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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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