correlation patterns
TRANSCRIPT
Correlation Patterns
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
Cross-Tabulation
• Analyze data by groups or categories
• Compare differences
• Contingency table
• Percentage cross-tabulations
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.
TwoTworatingratingscalesscales 4 quadrants4 quadrants
two-dimensionaltwo-dimensionaltabletable Importance-Importance-
PerformancePerformanceAnalysis)Analysis)
Quadrant Analysis
Calculating Rank Order
• Ordinal data
• Brand preferences
Correlation Coefficient• A statistical measure of the covariation or
association between two variables.
• Are dollar sales associated with advertising dollar expenditures?
The Correlation coefficient for two variables, X and Y is
xyr .
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
22YYiXXi
YYXXrr ii
yxxy
Simple Correlation Coefficient
22yx
xyyxxy rr
Simple Correlation Coefficient
= Variance of X
= Variance of Y
= Covariance of X and Y
2x2y
xy
Simple Correlation Coefficient Alternative Method
X
Y
NO CORRELATION
.
Correlation Patterns
X
Y
PERFECT NEGATIVECORRELATION - r= -1.0
.
Correlation Patterns
X
Y
A HIGH POSITIVE CORRELATIONr = +.98
.
Correlation Patterns
Pg 629
589.5837.17
3389.6r
712.99
3389.6 635.
Calculation of r
Coefficient of Determination
Variance
variance2
Total
Explainedr
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
Correlation Matrix
• The standard form for reporting correlational results.
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
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
Type of MeasurementDifferences between
two independent groups
Differences amongthree or more
independent groups
OrdinalMann-Whitney U-test
Wilcoxon testKruskal-Wallis test
Common Bivariate Tests
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
Type ofMeasurement
Differences between two independent groups
Nominal Chi-square test
Differences Between Groups
• Contingency Tables
• Cross-Tabulation
• Chi-Square allows testing for significant differences between groups
• “Goodness of Fit”
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
n
CRE ji
ij Ri = total observed frequency in the ith rowCj = total observed frequency in the jth columnn = sample size
Chi-Square Test
Degrees of Freedom
(R-1)(C-1)=(2-1)(2-1)=1
d.f.=(R-1)(C-1)
Degrees of Freedom
Men WomenTotalAware 50 10 60
Unaware 15 25 40
65 35 100
Awareness of Tire Manufacturer’s Brand
Chi-Square Test: Differences Among Groups Example
21
)2110(
39
)3950( 222
X
14
)1425(
26
)2615( 22
161.22
643.8654.4762.5102.32
2
1)12)(12(..
)1)(1(..
fd
CRfd
X2=3.84 with 1 d.f.
Type ofMeasurement
Differences between two independent groups
Interval andratio
t-test or Z-test
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
021
21
OR
Null Hypothesis About Mean Differences Between Groups
means random ofy Variabilit2mean - 1mean
t
t-Test for Difference of Means
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
21
21 XXS
t
t-Test for Difference of Means
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
Pooled Estimate of the Standard Error
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
Pooled Estimate of the Standard Error
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
Degrees of Freedom
• d.f. = n - k• where:
–n = n1 + n2
–k = number of groups
14
1
21
1
33
6.2131.220 22
21 XXS
797.
t-Test for Difference of Means Example
797.
2.125.16 t
797.
3.4
395.5
Type ofMeasurement
Differences between two independent groups
Nominal Z-test (two proportions)
Comparing Two Groups when Comparing Proportions
• Percentage Comparisons
• Sample Proportion - P
• Population Proportion -
Differences Between Two Groups when Comparing Proportions
The hypothesis is:
Ho: 1
may be restated as:
Ho: 1
21: oHor
0: 21 oH
Z-Test for Differences of Proportions
Z-Test for Differences of Proportions
21
2121
ppS
ppZ
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
Z-Test for Differences of Proportions
21
1121 nn
qpS pp
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
Z-Test for Differences of Proportions
21
2211
nn
pnpnp
100
1
100
1625.375.
21 ppS
068.
Z-Test for Differences of Proportions
100100
4.10035.100
p
375.
A Z-Test for Differences of Proportions
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”
i
ii )²( ²
E
EOx
Chi-Square Test
x² = chi-square statisticsOi = observed frequency in the ith cellEi = expected frequency on the ith cell
Chi-Square Test
n
CRE ji
ij
Chi-Square Test Estimation for Expected Number for Each Cell
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
Hypothesis Test of a Proportion
is the population proportion
p is the sample proportion
is estimated with p
5. :H
5. :H
1
0
Hypothesis Test of a Proportion
100
4.06.0pS
100
24.
0024. 04899.
pS
pZobs
04899.
5.6.
04899.
1. 04.2
0115.Sp
000133.Sp 1200
16.Sp
1200
)8)(.2(.Sp
n
pqSp
20.p 200,1n
Hypothesis Test of a Proportion: Another Example
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