Chap12 multiple regression

Statistics for
Business and Economics
7th Edition
Chapter 12
Multiple Regression
Ch. 12-1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Chapter Goals
After completing this chapter, you should be able to:
 Apply multiple regression analysis to business decision-
making situations
 Analyze and interpret the computer output for a multiple
regression model
 Perform a hypothesis test for all regression coefficients
or for a subset of coefficients
 Fit and interpret nonlinear regression models
 Incorporate qualitative variables into the regression
model by using dummy variables
 Discuss model specification and analyze residuals

The Multiple Regression Model
Idea: Examine the linear relationship between
1 dependent (Y) & 2 or more independent variables (Xi)
εXβXβXββY kk22110  
Multiple Regression Model with k Independent Variables:
Y-intercept Population slopes Random Error
Ch. 12-3
12.1
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Multiple Regression Equation
The coefficients of the multiple regression model are
estimated using sample data
kik2i21i10i xbxbxbby  ˆ
Estimated
(or predicted)
value of y
Estimated slope coefficients
Multiple regression equation with k independent variables:
Estimated
intercept
In this chapter we will always use a computer to obtain the
regression slope coefficients and other regression
summary measures.

Multiple Regression Equation
Two variable model
y
x1
x2
22110 xbxbby ˆ
(continued)

Multiple Regression Model
Two variable model
y
x1
x2
yi
yi
<
ei = (yi – yi)
<
x2i
x1i The best fit equation, y ,
is found by minimizing the
sum of squared errors, e2
<
Sample
observation

Standard Multiple Regression
Assumptions
 The values xi and the error terms εi are
independent
 The error terms are random variables with
mean 0 and a constant variance, 2.
(The constant variance property is called
homoscedasticity)
n),1,(iforσ]E[εand0]E[ε 22
ii 

 The random error terms, εi , are not correlated
with one another, so that
 It is not possible to find a set of numbers, c0,
c1, . . . , ck, such that
(This is the property of no linear relation for
the Xj’s)
(continued)
jiallfor0]εE[ε ji 
0xcxcxcc KiK2i21i10  
Ch. 12-8
Standard Multiple Regression
Assumptions

Example:
2 Independent Variables
 A distributor of frozen desert pies wants to
evaluate factors thought to influence demand
 Dependent variable: Pie sales (units per week)
 Independent variables: Price (in $)
Advertising ($100’s)
 Data are collected for 15 weeks

Pie Sales Example
Sales = b0 + b1 (Price)
+ b2 (Advertising)
Week
Pie
Sales
Price
($)
Advertising
($100s)
1 350 5.50 3.3
2 460 7.50 3.3
3 350 8.00 3.0
4 430 8.00 4.5
5 350 6.80 3.0
6 380 7.50 4.0
7 430 4.50 3.0
8 470 6.40 3.7
9 450 7.00 3.5
10 490 5.00 4.0
11 340 7.20 3.5
12 300 7.90 3.2
13 440 5.90 4.0
14 450 5.00 3.5
15 300 7.00 2.7
Multiple regression equation:

Estimating a Multiple Linear
Regression Equation
 Excel will be used to generate the coefficients and
measures of goodness of fit for multiple regression
 Data / Data Analysis / Regression
Ch. 12-11
12.2

Multiple Regression Output
Regression Statistics
Multiple R 0.72213
R Square 0.52148
Adjusted R Square 0.44172
Standard Error 47.46341
Observations 15
ANOVA df SS MS F Significance F
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
ertising)74.131(Advce)24.975(Pri-306.526Sales 

The Multiple Regression Equation
ertising)74.131(Advce)24.975(Pri-306.526Sales 
b1 = -24.975: sales
will decrease, on
average, by 24.975
pies per week for
each $1 increase in
selling price, net of
the effects of changes
due to advertising
b2 = 74.131: sales will
increase, on average,
by 74.131 pies per
week for each $100
increase in
advertising, net of the
effects of changes
due to price
where
Sales is in number of pies per week
Price is in $
Advertising is in $100’s.

Coefficient of Determination, R2
 Reports the proportion of total variation in y
explained by all x variables taken together
 This is the ratio of the explained variability to
total sample variability
squaresofsumtotal
squaresofsumregression
SST
SSR
R2

Ch. 12-14
12.3

Coefficient of Determination, R2
Multiple R 0.72213
R Square 0.52148
Observations 15
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
.52148
56493.3
29460.0
SST
SSR
R2

52.1% of the variation in pie sales
is explained by the variation in
price and advertising
(continued)

Estimation of Error Variance
 Consider the population regression model
 The unbiased estimate of the variance of the errors is
where
 The square root of the variance, se , is called the
standard error of the estimate
1Kn
SSE
1Kn
e
s
n
1i
2
i
2
e





iKiK2i21i10i εxβxβxββY  
iii yye ˆ

Standard Error, se
Multiple R 0.72213
R Square 0.52148
Observations 15
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
47.463se 
The magnitude of this
value can be compared to
the average y value

Adjusted Coefficient of
Determination,
 R2 never decreases when a new X variable is
added to the model, even if the new variable is
not an important predictor variable
 This can be a disadvantage when comparing
models
 What is the net effect of adding a new variable?
 We lose a degree of freedom when a new X
variable is added
 Did the new X variable add enough
explanatory power to offset the loss of one
degree of freedom?
2
R

Adjusted Coefficient of
Determination,
 Used to correct for the fact that adding non-relevant
independent variables will still reduce the error sum of
squares
(where n = sample size, K = number of independent variables)
 Adjusted R2 provides a better comparison between
multiple regression models with different numbers of
independent variables
 Penalize excessive use of unimportant independent
variables
 Smaller than R2
(continued)
2
R
1)(n/SST
1)K(n/SSE
1R2




Multiple R 0.72213
R Square 0.52148
Observations 15
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
.44172R2

44.2% of the variation in pie sales is
explained by the variation in price and
advertising, taking into account the sample
size and number of independent variables
2
R

Coefficient of Multiple
Correlation
 The coefficient of multiple correlation is the correlation
between the predicted value and the observed value of
the dependent variable
 Is the square root of the multiple coefficient of
determination
 Used as another measure of the strength of the linear
relationship between the dependent variable and the
independent variables
 Comparable to the correlation between Y and X in
simple regression
2
Ry),yr(R  ˆ

Evaluating Individual
Regression Coefficients
 Use t-tests for individual coefficients
 Shows if a specific independent variable is
conditionally important
 Hypotheses:
 H0: βj = 0 (no linear relationship)
 H1: βj ≠ 0 (linear relationship does exist
between xj and y)
Ch. 12-22
12.4

H0: βj = 0 (no linear relationship)
H1: βj ≠ 0 (linear relationship does exist
between xi and y)
Test Statistic:
(df = n – k – 1)
jb
j
S
0b
t


(continued)

Multiple R 0.72213
R Square 0.52148
Observations 15
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
t-value for Price is t = -2.306, with
p-value .0398
t-value for Advertising is t = 2.855,
with p-value .0145
(continued)

H0: βj = 0
H1: βj  0
d.f. = 15-2-1 = 12
 = .05
t12, .025 = 2.1788
The test statistic for each variable falls
in the rejection region (p-values < .05)
There is evidence that both
Price and Advertising affect
pie sales at  = .05
From Excel output:
Reject H0 for each variable
Coefficients Standard Error t Stat P-value
Price -24.97509 10.83213 -2.30565 0.03979
Advertising 74.13096 25.96732 2.85478 0.01449
Decision:
Conclusion:
Reject H0Reject H0
/2=.025
-tα/2
Do not reject H0
0
tα/2
/2=.025
-2.1788 2.1788
Example: Evaluating Individual

Confidence Interval Estimate
for the Slope
Confidence interval limits for the population slope βj
Example: Form a 95% confidence interval for the effect of
changes in price (x1) on pie sales:
-24.975 ± (2.1788)(10.832)
So the interval is -48.576 < β1 < -1.374
jbα/21,Knj Stb 
Coefficients Standard Error
Intercept 306.52619 114.25389
Price -24.97509 10.83213
Advertising 74.13096 25.96732
where t has
(n – K – 1) d.f.
Here, t has
(15 – 2 – 1) = 12 d.f.

Confidence Interval Estimate
for the Slope
Confidence interval for the population slope βi
Example: Excel output also reports these interval endpoints:
Weekly sales are estimated to be reduced by between 1.37 to
48.58 pies for each increase of $1 in the selling price
Coefficients Standard Error … Lower 95% Upper 95%
Intercept 306.52619 114.25389 … 57.58835 555.46404
Price -24.97509 10.83213 … -48.57626 -1.37392
Advertising 74.13096 25.96732 … 17.55303 130.70888
(continued)

Test on All Coefficients
 F-Test for Overall Significance of the Model
 Shows if there is a linear relationship between all
of the X variables considered 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)
Ch. 12-28
12.5

F-Test for Overall Significance
 Test statistic:
where F has k (numerator) and
(n – K – 1) (denominator)
degrees of freedom
 The decision rule is
1)KSSE/(n
SSR/K
s
MSR
F 2
e 

α1,Knk,0 FFifHReject 

6.5386
2252.8
14730.0
MSE
MSR
F 
Multiple R 0.72213
R Square 0.52148
Observations 15
Regression 2 29460.027 14730.013 6.53861 0.01201
Residual 12 27033.306 2252.776
Total 14 56493.333
Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404
Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392
Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888
(continued)
With 2 and 12 degrees
of freedom
P-value for
the F-Test

H0: β1 = β2 = 0
H1: β1 and β2 not both zero
 = .05
df1= 2 df2 = 12
Test Statistic:
Decision:
Conclusion:
Since F test statistic is in
the rejection region (p-
value < .05), reject H0
There is evidence that at least one
independent variable affects Y
0
 = .05
F.05 = 3.885
Reject H0Do not
reject H0
6.5386
MSE
MSR
F 
Critical
Value:
F = 3.885
(continued)
F

Tests on a Subset of
 Consider a multiple regression model involving
variables xj and zj , and the null hypothesis that the z
variable coefficients are all zero:
r)1,...,(j0αofoneleastat:H
0ααα:H
j1
r210

 
irir1i1KiK1i10i εzαzαxβxββy  

Tests on a Subset of
 Goal: compare the error sum of squares for the
complete model with the error sum of squares for the
restricted model
 First run a regression for the complete model and obtain SSE
 Next run a restricted regression that excludes the z variables
(the number of variables excluded is r) and obtain the
restricted error sum of squares SSE(r)
 Compute the F statistic and apply the decision rule for a
significance level 
(continued)
α1,rKnr,2
e
0 F
s
r/)SSESSE(r)(
FifHReject 



Prediction
 Given a population regression model
 then given a new observation of a data point
(x1,n+1, x 2,n+1, . . . , x K,n+1)
the best linear unbiased forecast of yn+1 is
 It is risky to forecast for new X values outside the range of the data used
to estimate the model coefficients, because we do not have data to
support that the linear model extends beyond the observed range.
n),1,2,(iεxβxβxββy iKiK2i21i10i  
1nK,K1n2,21n1,101n xbxbxbby   ˆ
^
Ch. 12-34
12.6

Using The Equation to Make
Predictions
Predict sales for a week in which the selling
price is $5.50 and advertising is $350:
Predicted sales
is 428.62 pies
428.62
(3.5)74.131(5.50)24.975-306.526
ertising)74.131(Advce)24.975(Pri-306.526Sales



Note that Advertising is
in $100’s, so $350
means that X2 = 3.5

Nonlinear Regression Models
 The relationship between the dependent
variable and an independent variable may
not be linear
 Can review the scatter diagram to check for
non-linear relationships
 Example: Quadratic model
 The second independent variable is the square
of the first variable
εXβXββY 2
12110 
Ch. 12-36
12.7

Quadratic Regression Model
 where:
β0 = Y intercept
β1 = regression coefficient for linear effect of X on Y
β2 = regression coefficient for quadratic effect on Y
εi = random error in Y for observation i
i
2
1i21i10i εxβxββy 
Model form:

Linear vs. Nonlinear Fit
Linear fit does not give
random residuals
Nonlinear fit gives
random residuals

X
residuals
X
Y
X
residuals
Y
X

Quadratic Regression Model
Quadratic models may be considered when the scatter
diagram takes on one of the following shapes:
X1
Y
X1X1
YYY
β1 < 0 β1 > 0 β1 < 0 β1 > 0
β1 = the coefficient of the linear term
β2 = the coefficient of the squared term
X1
β2 > 0 β2 > 0 β2 < 0 β2 < 0
i
2
1i21i10i εXβXββY 

Testing for Significance:
Quadratic Effect
 Testing the Quadratic Effect
 Compare the linear regression estimate
 with quadratic regression estimate
 Hypotheses
 (The quadratic term does not improve the model)
 (The quadratic term improves the model)
2
110 xbby ˆ
H0: β2 = 0
H1: β2  0

Quadratic Effect
Hypotheses
 (The quadratic term does not improve the model)
 (The quadratic term improves the model)
 The test statistic is
H0: β2 = 0
H1: β2  0
(continued)
2b
22
s
βb
t


3nd.f. 
where:
b2 = squared term slope
coefficient
β2 = hypothesized slope (zero)
Sb = standard error of the slope
2

Quadratic Effect
Compare R2 from simple regression to
R2 from the quadratic model
 If R2 from the quadratic model is larger than
R2 from the simple model, then the
quadratic model is a better model
(continued)

Example: Quadratic Model
 Purity increases as filter time
increases:
Purity
Filter
Time
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 17
Purity vs. Time
0
20
40
60
80
100
0 5 10 15 20
Time
Purity

 Simple regression results:
y = -11.283 + 5.985 Time
(continued)
R Square 0.96888
Coefficients
Standard
Error t Stat P-value
Intercept -11.28267 3.46805 -3.25332 0.00691
Time 5.98520 0.30966 19.32819 2.078E-10
F Significance F
373.57904 2.0778E-10
^
Time Residual Plot
-10
-5
0
5
10
0 5 10 15 20
Time
Residuals
t statistic, F statistic, and
R2 are all high, but the
residuals are not random:

Coefficients
Standard
Error t Stat P-value
Intercept 1.53870 2.24465 0.68550 0.50722
Time 1.56496 0.60179 2.60052 0.02467
Time-squared 0.24516 0.03258 7.52406 1.165E-05
R Square 0.99494
F Significance F
1080.7330 2.368E-13
 Quadratic regression results:
y = 1.539 + 1.565 Time + 0.245 (Time)2^
(continued)
Time Residual Plot
-5
0
5
10
0 5 10 15 20
Time
Residuals
Time-squared Residual Plot
-5
0
5
10
0 100 200 300 400
Time-squared
Residuals
The quadratic term is significant and
improves the model: R2 is higher and se is
lower, residuals are now random

The Log Transformation
 Original multiplicative model
 Transformed multiplicative model
The Multiplicative Model:
εXXβY 21 β
2
β
10
)log(ε)log(Xβ)log(Xβ)log(βlog(Y) 22110 

Interpretation of coefficients
For the multiplicative model:
 When both dependent and independent
variables are logged:
 The coefficient of the independent variable Xk can
be interpreted as
a 1 percent change in Xk leads to an estimated bk
percentage change in the average value of Y
 bk is the elasticity of Y with respect to a change in Xk
i1i10i εlogXlogββlogYlog 

Dummy Variables
 A dummy variable is a categorical independent
variable with two levels:
 yes or no, on or off, male or female
 recorded as 0 or 1
 Regression intercepts are different if the
variable is significant
 Assumes equal slopes for other variables
 If more than two levels, the number of dummy
variables needed is (number of levels - 1)
Ch. 12-48
12.8

Dummy Variable Example
Let:
y = Pie Sales
x1 = Price
x2 = Holiday (X2 = 1 if a holiday occurred during the week)
(X2 = 0 if there was no holiday that week)
210 xbxbby 21
ˆ

Dummy Variable Example
Same
slope
(continued)
x1 (Price)
y (sales)
b0 + b2
b0
1010
12010
xbb(0)bxbby
xb)b(b(1)bxbby
121
121


ˆ
ˆ Holiday
No Holiday
Different
intercept
If H0: β2 = 0 is
rejected, then
“Holiday” has a
significant effect
on pie sales

Interpreting the
Dummy Variable Coefficient
Sales: number of pies sold per week
Price: pie price in $
Holiday:
Example:
1 If a holiday occurred during the week
0 If no holiday occurred
b2 = 15: on average, sales were 15 pies greater in
weeks with a holiday than in weeks without a
holiday, given the same price
)15(Holiday30(Price)-300Sales 

Interaction Between
Explanatory Variables
 Hypothesizes interaction between pairs of x
variables
 Response to one x variable may vary at different
levels of another x variable
 Contains two-way cross product terms

)x(xbxbxbb
xbxbxbby
21322110
3322110

ˆ

Effect of Interaction
 Given:
 Without interaction term, effect of X1 on Y is
measured by β1
 With interaction term, effect of X1 on Y is
measured by β1 + β3 X2
 Effect changes as X2 changes
21322110
1231220
XXβXβXββ
)XXβ(βXββY



Interaction Example
x2 = 1:
y = 1 + 2x1 + 3(1) + 4x1(1) = 4 + 6x1
x2 = 0:
y = 1 + 2x1 + 3(0) + 4x1(0) = 1 + 2x1
Slopes are different if the effect of x1 on y depends on x2 value
x1
4
8
12
0
0 10.5 1.5
y
Suppose x2 is a dummy variable and the estimated
regression equation is 2121 x4x3x2x1y ˆ
^
^

Significance of Interaction Term
 The coefficient b3 is an estimate of the difference
in the coefficient of x1 when x2 = 1 compared to
when x2 = 0
 The t statistic for b3 can be used to test the
hypothesis
 If we reject the null hypothesis we conclude that there is
a difference in the slope coefficient for the two
subgroups
0β0,β|0β:H
0β0,β|0β:H
2131
2130



Multiple Regression Assumptions
Assumptions:
 The errors are normally distributed
 Errors have a constant variance
 The model errors are independent
ei = (yi – yi)
<
Errors (residuals) from the regression model:
Ch. 12-56
12.9

Analysis of Residuals
in Multiple Regression
 These residual plots are used in multiple
regression:
 Residuals vs. yi
 Residuals vs. x1i
 Residuals vs. x2i
 Residuals vs. time (if time series data)
<
Use the residual plots to check for
violations of regression assumptions

Chapter Summary
 Developed the multiple regression model
 Tested the significance of the multiple regression model
 Discussed adjusted R2 ( R2 )
 Tested individual regression coefficients
 Tested portions of the regression model
 Used quadratic terms and log transformations in
regression models
 Explained dummy variables
 Evaluated interaction effects
 Discussed using residual plots to check model
assumptions

Chap12 multiple regression

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Chap12 multiple regression