1 Multiple Regression. 2 Introduction In this chapter we extend the simple linear regression model, and allow for any number of independent variables

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Multiple RegressionMultiple Regression

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Introduction

• In this chapter we extend the simple linear regression model, and allow for any number of independent variables.

• We expect to build a model that fits the data better than the simple linear regression model.

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• We shall use computer printout to – Assess the model

• How well it fits the data• Is it useful• Are any required conditions violated?

– Employ the model• Interpreting the coefficients• Predictions using the prediction equation• Estimating the expected value of the dependent variable

Introduction

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Coefficients

Dependent variable Independent variables

Random error variable

Model and Required Conditions

• We allow for k independent variables to potentially be related to the dependent variable

y = 0 + 1x1+ 2x2 + …+ kxk +

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Multiple Regression for k = 2, Graphical Demonstration - I

y = 0 + 1xy = 0 + 1xy = 0 + 1xy = 0 + 1x

X

y

X2

1

The simple linear regression modelallows for one independent variable, “x”

y =0 + 1x +

The multiple linear regression modelallows for more than one independent variable.Y = 0 + 1x1 + 2x2 +

Note how the straight line becomes a plain, and...

y = 0 + 1x1 + 2x2

y = 0 + 1x1 + 2x2

y = 0 + 1x1 + 2x2

y = 0 + 1x1 + 2x2y = 0 + 1x1 + 2x2

y = 0 + 1x1 + 2x2

y = 0 + 1x1 + 2x2

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Multiple Regression for k = 2, Graphical Demonstration - II

Note how a parabola becomes a parabolic Surface.

X

y

X2

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y= b0+ b1x2

y = b0 + b1x12 + b2x2

b0

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• The error is normally distributed.• The mean is equal to zero and the standard

deviation is constant ( for all values of y. • The errors are independent.

Required conditions for the error variable

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– If the model assessment indicates good fit to the data, use it to interpret the coefficients and generate predictions.

– Assess the model fit using statistics obtained from the sample.

– Diagnose violations of required conditions. Try to remedy problems when identified.

Estimating the Coefficients and Assessing the Model

• The procedure used to perform regression analysis:– Obtain the model coefficients and statistics using a statistical

software.

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• Example 18.1 Where to locate a new motor inn?– La Quinta Motor Inns is planning an expansion.– Management wishes to predict which sites are likely to be

profitable.– Several areas where predictors of profitability can be

identified are:• Competition• Market awareness• Demand generators• Demographics• Physical quality

Estimating the Coefficients and Assessing the Model, Example

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Profitability

Competition Market awareness Customers Community Physical

Margin

Rooms Nearest Officespace

Collegeenrollment

Income Disttwn

Distance to downtown.

Medianhouseholdincome.

Distance tothe nearestLa Quinta inn.

Number of hotels/motelsrooms within 3 miles from the site.

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Profitability

Competition Market awareness Customers Community Physical

Operating Margin

Rooms Nearest Officespace

Collegeenrollment

Income Disttwn

Distance to downtown.

Medianhouseholdincome.

Distance tothe nearestLa Quinta inn.

Number of hotels/motelsrooms within 3 miles from the site.

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• Data were collected from randomly selected 100 inns that belong to La Quinta, and ran for the following suggested model:

Margin = Rooms NearestOfficeCollege + 5Income + 6Disttwn


Margin Number Nearest Office Space Enrollment Income Distance55.5 3203 4.2 549 8 37 2.733.8 2810 2.8 496 17.5 35 14.449 2890 2.4 254 20 35 2.6

31.9 3422 3.3 434 15.5 38 12.157.4 2687 0.9 678 15.5 42 6.949 3759 2.9 635 19 33 10.8

Xm18-01

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This is the sample regression equation (sometimes called the prediction equation)This is the sample regression equation (sometimes called the prediction equation)

Regression Analysis, Excel OutputSUMMARY OUTPUT

Regression StatisticsMultiple R 0.7246R Square 0.5251Adjusted R Square 0.4944Standard Error 5.51Observations 100

ANOVAdf SS MS F Significance F

Regression 6 3123.8 520.6 17.14 0.0000Residual 93 2825.6 30.4Total 99 5949.5

Coefficients Standard Error t Stat P-valueIntercept 38.14 6.99 5.45 0.0000Number -0.0076 0.0013 -6.07 0.0000Nearest 1.65 0.63 2.60 0.0108Office Space 0.020 0.0034 5.80 0.0000Enrollment 0.21 0.13 1.59 0.1159Income 0.41 0.14 2.96 0.0039Distance -0.23 0.18 -1.26 0.2107

Margin = 38.14 - 0.0076Number +1.65Nearest + 0.020Office Space +0.21Enrollment + 0.41Income - 0.23Distance

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Model Assessment

• The model is assessed using three tools:– The standard error of estimate – The coefficient of determination– The F-test of the analysis of variance

• The standard error of estimates participates in building the other tools.

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• The standard deviation of the error is estimated by the Standard Error of Estimate:

• The magnitude of s is judged by comparing it to

1knSSE

s

Standard Error of Estimate

.y

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• From the printout, s = 5.51 • Calculating the mean value of y we have• It seems s is not particularly small. • Question:

Can we conclude the model does not fit the data well?

739.45y

Standard Error of Estimate

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• The definition is

• From the printout, R2 = 0.5251• 52.51% of the variation in operating margin is explained by

the six independent variables. 47.49% remains unexplained.• When adjusted for degrees of freedom,

Adjusted R2 = 1-[SSE/(n-k-1)] / [SS(Total)/(n-1)] = = 49.44%

2i

2

)yy(SSE

1R

Coefficient of Determination

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• We pose the question:Is there at least one independent variable linearly related to the dependent variable?

• To answer the question we test the hypothesis

H0: 0 = 1 = 2 = … = k

H1: At least one i is not equal to zero.

• If at least one i is not equal to zero, the model has some validity.

Testing the Validity of the Model

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• The hypotheses are tested by an ANOVA procedure ( the Excel output)

Testing the Validity of the La Quinta Inns Regression Model

MSE=SSE/(n-k-1)

MSR=SSR/k

MSR/MSE

SSE

SSR

k =n–k–1 = n-1 =



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[Variation in y] = SSR + SSE. Large F results from a large SSR. Then, much of the variation in y is explained by the regression model; the model is useful, and thus, the null hypothesis should be rejected. Therefore, the rejection region is…

Rejection region

F>F,k,n-k-1


1knSSE

kSSR

F

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F,k,n-k-1 = F0.05,6,100-6-1=2.17F = 17.14 > 2.17

Also, the p-value (Significance F) = 0.0000Reject the null hypothesis.




Conclusion: There is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. At least one of the i is not equal to zero. Thus, at least one independent variable is linearly related to y. This linear regression model is valid

Conclusion: There is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. At least one of the i is not equal to zero. Thus, at least one independent variable is linearly related to y. This linear regression model is valid

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• b0 = 38.14. This is the intercept, the value of y when all

the variables take the value zero. Since the data range of all the independent variables do not cover the value zero, do not interpret the intercept.

• b1 = – 0.0076. In this model, for each additional room

within 3 mile of the La Quinta inn, the operating margin

decreases on average by .0076% (assuming the other

variables are held constant).

Interpreting the Coefficients

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• b2 = 1.65. In this model, for each additional mile that the nearest competitor is to a La Quinta inn, the operating margin increases on average by 1.65% when the other variables are held constant.

• b3 = 0.020. For each additional 1000 sq-ft of office space, the operating margin will increase on average by .02% when the other variables are held constant.

• b4 = 0.21. For each additional thousand students the operating margin increases on average by .21% when the other variables are held constant.


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• b5 = 0.41. For additional $1000 increase in median household income, the operating margin increases on average by .41%, when the other variables remain constant.

• b6 = -0.23. For each additional mile to the downtown

center, the operating margin decreases on average

by .23% when the other variables are held constant.


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• The hypothesis for each i is

• Excel printout

H0: i 0H1: i 0 d.f. = n - k -1

Test statistic

ib

iis

bt

Testing the Coefficients

Coefficients Standard Error t Stat P-valueIntercept 38.14 6.99 5.45 0.0000Number -0.0076 0.0013 -6.07 0.0000Nearest 1.65 0.63 2.60 0.0108Office Space 0.020 0.0034 5.80 0.0000Enrollment 0.21 0.13 1.59 0.1159Income 0.41 0.14 2.96 0.0039Distance -0.23 0.18 -1.26 0.2107

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• The model can be used for making predictions by– Producing prediction interval estimate for the particular

value of y, for a given values of xi.– Producing a confidence interval estimate for the

expected value of y, for given values of xi.

• The model can be used to learn about relationships between the independent variables xi, and the dependent variable y, by interpreting the coefficients i

Using the Linear Regression Equation

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• Predict the average operating margin of an inn at a site with the following characteristics:– 3815 rooms within 3 miles,– Closet competitor .9 miles away,– 476,000 sq-ft of office space,– 24,500 college students,– $35,000 median household income,– 11.2 miles distance to downtown center.

MARGIN = 38.14 - 0.0076(3815) +1.65(.9) + 0.020(476) +0.21(24.5) + 0.41(35) - 0.23(11.2) = 37.1%

La Quinta Inns, Predictions

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Assessment and Interpretation:MBA Program Admission Policy

• The dean of a large university wants to raise the admission standards to the popular MBA program.

• She plans to develop a method that can predict an applicant’s performance in the program.

• She believes a student’s success can be predicted by:– Undergraduate GPA– Graduate Management Admission Test (GMAT) score– Number of years of work experience

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MBA Program Admission Policy

• A randomly selected sample of students who completed the MBA was selected.

• Develop a plan to decide which applicant to admit.

MBA GPA UnderGPA GMAT Work8.43 10.89 584 96.58 10.38 483 78.15 10.39 484 48.88 10.73 646 6

. . . .

. . . .

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MBA Program Admission Policy

• Solution – The model to estimate is:

y = 0 +1x1+ 2x2+ 3x3+

y = MBA GPAx1 = undergraduate GPA [UnderGPA]x2 = GMAT score [GMAT]x3 = years of work experience [Work]

– The estimated model:MBA GPA = b0 + b1UnderGPA + b2GMAT + b3Work

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SUMMARY OUTPUT

Regression StatisticsMultiple R 0.6808R Square 0.4635Adjusted R Square0.4446Standard Error 0.788Observations 89



CoefficientsStandard Error t Stat P-valueIntercept 0.466 1.506 0.31 0.7576UnderGPA 0.063 0.120 0.52 0.6017GMAT 0.011 0.001 8.16 0.0000Work 0.093 0.031 3.00 0.0036

MBA Program Admission Policy – Model Diagnostics

Standardized residuals

0

10

20

30

40

-2.5 -1.5 -0.5 0.5 1.5 2.5 More

We estimate the regression model then we check:

Normality of errors

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SUMMARY OUTPUT






We estimate the regression model then we check:

The variance of the error variable

Residuals

-3

-2

-1

0

1

2

6 7 8 9 10

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SUMMARY OUTPUT

Regression StatisticsMultiple R 0.6808R Square 0.4635Adjusted R Square 0.4446Standard Error 0.788Observations 89



Coefficients Standard Error t Stat P-valueIntercept 0.466 1.506 0.31 0.7576UnderGPA 0.063 0.120 0.52 0.6017GMAT 0.011 0.001 8.16 0.0000Work 0.093 0.031 3.00 0.0036


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MBA Program Admission Policy – Model Assessment

SUMMARY OUTPUT





• The model is valid (p-value = 0.0000…)

• 46.35% of the variation in MBA GPA is explained by the model.

• GMAT score and years of work experience are linearly related to MBA GPA.

• Insufficient evidence of linear relationship between undergraduate GPA and MBA GPA.

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• The conditions required for the model assessment to apply must be checked.

– Is the error variable normally distributed?

– Is the error variance constant?

– Are the errors independent?

– Can we identify outlier?– Is multicolinearity (intercorrelation)a problem?

Regression Diagnostics - II

Draw a histogram of the residuals

Plot the residuals versus y

Plot the residuals versus the time periods

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Diagnostics: Multicolinearity

• Example: Predicting house price (Xm18-02) – A real estate agent believes that a house selling price can be

predicted using the house size, number of bedrooms, and lot size. – A random sample of 100 houses was drawn and data recorded.

– Analyze the relationship among the four variables

Price Bedrooms H Size Lot Size124100 3 1290 3900218300 4 2080 6600117800 3 1250 3750

. . . .

. . . .

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SUMMARY OUTPUT

Regression StatisticsMultiple R 0.7483R Square 0.5600Adjusted R Square0.5462Standard Error 25023Observations 100


Regression 3 76501718347 25500572782 40.73 0.0000Residual 96 60109046053 626135896Total 99 136610764400

Coefficients Standard Error t Stat P-valueIntercept 37718 14177 2.66 0.0091Bedrooms 2306 6994 0.33 0.7423House Size 74.30 52.98 1.40 0.1640Lot Size -4.36 17.02 -0.26 0.7982

• The proposed model isPRICE = 0 + 1BEDROOMS + 2H-SIZE +3LOTSIZE +

The model is valid, but no variable is significantly relatedto the selling price ?!


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• Multicolinearity is found to be a problem.Price Bedrooms H Size Lot Size

Price 1Bedrooms 0.6454 1H Size 0.7478 0.8465 1Lot Size 0.7409 0.8374 0.9936 1


• Multicolinearity causes two kinds of difficulties:– The t statistics appear to be too small.– The coefficients cannot be interpreted as “slopes”.

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Remedying Violations of the Required Conditions

• Nonnormality or heteroscedasticity can be remedied using transformations on the y variable.

• The transformations can improve the linear relationship between the dependent variable and the independent variables.

• Many computer software systems allow us to make the transformations easily.

Documents

1 Multiple Regression. 2 Introduction In this chapter we extend the simple linear regression model, and allow for any number of independent variables