8 Correlations [Session 8]

8/3/2019 8 Correlations [Session 8]

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CoCo--relation Analysisrelation Analysis

Session 10Session 10


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Introduction

Is there an association between two or more

variables? If yes, what is form and degree of that

relationship?Is the relationship strong or significant enough to

be useful to arrive at a desirable conclusion?

Can the relationship be used for predictive

purposes, that is, to predict the most likely value

of a dependent variable corresponding to the

given value of independent variable or variables?


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DefinitionDefinition

CorrelationCorrelation

existsexists betweenbetween twotwo variables variables whenwhenoneone ofof themthem isis relatedrelated toto thethe otherother inin

somesome wayway


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AssumptionsAssumptions

1)1) TheThe samplesample ofof paired paired datadata ((x,yx,y)) isis aarandomrandom samplesample..

2)2) TheThe pairspairs ofof ((x,yx,y)) datadata havehave aa bivariatebivariatenormalnormal distributiondistribution..


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Methods of Correlation Analysis

In this chapter, the following methods of finding the correlationcoefficient between two variables xandyare discussed:

Scatter Diagram method

Karl Pearsons Coefficient of Correlation method

Spearmans Rank Correlation method

Method of Least-squares

Figure shows how the strength of the association between two

variables is represented by the coefficient of correlation.

Negative Correlation Positive Correlation

1.00 0.50 0 + 0.50 + 1.00

Perfect negative

correlation

Moderate negative

correlation

No correlation Moderate positive

correlation

Strong negativecorrelation

Weak negativecorrelation

Weak positivecorrelation

Strong positivecorrelation

Perfect positivecorrelation


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DefinitionDefinition

Scatterplot (or scatter diagram)Scatterplot (or scatter diagram)

isis aa graphgraph inin whichwhich thethe pairedpaired ((x,yx,y))samplesample datadata areare plotted plotted with with aahorizontalhorizontal xxaxisaxis andand aa verticalverticalyyaxisaxis..

EachEach individualindividual ((x,yx,y)) pairpair isis plottedplottedasas aa singlesingle pointpoint..


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Scatter Diagram of Paired DataScatter Diagram of Paired Data


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Scatter Diagram of Paired Data


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Positive Linear CorrelationPositive Linear Correlation

x x

yy y

x

Scatter Plots

(a) Positive (b) Strongpositive

(c) Perfectpositive


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Negative Linear CorrelationNegative Linear Correlation

x x

yy y

x(d) Negative (e) Strong

negative(f) Perfect

negative

Scatter Plots


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No Linear CorrelationNo Linear Correlation

xx

yy

(g) No Correlation (h) Nonlinear Correlation

Scatter Plots


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Karl Pearson's Correlation

Coefficient


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7fdxdy - (7fdx)(7fdy)/N

(SDx) (SDy)r=

Definition Karl Pearson[For Classified data]

Correlation Coefficient r

SDx = fdx (fdx)/N

SDy = fdy (fdy)/N

dx = Xi A

dy = Yi A


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Example 1Example 1

FindFind coefficientcoefficient ofof correlationcorrelation betweenbetween heightheight (X)(X) andand

weightweight (Y)(Y) fromfrom thethe followingfollowing datadata.. Also,Also, obtainobtain thethe

twotwo regressionregression lineline..

HeightHeight 6161 6565 6868 6262 6060

Weight Weight 6262 5555 7070 6060 5353


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Answer 1Answer 1

rr == 00..6565

XX 6363..22 == 00..3232(Y(Y 6060))

YY 6060 == 11..3333(X(X 6363..22))


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Example 2Example 2

GivenGiven thethe twotwo regressionregression lineslines

44xx 55yy ++ 3333 == 00

2020xx 99yy 107107 == 00AndAnd variancevariance ofofxx beingbeing99,, calculatecalculate

1)1) MeanMean xx andand MeanMean yy

2)2) CorrelationCorrelation CoefficientCoefficient ofofxx && yy3)3) SDSD ofof yy


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Answer 2Answer 2

MeanMean XX == 1313,, MeanMean YY == 1717

rr == 00..66

VarianceVariance ofof y y ==1616


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Round toRound to threethree decimal placesdecimal places

Use calculator or computer if possibleUse calculator or computer if possible

Rounding the

Linear Correlation Coefficient r


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Properties of theProperties of the

Linear Correlation CoefficientLinear Correlation Coefficientrr

1.1. --11ee rree 11

2. Value of2. Value ofrrdoes not change if all values ofdoes not change if all values ofeither variable are converted to a differenteither variable are converted to a different

scale.scale.

3.3. TheThe rris not affected by the choice ofis not affected by the choice ofxxandandyy..InterchangeInterchange xxandandyyand the value ofand the value ofrrwill notwill notchange.change.

4.4. rr measures strength of ameasures strength of a linearlinear relationship.relationship.


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Interpreting the Linear CorrelationInterpreting the Linear Correlation

CoefficientCoefficient rr == ++ 11 :: PerfectPerfect PositivePositive CorrelationCorrelation

rr == 11 :: PerfectPerfect NegativeNegative CorrelationCorrelation

rr == 00 :: UncorrelatedUncorrelated CorrelationCorrelation

StandardStandard ErrorError (S(S..EE..)) == ((11 rr22)/)/N,N, NN == pair pair ofofobservationsobservations

ProbableProbable ErrorError == 00..67456745 XX SS..EE..


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Spearmans Rank Correlation Coefficient

This method is applied to measure the association between two variables when

only ordinal (or rank) data are available. In other words, this method is applied in a

situation in which quantitative measure of certain qualitative factors such as

judgment, brands personalities, TV programmes, leadership, colour, taste, cannot

be fixed, but individual observations can be arranged in a definite order(also called

rank). The ranking is decided by using a set of ordinal rank numbers, with 1 for the

individual observation ranked first either in terms ofquantity orquality; and n for the

individual observation ranked last in a group of n pairs of observations.

Mathematically, Spearmans rank correlation coefficient is defined as:

where R = rank correlation coefficient

R1 = rank of observations with respect to first variable

R2 = rank of observations with respect to second variable

d = R1 R2, difference in a pair of ranksn = number of paired observations or individuals being ranked

The number 6 is placed in the formula as a scaling device, it ensures that the

possible range of R is from 1 to 1. While using this method we may come across

three types of cases.


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Advantages

This method is easy to understand and its application issimpler than Pearsons method.

This method is useful for correlation analysis when variables

are expressed in qualitative terms like beauty, intelligence,

honesty, efficiency, and so on.This method is appropriate to measure the association between

two variables if the data type is at least ordinal scaled (ranked)

The sample data of values of two variables is converted into

ranks either in ascending order or descending order forcalculating degree of correlation between two variables.


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Disadvantages

Values of both variables are assumed to be normallydistributed and describing a linear relationship rather than non-

linear relationship.

A large computational time is required when number of pairs

of values of two variables exceed 30.This method cannot be applied to measure the association

between two variable grouped data.


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Rank Order CorrelationRank Order Correlation

HitsHits RankRank HRHR RankRank DD DD22

11 1010 33 88 22 44

22 99 44 77 22 44

33 88 55 66 22 44

44 77 11 1010 --33 9955 66 77 44 22 44

66 55 66 55 00 00

77 44 22 99 --55 2525

88 33 1010 11 22 44

99 22 99 22 00 00

1010 11 88 33 22 44


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Rank Order Correlation, contRank Order Correlation, cont

HitsHits RankRank HRHR RankRank DD DD22

11 1010 33 88 22 44

22 99 44 77 22 44

33 88 55 66 22 44

44 77 11 1010 --33 99

55 66 77 44 22 44

66 55 66 55 00 00

77 44 22 99 --55 2525

88 33 1010 11 22 44

99 22 99 22 00 00

1010 11 88 33 22 44

Rho = 1- [6 (D2) / N (N2-1)]

Rho = 1- [6(58)/10(102-1)]

Rho = 1- [348 / 10 (100 -1)]

Rho = 1- [348 / 990]

Rho = 1- 0.352

Rho = 0.648

(D2

= 58)N=10


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PearsonsPearsons rrH

itsH

itsHRHR 77

xyxy11 33 33

22 44 88

33 55 1515

44 11 44

55 77 3535

66 66 3636

77 22 1414

88 1010 8080

99 99 8181

1010 88 8080

77x/nx/n

=5.5=5.5

77x/nx/n

= 5.5= 5.577xy/nxy/n

==32.8632.86

7xy/n - (7x/n)(7y/n)

(SDx) (SDy)r=

r= 32.86 - (5.5) (5.5)/(3.03) (3.03)

r= 35.86 - 30.25 / 9.09

r= 5.61 / 9.09

r= 0.6172


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Example 3Example 3 Compute the correlation coefficient:Compute the correlation coefficient:

Age ofAge of

husbandshusbands

Age of wivesAge of wives

1515--2525 2525--3535 3535--4545 4545--5555 5555--6565 6565--7575 TotalTotal

1515--2525 11 11 -- -- -- -- 222525--3535 22 1212 11 -- -- -- 1515

3535--4545 -- 44 1010 11 -- -- 1515

4545--5555 -- -- 33 66 11 -- 1010

5555--6565 -- -- -- 22 44 22 88

6565--7575 -- -- -- -- 11 22 33

TotalTotal 33 1717 1414 99 66 44 5353


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Sol.3Sol.3 1515--2525 2525--3535 3535--4545 4545--5555 5555--6565 6565--7575

ff fdfdyy fdfdyy fdfdxxddyy

1515--2525

2525--3535

3535--4545

4545--5555

5555--6565

6565--7575

ff

fdfdxx

fdfdxx

fdfdxxdd

yy

X

dx

dyY


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Sol.3Sol.3 1515--2525 2525--3535 3535--4545 4545--5555 5555--6565 6565--7575

--22 --11 00 +1+1 +2+2 +3+3 ff fdfdyy fdfdyy fdfdxxddyy

1515--2525 --22 11 11 -- -- -- -- 22

2525--3535 --11 22 1212 11 -- -- -- 1515

3535--4545 00 -- 44 1010 11 -- -- 1515

4545--5555 +1+1 -- -- 33 66 11 -- 1010

5555--6565 +2+2 -- -- -- 22 44 22 88

6565--7575 +3+3 -- -- -- -- 11 22 33

ff 33 1717 1414 99 66 44 5353

fdfdxx

fdfdxx

fdfdxxdd

yy

X

dx

dyY


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Sol.3Sol.3 1515--2525 2525--3535 3535--4545 4545--5555 5555--6565 6565--7575


1515--2525 --22 11 11 -- -- -- -- 22 --44 88 66

2525--3535 --11 22 1212 11 -- -- -- 1515 --1515 1515 1616

3535--4545 00 -- 44 1010 11 -- -- 1515 00 00 00

45

45--5555 +1+1 -- -- 33 66 11 -- 1010 +10+10 1010 88

5555--6565 +2+2 -- -- -- 22 44 22 88 +16+16 3232 3232

6565--7575 +3+3 -- -- -- -- 11 22 33 +9+9 2727 2424

ff

33 1717 14

14 99

6644

5353 +16+169

29

2 8686fdfd

xx --66 --1717 00 99 1212 1212 +1+1

00

fdfdxx 1212 1717 00 99 2424 3636 9898

fdfdxxddyy 88 1414 00 1010 2424 3030 8686

X

dx

dyY

4

4

2

12 0

0 0 0

0 6 2

4 16 12

6 18


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Sol.3Sol.3 1515--2525 2525--3535 3535--4545 4545--5555 5555--6565 6565--7575


1515--2525 --22 11 11 -- -- -- -- 22 --44 88 66

2525--3535 --11 22 1212 11 -- -- -- 1515 --1515 1515 1616

3535--4545 00 -- 44 1010 11 -- -- 1515 00 00 00

4545--5555 +1+1 -- -- 33 66 11 -- 1010 +10+10 1010 885555--6565 +2+2 -- -- -- 22 44 22 88 +16+16 3232 3232

6565--7575 +3+3 -- -- -- -- 11 22 33 +9+9 2727 2424

ff

33 1717 1414 99 66 44 5353 +16+16 9292 8686fdfd

xx --66 --1717 00 99 1212 1212 +1+1

00

fdfdxx 1212 1717 00 99 2424 3636 9898

fdfdxxddyy 88 1414 00 1010 2424 3030 8686

X

dx

dyY

4

4

2

12 0

0 0 0

0 6 2

416 12

6 18

r= 0.907


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0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5

Data from the Garbage Project

x Plastic (lb)

y Household

Is there a significant linear correlation?


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0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5


x Plastic (lb)

y Household


Plastic Household

0.27 2

1.41 3

2.19 3

2.83 6

2.19 41.81 2

0.85 1

3.05 5


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0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5


x Plastic (lb)

y Household


r= 0.842

R2 = 0.71


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Correlation AnalysisCorrelation Analysis

Vs. Regression AnalysisVs. Regression Analysis

CorrelationCorrelation meansmeans thethe relationshiprelationship betweenbetween

twotwo or or moremore variablesvariables toto measuremeasure thethedirectiondirection andand degreedegree ofof linearlinear relationshiprelationship..

RegressionRegression analysisanalysis aimsaims atat establishingestablishing thethe

functionalfunctional relationshiprelationship..


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Correlation does not imply causationCorrelation does not imply causation

CorrelationCorrelation doesdoes notnot implyimply causationcausation isis aa phrasephraseusedused inin thethe sciencessciences andand statisticsstatistics toto emphasizeemphasizethatthat correlationcorrelation betweenbetween twotwo variables variables doesdoes notnotimplyimply therethere isis aa causecause--andand--effecteffect relationshiprelationship

betweenbetween thethe twotwo.. ItsIts converse,converse, correlationcorrelation provesprovescausation,causation, isis aa logicallogical fallacyfallacy byby whichwhich twotwo eventseventsthatthat occuroccur togethertogether areare claimedclaimed toto havehave aa causecause--andand--effecteffect relationshiprelationship.. ForFor example,example,

AA occursoccurs inin correlationcorrelation withwith BB..

Therefore,Therefore, AA causescauses BB..

ThisThis isis aa logicallogical fallacyfallacy becausebecause therethere areare atat leastleastfourfour otherother possibilitiespossibilities::


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Correlation does not imply causationCorrelation does not imply causation

1.1. BB maymay bebe thethe causecause ofof A,A, oror

2.2. somesome unknownunknown thirdthird factorfactor isis actuallyactually thethe causecause ofof

thethe relationshiprelationship betweenbetween AA andand B,B, oror3.3. thethe "relationship""relationship" isis soso complexcomplex itit cancan bebe labeledlabeled

coincidentalcoincidental (i(i..ee..,, twotwo eventsevents occurringoccurring atat thethe samesametimetime thatthat havehave nono simplesimple relationshiprelationship toto eacheach otherotherbesidesbesides thethe factfact thatthat theythey areare occurringoccurring atat thethe samesame

time)time)..4.4. BB maymay bebe thethe causecause ofof AA atat thethe samesame timetime asas AA isis

thethe causecause ofof BB (contradicting(contradicting thatthat thethe onlyonlyrelationshiprelationship betweenbetween AA andand BB isis thatthat AA causescauses B)B)..ThisThis describesdescribes aa selfself--reinforcingreinforcing systemsystem..

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8 Correlations [Session 8]