© 2001 prentice-hall, inc.chap 14-1 ba 201 lecture 23 correlation analysis and introduction to...

© 2001 Prentice-Hall, Inc. Chap 14-1

BA 201

Lecture 23Correlation Analysis

AndIntroduction to Multiple

Regression(Data)


Topics Correlation - Measuring the Strength of

the Association The Multiple Regression Model


Correlation Example: Mid-term Scores

Here is an Excel Workbook that contains the correlation analysis between Mid-term Scores and each of the various components of this course.

Microsoft Excel Worksheet


Purpose of Correlation Analysis

Correlation Analysis is Used to Measure Strength of Association (Linear Relationship) Between 2 Numerical Variables Only Strength of the Relationship is Concerned No Causal Effect is Implied

Population Correlation Coefficient (Rho) is Used to Measure the Strength between the Variables

XY

X Y


Purpose of Correlation Analysis

Sample Correlation Coefficient r is an Estimate of and is Used to Measure the Strength of the Linear Relationship in the Sample Observations

(continued)

1

2 2

1 1

n

i ii

n n

i ii i

X X Y Yr

X X Y Y

© 2001 Prentice-Hall, Inc. Chap 14-6r = .6 r = 1

Sample of Observations from Various r Values

Y

X

Y

X

Y

X

Y

X

Y

X

r = -1 r = -.6 r = 0


Features of and r

Unit Free Range between -1 and 1 The Closer to -1, the Stronger the

Negative Linear Relationship The Closer to 1, the Stronger the

Positive Linear Relationship The Closer to 0, the Weaker the Linear

Relationship


Sample Correlation Coefficient: Example

You wish to examine the relationship between the annual sales of produce stores and their sizes in square footage. Sample data for 7 stores were obtained. Find the sample correlation coefficient.

Annual Store Square Sales

Feet ($1000)

1 1,726 3,681

2 1,542 3,395

3 2,816 6,653

4 5,555 9,543

5 1,292 3,318

6 2,208 5,563

7 1,313 3,760


Solution: Produce Stores In PHStat, the

sample correlation coefficient (r ) is the “signed” multiple R The sign of r is the

same as the sign of the estimated slope coefficient

The absolute value of r is the same as the value of multiple R

1b

Regression StatisticsMultiple R 0.9705572R Square 0.94198129Adjusted R Square 0.93037754Standard Error 611.751517Observations 7

From Excel Printout

Coefficients Standard Error t StatIntercept 1636.4147 451.4953308 3.62443Footage 1.4866337 0.164999212 9.00994

1b

0.97055r


Test if There is a Linear Relationship

Hypotheses H0: = 0 (No Correlation) H1: 0 (Correlation)

Test Statistic

2

2 1

2 2

1 1

where

2n

i ii

n n

i ii i

rt

rn

X X Y Yr r

X X Y Y


Example: Produce Stores

Is there any evidence of linear relationship between Annual Sales of a store and its Square Footage at .05 level of significance?

H0: = 0 (No association)

H1: 0 (Association)

.05 n = 7df 7 - 2 = 5

Solution:


Example: Produce Stores Solution

0 2.5706-2.5706

.025

Reject Reject

.025

Critical Value(s):

Conclusion:There is evidence of a linear relationship at 5% level of significance

Decision:Reject H0

2

.97069.0099

1 .942052

rt

rn

The value of the t statistic is exactly the same as the t statistic value for test on the slope coefficient

r0

t


Simple Linear Regression in PHStat

In Excel, use PHStat | Regression | Simple Linear Regression …

EXCEL Spreadsheet of Regression Sales on Footage



0 1 1 2 2i i i k ki iY b b X b X b X e

Population Y-intercept

Population slopes Random Error

The Multiple Regression Model

Relationship between 1 dependent & 2 or more independent variables is a linear

function

Dependent (Response) variable for sample

Independent (Explanatory) variables for sample model

1 2i i i k ki iY X X X

Residual


Simple Linear Regression Model Revisited

Y

XObserved Value

|Y X iX

i

ii iY X

0 1i iY b b X

ie

0 1i iib bY X e 1b

0b


Population Multiple Regression Model

X2

Y

X1Y|X = 0 + 1X 1i + 2X 2i

0

Y i = 0 + 1X 1i + 2X 2i + i

ResponsePlane

(X 1i,X 2i)

(O bserved Y )

i

X2

Y

X1Y|X = 0 + 1X 1i + 2X 2i

0

Y i = 0 + 1X 1i + 2X 2i + i

ResponsePlane

(X 1i,X 2i)

(O bserved Y )

i

Bivariate model(2 Independent Variables: X1 and X2)

i


Sample Multiple Regression Model

X2

Y

X1

b0

Y i = b0 + b1X 1 i + b2X 2 i + e i

ResponsePlane

(X 1i, X 2i)

(O bserved Y)

^

e i

Y i = b0 + b1X 1 i + b2X 2 i

X2

Y

X1

b0

Y i = b0 + b1X 1 i + b2X 2 i + e i

ResponsePlane

(X 1i, X 2i)

(O bserved Y)

^

e i

Y i = b0 + b1X 1 i + b2X 2 i

Bivariate model

Sample Regression PlaneSample Regression Plane


Multiple Linear Regression Equation

Too complicated

by hand! Ouch!


Multiple Regression Model: Example

Oil (Gal) Temp Insulation275.30 40 3363.80 27 3164.30 40 1040.80 73 694.30 64 6

230.90 34 6366.70 9 6300.60 8 10237.80 23 10121.40 63 331.40 65 10

203.50 41 6441.10 21 3323.00 38 352.50 58 10

(0F)

Develop a model for estimating heating oil used for a single family home in the month of January based on average temperature and amount of insulation in inches.


1 2ˆ 562.151 5.437 20.012i i iY X X

Sample Multiple Regression Equation: Example

CoefficientsIntercept 562.1510092X Variable 1 -5.436580588X Variable 2 -20.01232067

Excel Output

For each degree increase in temperature, the estimated average amount of heating oil used is decreased by 5.437 gallons, holding insulation constant.

For each increase in one inch of insulation, the estimated average use of heating oil is decreased by 20.012 gallons, holding temperature constant.

0 1 1 2 2i i i k kiY b b X b X b X


Interpretation of Estimated Coefficients

Slope (bi) Estimated that the average value of Y changes

by bi for each 1 unit increase in Xi holding all other variables constant (ceterus paribus)

Example: If b1 = -2, then fuel oil usage (Y) is expected to decrease by an estimated 2 gallons for each 1 degree increase in temperature (X1) given the inches of insulation (X2)

Y-Intercept (b0) The estimated average value of Y when all Xi = 0


Multiple Regression in PHStat

PHStat | Regression | Multiple Regression …

EXCEL spreadsheet for the heating oil example.



Summary

Discussed Correlation - Measuring the Strength of the Association

Developed The Multiple Regression Model

© 2001 prentice-hall, inc.chap 14-1 ba 201 lecture 23 correlation analysis and introduction to...

Documents

correlation h

correlation example

footage slide

sign of r

slope coefficient r

topics correlation

value of multiple r

negative linear relationship