Linear Regression Models
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Answering the questions you
have asked.
• What is the influence of a policy variable
(price, advertising, and etc.) on a market
outcome (market shares, sales, overall
satisfaction)?
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Applications of Linear Regression
Forecasting Sales Response to Marketing Actions:
Linear Model
Yt   price  X price,t   t
Linear Relation Between Sales and Price
3500
Sales
3000
2500
2000
1500
$15.00
$17.00
$19.00
$21.00
$23.00
$25.00
Price
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SUMMARY OUTPUT
Regression Statistics
Multiple R
0.898
R Square
0.806
Adjusted R Square
0.799
Standard Error
144.74
Observations
29
ANOVA
df
SS
2346991.779
565650.3691
2912642.148
MS
2346992
20950.01
Coefficients Standard Error
5126
240
-123.24
11.64
t Stat
21.34
-10.58
Regression
Residual
Total
Intercept
Price
1
27
28
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F
Significance F
112.03 4.14101E-11
P-value
1.9601E-18
4.141E-11
Lower 95%
Upper 95%
4633
5619
-147.13
-99.35
4
Log-Linear Model
log Yt   price  X price,t   t
Sales
Log Linear Relation Between Sales and Price
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
$15.00
$16.00
$17.00
$18.00
$19.00
$20.00
Price
What if you try to fit a linear model?
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SUMMARY OUTPUT
Regression Statistics
Multiple R
0.807358
R Square
0.651827
Adjusted R
0.645719
Square
Standard Error 5976.594
Observations
59
ANOVA
df
Regression
Residual
Total
Intercept
Price
SS
MS
F
Significance F
1 3.81E+09 3.81E+09 106.7118 1.13E-14
57 2.04E+09 35719681
58 5.85E+09
Coefficie Standard
nts
Error
118158.3 10644.3
-6249.31 604.959
t Stat
P-value
Lower
Upper
95%
95%
11.10061 7.13E-16 96843.45 139473.2
-10.3301 1.13E-14 -7460.72 -5037.9
Good fit alone is no guarantee that the model is apt.
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Look at the residuals!
Yt   price  X price,t   t
Price Residual Plot
25000
20000
Residuals
15000
10000
5000
0
$15.00
-5000
$16.00
$17.00
$18.00
$19.00
$20.00
-10000
Price
The residuals should be random noise
 Homoscedastic, not heteroscedastic
 No pattern.
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Try the right model.
log Yt   price  X price,t   t
Log of Sales and Price
11.00
Log of Sales
10.00
9.00
8.00
7.00
6.00
$15.00
$16.00
$17.00
$18.00
$19.00
$20.00
Price
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SUMMARY
OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.96
0.92
0.92
0.30
59
ANOVA
df
Regression
Residual
Total
Intercept
Price
1
57
58
SS
59.24
5.15
64.40
Coefficients Standard
Error
22.17
0.54
-0.78
0.03
MS
59.24
0.09
t Stat
41.40
-25.59
F
Significance F
655.09
6.09178E-33
P-value
3.15E-44
6.09E-33
Lower 95%
21.10
-0.84
Upper
95%
23.24
-0.72
It looks better, but is it apt?
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Look at the residuals from this model!
Price Residual Plot
0.6
Residuals
0.4
0.2
0
$15.00
-0.2
$16.00
$17.00
$18.00
$19.00
$20.00
-0.4
-0.6
Price


Homoscedastic.
No pattern.
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If the prices have a time order, check for serial
dependency.

Serial Dependency Among Residuals?
0.6
0.4
Residual
0.2
0
-0.2
-0.4
-0.6
1
4
7
10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58
Week No.
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Most data show a log-log relation between prices and sales.
log Yt   price  log X price,t   t
Log-Log Relation Between Sales and Prices
Sales
1200
1100
1000
900
800
700
600
500
400
300
$15.00
$20.00
$25.00
$30.00
$35.00
$40.00
Prices
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What happens if you fit a linear model?
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.964864
R Square 0.930962
Adjusted
0.929751
R Square
Standard
49.31677
Error
Observati
59
ons
ANOVA
df
Regressio
n
Residual
Total
Intercept
Price
1
SS
1869427
MS
F
1869427 768.6334
Significance F
8.93E-35
57 138632.2 2432.143
58 2008059
Coefficient Standard
t Stat
P-value
Lower
Upper
s
Error
95%
95%
1344.148 25.87085 51.95606 1.04E-49 1292.342 1395.953
-25.3197 0.913271 -27.7242 8.93E-35 -27.1485 -23.4909
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Look at the residuals!
Price Residual Plot
150
Residuals
100
50
0
-50
-100
$15.00
$20.00
$25.00
$30.00
$35.00
$40.00
Price
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Try the right model.
Plot of Log Sales and Log Prices
7.20
Log of Sales
7.00
6.80
6.60
6.40
6.20
6.00
2.70
2.90
3.10
3.30
3.50
3.70
Log of Prices
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SUMMARY OUTPUT
Regression Statistics
Multiple R
0.994507
R Square
0.989044
Adjusted R
0.988852
Square
Standard Error
0.028329
Observations
59
ANOVA
df
Regression
1
Residual
57
Total
58
Coefficients
Intercept
Log Price
SS
4.129669
0.045744
4.175412
Standard
Error
9.55979
0.04366
-0.9527 0.013281
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MS
4.129669
0.000803
t Stat
218.9615
-71.7346
F
5145.847
P-value
4.46E-85
1.43E-57
Significance F
1.43E-57
Lower 95%
9.472363
-0.9793
Upper 95%
9.647217
-0.92611
16
Log Price Residual Plot
0.06
Residuals
0.04
0.02
0
-0.02
-0.04
-0.06
2.70
2.90
3.10
3.30
3.50
3.70
Log Price
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Tracking the Components of Customer Satisfaction
16 SPECIFIC CRITERIA
The value received for the price.
Product reliability
Product capability
How well the software utilizes the available computer
system resources.
How easy the product is to use.
How trouble free is the installation process.
The quality of technical support from the system
engineers in the field.
The quality of technical support from the Customer
Support Center.
The quality of the technical documentation.
How easy is it to acquire maintenance for the product.
How easy is it to apply the required maintenance.
An evaluation of the company's sales representative.
An evaluation of the company's billing services.
An evaluation of the company's contracts services.
An evaluation of the company's complaint-resolution
process.
An evaluation of the company's handling of customers'
phone calls.
And a measure of overall satisfaction.
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MEASURING PERFORMANCE
How well is the company doing on each aspect?
How important is it to the customer that the company performs
well on each aspect?
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WHAT DRIVES OVERALL SATISFACTION?
Yi  0  1 X1i  2 X2i ... 16 X16i  i
where:
Yi is the overall satisfaction rating given by customer i,
X 1i is the first rating given by customer i,
 1 is the statistical importance weight for the first rating,
 0 is the intercept term that adjusts the mean rating on the
criterion (overall satisfaction) to match the mean ratings on the
linear combination of rating scales,
i
is error.
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  so as to
Multiple regression attempts to estimate the weights
minimize the error (i.e. the sum of squared errors).
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Ta b le 2. Rela tio n o f Perfo rm a nc e Sc a les to Overa ll Sa tisfa c tio n
De p e nd e nt Va ria b le : O VERALL
O ve ra ll sa tisfa c tio n w ith C o m p a ny
So urc e
DF
Sum o f
Sq ua re s
Mea n
Sq ua re
Mod el
Erro r
C To ta l
16
1821
1837
470.78
423.91
894.69
Ro o t M SE
R-sq ua re
Ad j R-sq
F Va lue
Pro b >F
29.42
0.23
126.40
0.0001
0.48
0.53
0.52
De p M e a n
C .V.
4.04
11.94
Pa ra m eter Estim a tes
Va ria b le
DF
INTERC EP
VALUE
RELIABLE
C APABLE
RESO URC E
EASE
INSTALL
TS_FES
TS_C SC
DO C S
M AIN_AC Q
M AIN_APP
SALESREP
BILLING
C O NTRAC T
C O M PLAIN
PHO NEC AL
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Pa ra m e te r Sta nd a rd
Estim a te
Erro r
0.29
0.19
0.20
0.16
0.02
0.09
0.00
0.01
0.14
0.05
0.04
-0.05
0.06
-0.04
-0.01
0.02
0.07
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0.10
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.03
0.02
0.02
0.03
0.02
0.04
0.04
0.03
0.02
T fo r H0:
Sta nd a rd ize d
Pa ra m e te r=0 Pro b > | T|
Estim a te To le ra nc e La b e l
2.93
9.66
9.77
7.25
0.96
5.79
-0.07
0.38
5.73
3.16
1.77
-1.76
2.94
-1.07
-0.17
0.64
2.85
0.01
0.01
0.01
0.01
0.34
0.01
0.95
0.71
0.01
0.01
0.08
0.08
0.01
0.29
0.87
0.52
0.01
0.00
0.20
0.20
0.15
0.02
0.12
0.00
0.01
0.15
0.06
0.05
-0.05
0.07
-0.03
-0.01
0.02
0.07
.
0.61
0.59
0.57
0.65
0.65
0.54
0.50
0.39
0.62
0.40
0.36
0.52
0.25
0.24
0.39
0.43
Inte rc e p t
Va lue re c e ive d fo r p ric e
Pro d uc t re lia b ility
Pro d uc t c a p a b ility
Use o f syste m re so urc e s
Pro d uc t e a se o f use
Pro d uc t insta lla tio n
Te c h sup p o rt - Fie ld SEs
Te c h sup p o rt - C usto m e r Sup p o rt C e nte r
Te c hnic a l d o c um e nta tio n
Ea se o f a c q uiring m a inte na nc e
Ea se o f a p p lying m a inte na nc e
C o m p a ny's sa le s re p re se nta tive
Billing se rvic e
C o ntra c ts se rvic e s
C o m p la int re so lutio n
Ha nd ling o f yo ur c a lls
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Linear Regression Models are
Powerful
• Powerful models can be powerfully abused.
• Check to see if the model not only fits, but
is also apt.
• Think about what validity means for your
model.
– Forecasting on new data.
– Relates to other concepts as expected.
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