Business statistics, 6e, 2005, groebner CH14

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Prentice-Business Statistics:

A Decision-Making Approach

6 th Edition

Chapter 14

Multiple Regression Analysis

and Model Building

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multiple regression model

in a multiple regression model

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regression analysis and take the steps to correct the problems.

regression model by using dummy variables.

(continued)

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Prentice-The Multiple Regression

Model

Idea: Examine the linear relationship between

å x

â x

â â

k k

2 2

1 1

Estimated slope coefficients

Estimated multiple regression model:

Estimated intercept

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Prentice-Multiple Regression Model

Two variable model

y

2 2 1

1

0 b x b x b

Sl op

e f or

va ria ble x

1

Slope fo r variab

le x2

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Prentice-Multiple Regression Model

Two variable model

y

2 2 1

1

0 b x b x b

x 1i The best fit equation, y ,

is found by minimizing the sum of squared errors, e 2

Sample observation

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Prentice-Multiple Regression

Assumptions

e = (y – y)

Errors (residuals) from the regression model:

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Prentice-The Correlation Matrix

selected independent variables can be found using Excel:

 Tools / Data Analysis… / Correlation

with a t test

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Prentice-Example

evaluate factors thought to influence demand

 Dependent variable: Pie sales (units per week)

 Independent variables: Price (in $)

Advertising ($100’s)

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Prentice-Pie Sales Model

•Week

•Pie Sales

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 Example: if b 1 = -20, then sales (y) is expected to decrease

by an estimated 20 pies per week for each $1 increase in selling price (x 1 ), net of the effects of changes due to

advertising (x 2 )

 y-intercept (b 0 )

 The estimated average value of y when all x i = 0 (assuming all x i = 0 is within the range of observed values)

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Prentice-Pie Sales Correlation Matrix

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Prentice-Estimating a Multiple Linear

Regression Equation

generate the coefficients and measures of goodness of fit for multiple regression

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Prentice-Multiple Regression Output

ce) 24.975(Pri -

306.526

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Prentice-The Multiple Regression

Equation

ertising) 74.131(Adv

ce) 24.975(Pri

306.526

b 1 = -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

b 2 = 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.

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Prentice-Using The Model 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(Adv

ce) 24.975(Pri

306.526 Sales

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Prentice-Predictions in PHStat

 PHStat | regression | multiple regression …

Check the

“confidence and prediction interval estimates” box

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Prediction interval for an individual y value, given these x’s

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Prentice-Multiple Coefficient of

Determination

explained by all x variables taken together

squares of

sum Total

regression squares

of

Sum SST

SSR

R 2 = =

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29460.0 SST

SSR

R 2 = = =

52.1% of the variation in pie sales

is explained by the variation in price and advertising

Multiple Coefficient of

Determination

(continued)

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added to the model

models

 What is the net effect of adding a new variable?

variable is added

explanatory power to offset the loss of one degree of freedom?

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all x variables adjusted for the number of x

variables used

(where n = sample size, k = number of independent variables)

 Penalize excessive use of unimportant independent variables

Ë

Ê

-

-=

1 k

n

1

n )

R 1

( 1

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Multiple Coefficient of

Determination

(continued)

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Prentice-Is the Model Significant?

 F-Test for Overall Significance of the Model

of the x variables considered together and y

 H 0 : β 1 = β 2 = … = β k = 0 (no linear relationship)

 H A : at least one β i ≠ 0 (at least one independent

variable affects y)

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Prentice-F-Test for Overall

SSE k

SSR

-

-=

1

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Prentice-6.5386 2252.8

14730.0 MSE

With 2 and 12 degrees

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The regression model does explain

a significant portion of the variation in pie sales

(There is evidence that at least one independent variable affects y)

MSR

F = =

Critical Value:

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Prentice-Are Individual Variables

Significant?

 Use t-tests of individual variable slopes

variable x i and y

 H 0 : β i = 0 (no linear relationship)

 H A : β i ≠ 0 (linear relationship does exist

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Prentice-Are Individual Variables

Significant?

H 0 : β i = 0 (no linear relationship)

H A : β i ≠ 0 (linear relationship does exist

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The test statistic for each variable falls

in the rejection region (p-values < 05)

There is evidence that both Price and Advertising affect

From Excel output:

• •Coefficients •Standard Error •t Stat •P-value

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Prentice-Confidence Interval Estimate

for the Slope

(the effect of changes in price on pie sales):

by between 1.37 to 48.58 pies for each increase of $1

in the selling price

i

b 2

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Prentice-Standard Deviation of the

Regression Model

regression model is:

MSE k

n

SSE

-

-=

e

1

mean size of y for comparison

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Prentice- The standard deviation of the regression model is 47.46

week is

per week range, so this range is probably too large to be acceptable The analyst may want to look for additional variables that can explain more

of the variation in weekly sales

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Prentice-Multicollinearity

 Multicollinearity: High correlation exists

between two independent variables

redundant information to the multiple regression model

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Prentice-Multicollinearity

variables can adversely affect the regression results

standard error and low t-values)

expectations

(continued)

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Prentice-Some Indications of Severe

Multicollinearity

coefficient when a new variable is added to the model

insignificant when a new independent variable

is added

model increases when a variable is added to the model

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Prentice-Detect Collinearity (Variance Inflationary

Factor)

the other explanatory variables

R 2

j is the coefficient of determination when the j th independent variable is regressed against the remaining k – 1 independent variables

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Prentice-Detect Collinearity in PHStat

Output for the pie sales example:

 Since there are only two explanatory variables, only one VIF

PHStat / regression / multiple regression …

Check the “variance inflationary factor (VIF)” box

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Prentice-Qualitative (Dummy)

Variables

variable) with two or more levels:

 yes or no, on or off, male or female

 coded as 0 or 1

is significant

(number of levels - 1)

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Prentice-Dummy-Variable Model Example (with 2 Levels)

Let:

y = pie sales

x 1 = price

x 2 = holiday (X 2 = 1 if a holiday occurred during the week)

2 1

b

yˆ = + 1 + 2

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Prentice-Same slope

Dummy-Variable Model

Example (with 2 Levels)

1 0

1 2

0 1

0

x b

b (0)

b x

b b

yˆ

x b )

b (b

(1) b

x b b

yˆ

1 2

1

1 2

1

+

= +

+

=

+ +

= +

+

No Holiday

Different intercept

on pie sales

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Prentice-Sales: number of pies sold per week

Price: pie price in $

weeks with a holiday than in weeks without a

holiday, given the same price

) 15(Holiday 30(Price)

300

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Prentice-Dummy-Variable Models

(more than 2 Levels)

the number of levels

Three levels, so two dummy

variables are needed

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Prentice-Dummy-Variable Models

(more than 2 Levels)

Ô Ó

Ô Ì

Ï

= Ô

Ó

Ô Ì

Ï

=

not if

0

level split

if

1 x

not

if 0

ranch if

1

3 2

1

b

yˆ = + 1 + 2 + 3

(continued)

Let the default category be “condo”

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Prentice-Interpreting the Dummy Variable Coefficients (with 3

With the same square feet, a ranch will have an estimated average price of 23.53

thousand dollars more than a condo.

Suppose the estimated equation is

3 2

0.045x 20.43

18.84 0.045x

20.43

23.53 0.045x

20.43

1

0.045x 20.43

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variable and an independent variable may not

â x

â â

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Prentice-Polynomial Regression

Model

 where:

β0 = Population regression constant

βi = Population regression coefficient for variable xj : j = 1, 2, …k

p = Order of the polynomial

i = Model error

å x

â x

â â

å x

â x

â â

y = 0 + 1 j + 2 2 j + K + p p j +

If p = 2 the model is a quadratic model:

General form:

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Linear fit does not give random residuals

Linear vs Nonlinear Fit

Nonlinear fit gives random residuals

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Prentice-Quadratic Regression Model

Quadratic models may be considered when scatter

diagram takes on the following shapes:

y

β1 = the coefficient of the linear term

β2 = the coefficient of the squared term

x 1

å x

â x

â â

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Prentice-Testing for Significance:

Quadratic Model

with the linear model

 (No 2 nd order polynomial term)

 (2 nd order polynomial term is needed)

å x

â x

â â

y = 0 + 1 j + 2 2 j +

å x

â â

H0: β2 = 0

HA: β2  0

MSE MSR

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Prentice-Higher Order Models

y

x

å x

â x

â â

If p = 3 the model is a cubic form:

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Prentice-Interaction Effects

variables

levels of another x variable

2

2 1 5 2

1 4 3

3

2 1 2 1

1

0 â x â x â x â x x â x x â

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x â x

â x

â â

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Prentice-Interaction Regression Model

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Prentice-å x

x â x

â x

â â

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 Stepwise regression procedure

are added

 Best-subset approach

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selection , backward elimination , or through

standard stepwise regression

 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

Stepwise Regression

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Prentice-Best Subsets Regression

variables

adjusted R 2 and lowest standard error s ε

Stepwise regression and best subsets regression can be performed using PHStat, Minitab, or other statistical software packages

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Prentice-Aptness of the Model

verifying the assumptions of multiple regression:

 Each x i is linearly related to y

) yˆ y

(

-Errors (or Residuals) are given by

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Prentice-The Normality Assumption

computer

of the standardized residuals to check for normality

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Prentice-Chapter Summary

regression model

model

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assumptions

(continued)

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