Bivariate Analysis Final

7/31/2019 Bivariate Analysis Final

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Bivariate Analysis:Measures of Association


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Measures of Association

Refers to bivariate statistical techniques

used to measure the strength of a

relationship between two variables. The chi-square ( 2) test provides information about

whether two or more less-than interval variables areinterrelated.

Correlation analysis is most appropriate for interval orratio variables.

Regression can accommodate either less-than interval

independent variables, but the dependent variablemust be continuous.

CovarianceExtent to which two variables are

associated systematically with each other.


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Type of

Measurement

Measure of

Association

Interval andRatio Scales

Correlation CoefficientBivariate Regression

Ordinal ScalesChi-square

Rank Correlation

Nominal

Chi-Square

Phi Coefficient

Contingency Coefficient

Common Bivariate Tests


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Walkups First Laws of Statistics Law No. 1

Everything correlates with everything,

especially when the same individual definesthe variables to be 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.


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Walkups

First Laws of Statistics

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 acorrelation between them.

In Columbus, Ohio, the mean monthly rainfall

correlates very nicely with the number of letters in

the names of the months!


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The correlation coefficient (r) for two variables

(X,Y) is which ranges from +1 to -1.xyr

Correlation.. is the measure of associationbetween two at least interval scaled variables suchas age and income, sales and selling expenses.

Correlation ... is a mathematical relationship. It cannever prove a casual connection. It does however givesupport to an explanation based on logic.


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Simple Correlation Coefficient

The correlation coefficient. . . . (R) is a

measure of strength and direction of association

R ranges between -1 (perfect negative linear

relationship) to +1 (perfect positive linearrelationship). R near zero reflects the absence

of linear association

xyr

-1 0 +1

22YYiXXi

YYXXrr

ii

yxxy


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X

Y

Correlation Patterns

NO CORRELATION R=0

Simple Correlation Coefficient xyr


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X

Y

PERFECT NEGATIVE

CORRELATION -

R = -1.0

Correlation Patterns

Negative correlation . . . The variables move inopposite directions. A high value on onevariable will be associated with a low value on a2nd variable


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Positive Correlation

Market

Share(y)

Brand Awareness (x)Positive correlation:

As brand awareness , market increases

. . . As one variable (x) increases or decreases,

the second variable (y) increases or decreases.The variables move in the same direction.


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Correlation Coefficient

-1 0 1

{

{

There is

linear correlation

No linear correlationThere is

linear correlation

Decision points

xy

r


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Correlation Coefficient

SAMPLE SIZE n

For P


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Statistical Significance

xy

rt =

Ho: r = 0

Square root of n-2 divided by 1-r squared


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Correlation Coefficient Interpretation

Strongly Disagree NeutralStrongly Agree

Good Taste 1 2 3 4* 5

Strongly Disagree NeutralStrongly Agree

High price 1 2 3 4* 5

Strongly Strongly

Disagree Neutral Agree

Statistical Results: r = -.61, p = .07, n =100

As the taste of seven up increases, the price


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Pg 629

589.5837.173389.6

r

712.99

3389.6 635.

Calculation of rxyr

Simple Correlation Coefficient


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Coefficient of Determination

Coefficient of Determination (R

2

) A measure obtained by squaring the correlation

coefficient; the proportion of the total variance of

a variable accounted for by another value of

another variable. Measures that part of the total variance ofYthat

is accounted for by knowing the value ofX.

VarianceTotal

varianceExplained2R


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EXHIBIT 23.3

Correlation Analysisof Number of Hours

Worked inManufacturingIndustrieswith UnemploymentRate


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Correlation Matrix

Correlation matrix -The standard form forreporting correlation coefficients for more than

two variables.

The Significance of the Correlation- Theprocedure for determining statistical significance

of a correlation coefficient is the t-test.


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Correlation Matrix for 3 variables

Var1 Var2 Var3

Var1 1.0 0.45 0.31

Var2 0.45 1.0 0.10

Var3 0.31 0.10 1.0

The standard form for reporting correlationresults.


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EXHIBIT 23.4 Pearson Product-Moment Correlation Matrixfor Salesperson Examplea

aNumbers below the diagonal are for the sample; those above the diagonal are omitted.bp


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Correlation Does Not Mean

Causation

When two variables covary, they display

concomitant variation.

Systematic covariation (a high correlation) does

not in and of itself establish causality Roosters crow and the rising of the sun

Rooster does not cause the sun to rise.

Teachers salaries and the consumption of liquor

Variables covary because they are both influenced by a

third variable


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Excel Spreadsheet

Fx = correl (col2:col22:col3,col32)

The Correlation coefficient for twovariables, X and Y is computed by

the following excel instruction.

Where Xs data is in column 2 and

Ys data is in column 3.

C l ti


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Correlation


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Correlation Coefficient, r = .75

Correlation: Player Salary and Ticket

Price

-20

-10

0

10

20

30

1995 1996 1997 1998 1999 2000 2001

Change in Ticket

Price

Change in

Player Salary

Regression Analysis


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Regression Analysis


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Bivariate Analysis Final