Matakuliah
Tahun
Versi
: I0284 - Statistika
: 2008
: Revisi
Pertemuan 23 dan 24
Regresi dan Korelasi Ganda
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Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menghitung
koefisien regresi, korelasi dan determinasi
ganda.
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Outline Materi
•
•
•
•
Model regresi ganda
Persamaan normal regresi ganda
Persamaan regresi dugaan
Koefisien korelasi dan determinasi ganda
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Multiple Regression
•
•
•
•
•
•
Multiple Regression Model
Least Squares Method
Multiple Coefficient of Determination
Model Assumptions
Testing for Significance
Using the Estimated Regression Equation
for Estimation and Prediction
• Qualitative Independent Variables
• Residual Analysis
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The Multiple Regression Model
• The Multiple Regression Model
y = 0 + 1x1 + 2x2 + . . . + pxp + 
• The Multiple Regression Equation
E(y) = 0 + 1x1 + 2x2 + . . . + pxp
• The Estimated Multiple Regression Equation
y^ = b0 + b1x1 + b2x2 + . . . + bpxp
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The Least Squares Method
• Least Squares Criterion
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min  ( y i  y i )
^
• Computation of Coefficients’ Values
The formulas for the regression coefficients b0,
b1, b2, . . . bp involve the use of matrix algebra. We
will rely on computer software packages to perform
the calculations.
• A Note on Interpretation of Coefficients
bi represents an estimate of the change in y
corresponding to a one-unit change in xi when all
other independent variables are held constant.
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The Multiple Coefficient of
Determination
• Relationship Among SST, SSR, SSE
SST = SSR + SSE
2
2
2
 ( y i  y^ )   ( y i  y^)   ( y i  y i )
• Multiple Coefficient of Determination
R 2 = SSR/SST
Ra2
n1
 1  (1  R )
np1
2
• Adjusted Multiple Coefficient of
Determination
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Model Assumptions
• Assumptions About the Error Term 
–
–
–
–
The error  is a random variable with mean of
zero.
The variance of  , denoted by 2, is the
same for all values of the independent
variables.
The values of  are independent.
The error  is a normally distributed random
variable reflecting the deviation between the y
value and the expected value of y given by
0 + 1 x 1 + 2 x 2 + . . . + p x p
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Testing for Significance: F Test
• Hypotheses
H0: 1 = 2 = . . . = p = 0
Ha: One or more of the parameters
is not equal to zero.
• Test Statistic
F = MSR/MSE
• Rejection Rule
Reject H0 if F > F
where F is based on an F distribution with p d.f.
in
the numerator and n - p - 1 d.f. in the
denominator.
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Testing for Significance: t Test
• Hypotheses
H0: i = 0
Ha: i ≠ 0
• Test Statistic
bi
t 
sbii
• Rejection Rule
Reject H0 if t < -tor t > t
where t is based on a t distribution with
n - p - 1 degrees of freedom.
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Testing for Significance:
Multicollinearity
• The term multicollinearity refers to the
correlation among the independent variables.
• When the independent variables are highly
correlated (say, |r | > .7), it is not possible to
determine the separate effect of any particular
independent variable on the dependent variable.
• If the estimated regression equation is to be
used only for predictive purposes,
multicollinearity is usually not a serious problem.
• Every attempt should be made to avoid including
independent variables that are highly correlated.
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• Selamat Belajar Semoga Sukses.
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