Correlation Patterns.
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![Page 1: Correlation Patterns.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649ed15503460f94bdf58f/html5/thumbnails/1.jpg)
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Correlation Patterns
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Cross-Tabulation
• A technique for organizing data by groups, categories, or classes, thus facilitating comparisons; a joint frequency distribution of observations on two or more sets of variables
• Contingency table- The results of a cross-tabulation of two variables, such as survey questions
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Cross-Tabulation
• Analyze data by groups or categories
• Compare differences
• Contingency table
• Percentage cross-tabulations
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Elaboration and Refinement
• Moderator variable– A third variable that, when introduced into an
analysis, alters or has a contingent effect on the relationship between an independent variable and a dependent variable.
– Spurious relationship• An apparent relationship between two variables
that is not authentic.
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TwoTworatingratingscalesscales 4 quadrants4 quadrants
two-dimensionaltwo-dimensionaltabletable Importance-Importance-
PerformancePerformanceAnalysis)Analysis)
Quadrant Analysis
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Calculating Rank Order
• Ordinal data
• Brand preferences
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Correlation Coefficient• A statistical measure of the covariation or
association between two variables.
• Are dollar sales associated with advertising dollar expenditures?
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The Correlation coefficient for two variables, X and Y is
xyr .
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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
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22YYiXXi
YYXXrr ii
yxxy
Simple Correlation Coefficient
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22yx
xyyxxy rr
Simple Correlation Coefficient
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= Variance of X
= Variance of Y
= Covariance of X and Y
2x2y
xy
Simple Correlation Coefficient Alternative Method
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X
Y
NO CORRELATION
.
Correlation Patterns
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X
Y
PERFECT NEGATIVECORRELATION - r= -1.0
.
Correlation Patterns
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X
Y
A HIGH POSITIVE CORRELATIONr = +.98
.
Correlation Patterns
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Pg 629
589.5837.17
3389.6r
712.99
3389.6 635.
Calculation of r
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Coefficient of Determination
Variance
variance2
Total
Explainedr
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Correlation Does Not Mean Causation
• High correlation
• Rooster’s 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
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Correlation Matrix
• The standard form for reporting correlational results.
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Correlation Matrix
Var1 Var2 Var3
Var1 1.0 0.45 0.31
Var2 0.45 1.0 0.10
Var3 0.31 0.10 1.0
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Type of MeasurementDifferences between
two independent groups
Differences amongthree or more
independent groups
Interval and ratioIndependent groups:
t-test or Z-testOne-wayANOVA
Common Bivariate Tests
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Type of MeasurementDifferences between
two independent groups
Differences amongthree or more
independent groups
OrdinalMann-Whitney U-test
Wilcoxon testKruskal-Wallis test
Common Bivariate Tests
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Type of MeasurementDifferences between
two independent groups
Differences amongthree or more
independent groups
NominalZ-test (two proportions)
Chi-square testChi-square test
Common Bivariate Tests
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Type ofMeasurement
Differences between two independent groups
Nominal Chi-square test
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Differences Between Groups
• Contingency Tables
• Cross-Tabulation
• Chi-Square allows testing for significant differences between groups
• “Goodness of Fit”
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Chi-Square Test
i
ii )²( ²
E
EOx
x² = chi-square statisticsOi = observed frequency in the ith cellEi = expected frequency on the ith cell
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n
CRE ji
ij Ri = total observed frequency in the ith rowCj = total observed frequency in the jth columnn = sample size
Chi-Square Test
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Degrees of Freedom
(R-1)(C-1)=(2-1)(2-1)=1
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d.f.=(R-1)(C-1)
Degrees of Freedom
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Men WomenTotalAware 50 10 60
Unaware 15 25 40
65 35 100
Awareness of Tire Manufacturer’s Brand
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Chi-Square Test: Differences Among Groups Example
21
)2110(
39
)3950( 222
X
14
)1425(
26
)2615( 22
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161.22
643.8654.4762.5102.32
2
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1)12)(12(..
)1)(1(..
fd
CRfd
X2=3.84 with 1 d.f.
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Type ofMeasurement
Differences between two independent groups
Interval andratio
t-test or Z-test
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Differences Between Groups when Comparing Means
• Ratio scaled dependent variables
• t-test – When groups are small– When population standard deviation is
unknown
• z-test – When groups are large
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021
21
OR
Null Hypothesis About Mean Differences Between Groups
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means random ofy Variabilit2mean - 1mean
t
t-Test for Difference of Means
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21
21 XXS
t
X1 = mean for Group 1X2 = mean for Group 2SX1-X2 = the pooled or combined standard error of difference between means.
t-Test for Difference of Means
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21
21 XXS
t
t-Test for Difference of Means
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X1 = mean for Group 1X2 = mean for Group 2SX1-X2
= the pooled or combined standard error
of difference between means.
t-Test for Difference of Means
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Pooled Estimate of the Standard Error
2121
222
211 11
2
))1(121 nnnn
SnSnS XX
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S12 = the variance of Group 1
S22
= the variance of Group 2n1 = the sample size of Group 1n2 = the sample size of Group 2
Pooled Estimate of the Standard Error
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Pooled Estimate of the Standard Error
t-test for the Difference of Means
2121
222
211 11
2
))1(121 nnnn
SnSnS XX
S12 = the variance of Group 1
S22
= the variance of Group 2n1 = the sample size of Group 1n2 = the sample size of Group 2
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Degrees of Freedom
• d.f. = n - k• where:
–n = n1 + n2
–k = number of groups
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14
1
21
1
33
6.2131.220 22
21 XXS
797.
t-Test for Difference of Means Example
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797.
2.125.16 t
797.
3.4
395.5
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Type ofMeasurement
Differences between two independent groups
Nominal Z-test (two proportions)
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Comparing Two Groups when Comparing Proportions
• Percentage Comparisons
• Sample Proportion - P
• Population Proportion -
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Differences Between Two Groups when Comparing Proportions
The hypothesis is:
Ho: 1
may be restated as:
Ho: 1
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21: oHor
0: 21 oH
Z-Test for Differences of Proportions
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Z-Test for Differences of Proportions
21
2121
ppS
ppZ
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p1 = sample portion of successes in Group 1p2 = sample portion of successes in Group 21 1)= hypothesized population proportion 1
minus hypothesized populationproportion 1 minus
Sp1-p2 = pooled estimate of the standard errors of difference of proportions
Z-Test for Differences of Proportions
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Z-Test for Differences of Proportions
21
1121 nn
qpS pp
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p = pooled estimate of proportion of success in a sample of both groupsp = (1- p) or a pooled estimate of proportion of failures in a sample of both groupsn= sample size for group 1 n= sample size for group 2
p
q p
Z-Test for Differences of Proportions
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Z-Test for Differences of Proportions
21
2211
nn
pnpnp
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100
1
100
1625.375.
21 ppS
068.
Z-Test for Differences of Proportions
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100100
4.10035.100
p
375.
A Z-Test for Differences of Proportions
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Testing a Hypothesis about a Distribution
• Chi-Square test
• Test for significance in the analysis of frequency distributions
• Compare observed frequencies with expected frequencies
• “Goodness of Fit”
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i
ii )²( ²
E
EOx
Chi-Square Test
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x² = chi-square statisticsOi = observed frequency in the ith cellEi = expected frequency on the ith cell
Chi-Square Test
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n
CRE ji
ij
Chi-Square Test Estimation for Expected Number for Each Cell
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Chi-Square Test Estimation for Expected Number for Each Cell
Ri = total observed frequency in the ith rowCj = total observed frequency in the jth columnn = sample size
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Hypothesis Test of a Proportion
is the population proportion
p is the sample proportion
is estimated with p
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5. :H
5. :H
1
0
Hypothesis Test of a Proportion
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100
4.06.0pS
100
24.
0024. 04899.
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pS
pZobs
04899.
5.6.
04899.
1. 04.2
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0115.Sp
000133.Sp 1200
16.Sp
1200
)8)(.2(.Sp
n
pqSp
20.p 200,1n
Hypothesis Test of a Proportion: Another Example
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Indeed .001 the beyond t significant is it
level. .05 the at rejected be should hypothesis null the so 1.96, exceeds value Z The
348.4Z0115.05.
Z
0115.15.20.
Z
Sp
Zp
Hypothesis Test of a Proportion: Another Example