Lecture 22

• Multiple Regression (Sections 19.3-19.4)

19.1 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.

Examples of Multiple Regression

• Business decisionmaking: La Quinta Inns wants to decide where to locate new inns. It wants to predict operating margin based on variables related to competition, market awareness, demand generators, demographics and physical location.

• College admissions: The admissions officer wants to predict which students will be most successful. She wants to predict college GPA based on GPA from high school, SAT score and amount of time participating in extracurricular activities.

More Examples

• Improving operations: A parcel delivery service would like to increase the number of packages that are sorted in each of its hub locations. Three factors that the company can control and that influence sorting performance are the number of sorting lines, the number of sorting workers, and the number of truck drivers. What can the company do to improve sorting performance?

• Understanding relationships: Executive compensation. Does it matter how long the executive has been at the firm controlling for other factors? Do CEOs pay themselves less if they have a large stake in the stock of the company controlling for other factors? Does having an MBA increase executive salary controlling for other factors?

• 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

• 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

Example 19.1

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.

Coefficients

Dependent variable Independent variables

Random error variable

19.2 Model and Required Conditions• We allow for k independent variables to

potentially be related to the dependent variable:

y = 0 + 1x1+ 2x2 + …+ kxk +

kkk xxxxyE 111 ),,|(

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 plane, 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

Multiple Regression for k = 2, Graphical Demonstration - II

Note how a parabola becomes a parabolic Surface.

X

y

X2

1

y= b0+ b1x2

y = b0 + b1x12 + b2x2

b0

• The error is normally distributed.

• The mean of the error is equal to zero for each combination of x’s, i.e., .

• The standard deviation is constant ( for all values of x’s.

• The errors are independent.

Required conditions for the error variable

0),,|( 1 kxxE

• 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

Estimating the Coefficients and Assessing the Model, Example

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

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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.

19.3 Estimating the Coefficients and Assessing the Model

• The procedure used to perform regression analysis:– Estimate the model coefficients and statistics using least

squares using JMP.

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.

• 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

• From the printout, s = 5.51

• Calculating the mean value of y,

• It seems s is not particularly small relative to y.

• Question:Can we conclude the model does not fit the data well?

739.45y

Standard Error of Estimate

• 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

• 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

• The hypotheses are tested by an ANOVA procedure.

Testing the Validity of the La Quinta Inns Regression Model

Analysis of Variance Source DF Sum of

Squares Mean

Square F Ratio

Model 6 3123.8320 520.639 17.1358 Error 93 2825.6259 30.383 Prob > F C. Total

99 5949.4579 <.0001

[Variation in y] = SSR + SSE. If SSR is large relative to SSE, much of the variation in y is explained by the regression model; the model is useful and thus, the null hypothesis should be rejected. Thus, we reject for large F.

Rejection region

F>F,k,n-k-1

Testing the Validity of the La Quinta Inns Regression Model

1knSSE

kSSR

F

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.

Testing the Validity of the La Quinta Inns Regression Model

ANOVAdf SS MS F Significance F

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

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

• 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

• 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.

Interpreting the Coefficients

• 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.

Interpreting the Coefficients

• The hypothesis for each i is

• JMP printout

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

Test statistic

ib

iis

bt

Testing the Coefficients

Parameter Estimates Term Estimate Std Error t Ratio Prob>|t|

Intercept 38.138575 6.992948 5.45 <.0001 Number -0.007618 0.001255 -6.07 <.0001 Nearest 1.6462371 0.632837 2.60 0.0108 Office Space 0.0197655 0.00341 5.80 <.0001 Enrollment 0.2117829 0.133428 1.59 0.1159 Income 0.4131221 0.139552 2.96 0.0039 Distance -0.225258 0.178709 -1.26 0.2107

• 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%

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La Quinta Inns, Predictions