Regresi dan Rancangan Faktorial Pertemuan 23 Matakuliah: I0174
Analisis Regresi Tahun: Ganjil 2007/2008
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Bina Nusantara Regresi dan Rancangan Faktorial Penyandian
Ortogonal pada rancangan faktorial. Persamaan Regresi pada
rancangan faktorial
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Bina Nusantara Population Y-intercept Population slopesRandom
error The Multiple Regression Model Relationship between 1
dependent & 2 or more independent variables is a linear
function Dependent (Response) variable Independent (Explanatory)
variables
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Bina Nusantara Multiple Regression Model Bivariate model
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Bina Nusantara Multiple Regression Equation Bivariate model
Multiple Regression Equation
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Bina Nusantara Interpretation of Estimated Coefficients Slope (
b j ) Estimated that the average value of Y changes by b j for each
1 unit increase in X j, holding all other variables constant
(ceterus paribus) Example: If b 1 = -2, then fuel oil usage ( Y )
is expected to decrease by an estimated 2 gallons for each 1 degree
increase in temperature ( X 1 ), given the inches of insulation ( X
2 ) Y-Intercept ( b 0 ) The estimated average value of Y when all X
j = 0
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Bina Nusantara Multiple Regression Model: Example ( 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.
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Bina Nusantara Multiple Regression Equation: Example 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.
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Bina Nusantara Simple and Multiple Regression Compared simple
The slope coefficient in a simple regression picks up the impact of
the independent variable plus the impacts of other variables that
are excluded from the model, but are correlated with the included
independent variable and the dependent variable multiple
Coefficients in a multiple regression net out the impacts of other
variables in the equation Hence, they are called the net regression
coefficients They still pick up the effects of other variables that
are excluded from the model, but are correlated with the included
independent variables and the dependent variable
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Bina Nusantara Simple and Multiple Regression Compared: Example
Two Simple Regressions: Multiple Regression:
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Bina Nusantara Simple and Multiple Regression Compared: Slope
Coefficients
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Bina Nusantara Simple and Multiple Regression Compared: r
2
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Bina Nusantara Example: Adjusted r 2 Can Decrease Adjusted r 2
decreases when k increases from 2 to 3 Color is not useful in
explaining the variation in oil consumption.
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Bina Nusantara Using the Regression Equation to Make
Predictions Predict the amount of heating oil used for a home if
the average temperature is 30 0 and the insulation is 6 inches. The
predicted heating oil used is 278.97 gallons.
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Bina Nusantara Predictions in PHStat PHStat | Regression |
Multiple Regression Check the Confidence and Prediction Interval
Estimate box Excel spreadsheet for the heating oil example
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Bina Nusantara 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
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Bina Nusantara Residual Plots: Example No Discernable Pattern
Maybe some non- linear relationship
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Bina Nusantara Testing for Overall Significance Shows if Y
Depends Linearly on All of the X Variables Together as a Group Use
F Test Statistic Hypotheses: H 0 : k = 0 (No linear relationship) H
1 : At least one i ( At least one independentvariable affects Y )
The Null Hypothesis is a Very Strong Statement The Null Hypothesis
is Almost Always Rejected
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Bina Nusantara Testing for Overall Significance Test Statistic:
Where F has k numerator and ( n-k-1 ) denominator degrees of
freedom (continued)
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Bina Nusantara Test for Overall Significance Excel Output:
Example k = 2, the number of explanatory variables n - 1 p
-value
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Bina Nusantara Test for Overall Significance: Example Solution
F 03.89 H 0 : 1 = 2 = = k = 0 H 1 : At least one j 0 =.05 df = 2
and 12 Critical Value : Test Statistic: Decision: Conclusion:
Reject at = 0.05. There is evidence that at least one independent
variable affects Y. = 0.05 F 168.47 (Excel Output)
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Bina Nusantara Test for Significance: Individual Variables Show
If Y Depends Linearly on a Single X j Individually While Holding
the Effects of Other X s Fixed Use t Test Statistic Hypotheses: H 0
: j 0 (No linear relationship) H 1 : j 0 (Linear relationship
between X j and Y )
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Bina Nusantara t Test Statistic Excel Output: Example t Test
Statistic for X 1 (Temperature) t Test Statistic for X 2
(Insulation)
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Bina Nusantara t Test : Example Solution H 0 : 1 = 0 H 1 : 1 0
df = 12 Critical Values: Test Statistic: Decision: Conclusion:
Reject H 0 at = 0.05. There is evidence of a significant effect of
temperature on oil consumption holding constant the effect of
insulation. t 0 2.1788 -2.1788.025 Reject H 0 0.025 Does
temperature have a significant effect on monthly consumption of
heating oil? Test at = 0.05. t Test Statistic = -16.1699
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Bina Nusantara Venn Diagrams and Estimation of Regression Model
Oil Temp Insulation Only this information is used in the estimation
of This information is NOT used in the estimation of nor
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Bina Nusantara Confidence Interval Estimate for the Slope
Provide the 95% confidence interval for the population slope 1 (the
effect of temperature on oil consumption). -6.169 1 -4.704 We are
95% confident that the estimated average consumption of oil is
reduced by between 4.7 gallons to 6.17 gallons per each increase of
1 0 F holding insulation constant. We can also perform the test for
the significance of individual variables, H 0 : 1 = 0 vs. H 1 : 1
0, using this confidence interval.
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Bina Nusantara Contribution of a Single Independent Variable
Let X j Be the Independent Variable of Interest Measures the
additional contribution of X j in explaining the total variation in
Y with the inclusion of all the remaining independent
variables
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Bina Nusantara Contribution of a Single Independent Variable
Measures the additional contribution of X 1 in explaining Y with
the inclusion of X 2 and X 3. From ANOVA section of regression
for
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Bina Nusantara Coefficient of Partial Determination of Measures
the proportion of variation in the dependent variable that is
explained by X j while controlling for (holding constant) the other
independent variables
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Bina Nusantara Coefficient of Partial Determination for
(continued) Example: Model with two independent variables
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Bina Nusantara Venn Diagrams and Coefficient of Partial
Determination for Oil Temp Insulation =
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Bina Nusantara Coefficient of Partial Determination in PHStat
PHStat | Regression | Multiple Regression Check the Coefficient of
Partial Determination box Excel spreadsheet for the heating oil
example
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Bina Nusantara Contribution of a Subset of Independent
Variables Let X s Be the Subset of Independent Variables of
Interest Measures the contribution of the subset X s in explaining
SST with the inclusion of the remaining independent variables
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Bina Nusantara Contribution of a Subset of Independent
Variables: Example Let X s be X 1 and X 3 From ANOVA section of
regression for
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Bina Nusantara Testing Portions of Model Examines the
Contribution of a Subset X s of Explanatory Variables to the
Relationship with Y Null Hypothesis: Variables in the subset do not
improve the model significantly when all other variables are
included Alternative Hypothesis: At least one variable in the
subset is significant when all other variables are included
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Bina Nusantara Testing Portions of Model One-Tailed Rejection
Region Requires Comparison of Two Regressions One regression
includes everything Another regression includes everything except
the portion to be tested (continued)
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Bina Nusantara Partial F Test for the Contribution of a Subset
of X Variables Hypotheses: H 0 : Variables X s do not significantly
improve the model given all other variables included H 1 :
Variables X s significantly improve the model given all others
included Test Statistic: with df = m and ( n-k-1 ) m = # of
variables in the subset X s
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Bina Nusantara Partial F Test for the Contribution of a Single
Hypotheses: H 0 : Variable X j does not significantly improve the
model given all others included H 1 : Variable X j significantly
improves the model given all others included Test Statistic: with
df = 1 and ( n-k-1 ) m = 1 here
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Bina Nusantara Testing Portions of Model: Example Test at the
=.05 level to determine if the variable of average temperature
significantly improves the model, given that insulation is
included.
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Bina Nusantara Testing Portions of Model: Example H 0 : X 1
(temperature) does not improve model with X 2 (insulation) included
H 1 : X 1 does improve model =.05, df = 1 and 12 Critical Value =
4.75 (For X 1 and X 2 )(For X 2 ) Conclusion: Reject H 0 ; X 1 does
improve model.
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Bina Nusantara Dummy-Variable Models Categorical Explanatory
Variable with 2 or More Levels Yes or No, On or Off, Male or
Female, Use Dummy-Variables (Coded as 0 or 1) Only Intercepts are
Different Assumes Equal Slopes Across Categories The Number of
Dummy-Variables Needed is (# of Levels - 1) Regression Model Has
Same Form:
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Bina Nusantara Dummy-Variable Models (with 2 Levels) Given: Y =
Assessed Value of House X 1 = Square Footage of House X 2 =
Desirability of Neighborhood = Desirable ( X 2 = 1) Undesirable ( X
2 = 0) 0 if undesirable 1 if desirable Same slopes
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Bina Nusantara Undesirable Desirable Location Dummy-Variable
Models (with 2 Levels) (continued) X 1 (Square footage) Y (Assessed
Value) b 0 + b 2 b0b0 Same slopes Intercepts different
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Bina Nusantara Interpretation of the Dummy-Variable Coefficient
(with 2 Levels) Example: : GPA 0 non-business degree 1 business
degree : Annual salary of college graduate in thousand $ With the
same GPA, college graduates with a business degree are making an
estimated 6 thousand dollars more than graduates with a
non-business degree, on average. :
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Bina Nusantara Dummy-Variable Models (with 3 Levels)
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Bina Nusantara Interpretation of the Dummy-Variable
Coefficients (with 3 Levels) With the same footage, a Split- level
will have an estimated average assessed value of 18.84 thousand
dollars more than a Condo. With the same footage, a Ranch will have
an estimated average assessed value of 23.53 thousand dollars more
than a Condo.
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Bina Nusantara Regression Model Containing an Interaction Term
Hypothesizes Interaction between a Pair of X Variables Response to
one X variable varies at different levels of another X variable
Contains a Cross-Product Term Can Be Combined with Other Models
E.g., Dummy-Variable Model
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Bina Nusantara Effect of Interaction Given: Without Interaction
Term, Effect of X 1 on Y is Measured by 1 With Interaction Term,
Effect of X 1 on Y is Measured by 1 + 3 X 2 Effect Changes as X 2
Changes
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Bina Nusantara Y = 1 + 2X 1 + 3(1) + 4X 1 (1) = 4 + 6X 1 Y = 1
+ 2X 1 + 3(0) + 4X 1 (0) = 1 + 2X 1 Interaction Example Effect
(slope) of X 1 on Y depends on X 2 value X1X1 4 8 12 0 010.51.5 Y Y
= 1 + 2X 1 + 3X 2 + 4X 1 X 2
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Bina Nusantara Interaction Regression Model Worksheet Multiply
X 1 by X 2 to get X 1 X 2 Run regression with Y, X 1, X 2, X 1 X 2
Case, iYiYi X 1i X 2i X 1i X 2i 11133 248540 31326 435630
:::::
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Bina Nusantara Interpretation When There Are 3+ Levels MALE = 0
if female and 1 if male MARRIED = 1 if married; 0 if not DIVORCED =
1 if divorced; 0 if not MALEMARRIED = 1 if male married; 0
otherwise = (MALE times MARRIED) MALEDIVORCED = 1 if male divorced;
0 otherwise = (MALE times DIVORCED)
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Bina Nusantara Interpretation When There Are 3+ Levels
(continued)
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Bina Nusantara Interpreting Results FEMALE Single: Married:
Divorced: MALE Single: Married: Divorced: Main Effects : MALE,
MARRIED and DIVORCED Interaction Effects : MALEMARRIED and
MALEDIVORCED Difference
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Bina Nusantara Suppose X 1 and X 2 are Numerical Variables and
X 3 is a Dummy-Variable To Test if the Slope of Y with X 1 and/or X
2 are the Same for the Two Levels of X 3 Model: Hypotheses: H 0 : =
= 0 (No Interaction between X 1 and X 3 or X 2 and X 3 ) H 1 : 4
and/or 5 0 ( X 1 and/or X 2 Interacts with X 3 ) Perform a Partial
F Test Evaluating the Presence of Interaction with Dummy-
Variable
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Bina Nusantara Evaluating the Presence of Interaction with
Numerical Variables Suppose X 1, X 2 and X 3 are Numerical
Variables To Test If the Independent Variables Interact with Each
Other Model: Hypotheses: H 0 : = = = 0 (no interaction among X 1, X
2 and X 3 ) H 1 : at least one of 4, 5, 6 0 (at least one pair of X
1, X 2, X 3 interact with each other) Perform a Partial F Test