OM Forecasting GLM(en)

8/3/2019 OM Forecasting GLM(en)

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Slide

Forecasting MethodsForecasting

Methods

Quantitative Qualitative

Causal Time Series

Smoothing TrendProjectionTrend Projection

Adjusted forSeasonal Influence


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2006 by Thomson Learning, a division of Thomson Asia Pte Ltd.. #

Slide

Models in which the parameters ( 0, 1, . . . , p ) allhave exponents of one are called linear models.It does not imply that the relationship between y andthe x is is linear.

General Linear Model

A general linear model involving p independentvariables is

0 1 1 2 2

p py z z z

Each of the independent variables z is a function of x1, x2, ... , x k (the variables for which data have beencollected).


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2006 by Thomson Learning, a division of Thomson Asia Pte Ltd.. #

Slide

General Linear Model

y x 0 1 1y x 0 1 1

The simplest case is when we have collected data for just one variable x1 and want to estimate y by using astraight-line relationship. In this case z1 = x1.

This model is called a simple first-order model with onepredictor variable.


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Estimation Process

Multiple Regression ModelE(y) = 0 + 1x1 + 2x2 +. . .+ px p + Multiple Regression Equation

E(y) = 0 + 1x1 + 2x2 +. . .+ px p Unknown parameters are

0, 1, 2, . . . , p

Sample Data:x1 x2 . . . x p y. . . .. . . .

0 1 1 2 2

... p py b b x b x b x

Estimated MultipleRegression Equation

Sample statistics areb0, b1, b2 , . . . , b p

b0, b1, b2 , . . . , b p provide estimates of 0, 1, 2, . . . , p


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Slide

Least Squares Method

Least Squares Criterion

2min ( )i iy y

Computation of Coefficient Values

The formulas for the regression coefficients b0, b1, b2, . . ., b p involve the use of matrix algebra.We will rely on computer software packages to

perform the calculations.


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Multiple Regression Equation

Example: Butler Trucking CompanyTo develop better work schedules, the managers wantto estimate the total daily travel time for their driversData


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Multiple Regression EquationMINITAB Output


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The years of experience, score on the aptitude

test, and corresponding annual salary ($1000s) for asample of 20 programmers is shown on the nextslide.

Example: Programmer Salary Survey

Multiple Regression Model

A software firm collected data for a sampleof 20 computer programmers. A suggestionwas made that regression analysis couldbe used to determine if salary was relatedto the years of experience and the scoreon the firms programmer aptitude test.


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47158100

166

92105684

633

781008682868475

808391

88737581748779

947089

2443

23.734.335.838

22.2

23.13033

3826.636.231.62934

30.1

33.928.230

Exper. Score ScoreExper.Salary Salary



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Suppose we believe that salary ( y) isrelated to the years of experience ( x1) and the score onthe programmer aptitude test ( x2) by the followingregression model:


where y = annual salary ($1000)

x1 = years of experience x2 = score on programmer aptitude test

y = 0 + 1x1 + 2x2 +


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Solving for the Estimates of 0 , 1 , 2

Input DataLeast Squares

Outputx1 x2 y

4 78 247 100 43. . .

. . .3 89 30

ComputerPackage

for SolvingMultiple

RegressionProblems

b0

=b1 =b2 =R2 =

etc.


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Excels Regression Dialog Box



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Excels Regression Equation Output

A B C D E

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39 Coeffic. Std. Err. t Stat P-value 40 Intercept 3.17394 6.15607 0.5156 0.6127941 Experience 1.4039 0.19857 7.0702 1.9E-0642 Test Score 0.25089 0.07735 3.2433 0.0047843

Note: Columns F-I are not shown.



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Estimated Regression Equation

SALARY = 3.174 + 1.404(EXPER) + 0.251(SCORE)

Note: Predicted salary will be in thousands of dollars.


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Interpreting the Coefficients

In multiple regression analysis, we interpret each

regression coefficient as follows:

bi represents an estimate of the change in ycorresponding to a 1-unit increase in xi when all

other independent variables are held constant.


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Salary is expected to increase by $1,404 foreach additional year of experience (when the variable

score on programmer attitude test is held constant).

b1 = 1. 404



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Salary is expected to increase by $251 for eachadditional point scored on the programmer aptitude

test (when the variable years of experience is heldconstant).

b2 = 0.251



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Multiple Coefficient of Determination

Relationship Among SST, SSR, SSE

where:SST = total sum of squaresSSR = sum of squares due to regression

SSE = sum of squares due to error

SST = SSR + SSE

2( )iy y2( )iy y

2( )i iy y


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Excels ANOVA Output

A B C D E F3233 ANOVA34 df SS MS F Significance F 35 Regression 2 500.3285 250.1643 42.76013 2.32774E-0736 Residual 17 99.45697 5.8504137 Total 19 599.785538

SST SSR



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R2 = 500.3285/599.7855 = .83418

R2 = SSR/SST

In general, R 2 always increases as independent variables areadded to the model.

adjusting R 2 for the number of independent variables to avoidoverestimating the impact of adding an independent variable


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d d l l ff


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Excels Regression Statistics

A B C

23 24 SUMMARY OUTPUT25

26 Regression Statistics 27 Multiple R 0.91333405928 R Square 0.83417910329 Adjus ted R Square 0.81467076230 Standard Error 2.41876207631 Observations 2032

Adjusted Multiple Coefficientof Determination


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The variance of , denoted by 2, is the same for allvalues of the independent variables.

The error is a normally distributed random variable

reflecting the deviation between the y value and theexpected value of y given by 0 + 1x1 + 2x2 + ... + px p.

Assumptions About the Error Term

The error is a random variable with mean of zero.

The values of are independent.


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In simple linear regression, the F and t tests providethe same conclusion.

Testing for Significance

In multiple regression, the F and t tests have different

purposes.


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Testing for Significance: F Test

The F test is referred to as the test for overallsignificance.

The F test is used to determine whether a significantrelationship exists between the dependent variableand the set of all the independent variables.


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Testing for Significance: F Test

Hypotheses

Rejection Rule

Test Statistics

H 0: 1

= 2

= . . . = p

= 0 H a: One or more of the parameters

is not equal to zero.

F = MSR/MSE

Reject H 0 if p-value < a or if F > F a ,where F a is based on an F distribution

with p d.f. in the numerator and n - p - 1 d.f. in the denominator.


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Testing for Significance: F TestANOVA Table for A Multiple Regression Model with pIndependent Variables


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F Test for Overall Significance

Hypotheses H 0: 1 = 2 = 0 H a: One or both of the parametersis not equal to zero.

Rejection Rule For a = .05 and d.f. = 2, 17; F .05 = 3.59Reject H 0 if p-value < .05 or F > 3.59


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Excels ANOVA Output

A B C D E F3233 ANOVA34 df SS MS F Significance F 35 Regression 2 500.3285 250.1643 42.76013 2.32774E-0736 Residual 17 99.45697 5.8504137 Total 19 599.785538


p-value used to test foroverall significance


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Test Statistics F = MSR/MSE

= 250.16/5.85 = 42.76

Conclusion p-value < .05, so we can reject H 0.(Also, F = 42.76 > 3.59)


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Testing for Significance: t Test

Hypotheses

Rejection Rule

Test Statistics

Reject H 0 if p-value < a orif t < -ta or t > ta where ta

is based on a t distributionwith n - p - 1 degrees of freedom.

t bs

i

bi

t bs

i

bi

0 : 0iH

: 0a iH

t Test for Significance


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t Test for Significanceof Individual Parameters

Hypotheses

Rejection Rule For a = .05 and d.f. = 17, t.025 = 2.11

Reject H 0 if p-value < .05 or if t > 2.11

0 : 0iH

: 0a iH



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A B C D E

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t statistic and p-value used to test for theindividual significance of Experience



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A B C D E

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t statistic and p-value used to test for theindividual significance of Test Score



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bsb11

1 40391986 7 07 .. .bsb11

1 40391986 7 07 .. .

bsb

2

2

2508907735

3 24 ..

.bsb

2

2

2508907735

3 24 ..

.

Test Statistics

Conclusions Reject both H 0: 1 = 0 and H 0: 2 = 0.Both independent variables aresignificant.


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Testing for Significance: Multicollinearity

The term multicollinearity refers to the correlationamong the independent variables.

When the independent variables are highly correlated

(say, | r | > .7), it is not possible to determine theseparate effect of any particular independent variableon the dependent variable.


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Testing for Significance: Multicollinearity

Every attempt should be made to avoid includingindependent variables that are highly correlated.

If the estimated regression equation is to be used onlyfor predictive purposes, multicollinearity is usuallynot a serious problem.


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Modeling Curvilinear Relationships

This model is called a second-order model with onepredictor variable.

y x x 0 1 1 2 12y x x 0 1 1 2 12

To account for a curvilinear relationship, we might set z1 = x1 and z2 = .21 x


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Modeling Curvilinear RelationshipsExample: Reynolds, Inc.,

Managers at Reynolds want toinvestigate the relationshipbetween length of employmentof their salespeople and thenumber of electronic laboratoryscales sold.Data


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Modeling Curvilinear RelationshipsScatter Diagram for the Reynolds Example


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Modeling Curvilinear RelationshipsLet us consider a simple first-order model and the

estimated regression is

Sales = 111 + 2.38 Months,

where :Sales = number of electronic laboratory scales sold,Months = the number of months the salesperson

has been employed


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Modeling Curvilinear RelationshipsMINITAB output first-order model


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Modeling Curvilinear RelationshipsStandardized Residual plot first-order model

The standardized residual plot suggests that acurvilinear relationship is needed


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Modeling Curvilinear RelationshipsReynolds Example : The second-order model

The estimated regression equation isSales = 45.3 + 6.34 Months + .0345 MonthsSq

where :Sales = number of electronic laboratory scales sold,MonthsSq = the square of the number of months the

salesperson has been employed


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Modeling Curvilinear RelationshipsMINITAB output second-order model


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Modeling Curvilinear RelationshipsStandardized Residual plot second-order model

V i bl S l ti P d


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Variable Selection Procedures

Stepwise Regression

Forward SelectionBackward Elimination

Iterative; one independentvariable at a time is added or

deleted based on the F statistic

Variable Selection: Stepwise


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Variable Selection: StepwiseRegression

Compute F stat. and

p-value for each indep.variable not in model

Start with no indep.variables in model

Any p-value > alpha

to remove?

Stop

Indep. variablewith largest

p-value isremovedfrom model

Compute F stat. and

p-value for each indep.variable in model

Any p-value < alpha

to enter ?

Indep. variable with

smallest p-value isentered into model

NoNo

YesYes

nextiteration


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Variable Selection: Backward


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Variable Selection: BackwardElimination

Stop

Compute F stat. and p-value for each indep.

variable in model

Any p-value > alpha

to remove?

Indep. variable withlargest p-value is

removed from model

No

Yes

Start with all indep.

variables in model

Qualitative Independent Variables


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In many situations we must work with qualitativeindependent variables such as gender (male, female),method of payment (cash, check, credit card), etc.

For example, x2

might represent gender where x2

= 0indicates male and x2 = 1 indicates female.


In this case, x2 is called a dummy or indicator variable.


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471

58100166

9210

5684633

7810086

82868475808391

887375

81748779947089

2443

23.7

34.335.838

22.223.13033

3826.636.2

31.62934

30.133.928.230

Exper. Score ScoreExper.Salary SalaryDegr.NoYesNo

YesYesYesNoNoNoYes

Degr.YesNoYes

NoNoYesNoYesNoNo




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y = b0 + b1x1 + b2x2 + b3x3

^where:

y = annual salary ($1000)x1 = years of experiencex2 = score on programmer aptitude testx3 = 0 if individual does not have a graduate degree

1 if individual does have a graduate degree

x3 is a dummy variable



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Excels Regression Statistics

A B C

23 24 SUMMARY OUTPUT25

26 Regression Statistics 27 Multiple R 0.92021523928 R Square 0.84679608529 Adjus ted R Square 0.81807035130 Standard Error 2.39647510131 Observations 2032



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A B C D E

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39 Coeffic. Std. Err. t Stat P-value 40 Intercept 7.94485 7.3808 1.0764 0.297741 Experience 1.14758 0.2976 3.8561 0.001442 Test Score 0.19694 0.0899 2.1905 0.0436443 Grad. Degr. 2.28042 1.98661 1.1479 0.2678944


Not significant


More Complex Qualitative Variables


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More Complex Qualitative Variables

If a qualitative variable has k levels, k - 1 dummyvariables are required, with each dummy variablebeing coded as 0 or 1.

For example, a variable with levels A, B, and C couldbe represented by x1 and x2 values of (0, 0) for A, (1, 0)for B, and (0,1) for C.

Care must be taken in defining and interpreting the

dummy variables.


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Interaction


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Interaction

y x x x x x x 0 1 1 2 2 3 12

4 22

5 1 2y x x x x x x 0 1 1 2 2 3 12

4 22

5 1 2

This type of effect is called interaction.

In this model, the variable z5 = x1 x2 is added to accountfor the potential effects of the two variables actingtogether.

If the original data set consists of observations for y and

two independent variables x1 and x2 we might develop asecond-order model with two predictor variables.

Interaction


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Interaction

Example: Tyler Personal Care

New shampoo products, two factors believed to havethe most influence on sales are unit selling price andadvertising expenditure.Data


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I t ti


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InteractionTo account for the effect of interaction, use the

following regression model

where : y = unit sales (1000s), x1 = price ($), x2 = advertising expenditure ($1000s).

21322110 x x x x y


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InteractionGeneral Linear Model involving three independent

variables ( z1, z2, and z3)

where : y= Sales = unit sales (1000s) z1 = x1 (price) = price of the product ($) z2 = x2 (AdvExp) = advertising expenditure ($1000s)

z3 = x1 x 2 (PriceAdv) = interaction term(Price times AdvExp)

3322110 z z z y

I i


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InteractionMINITAB Output for the Tyler Personal Care Example

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