23. Research Methodology - I

8/7/2019 23. Research Methodology - I

1/66

Business

Research Methods

William G. Zikmund

Chapter 23

Bivariate Analysis: Measures of

Associations


2/66

Measures of Association A general term that refers to a number of

bivariate statistical techniques used to

measure the strength of a relationship

between two variables.


3/66

Relationships Among Variables Correlation analysis

Bivariate regression analysis


4/66


5/66

Type ofMeasurement

Measure ofAssociation

Ordinal Scales Chi-squareRank Correlation


6/66

Type ofMeasurement Measure ofAssociation

Nominal

Chi-Square

Phi CoefficientContingency Coefficient


7/66

Correlation Coefficient A statistical measure of the covariation or

association between two variables.

Are dollar sales associated with advertising

dollar expenditures?


8/66

The Correlation coefficient for two

variables, X and Y is

xyr.


9/66

Correlation Coefficient r

r ranges from +1 to -1

r = +1 a perfect positive linear relationship

r = -1 a perfect negative linear relationship

r = 0 indicates no correlation


10/66

!!

22

YYiXXi

YYXXrr

ii

yxxy

Simple Correlation Coefficient


11/66

22

yx

xy

yxxyrr

WW

W

!!



12/66

= Variance of X= Variance of Y

= Covariance of X and Y

2

xW2

yW

xyW


Alternative Method


13/66

X

Y

NO CORRELATION

.

Correlation Patterns


14/66

X

Y

PERFECT NEGATIVE

CORRELATION -

r= -1.0

.



15/66

X

Y

A HIGH POSITIVE CORRELATION

r = +.98

.



16/66

Pg 629

589.5837.173389.6

!r

712.993389

.6! 635.!

Calculation of r


17/66


18/66

Correlation Does Not Mean

Causation High correlation

Roosters crow and the rising of the sun

Rooster does not cause the sun to rise.

Teachers salaries and the consumption of

liquor

Covary because they are both influenced by a

third variable


19/66

Correlation Matrix

The standard form for reporting

correlational results.


20/66

Correlation MatrixVar1 Var2 Var3

Var1 1.0 0.45 0.31

Var2 0.45 1.0 0.10

Var3 0.31 0.10 1.0


21/66

Walkups

First Laws ofS

tatistics Law No. 1

Everything correlates with everything, especially

when the same individual defines the variables tobe correlated.

Law No. 2

It wont help very much to find a good correlation

between the variable you are interested in and some

other variable that you dont understand any better.


22/66

Law No. 3

Unless you can think of a logical reason why

two variables should be connected as cause and

effect, it doesnt help much to find a correlation

between them. In Columbus, Ohio, the mean

monthly rainfall correlates very nicely with the

number of letters in the names of the months!

Walkups

First Laws ofS

tatistics


23/66

Going back to previous conditions

Going back to previous conditions

Tall mens sonsTall mens sons

DICTIONARYDICTIONARY

DEFINITIONDEFINITION

GOING ORGOING OR

MOVINGMOVINGBACKWARDBACKWARD

Regression


24/66

Bivariate Regression A measure of linear association that

investigates a straight line relationship

Useful in forecasting


25/66

Bivariate Linear Regression

A measure of linear association that

investigates a straight-line relationship

Y = a + bX

where

Y is the dependent variable

X is the independent variable

a and b are two constants to be estimated


26/66

Y intercept a

An intercepted segment of a line

The point at which a regression line

intercepts the Y-axis


27/66

Slope

b

The inclination of a regression line as

compared to a base line

Rise over run

D - notation for a change in


28/66

Y

160

150

140

130

120

110

100

90

80

70 80 90 100 110 120 130 140 150 160 170 180 190

X

My line

Your line

.

Scatter Diagram

and Eyeball Forecast


29/66


30/66

X

Y

160

150

140

130

120

110

100

90

80

70 80 90 100 110 120 130 140 150 160 170 180 190

Y hat for

Dealer 3

Actual Y for

Dealer 7

Y hat for Dealer 7

Actual Y for

Dealer 3

Least-Squares

Regression Line


31/66

130

120

110

100

90

80

80 90 100 110 120 130 140 150 160 170 180 190

X

Y

}}{

Deviation not explained

Total deviation

Deviation explained by the regression

Y

.

Scatter Diagram of Explained

and Unexplained Variation


32/66


33/66

The Least-

Square Method

A relatively simple mathematical technique

that ensures that the straight line will most

closely represent the relationship between Xand Y.


34/66

Regression - Least-Square

Method

!

n

i

ie

1

2 minimumis


35/66

= - (The residual)

= actual value of the dependent variable

= estimated value of the dependent variable (Y hat)

n = number of observations

i = number of the observation

ie

iY

iY

iY

iY


36/66

The Logic behind the Least-

Squares Technique No straight line can completely represent

every dot in the scatter diagram

There will be a discrepancy between most

of the actual scores (each dot) and the

predicted score

Uses the criterion of attempting to make theleast amount of total error in prediction of Y

from X


37/66

XYa F !

Bivariate Regression


38/66

22

!XXn

YXXYnF

Bivariate Regression


39/66

= estimated slope of the line (the regression coefficient)

= estimated intercept of the y axis= dependent variable

= mean of the dependent variable

= independent variable

= mean of the independent variable

= number of observations

F

X

Y

n

a

Y

X


40/66

625,515,3759,24515

875,806,2345,19315

!F

625

,515

,3385

,686,3

875,806,2175,900,2

!

760,170

300,93! 54638.!


41/66

12554638.8.99 !a3.688.99 !

5.31!


42/66

12554638.8.99 !a3.688.99 !

5.31!


43/66

XY 546.5.31 !

89546.5.31 !

6.485.31 !

1.80!


44/66

XY 546.5.31 !

89546.5.31 !

6.485.31 !

1.80!


45/66

165546.5.31

129)value( ctual7ealer

7 !

!

Y

6.121!

95546.5.31)

80valueY(Actual3Dealer

3 !!

Y

4.83!


46/66

99

YYei !

5.9697 !

5.0!


47/66

165546.5.31

129)value( ctual7ealer

7 !

!

Y6.121!

95546.5.31)

80valueY(Actual3Dealer

3 !!

Y

4.83!


48/66


49/66

119546.5.319 !Y


50/66

F-Test (Regression)

A procedure to determine whether there is

more variability explained by the regression

or unexplained by the regression.

Analysis of variance summary table


51/66

Total Deviation can be

Partitioned into Two Parts Total deviation equals

Deviation explained by the regression plus

Deviation unexplained by the regression


52/66


53/66

iiii

YYYYYY !

Partitioning the Variance

Total

deviation=

Deviation

explained by the

regression

Deviation

unexplained by

the regression(Residual

error)

+


54/66

= Mean of the total group= Value predicted with regression equation

= Actual value

YY

iY


55/66

!

222

iiiiYYYYYY

Total

variation

explained=

Explained

variation

Unexplained

variation

(residual)+


56/66

SSeSSrSSt !

Sum ofSquares


57/66


58/66

Coefficient of Determination

r2

SSt

SSe

SSt

SSrr !! 12


59/66

Source of Variation

Explained by Regression

Degrees of Freedom

k-1 where k= number of estimated constants

(variables)

Sum ofSquares

SSr

Mean Squared

SSr/k-1


60/66

Source of Variation

Unexplained by Regression

Degrees of Freedom

n-k where n=number of observations

Sum ofSquares

SSe

Mean Squared

SSe/n-k


61/66


62/66

Multiple Regression

Extension of Bivariate Regression

Multidimensional when three or more

variables are involved

Simultaneously investigates the effect of

two or more variables on a single dependent

variable Discussed in Chapter 24


63/66


64/66


65/66


66/66

23. Research Methodology - I

Documents

Transcript of 23. Research Methodology - I