Correlation Patterns

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

.


X

Y

PERFECT NEGATIVECORRELATION - r= -1.0

.


X

Y

A HIGH POSITIVE CORRELATIONr = +.98

.


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




independent groups

OrdinalMann-Whitney U-test

Wilcoxon testKruskal-Wallis test





independent groups

NominalZ-test (two proportions)

Chi-square testChi-square test


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


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.


21

21 XXS

t


X1 = mean for Group 1X2 = mean for Group 2SX1-X2

= the pooled or combined standard error

of difference between 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



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


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


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



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



21

2211

nn

pnpnp

100

1

100

1625.375.

21 ppS

068.


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

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Correlation Patterns