Matakuliah : I0174 – Analisis Regresi
Tahun
: Ganjil 2007/2008
Uji Kelinearan dan Keberartian Regresi
Pertemuan 02
Uji Kelinieran dan Keberartian Regresi
• Anova pada regresi Sederhana
• Selang Kepercayaan Parameter Regresi
• Uji Independen Antar Peubah
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Measures of Variation:
The Sum of Squares
SST
=
Total
=
Sample
Variability
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SSR
Explained
Variability
+
SSE
+
Unexplained
Variability
Measures of Variation:
The Sum of Squares
(continued)
• SST = Total Sum of Squares
– Measures the variation of the Yi values around their mean, Y
• SSR = Regression Sum of Squares
– Explained variation attributable to the relationship between X and Y
• SSE = Error Sum of Squares
– Variation attributable to factors other than the relationship between X and Y
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Measures of Variation:
The Sum of Squares
(continued)

SSE =(Yi - Yi )2
Y
_
SST = (Yi - Y)2
 _
SSR = (Yi - Y)2
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Xi
_
Y
X
Venn Diagrams and Explanatory Power of Regression
Variations in
store Sizes not
used in
explaining
variation in
Sales
Sizes
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Sales
Variations in Sales
explained by the
error term or
unexplained by
Sizes  SSE 
Variations in Sales
explained by Sizes
or variations in Sizes
used in explaining
variation in Sales
 SSR 
The ANOVA Table in Excel
ANOVA
df
SS
Regression
k
Residuals
n-k-1 SSE
Total
n-1
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SSR
SST
MS
MSR
=SSR/k
MSE
=SSE/(n-k-1)
F
MSR/MSE
Significance
F
P-value of
the F Test
Measures of Variation
The Sum of Squares: Example
Excel Output for Produce Stores
Degrees of freedom
ANOVA
df
SS
MS
Regression
1
30380456.12
30380456
Residual
5
1871199.595 374239.92
Total
6
32251655.71
F
81.17909
Regression (explained) df
Error (residual) df
Total df
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SSE
SSR
Significance F
0.000281201
SST
The Coefficient of Determination
•
SSR Regression Sum of Squares
r 

SST
Total Sum of Squares
2
• Measures the proportion of variation in Y that is explained by the
independent variable X in the regression model
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Venn Diagrams and
Explanatory Power of Regression
r 
2
Sales
Sizes
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SSR

SSR  SSE
Coefficients of Determination (r 2) and Correlation (r)
Y r2 = 1, r = +1
Y r2 = 1, r = -1
^=b +b X
Y
i
^=b +b X
Y
i
0
1 i
0
X
Y r2 = .81,r = +0.9
X
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X
Y
^=b +b X
Y
i
0
1 i
1 i
r2 = 0, r = 0
^=b +b X
Y
i
0
1 i
X
Standard Error of Estimate

n
•
SYX
SSE


n2
i 1
Y  Yˆi

2
n2
• Measures the standard deviation (variation) of the Y values around
the regression equation
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Measures of Variation:
Produce Store Example
Excel Output for Produce Stores
R e g r e ssi o n S ta ti sti c s
M u lt ip le R
R S q u a re
0 .9 4 1 9 8 1 2 9
A d ju s t e d R S q u a re
0 .9 3 0 3 7 7 5 4
S t a n d a rd E rro r
6 1 1 .7 5 1 5 1 7
O b s e r va t i o n s
r2 = .94
0 .9 7 0 5 5 7 2
n
7
94% of the variation in annual sales can be
explained by the variability in the size of the
store as measured by square footage.
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Syx
Linear Regression Assumptions
• Normality
– Y values are normally distributed for each X
– Probability distribution of error is normal
• Homoscedasticity (Constant Variance)
• Independence of Errors
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Consequences of Violation
of the Assumptions
• Violation of the Assumptions
– Non-normality (error not normally distributed)
– Heteroscedasticity (variance not constant)
• Usually happens in cross-sectional data
– Autocorrelation (errors are not independent)
• Usually happens in time-series data
• Consequences of Any Violation of the Assumptions
– Predictions and estimations obtained from the sample regression line
will not be accurate
– Hypothesis testing results will not be reliable
• It is Important to Verify the Assumptions
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Variation of Errors Around
the Regression Line
f(e)
• Y values are normally distributed
around the regression line.
• For each X value, the “spread” or
variance around the regression line is
the same.
Y
X2
X1
X
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Sample Regression Line
Residual Analysis
• Purposes
– Examine linearity
– Evaluate violations of assumptions
• Graphical Analysis of Residuals
– Plot residuals vs. X and time
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