Model Building

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Model Building

• Chapter 19

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19.1 Introduction

• Regression analysis is one of the most commonly
  used techniques in statistics.
• It is considered powerful for several reasons
• It can cover variety of mathematical models
• linear relationships.
• non - linear relationships.
• qualitative variables.
• It provides efficient methods for model building,
  to select the best fitting set of variables.

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19.2 Polynomial Models

• The independent variables may appear as functions
  of a number of predictor variables.
• Polynomial models of order p with one predictor
  variable y b0 b1x b2x2 bpxp e
• Polynomial models with two predictor
  variablesFor exampley b0 b1x1 b2x2 ey
  b0 b1x1 b2x2 b3x1x2 e

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Polynomial models with one predictor variable

• First order model (p 1)

y b0 b1x e

• Second order model (p2)

y b0 b1x e
b2x2 e
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• Third order model (p3)

y b0 b1x b2x2 e
b3x3 e
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Polynomial models with two predictor variables
y
b1 gt 0

• First order modely b0 b1x1 e

b2x2 e
b1 lt 0
x1
x2
b2 gt 0
b2 lt 0
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Polynomial models with two predictor variables

• First order modely b0 b1x1 b2x2 e

The effect of one predictor variable on y is
independent of the effect of the other predictor
variable on y.
X2 3
b0b2(3) b1x1
X2 2
b0b2(3) (b1b3(3))x1
b0b2(2) b1x1
X2 1
b0b2(1) b1x1
b0b2(2) (b1b3(2))x1
b0b2(1) (b1b3(1))x1
x1
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• Second order modely b0 b1x1 b2x2 b3x12
  b4x22 e

b5x1x2 e
X2 3
X2 3
y b0b2(3)b4(32) b1x1 b3x12 e
X2 2
X2 2
X2 1
y b0b2(2)b4(22) b1x1 b3x12 e
X2 1
y b0b2(1)b4(12) b1x1 b3x12 e
x1
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Example 19.1 Location for a new restaurant

• A fast food restaurant chain tries to identify
  new locations that are likely to be profitable.
• The primary market for such restaurants is
  middle-income adults and their children (between
  the age 5 and 12).
• Which regression model should be proposed to
  predict the profitability of new locations?

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• Solution
• The dependent variable will be Gross Revenue

• There are quadratic relationships between Revenue
  and each predictor variable. Why?

• Members of middle-class families are more likely
  to visit a fast food family than members of poor
  or wealthy families.

Revenue b0 b1Income b2Age b3Income2
b4Age2 b5(Income)(Age) e

• Families with very young or older kids will not
  visit the restaurant as frequent as families with
  mid-range ages of kids.

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Example 19.2

• To verify the validity of the model proposed in
  example 19.1, 25 areas with fast food restaurants
  were randomly selected.
• Data collected included (see Xm19-02.xls)
• Previous years annual gross sales.
• Mean annual household income.
• Mean age of children

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The model provides a good fit
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The model can be used to make predictions.
However, do not interpret the coefficients or
test them. Multicollinearity is a problem!!
In excel Tools gt Data Analysis gt Correlation
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The multicollinearity can be reduced by
modifying the original predictor variables
Income Income - average income Age Age -
average Age
Income Income - 24.2
Age Age - 8.392
(-.7)(2.11)
(-.7)2
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Regression results of the modified model
Multicolinearity is not a problem anymore
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19.3 Qualitative Independent Variables

• In many real-life situations one or more
  independent variables are qualitative.
• Including qualitative variables in a regression
  analysis model is done via indicator variables.
• An indicator variable (I) can assume one out of
  two values, zero or one.

1 if a degree earned is in Finance 0 if a
degree earned is not in Finance
1 if the temperature was below 50o 0 if the
temperature was 50o or more
1 if a first condition out of two is met 0 if a
second condition out of two is met
1 if data were collected before 1980 0 if data
were collected after 1980
I
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Example 17.1 - continued

• The dealer believes that color is a variable that
  affects a cars price.
• Three color categories are considered
• White
• Silver
• Other colors
• Note Color is a qualitative variable.

And what about Other colors? Set I1 0 and I2
0
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• Solution
• the proposed model is y b0 b1(Odometer)
  b2I1 b3I2 e
• The data

White car
Other color
Silver color
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From Excel we get the regression equation PRICE
6350-.0278(ODOMETER)45.2I1148I2
For one additional mile the auction
price decreases by 2.78 cents.
A white car sells, on the average, for 45.2
more than a car of the Other color category
A silver color car sells, on the average, for
148 more than a car of the Other color
category
The equation for a car of silver color
Price 6350 - .0278(Odometer) 45.2(0) 148(1)
The equation for a car of white color
The equation for a car of the Other
color category.
Price 6350 - .0278(Odometer) 45.2(1) 148(0)
Price 6350 - .0278(Odometer) 45.2(0) 148(0)
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There is insufficient evidence to infer that a
white color car and a car of Other color sell
for a different auction price.
There is sufficient evidence to infer that a
silver color car sells for a larger price than
a car of the Other color category.
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19.4 Regression and the Analysis of Variance

• We can use regression analysis with indicator
  variables to conduct ANOVA.

ANOVA model
Regression model
Single factor independent samples.
y b0 b1I1 b2I2 e Example Compare 3
treatments y mi e (i 1,2,3) Single factor
randomized block design. y b0 b1 I1
Example Compare 2 treatments, 3 blocks
b2I2 b3I3 y Ti Bj e (i 1,2 and j
1,2,3)
Indicator variable for treatment 1
Indicator variables for blocks
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19.5 Stepwise Regression

• Multicollinearity may prevent the study of the
  relationship between dependent and independent
  variables.
• To reduce multicollinearity we can use stepwise
  regression.
• In stepwise regression variables are added or
  deleted one at a time, based on their
  contribution to the current model.

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19.6 Model Building

• Identify the dependent variable, and clearly
  define it.
• List potential predictors.
• Bear in mind the problem of multicolinearity.
• Consider the cost of gathering, processing and
  storing data.
• Be selective in your choice (try to use as little
  variables as possible).

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• Gather the required observations (have at least
  six observations for each independent variable).

• Identify several possible models.
• A scatter diagram of the dependent variables can
  be helpful in formulating the right model.
• If you are uncertain, start with first order and
  second order models, with and without
  interaction.
• Try other relationships (transformations) if the
  polynomial models fail to provide a good fit.
• Use statistical software to estimate the model.

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• Determine whether the required conditions are
  satisfied. If not, attempt to correct the
  problem.
• Select the best model.
• Use the statistical output.
• Use your judgment!!