for Statistics Managers
Using Microsoft Excel
5th Edition
Chapter 15
Multiple Regression Model Building
Statistics for Managers Using Microsoft Excel, 5e 2008 Prentice-Hall, Inc.
Chap 15-1
Learning Objectives
In this chapter, you learn:
To use quadratic terms in a regression model
To use transformed variables in a regression model
To measure the correlation among independent
variables
To build a regression model, using either the
stepwise or best-subsets approach
To avoid the pitfalls involved in developing a
multiple regression model
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-2
Nonlinear Relationships
The relationship between the dependent variable
and an independent variable may not be linear
Can review the scatter plot to check for nonlinear relationships
Example: Quadratic model
2
Yi = 0 + 1X1i + 2 X1i + i
The second independent variable is the square of the
first variable
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-3
Quadratic Regression Model
Model form:
2
Yi = 0 + 1X1i + 2 X1i + i
where:
0 = Y intercept
1 = regression coefficient for linear effect of X on Y
2 = regression coefficient for quadratic effect on Y
i = random error in Y for observation i
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-4
Quadratic Regression Model
Y
Y
X
X
residuals
residuals
X
Linear fit does not give
random residuals
Statistics for Managers Using Microsoft Excel, 5e
X
Nonlinear fit gives
random residuals
Chap 15-5
Quadratic Regression Model
2
Yi = 0 + 1X1i + 2 X1i + i
Quadratic models may be considered when the scatter
diagram takes on one of the following shapes:
Y
Y
1 < 0
2 > 0
X1
Y
1 > 0
2 > 0
X1
Y
1 < 0
2 < 0
X1
1 > 0
2 < 0
1 = the coefficient of the linear term
2 = the coefficient of the squared term
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-6
X1
Quadratic Regression
Equation
Collect data and use Excel to calculate the
regression equation:
= b +b X +b X2
Yi
0
1 1i
2 1i
Test for Overall Relationship
H0: 1 = 2 = 0 (no overall relationship between X and Y)
H1: 1 and/or 2 0 (there is a relationship between X and Y)
F-test statistic =
MSR
MSE
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-7
Testing for Significance
Quadratic Effect
Testing the Quadratic Effect
Compare quadratic regression equation
^
Yi = b 0 + b1X1i + b 2 X12
i
with the linear regression equation
^
Yi = b 0 + b1X1i
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-8
Testing for Significance
Quadratic Effect
Testing the Quadratic Effect
Consider the quadratic regression equation
^
Yi = b 0 + b1X1i + b 2 X12
i
Hypotheses
H0: 2 = 0
(The quadratic term does not improve the
model)
H1: 2 0 (The quadratic term improves the model)
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-9
Testing for Significance
Quadratic Effect
Testing the Quadratic Effect
Hypotheses
H0: 2 = 0
(The quadratic term does not improve the model)
H1: 2 0 (The quadratic term improves the model)
The test statistic is
b2 2
t=
Sb 2
d.f. = n 3
where:
b2 = estimated slope
2 = hypothesized slope (zero)
Sb2 = standard error of the slope
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-10
Testing for Significance
Quadratic Effect
Testing the Quadratic Effect
If the t test for the quadratic effect is
significant, keep the quadratic term in the
model.
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-11
Quadratic Regression
Example
3
1
7
2
8
3
15
5
22
7
33
8
40
10
54
12
67
13
70
14
78
15
85
15
87
16
99
Purity increases as filter time increases:
17
Purity vs. Time
100
80
Purity
Purity
Filter
Time
60
40
20
0
0
5
Statistics for Managers Using Microsoft Excel, 5e
10
15
20
Time
Chap 15-12
Quadratic Regression
Example
Simple (linear) regression results:
^
t statistic, F statistic, and
Y = -11.283 + 5.985 Time
2
Coefficients
Intercept
adjusted r are all high, but
the residuals are not random:
Standard
Error
t Stat
P-value
-11.28267
-3.25332
0.00691
5.98520
Time
3.46805
0.30966
19.32819
2.078E-10
Time Residual Plot
10
5
Regression Statistics
0.96888
Adjusted R Square
0.96628
Standard Error
6.15997
373.57904
Significance F
2.0778E-10
Residuals
R Square
F
0
-5 0
5
10
15
-10
Time
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-13
20
Quadratic Regression
Example
Quadratic regression results:
^
Y = 1.539 + 1.565 Time + 0.245 (Time)2
Coefficients
t Stat
P-value
Intercept
1.53870
2.24465
0.68550
0.50722
Time
1.56496
0.60179
2.60052
0.24516
0.03258
7.52406
5
0
0
5
0.02467
Time-squared
10
Residuals
Standard
Error
Time Residual Plot
10
15
20
1.165E-05
-5
Time
Time-squared Residual Plot
F
R Square
0.99494
Adjusted R Square
0.99402
Standard Error
Significance F
2.59513
1080.7330
2.368E-13
The quadratic term is significant and improves
the model: adjusted r2 is higher and SYX is
lower, residuals are now random.
Statistics for Managers Using Microsoft Excel, 5e
10
Residuals
Regression Statistics
5
0
-5
0
100
200
Time-squared
Chap 15-14
300
400
Using Transformations in
Regression Analysis
Idea:
Non-linear models can often be transformed to a
linear form
Can be estimated by least squares if
transformed
Transform X or Y or both to get a better fit or to
deal with violations of regression assumptions
Can be based on theory, logic or scatter plots
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-15
The Square Root
Transformation
The square-root transformation
Yi = 0 + 1 X1i + i
Used to
overcome violations of the equal variance assumption
fit a non-linear relationship
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-16
The Square Root
Transformation
Yi = 0 + 1 X1i + i
Yi = 0 + 1X1i + i
Shape of original relationship
Y
Relationship when transformed
Y
X
X
Y
b1 > 0
Y
b1 < 0
X
Statistics for Managers Using Microsoft Excel, 5e
X
Chap 15-17
The Log Transformation
The Multiplicative Model:
Original multiplicative model
Transformed multiplicative model
log Yi = log 0 + 1 log X1i + log i
Yi = 0 X1i1 i
The Exponential Model:
Original multiplicative model
Yi = e
0 +1X1i + 2 X 2i
i
Transformed exponential model
ln Yi = 0 + 1X1i + 2 X 2i + ln i
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-18
Interpretation of Coefficients
For the multiplicative model:
log Yi = log 0 + 1 log X1i + log i
When both dependent and independent variables are
transformed:
coefficient The of the independent variable Xk can be
interpreted as follows: a 1 percent change in Xk leads
to an estimated bk percentage change in the mean value
of Y.
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-19
Collinearity
Collinearity: High correlation exists among
two or more independent variables
The correlated variables contribute redundant
information to the multiple regression model
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-20
Collinearity
Including two highly correlated independent
variables can adversely affect the regression
results
No new information provided
Can lead to unstable coefficients (large
standard error and low t-values)
Coefficient signs may not match prior
expectations
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-21
Some Indications of Strong
Collinearity
Incorrect signs on the coefficients
Large change in the value of a previous coefficient
when a new variable is added to the model
A previously significant variable becomes
insignificant when a new independent variable is
added
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-22
Detecting Collinearity
Variance Inflationary Factor
VIFj is used to measure collinearity:
1
VIFj =
2
1 R j
where R2j is the coefficient of determination from a regression
model that uses Xj as the dependent variable and all other X
variables as the independent variables
If VIFj > 5, Xj is highly correlated with the
other independent variables
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-23
Model Building
Goal is to develop a model with the best set of
independent variables
Easier to interpret if unimportant variables are removed
Lower probability of collinearity
Stepwise regression procedure
Provide evaluation of alternative models as variables are
added
Best-subset approach
Try all combinations and select the model with the
highest adjusted r2
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-24
Stepwise Regression
Idea: develop the least squares regression
equation in steps, adding one independent
variable at a time and evaluating whether
existing variables should remain or be
removed
The coefficient of partial determination is the
measure of the marginal contribution of each
independent variable, given that other
independent variables are in the model
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-25
Best Subsets Regression
Idea: estimate all possible regression equations using all
possible combinations of independent variables
Choose the best model by looking for the highest adjusted r 2
The model with the largest adjusted r2 will also have the
smallest SYX
Stepwise regression and best subsets regression can
be performed using Excel with PHStat add in
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-26
Alternative Best Subsets
Criterion
Calculate the value Cp for each potential
regression model
Consider models with Cp values close to or
below k + 1
k is the number of independent variables in the
model under consideration
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-27
Alternative Best Subsets
Criterion
The Cp Statistic
2
(1 Rk )(n T )
Cp =
(n 2(k + 1))
2
1 RT
Where
k = number of independent variables included in a
particular regression model
T = total number of parameters to be estimated in the
full regression model
2
Rk = coefficient of multiple determination for model with k
independent variables
R 2 = coefficient of multiple determination for full model with
T
all T estimated parameters
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-28
8 Steps in Model Building
1. Choose independent variables to include in the model
2. Estimate full model and check VIFs and check if any VIFs > 5
If no VIF > 5, go to step 3
If one VIF > 5, remove this variable
If more than one, eliminate the variable with the highest VIF
and repeat step 2
3. Perform best subsets regression with remaining variables.
4. List all models with Cp close to or less than (k + 1).
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-29
8 Steps in Model Building
5. Choose the best model.
Consider parsimony.
Do extra variables make a significant contribution?
6. Perform complete analysis with chosen model, including
residual analysis.
7. Transform the model if necessary to deal with violations of
linearity or other model assumptions.
8. Use the model for prediction and inference.
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-30
Model Validation
The final step in the model-building process is
to validate the selected regression model.
Collect new data and compare the results.
Compare the results of the regression model
to previous results.
If the data set is large, split the data into two
parts and cross-validate the results.
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-31
Model Building Flowchart
Choose X1,X2,,Xk
Run regression to
find VIFs
Any
VIF>5?
Run best subsets
regression to obtain
best models in terms
of Cp
No
Yes
Remove
variable with
highest
VIF
Yes
More than
one?
Do complete analysis
Add quadratic term and/or
transform variables as indicated
No
Remove this X
Statistics for Managers Using Microsoft Excel, 5e
Perform predictions
Chap 15-32
Pitfalls and Ethical
Considerations
To avoid pitfalls and address ethical considerations:
Understand that interpretation of the estimated
regression coefficients are performed holding all
other independent variables constant
Evaluate residual plots for each independent
variable
Evaluate interaction terms
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-33
Pitfalls and Ethical
Considerations
To avoid pitfalls and address ethical considerations:
Obtain VIFs for each independent variable before
determining which variables should be included in the
model.
Examine several alternative models using best-subsets
regression.
Use other methods when the assumptions necessary for
least-squares regression have been seriously violated.
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-34
Chapter Summary
In this chapter, we have
Developed the quadratic regression model
Discussed using transformations in regression
models
The multiplicative model
The exponential model
Described collinearity
Discussed model building
Stepwise regression
Best subsets
Addressed pitfalls in multiple regression and ethical
considerations
Statistics for Managers Using Microsoft Excel, 5e
Chap 15-35