Ch18 Multiple Regression

8/12/2019 Ch18 Multiple Regression

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MultipleRegression

Chapter 17


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Introduction

In this chapter we extend the simple linearregression 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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Weight

Calories consumed

Introduction We all believe that weight is affected by the amount of calories

consumed. Yet, the actual effect is different from one individual toanother.

Therefore, a simple linear relationship leaves much unexplained error.


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Weight

Calories consumed

Introduction

Click to to continue

In an attempt to reduce the unexplained errors, well adda second explanatory (independent) variable


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Weight

Calories consumed

Weight = b0+ b1Calories+ b2Height + e

Introduction

If we believe a persons height explains his/her weight too, we can add this

variable to our model. The resulting Multiple regression model is shown:


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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 Making predictions using the prediction equation

Estimating the expected value of the dependent variable

Introduction


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Dependent variable Independent variables

Random error variable

17.1 Model and Required Conditions

Coefficients

We allow k independent variables to potentiallyexplain the dependent variable

y = b0+ b1x1+ b2x2+ + bkxk+ e


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The erroreis normally distributed.

The mean is equal to zero and the standard deviation isconstant (se)for all values of y.

The errors are independent.

Model AssumptionsRequired conditions for e


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

Assess the model fit using statistics obtained from the

sample.

Diagnose violations of required conditions. Try to remedyproblems when identified.

17.2 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 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 beprofitable.

Several areas where predictors of profitability can be identifiedare:

Competition Market awareness

Demand generators

Demographics

Physical quality

Estimating the Coefficients and

Assessing the Model, Example


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Profitability

Competition Marketawareness Customers Community Physical

Operating Margin

Rooms Nearest Officespace

Enrollment Income Distance

Distance todowntown.

Medianhouseholdincome.

Distance tothe nearestLa Quinta inn.

Number ofhotels/motelsrooms within3 miles from

the site.

X1 x2 x3 x4 x5 x6

CollegeEnrollment




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

Margin = b0 b1Rooms b2Nearest b3Officeb4College + b5Income + b6Disttwn + e

INN MARGIN ROOMS NEAREST OFFICE COLLEGE INCOME DISTTWN1 55.5 3203 4.2 549 8 37 2.7

2 33.8 2810 2.8 496 17.5 35 14.4

3 49 2890 2.4 254 20 35 2.6

4 31.9 3422 3.3 434 15.5 38 12.1

5 57.4 2687 0.9 678 15.5 42 6.9

6 49 3759 2.9 635 19 33 10.8




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SUMMARY OUTPUT

Regression Statistics

Multiple R 0.724611

R Square 0.525062

Adjusted 0.49442

Standard 5.512084

Observatio 100

ANOVA

df SS MS F gnificance F

Regressio 6 3123.832 520.6387 17.13581 3.03E-13

Residual 93 2825.626 30.38307

Total 99 5949.458

Coeff icient andard Err t Stat P-value Lower 95%Upper 95%

Intercept 38.13858 6.992948 5.453862 4.04E-07 24.25197 52.02518

Number -0.00762 0.001255 -6.06871 2.77E-08 -0.01011 -0.00513

Nearest 1.646237 0.632837 2.601361 0.010803 0.389548 2.902926

Office Spa 0.019766 0.00341 5.795594 9.24E-08 0.012993 0.026538

Enrollment 0.211783 0.133428 1.587246 0.115851 -0.05318 0.476744

Income 0.413122 0.139552 2.960337 0.003899 0.135999 0.690246

Distance -0.22526 0.178709 -1.26048 0.210651 -0.58014 0.129622

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

MARGIN = 38.14 -0.0076ROOMS+1.65NEAREST

+ 0.02OFFICE+0.21COLLEGE+0.41INCOME - 0.23DISTTWN

Regression Analysis, Excel OutputLa Quinta
http://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/La%20Quinta.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/La%20Quinta.xls


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Model Assessment -

Standard Error of Estimate A small value of seindicates (by definition) a small

variation of the errors around their mean.

Since the mean is zero, small variation of the errorsmeans the errors are close to zero.

So we would prefer a model with a small standarddeviation of the error rather than a large one.

How can we determine whether the standard deviationof the error is small/large?


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The standard deviation of the error seis estimated bythe Standard Error of Estimate se:

1kn

SSEs

e

Model Assessment -

Standard Error of Estimate

.yThe magnitude of seis judged by comparing it to


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SUMMARY OUTPUT


Multiple R 0.724611

R Square 0.525062

Adjusted R S 0.49442

Standard Erro 5.512084

Observat ions 100

ANOVA


Regression 6 3123.832 520.6387 17.13581 3.03E-13

Residual 93 2825.626 30.38307

Total 99 5949.458

Coefficientsandard Err t Stat P-value Lower 95%Upper 95%

Intercept 38.13858 6.992948 5.453862 4.04E-07 24.25197 52.02518

Number -0.00762 0.001255 -6.06871 2.77E-08 -0.01011 -0.00513

Nearest 1.646237 0.632837 2.601361 0.010803 0.389548 2.902926

Office Space 0.019766 0.00341 5.795594 9.24E-08 0.012993 0.026538

Enrollment 0.211783 0.133428 1.587246 0.115851 -0.05318 0.476744

Income 0.413122 0.139552 2.960337 0.003899 0.135999 0.690246

Distance -0.22526 0.178709 -1.26048 0.210651 -0.58014 0.129622

From the printout, se= 5.5121

Calculating the mean value ofy we have 739.45y

Standard Error of Estimate


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Model Assessment

Coefficient of Determination In our example it seems seis not particularly small, or is it?

If seis small the model fits the data well, and is considered useful.The usefulness of the model is evaluated by the amount of variability

in the y values explained by the model. This is done by thecoefficient of determination.

The coefficient of determination is calculated by

As you can see, SSE (thus se) effects the value of r2.

SST

SSESST

SST

SSRR2


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SUMMARY OUTPUT


Multiple R 0.724611

R Square 0.525062

Adjusted 0.49442

Standard 5.512084

Observatio 100

ANOVA


Regressio 6 3123.832 520.6387 17.13581 3.03E-13

Residual 93 2825.626 30.38307

Total 99 5949.458

Coefficient andard Err t Stat P-value Lower 95% pper 95%

Intercept 72.45461 7.893104 9.179483 1.11E-14 56.78049 88.12874

ROOMS -0.00762 0.001255 -6.06871 2.77E-08 -0.01011 -0.00513

NEAREST -1.64624 0.632837 -2.60136 0.010803 -2.90292 -0.38955

OFFICE 0.019766 0.00341 5.795594 9.24E-08 0.012993 0.026538

COLLEGE 0.211783 0.133428 1.587246 0.115851 -0.05318 0.476744

INCOME -0.41312 0.139552 -2.96034 0.003899 -0.69025 -0.136

DISTTWN 0.225258 0.178709 1.260475 0.210651 -0.12962 0.580138

Coefficient of Determination

From the printout, R2= 0.5251that is, 52.51% of the variabilityin the margin values is explainedby this model.


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To answer the question we test the hypothesis

H0: b1= b2= = bk= 0H1: At least one biis not equal to zero.

If at least one biis not equal to zero, the model hassome validity.

We pose the question:

Is there at least one independent variable linearlyrelated to the dependent variable?

Testing the Validity of the Model


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Note, that if all the data points satisfy the linear equation without errors, yiandcoincide, and thus SSE = 0. In this case all the variation in y is explained bythe regression (SS(Total) = SSR).

The total variation in y (SS(Total)) can be explained in part by the regression(SSR) while the rest remains unexplained (SSE):SS(Total) = SSR + SSE or

iy

2

ii

2

i )y(y)yy( +2

i )y(y

If errors exist in small amounts, SSR will be close to SS(Total) and the ratio

SSR/SSE will be large. This leads to the F ratio test presented next.

Testing the Validity of the Model


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Testing for Significance

1kn

SSEMSEk

SSRMSR

1knSSE

kSSR

MSE

MSRF

Define the Mean of the Sum of Squares-Regression (MSR)Define the Mean of the Sum of Squares-Error (MSE)

The ratio MSR/MSE is F-distributed


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Rejection region

F>Fa,k,n-k-1

Testing for Significance

Note.

A Large Fresults from a large SSR, which indicates much of the

variation in y is explained by the regression model; this is when themodel is useful. Hence, the null hypothesis (which states that themodel is not useful) should be rejected when F is sufficiently large.Therefore, the rejection region has the form of F > Fa,k,n-k-1


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SUMMARY OUTPUT


Multiple R 0.724611

R Square 0.525062

Adjusted 0.49442

Standard 5.512084Observatio 100

ANOVA


Regressio 6 3123.832 520.6387 17.13581 3.03E-13

Residual 93 2825.626 30.38307

Total 99 5949.458

Coefficient andard Err t Stat P-value ower 95%Upper 95%

Intercept 72.45461 7.893104 9.179483 1.11E-14 56.78049 88.12874

ROOMS -0.00762 0.001255 -6.06871 2.77E-08 -0.01011 -0.00513

NEAREST -1.64624 0.632837 -2.60136 0.010803 -2.90292 -0.38955

OFFICE 0.019766 0.00341 5.795594 9.24E-08 0.012993 0.026538

COLLEGE 0.211783 0.133428 1.587246 0.115851 -0.05318 0.476744

INCOME -0.41312 0.139552 -2.96034 0.003899 -0.69025 -0.136DISTTWN 0.225258 0.178709 1.260475 0.210651 -0.12962 0.580138

ANOVA

df SS MS F Significance F

Regression 6 3123.832 520.6387 17.13581 3.03382E-13

Residual 93 2825.626 30.38307Total 99 5949.458

k =

nk1 =n1 =

Testing the Model Validity of the La Quinta

Inns Regression Model

MSE=SSE/(n-k-1)

MSR=SSR/k

MSR/MSE

SSE

SSR

The F ratio test is performed using the ANOVAportion of the regression output


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ANOVA

df SS MS F Significance F

Regression 6 3123.832 520.6387 17.13581 3.03382E-13

Residual 93 2825.626 30.38307

Total 99 5949.458

k =nk1 =

n1 =

If alpha = .05, the critical F isFa,k,n-k-1= F0.05,6,100-6-1=2.17

F = 17.14 > 2.17

Also, the p-value = 3.033(10)-13.Clearly,p-value=3.033(10)-13


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b0 = 38.14.This is the y 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 thevalue zero, do not interpret the intercept.

Interpreting the Coefficients

Interpreting the coefficients b1through bk

y = b0+ b1x1+ b2x2++bkxky = b0+ b1(x1+1)+ b2x2++bkxk

= b0+ b1x1+ b2x2++bkxk + b1


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

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

decreases on the average by .0076% (assuming the

other variables are held constant).


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b2= 1.65.In this model, for each additional mile that thenearest competitor is to a La Quinta inn, the average operating

margin increases by 1.65% when the other variables are held

constant.

b3 = 0.02. For each additional 1000 sq-ft of office space, theaverage increase in operating margin will be .02%.

b4= 0.21. For each additional thousand students the averageoperating margin increases by .21% when the othervariables

remain constant.



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b5= 0.41.For additional $1000 increase in medianhousehold income, the average operating marginincreases by .41%, when the other variables remainconstant.

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

center, the average operating margin decreases by

.23%.



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Test statistic

ib

ii

s

bt

b d.f. = n - k -1

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 38.13858 6.992948 5.453862 4.04E-07 24.25196697 52.02518Number -0.007618 0.00125527 -6.06871 2.77E-08 -0.010110585 -0.00513

Nearest 1.646237 0.63283691 2.601361 0.010803 0.389548431 2.902926

Office Spa 0.019766 0.00341044 5.795594 9.24E-08 0.012993078 0.026538

Enrollment 0.211783 0.13342794 1.587246 0.115851 -0.053178488 0.476744

Income 0.413122 0.1395524 2.960337 0.003899 0.135998719 0.690246

Distance -0.225258 0.17870889 -1.26048 0.210651 -0.580138524 0.129622

The hypothesis for each bi is

Excel printout

H0: bi0

H1: bi0

Testing the Coefficients

For example, a test for b1:t = (-.007618-0)/.001255 = -6.068Suppose alpha=.01. t.005,100-6-1=3.39There is sufficient evidence to rejectH

0at 1% significance level.

Moreover the p=value of the test is2.77(10-8). Clearly H0is stronglyrejected. The number of rooms islinearly related to the margin.


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Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 38.13858 6.992948 5.453862 4.04E-07 24.25196697 52.02518Number -0.007618 0.00125527 -6.06871 2.77E-08 -0.010110585 -0.00513

Nearest 1.646237 0.63283691 2.601361 0.010803 0.389548431 2.902926

Office Spa 0.019766 0.00341044 5.795594 9.24E-08 0.012993078 0.026538

Enrollment 0.211783 0.13342794 1.587246 0.115851 -0.053178488 0.476744

Income 0.413122 0.1395524 2.960337 0.003899 0.135998719 0.690246

Distance -0.225258 0.17870889 -1.26048 0.210651 -0.580138524 0.129622

The hypothesis for each bi is

Excel printout

H0: bi0

H1: bi0

Testing the Coefficients

See next the interpretationof the p-value results


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Interpretation

Interpretation of the regression results for this model

The number of hotel and motel rooms, distance to thenearest motel, the amount of office space, and the medianhousehold income are linearly related to the operating margin

Students enrollment and distance from downtown are notlinearly related to the margin

Preferable locations have only few other motels nearby,much office space, and the surrounding households areaffluent.


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

value of y, for given values of xi.

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

The model can be used to learn aboutrelationships between the independent variables xi,and the dependent variable y, by interpreting thecoefficients bi

Using the Regression Equation


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Predict the average operating margin of an inn at a sitewith the following characteristics:

3815 rooms within 3 miles, Closet competitor 3.4 miles away,

476,000 sq-ft of office space,

24,500 college students,

$39,000 median household income, 3.6 miles distance to downtown center.

MARGIN = 38.14 -0.0076(3815)-1.646(.9) + 0.02(476)

+0.212(24.5) -0.413(35) + 0.225(11.2) = 37.1%

La Quinta

La Quinta Inns, Predictions
http://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/La%20Quinta.xlshttp://f/Study%20Guide/Ch19-Multiple%20Regression/WINDOWS/Desktop/Power%20Point/Keller-Xms/Xm19-01.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/La%20Quinta.xlshttp://f/Study%20Guide/Ch19-Multiple%20Regression/WINDOWS/Desktop/Power%20Point/Keller-Xms/Xm19-01.xls


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Interval estimates by Excel (Data analysis plus)

Prediction Interval

Margin

Predicted value = 37.09149

Prediction Interval

Lower limit = 25.39527

Upper limit = 48.78771

Interval Estimate of Expected Value

Lower limit = 32.96972

Upper limit = 41.21326

It is predicted that the averageoperating margin will lie

within 25.4% and 48.8%,with 95% confidence.

It is expected the averageoperating margin of all sitesthat fit this category falls

within 33% and 41.2% with95% confidence.

The average inn would not beprofitable (Less than 50%).

La Quinta Inns, Predictions


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18.2 Qualitative Independent Variables

In many real-life situations one or moreindependent variables are qualitative.

Including qualitative variables in a regressionanalysis model is done via indicator variables.

An indicator variable (I) can assume one out of

two values, zero or one.

1 if a first condition out of two is met0 if a second condition out of two is met

I=1 if data were collected before 19800 if data were collected after 1980

1 if the temperature was below 50o0 if the temperature was 50o or more

1 if a degree earned is in Finance0 if a degree earned is not in Finance


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Qualitative Independent Variables;

Example: Auction Car Price (II) Example 2 - continued

Recall: A car dealer wants to predict the auction

price of a car. The dealer believes now that both odometer reading

and car colorare variables that affect a cars price.

Three color categories are considered: White Silver

Other colors

Note: Color is aqualitative variable.


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Example 2 - continued

I1 = 1 if the color is white0 if the color is not white

I2 =1 if the color is silver0 if the color is not silver

The category Other colors is defined by:

I1= 0; I2= 0


Example: Auction Car Price (II)


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Note: To represent the situation of three possiblecolors we need only two indicator variables.

Generally to represent a nominal variable with m

possible values, we must create m-1indicatorvariables.

How Many Indicator Variables?


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Solution

the proposed model is

y = b0+ b1(Odometer) + b2I1+ b3I2+ e The data

Price Odometer I-1 I-2

14636 37388 1 0

14122 44758 1 014016 45833 0 0

15590 30862 0 0

15568 31705 0 1

14718 34010 0 1

. . . .

. . . .

White color

Other color

Silver color



Enter the data in Excel as usual


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Odometer

Price

Price = 16.837 - .0591(Odometer) + .0911(0) + .3304(1)

Price=16.837 - .0591(Odometer) + .0911(1) + .3304(0)

Price = 16.837 - .0591(Odometer) + .0911(0) + .3304(0)

From Excel we get the regression equationPRICE = 16.837 - .0591(Odometer) + .0911(I-1) + .3304(I-2)


The Regression Equation


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From Excel we get the regression equation

PRICE = 16701-.0591(Odometer)+.0911(I-1)+.3304(I-2)

A white car sells, on the average,for $91.1 more than a car of the

Other color category

A silver color car sells, on the average,for $330.4 more than a car of theOther color category.

For one additional mile theauction price decreases by5.91 cents on the average.


The Regression EquationInterpreting the equation


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SUMMARY OUTPUT


Multiple R 0.837135

R Square 0.700794Adjusted R Square 0.691444

Standard Error 0.304258

Observations 100

ANOVA

df SS MS F ignificance F

Regression 3 20.814919 6.938306 74.9498 4.65E-25

Residual 96 8.8869809 0.092573

Total 99 29.7019

Coefficient tandard Err t Stat P-value Lower 95%Upper 95%

Intercept 16.83725 0.1971054 85.42255 2.28E-92 16.446 17.2285

Odometer -0.059123 0.0050653 -11.67219 4.04E-20 -0.069177 -0.049068

I-1 0.091131 0.0728916 1.250224 0.214257 -0.053558 0.235819

I-2 0.330368 0.0816498 4.046157 0.000105 0.168294 0.492442

There is insufficient evidenceto infer that a white color car anda car of other color sell for adifferent auction price.

There is sufficient evidenceto infer that a silver color carsells for a larger price than acar of the other color category.

Car Price-Dummy


The Regression Equation
http://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/CarPrice%20Dummy.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/CarPrice%20Dummy.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/CarPrice%20Dummy.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/CarPrice%20Dummy.xls


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Recall: The Dean wanted to evaluate applications for theMBA program by predicting future performance of the

applicants. The following three predictors were suggested:

Undergraduate GPA

GMAT score

Years of work experience

It is now believed that the type of undergraduate degreeshould be included in the model.


Example: MBA Program Admission (II)

Note: The undergraduate

degree is qualitative.


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Example: MBA Program Admission (II)

I1 =1 if B.A.0 otherwise

I2 =1 if B.B.A0 otherwise

The category Other group is defined by:

I1= 0; I2= 0; I3= 0

I3 =1 if B.Sc. or B.Eng.0 otherwise


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SUMMARY OUTPUT


Multiple R 0.746053

R Square 0.556595

Adjusted R Square 0.524151


Observations 89

ANOVA


Regression 6 54.75184 9.125307 17.15544 9.59E-13

Residual 82 43.61738 0.531919

Total 88 98.36922

Coeffic ientsandard Err t Stat P-value ower 95%Upper 95%

Intercept 0.189814 1.406734 0.134932 0.892996 -2.60863 2.988258

UnderGPA -0.00606 0.113968 -0.05317 0.957728 -0.23278 0.22066

GMAT 0.012793 0.001356 9.432831 9.92E-15 0.010095 0.015491

Work 0.098182 0.030323 3.237862 0.001739 0.03786 0.158504

I-1 -0.34499 0.223728 -1.54199 0.126928 -0.79005 0.100081

I-2 0.705725 0.240529 2.934058 0.004338 0.227237 1.184213

I-3 0.034805 0.209401 0.166211 0.8684 -0.38176 0.45137


Example: MBA Program Admission (II)MBA-II
http://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/MBA-II.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/MBA-II.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/MBA-II.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/MBA-II.xls


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Applications in Human Resources

Management: Pay-Equity Pay-equity can be handled in two different forms:

Equal pay for equal work

Equal pay for work of equal value.

Regression analysis is extensively employed incases of equal pay for equal work.


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Human Resources Management:

Pay-Equity Example 3

Is there sex discrimination against female managers

in a large firm? A random sample of 100 managers was selectedand data were collected as follows:

Annual salary

Years of education Years of experience

Gender


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Solution Construct the following multiple regression model:

y = b0+ b1Education + b2Experience + b3Gender + e Note the nature of the variables:

Educationquantitative Experiencequantitative

Genderqualitative (Gender = 1 if male; =0 otherwise).


Pay-Equity


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SolutionContinued (HumanResource)


Pay-Equity

SUMMARY OUTPUT


Multiple R 0.83256

R Square 0.693155



Observations 100

ANOVA


Regression 3 5.74E+10 1.91E+10 72.28735 1.55E-24Residual 96 2.54E+10 2.65E+08

Total 99 8.29E+10

Coeff ic ient andard Err t Stat P-value Lower 95%Upper 95%

Intercept -5835.1 16082.8 -0.36282 0.71754 -37759.2 26089.02

Education 2118.898 1018.486 2.08044 0.040149 97.21837 4140.578

Experience 4099.338 317.1936 12.92377 9.89E-23 3469.714 4728.963

Gender 1850.985 3703.07 0.499851 0.618323 -5499.56 9201.527

Analysis and Interpretation The model fits the data quite well.

The model is very useful. Experience is a variable strongly

related to salary. There is no evidence of sex discrimination.
http://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/HumanResource.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/HumanResource.xls


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SolutionContinued (HumanResource)


Pay-Equity

SUMMARY OUTPUT


Multiple R 0.83256

R Square 0.693155



Observations 100

ANOVA


Regression 3 5.74E+10 1.91E+10 72.28735 1.55E-24Residual 96 2.54E+10 2.65E+08

Total 99 8.29E+10

Coeff ic ient andard Err t Stat P-value Lower 95%Upper 95%

Intercept -5835.1 16082.8 -0.36282 0.71754 -37759.2 26089.02

Education 2118.898 1018.486 2.08044 0.040149 97.21837 4140.578

Experience 4099.338 317.1936 12.92377 9.89E-23 3469.714 4728.963

Gender 1850.985 3703.07 0.499851 0.618323 -5499.56 9201.527

Analysis and Interpretation Further studying the data we find:

Average experience (years) for women is 12.Average experience (years) for men is 17

Average salary for female manager is $76,189Average salary for male manager is $97,832
http://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/HumanResource.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_9/Exercises/HumanResource.xls


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Review problems
http://localhost/var/www/apps/conversion/tmp/Additional/Chapter%2019-20.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_9//Dads-moms/OnlineKeller/Review%20Problems/Chapter%2019-20.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_9//Dads-moms/OnlineKeller/Review%20Problems/Chapter%2019-20.ppthttp://localhost/var/www/apps/conversion/tmp/Additional/Chapter%2019-20.ppt

Documents

Ch18 Multiple Regression