Test for Overall Significance
Example Solution
F0 3.89
H0: β1 = β2 = … = βp = 0
H1: At least one βi ≠ 0
α = .05
df = 2 and 12
Critical Value(s):
Test statistic:
Decision:
Conclusion:
Reject at α = 0.05
There is evidence that
at least one independent
variable affects Y
α =
0.05
F = 168.47
(Excel Output)

Chap 14-33
Test for Significance:
Individual Variables
 Show whether there is a linear relationship
between the variable Xi and Y
 Use t Test Statistic
 Hypotheses:
 H0: βi = 0 (No linear relationship)
 H1: βi ≠ 0 (Linear relationship between Xi and Y)

Chap 14-34
t Test Statistic
Excel Output: Example
Coefficients Standard Error t Stat
Intercept 562.1510092 21.09310433 26.65093769
X Variable 1 -5.436580588 0.336216167 -16.16989642
X Variable 2 -20.01232067 2.342505227 -8.543127434
t Test Statistic for X1
(Temperature)
t Test Statistic for X2
(Insulation)
i
i
b
b
t
S
=

Chap 14-35
t Test : Example Solution
H0: β1 = 0
H1: β1 ≠ 0
df = 12
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Reject H0 at α = 0.05
There is evidence of a
significant effect of
temperature on oil
consumption.
t0 2.1788-2.1788
.025
Reject H0 Reject H0
.025
Does temperature have a significant effect on monthly
consumption of heating oil? Test at α = 0.05.
t Test Statistic = -16.1699

Chap 14-36
Venn Diagrams and
Estimation of Regression Model
Oil
Temp
Insulation
Only this
information is
used in the
estimation of 2β
Only this
information is
used in the
estimation of
1β This
information
is NOT used
in the
estimation of
nor1β 2β

Chap 14-37
Confidence Interval Estimate
for the Slope
Provide the 95% confidence interval for the population
slope β1 (the effect of temperature on oil consumption).
11 1n p bb t S− −±
Coefficients Lower 95% Upper 95%
Intercept 562.151009 516.1930837 608.108935
X Variable 1 -5.4365806 -6.169132673 -4.7040285
X Variable 2 -20.012321 -25.11620102 -14.90844
-6.169 ≤ β1 ≤ -4.704
The estimated average consumption of oil is reduced by
between 4.7 gallons to 6.17 gallons per each increase of 10
F.

Chap 14-38
Contribution of a Single
Independent Variable
 Let Xk be the independent variable of
interest

 Measures the contribution of Xk in explaining the
total variation in Y (SST)
kX
( )
( ) ( )
| all others except
all all others except
k k
k
SSR X X
SSR SSR X= −

Chap 14-39
Contribution of a Single
Independent Variable kX
( )
( ) ( )
1 2 3
1 2 3 2 3
| and
, and and
SSR X X X
SSR X X X SSR X X= −
Measures the contribution of in explaining SST1X
From ANOVA section
of regression for
From ANOVA section of
regression for
0 1 1 2 2 3 3
ˆ
i i i iY b b X b X b X= + + + 0 2 2 3 3
ˆ
i i iY b b X b X= + +

Chap 14-40
Coefficient of Partial
Determination of

 Measures the proportion of variation in the
dependent variable that is explained by Xk
while controlling for (holding constant) the
other independent variables
( )
( ) ( )
2
all others
| all others
all | all others
Yk
k
k
r
SSR X
SST SSR SSR X
• =
− +
kX

Chap 14-41
Coefficient of Partial
Determination for kX
(continued)
( )
( ) ( )
1 22
1 2
1 2 1 2
|
, |
Y
SSR X X
r
SST SSR X X SSR X X
• =
− +
Example: Two Independent Variable Model

Chap 14-42
Venn Diagrams and Coefficient of
Partial Determination for kX
Oil
Temp
Insulation
( )1 2|SSR X X
( )
( ) ( )
2
1 2
1 2
1 2 1 2
|
, |
Yr
SSR X X
SST SSR X X SSR X X
• =
− +
=

Chap 14-43
Contribution of a Subset of
Independent Variables
Let Xs be the subset of independent variables of
interest
Measures the contribution of the subset xs in
explaining SST
( )
( ) ( )
| all others except
all all others except
s s
s
SSR X X
SSR SSR X= −

Chap 14-44
Contribution of a Subset of
Independent Variables: Example
Let Xs be X1 and X3
( )
( ) ( )
1 3 2
1 2 3 2
and |
, and
SSR X X X
SSR X X X SSR X= −
From ANOVA section of
regression for
From ANOVA
section of
regression for
0 1 1 2 2 3 3
ˆ
i i i iY b b X b X b X= + + + 0 2 2
ˆ
i iY b b X= +

Chap 14-45
Testing Portions of Model
 Examines the contribution of a subset Xs
of explanatory variables to the
relationship with Y
 Null hypothesis:
 Variables in the subset do not significantly
improve the model when all other variables
are included
 Alternative hypothesis:
 At least one variable is significant

Chap 14-46
Testing Portions of Model
 Always one-tailed rejection region
 Requires comparison of two regressions
 One regression includes everything
 Another regression includes everything except the
portion to be tested
(continued)

Chap 14-47
Partial F Test For Contribution
of Subset of X variables
 Hypotheses:
 H0 : Variables Xs do not significantly improve the
model given all others variables included
 H1 : Variables Xs significantly improve the model
given all others included
 Test Statistic:

 with df = m and (n-p-1)
 m = # of variables in the subset Xs
( )
( )
| all others /
all
sSSR X m
F
MSE
=

Chap 14-48
Partial F Test For Contribution
of A Single
 Hypotheses:
 H0 : Variable Xj does not significantly improve
the model given all others included
 H1 : Variable Xj significantly improves the
model given all others included
 Test Statistic:

 With df = 1 and (n-p-1)
 m = 1 here
jX
( )
( )
| all others
all
jSSR X
F
MSE
=

Chap 14-49
Testing Portions of Model:
Example
Test at the α = .05 level to
determine whether the
variable of average
temperature significantly
improves the model given
that insulation is included.

Chap 14-50
Testing Portions of Model:
Example
H0: X1 (temperature) does
not improve model with X2
(insulation) included
H1: X1 does improve model
α = .05, df = 1 and 12
Critical Value = 4.75
ANOVA
SS
Regression 51076.47
Residual 185058.8
Total 236135.2
ANOVA
SS MS
Regression 228014.6263 114007.313
Residual 8120.603016 676.716918
Total 236135.2293
(For X1 and X2) (For X2)
Conclusion: Reject H0; X1 does improve model
( )
( )
( )1 2
1 2
| 228,015 51,076
261.47
, 676.717
SSR X X
F
MSE X X
−
= = =

Chap 14-51
When to Use the F test
 The F test for the inclusion of a single
variable after all other variables are included
in the model is IDENTICAL to the t test of the
slope for that variable
 The only reason to do an F test is to test
several variables together

Chap 14-52
Chapter Summary
 Developed the multiple regression
model
 Discussed residual plots
 Presented influence analysis
 Addressed testing the significance of
the multiple regression model
 Discussed inferences on population
regression coefficients
 Addressed testing portion of the multiple
regression model

Chap 14-53
nieXXY iippii ,,2,1,110  =++++= βββ
Multiple Linear Regression














=














=














=














=
npnpn
p
p
n e
e
e
e
XX
XX
XX
X
Y
Y
Y
whereY






2
1
1
0
1
221
111
2
1
,,
1
1
1
,
β
β
β
β
Data
Model:
Matrix
Model: eXY += β

Chap 14-54
( )
( ) ( )
( )ppp
n
i
ippiip
p
XXXXwhereXYXXY
XXYQ
Q
FittingSquaresLeast
,,,,1
,,,
,,,,min.1
:
21
22
110
1
2
11010
10



=−=−−−−=
−−−−= ∑=
ββββ
ββββββ
βββ
( )
ββββ
βββββ
ˆˆˆˆˆ
,,,minˆ,,ˆ.2
110
100
XXXY
valuefittedtheThen
QimizeLet
pp
pp
=+++= 

noningularisXXiff
YXXXXYXXYYSince
T
TTT
)(
)(ˆ0)ˆ(ˆ 1−
=⇒=−⇒⊥− ββ

Chap 14-55
YYXXY rR t ˆ,,,; 1
=
Multiple Correlation Coefficient:
Multiple Coefficient of Determination: may be interpreted
as the proportion of variance explained by the regression of
Y on X.
2
ˆ,
2
,,;
2
2
ˆ
2
ˆ
2
1
,
YYXXY
eYY
rRR
SSS
Moreover
t
==
+=

2
2
ˆ2
Y
Y
S
S
R =

Chap 14-56
( ) ( ) )
0
0
..(,0~
2
2
2
..










=
+=
σ
σ
σ
β



eVareiINeassume
eXYModel
dii
( ) ( )( )YXXXEESolve TT 1
ˆ:
−
=β
( )( )12
,~ˆ −
XXN T
σββTheorem:
( ) ( )YEXXX TT 1−
=
( ) ( )
( ) ( )
( )
β
β
β
β
=
=
=
+=
−
−
−
XXXX
XEXXX
eXEXXX
TT
TT
TT
1
1
1

Chap 14-57
( )( )12
,~ˆ −
XXN T
σββ
( ) ( )βββ ˆ,ˆˆ: CovVarSolve =
( ) ( )[ ]YXXXYXXXCov TTTT 11
,
−−
=
( ) ( ) ( )[ ]T
TTTT
XXXYYCovXXX
11
,
−−
=
( ) ( ) ( )[ ]T
TTTT
XXXeXVarXXX
11 −−
+= β
( ) ( )[ ]T
TTT
XXXIXXX
121 −−
⋅= σ
( ) ( )( ) 112 −−
= XXXXXX TTT
σ
( ) 12 −
= XX T
σ
( ) 1
ˆ
1
1
1
ˆ
ˆ
1
2
1
2
2
−−
=−
−−
=
−−
= ∑
∑
=
=
pn
RSS
YY
pnpn
e
where
n
i
ii
n
i
i
σ

Chap 14-58
DATA;
INPUT X1 X2 Y;
CARDS;
68 60 75
49 94 63
60 91 57
.
77 78 72
;
PROC PRINT;
PROC REG;
MODEL Y=X1 X2 / COVB CORRB R INFLUENCE;
RUN;

Chap 14-59
Model: MODEL1
Dependent Variable: Y Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 1966.20840 983.10420 14.86 0.0002
Error 17 1124.79160 66.16421
Corrected Total 19 3091.00000
Root MSE 8.13414 R-Square 0.6361
Dependent Mean 74.50000 Adj R-Sq 0.5933
Coeff Var 10.91831
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 14.49614 14.20435 1.02 0.3218
X1 1 0.56319 0.11801 4.77 0.0002
X2 1 0.26736 0.15704 1.70 0.1069

Chap 14-60
Covariance of Estimates
COVB Intercept X1 X2
Intercept 201.7635339 -0.635820247 -1.851491131
X1 -0.635820247 0.0139252459 -0.003440529
X2 -1.851491131 -0.003440529 0.0246625524
Correlation of Estimates
COVB Intercept X1 X2
Intercept 1.0000 -0.3793 -0.8300
X1 -0.3793 1.0000 -0.1857
X2 -0.8300 -0.1857 1.0000

Chap 14-61
Dep Var Predicted Std Error Std Error Student Cook's
Obs Y Value Predict Residual Residual Residual -2-1 0 1 2 D
1 75.0000 68.8346 4.2678 6.1654 6.925 0.890 | |* | 0.100
2 63.0000 67.2242 3.3214 -4.2242 7.425 -0.569 | *| | 0.022
3 57.0000 72.6172 2.2988 -15.6172 7.803 -2.002 | ****| | 0.116
4 88.0000 74.4491 1.9107 13.5509 7.907 1.714 | |*** | 0.057
5 88.0000 90.5143 4.2002 -2.5143 6.966 -0.361 | | | 0.016
6 79.0000 85.2747 2.6984 -6.2747 7.674 -0.818 | *| | 0.028
7 82.0000 67.5089 2.4898 14.4911 7.744 1.871 | |*** | 0.121
8 73.0000 66.4506 2.8567 6.5494 7.616 0.860 | |* | 0.035
9 90.0000 81.2755 2.5928 8.7245 7.710 1.132 | |** | 0.048
10 62.0000 59.7208 3.8097 2.2792 7.187 0.317 | | | 0.009
11 70.0000 77.4755 1.8990 -7.4755 7.909 -0.945 | *| | 0.017
12 96.0000 93.1309 3.8760 2.8691 7.151 0.401 | | | 0.016
13 76.0000 73.9825 2.5281 2.0175 7.731 0.261 | | | 0.002
14 75.0000 80.1776 2.3793 -5.1776 7.778 -0.666 | *| | 0.014
15 85.0000 84.6150 3.2590 0.3850 7.453 0.0517 | | | 0.000
16 40.0000 50.3917 5.9936 -10.3917 5.499 -1.890 | ***| | 1.414
17 74.0000 76.2637 2.1866 -2.2637 7.835 -0.289 | | | 0.002
18 70.0000 69.0846 2.0768 0.9154 7.865 0.116 | | | 0.000
19 75.0000 72.2929 2.6787 2.7071 7.680 0.352 | | | 0.005
20 72.0000 78.7158 2.5093 -6.7158 7.737 -0.868 | *| | 0.026
21 . 83.7560 3.0157 . . . .

Chap 14-62
Hat Diag
Obs Residual RStudent H
1 6.1654 0.8846 0.2753
2 -4.2242 -0.5572 0.1667
3 -15.6172 -2.2211 0.0799
4 13.5509 1.8281 0.0552
5 -2.5143 -0.3515 0.2666
6 -6.2747 -0.8094 0.1100
7 14.4911 2.0374 0.0937
8 6.5494 0.8530 0.1233
9 8.7245 1.1417 0.1016
10 2.2792 0.3086 0.2194
11 -7.4755 -0.9420 0.0545
12 2.8691 0.3911 0.2271
13 2.0175 0.2537 0.0966
14 -5.1776 -0.6543 0.0856
15 0.3850 0.0501 0.1605
16 -10.3917 -2.0627 0.5429
17 -2.2637 -0.2810 0.0723
18 0.9154 0.1130 0.0652
19 2.7071 0.3432 0.1084
20 -6.7158 -0.8613 0.0952

Chap 14-63
100 +
|
| o
| o o o o o
| o o o
90 + o
| o
| o
H |
O | o o
M 80 + o o
E | o
W |
O | o
R |
K 70 +
|
|
|
|
60 + o
|
|
|
|
50 + o
|
-+------------+------------+------------+------------+------------+------------+------------+
30 40 50 60 70 80 90 100

Chap 14-64



≠
=
++=
0
00
11 ,.1
rrH
rrH
hypothesisthetesttowantweccrLet
a
pp
：
：
ββ 
( ) ( )rdtspntr
rforICaConstruct
cccrLet pp
ˆˆ1ˆ
..%95
.2
025.0
1100
−−±⇒
+++= βββ 
( )
( )
( )
( ) ( ) ( ) T
ccVarcVarrVar
andpnt
rdts
rEr
tisstatistictesttheThen
ββ ˆˆˆ
,1~
ˆˆ
ˆˆ
==
−−
−
=
.
ˆ
ˆ
ˆ),,,(,ˆˆˆ
0
000










==++=
p
ppp cccwhereccrchooseWe
β
β
βββ 

Chap 14-65
setdataaforelbetterachoosetoHow mod
Goodness of Fit
( )1,~
1 0
−−−
−
⋅
−
−−
=⇒ pnqpF
RSS
RSSRSS
qp
pn
F
pqjoneleastatH
H
eldrestriceteatoeledunrestrictanreducetoHow
j
pq
,,1,0
0
?modmod
1
10


+=≠
===⇒ +
β
ββ
：
：
eXXY
elstricted
eXXY
eledUnrestrict
pqAssume
qq
pp
++++=
++++=
<
βββ
βββ


110
110
modRe
mod

Chap 14-66
?mod datathefittoelbetterachoosetoHow
eYelrestricted
eXYeledunrestrictEx
+=
++=
α
βα
mod
mod：
00 0 =⇒= ββ ：H
( )2,1~
1
2
0.0
20
10
−
−
⋅
−
=
≠=
n
RSS
RSSRSSn
F
HsvHSolve
χ
ββ ：：：
( )
( ) .ˆˆ
1
2
1
2
0
∑
∑
=
=
−−=
−=
n
i
ii
n
i
i
XYRSS
YYRSSwhere
βα

Chap 14-67
Regression Effect
eY
elstricted
eXXY
eledUnrestrict
pp
+=
++++=
0
110
modRe
mod
β
βββ 
2
ˆ
2
ˆ0 11
:
e
Y
S
S
p
pn
RSS
RSSRSS
p
pn
FSolve ⋅
−−
=
−
⋅
−−
=
,
0
0210



≠
===
⇒
ja
p
oneleastatH
H
β
βββ
：
： 
2
2
1
1
R
R
p
pn
Fthen
−
⋅
−−
=
22
ˆ
22
ˆ1
Ye
YY
SS
SS
p
pn
⋅
−−
=
2
2
1
1
R
R
p
pn
−
⋅
−−
=

Chap 14-68
( )
( )
i
ijiij
ijiij
njkifor
RSSeYeledUnrestrict
RSSeXYelstricted
,,2,1,,2,1
:mod
:modRe 0
 ==
+=
++=
；
α
βα
∑=
=
−
⋅
−
−
=⇒
k
i
inn
RSS
RSSRSS
k
kn
F
1
0
,
2



≠
=
==+++=
jiif
jiif
dkidddLet ijikkiii
,0
,1
,,,2,1,2211  αααα
Goodness of Fit for using replicate observations
)(
mod
2211 RSSedddY
asrewrittenbecaneledunrestricttheThen
ijikkiiij ++++= ααα 

12 introduction to multiple regression model

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