linear-correlation-1205885176993532-3

7/23/2019 linear-correlation-1205885176993532-3

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Lecture 4Survey Research & Design in Psychology

James Neill, 2012

Correlation


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1. Purpose of correlation2. ovariation

!. "inear correlation#. $ypes of correlation

%. nterpreting correlation

'.(ssumptions ) limitations*. Dealing +ith several correlations

Overview


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Howell (2010) h'ategorical Data & hiS-uare horrelation & Regression h10(lternative orrelational

$echni-ues

10.1Point/iserial orrelation an PhiPearson orrelation y (nother Name

10.!orrelation oefficients for Ran3e

Data

Readings


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Purpose of correlation


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$he unerlying purpose ofcorrelation is to help aress the-uestion

4hat is the5 relations!ipor5 egree of associationor5 amount of s!ared variance

et+een two varia"les6

Purpose of correlation


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Purpose of correlation7ther +ays of e8pressing the

unerlying correlational -uestioninclue

$o +hat e8tent5 o t+o variales covar$65 are t+o variales dependentor

independentof one another65 can one variale e predicted

from another6


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Covariation


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$he +orl is mae of

covariation


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4e oserve

covariations inthe psycho

social +orl.


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4e oserve

covariations inthe psycho

social +orl.

4e canmeasure ouroservations.

e.g., epictions ofviolence in the

environment.

e.g., psychological statessuch as stress

an epression.

Do they ten

to cooccur6


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Linear correlation


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Linear correlation

$he e8tent to +hich t+o varialeshave a simple linear9straightline:relationship.

"inear correlations provie theuiling loc3s for multivariatecorrelational analyses, such as

5 ;actor analysis

5 Reliaility

5


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Linear correlation

"inear relations et+een varialesare inicate y correlations

5 &irection'orrelation sign 9= ) :

inicates irection of linear relationship5 trengt!'orrelation si>e inicates

strength 9ranges from 1 to =1:

5 tatistical significance'pinicatesli3elihoo that oserve relationshipcoul have occurre y chance


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!at is t!e linear correlation*+$pes of answers

5 No relationship 9inepenence:5 "inear relationship

?(s one variale @s, so oes the other 9=ve:

?(s one variale @s, the other As 9ve:

5 Nonlinear relationship

5 Pay caution ue to? Beterosceasticity

? Restricte range

? Beterogeneous samples


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+$pes of correlation

$o ecie +hich type of

correlation to use, consierthe levels of ,easure,entfor each variale


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+$pes of correlation

5 Nominal y nominalPhi 9C: ) ramers V, his-uare

5 7rinal y orinalSpearmans ran3 ) Eenalls $au b

5 Dichotomous y interval)ratioPoint iserial r

pb

5 nterval)ratio y interval)ratioProuctmoment or Pearsons r


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1-

+$pes of correlation and LO.

Scatterplot

Product-

momentcorrelation r

Int/Ratio

Recode

Scatterplot or

clustered bar

chart

Spearman'sRho or

Kendall's Tau

Ordinal

Scatterplot,bar chart or

error-bar chart

Point bi-serial

correlation

(rpb)

RecodeClustered bar-chart,

Chi-square,

Phi () or

Cramer's V

Nominal

Int/RatioOrdinalNominal


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1/

o,inal "$ no,inal


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o,inal "$ no,inalcorrelational approac!es

5 ontingency 9or crossta: tales? 7serve

? F8pecte? Ro+ an)or column Gs

?


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ontingency tales

/ivariate fre-uency tales ell fre-uencies 9re:


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ontingency tale F8ample

RFD ontingency cells

/"KF


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ontingency tale F8ample

his-uare is ase on the ifferences et+een

the actual an e8pecte cell counts.


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Example

Ro+ an)or column cell percentages may alsoai interpretatione.g., L2)!rs of smo3ers snore, +hereas onlyL1)!rof nonsmo3ers snore.


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lustere ar graph/ivariate ar graph of fre-uencies or percentages.

$he categorya8is ars are

clustere 9ycolour or fillpattern: toinicate the the

secon varialescategories.


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L2)!rs ofsnorers aresmo3ers,+hereas onlyL1)!rof nonsnores are

smo3ers.


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Pearson chis-uare test

P hi t t


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284riteup 2 91, 1M': 10.2',p .001

Pearson chis-uare testF8ample

hi i t i ti F l


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his-uare istriution F8ample


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P!i () Cra,ers V

P!i ()

5 Kse for 282, 28!, !82 analysese.g., ener 92: & Pass);ail 92:

Cra,ers V

5 Kse for !8! or greater analysese.g., ;avourite Season 9#: 8 ;avouriteSense 9%:

9nonparametric measures of correlation:


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Phi 9: & ramers V F8ample

291, 1M': 10.2',p .001, .2#


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Ordinal "$ ordinal


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Ordinal "$ ordinalcorrelational approac!es

5 SpearmanHs rho 9rs:

5 Eenall tau 9:5(lternatively, use nominal y

nominal techni-ues 9i.e., treat aslo+er level of measurement:


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rap!ing ordinal "$ ordinal data

5 7rinal y orinal ata is ifficult tovisualise ecause its nonparametric,yet there may e many points.

5 onsier using

?Nonparametric approaches 9e.g.,

clustere ar chart:?Parametric approaches 9e.g.,

scatterplot +ith inning:

! ( )


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pear,ans r!o (rs) or

pear,ans ran5 order correlation

5 ;or ran3e 9orinal: ata

?e.g. 7lympic Placing correlate +ith

4orl Ran3ing5 Kses prouctmoment correlation

formula

5 nterpretation is aOuste to consierthe unerlying ran3e scales

6 d ll + ( )


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3#

6endalls +au ()

5 $au a? Does not ta3e Ooint ran3s into account

5 $au

? $a3es Ooint ran3s into account? ;or s-uare tales

5 $au c? $a3es Ooint ran3s into account

? ;or rectangular tales


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3%

&ic!oto,ous "$

interval7ratio


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Point8"iserial correlation (rp"

)

5 7ne ichotomous & onecontinuous variale

?e.g., elief in go 9yes)no: anamount of international travel

5 alculate as for PearsonHs

prouctmoment r,5(Oust interpretation to consier

the unerlying scales


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Pointiserial correlation 9rp

:

F8ample

$hose +ho report that they

elieve in o also reporthaving travelle to slightlyfe+er countries 9r

p .10: ut

this ifference coul haveoccurre y chance 9p .0%:,thus B

0is not reOecte.

Do not elieve /elieve


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Pointiserial correlation 9rp

:

F8ample

0 No1 Qes


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9nterval7ratio "$

9nterval7ratio


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catterplot

5 Plot each pair of oservations 9, Q:?8 preictor variale 9inepenent:

?y criterion variale 9epenent:

5 /y convention

?the I shoul e plotte on the 8

9hori>ontal: a8is?the DI on the y 9vertical: a8is.


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Scatterplot sho+ing relationship et+eenage & cholesterol +ith line of est fit

Li f " t fit


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5 $he correlation et+een 2

variales is a measure of theegree to +hich pairs of numers9points: cluster together aroun aestfitting straight line

5 "ine of est fit y a = 8

5 hec3 for?outliers

?linearity

Line of "est fit


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4hatHs +rong +ith this scatterplot6

I shoul

treate as an DI as Q,although this is

not al+aysistinct.

Scatterplot e8ample


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Scatterplot e8ampleStrong positive 9.M1:

4hy is infantmortality positivelylinearly associate+ith the numer of

physicians 9+ith theeffects of DPremove:6

( /ecause more

octors ten to eeploye to areas+ith infant mortality9socioeconomicstatus asie:.

Scatterplot e8ample


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Scatterplot e8ample4ea3 positive 9.1#:

Scatterplot e8ample


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Scatterplot e8ample


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Pearson product8,o,ent correlation (r)

$he prouctmomentcorrelation is the

standardised covariance.


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Covariance

5 Iariance share y 2 variales

5 ovariance reflects the

irection of the relationship=ve cov inicates = relationship

ve cov inicates relationship.

Cross products

1

))((

=

N

YYXXCov

XY


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ovariance rossproucts

vecross

proucts

X1

403020100

Y1

3

3

2

2

1

1

0

ve ev.proucts

ve ev.proucts

=ve ev.proucts

=ve ev.proucts

C i


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Covariance5 Depenent on the scale of

measurementT ant comparecovariance across ifferent scales ofmeasurement9e.g., age y +eight in 3ilos versus

age y +eight in grams:.

5 $herefore, standardisecovariance9ivie y the crossprouct of

the Ss: T correlation5 orrelation is an effect si>e? i.e.,

stanarise measure of strength of linear

relationship

Covariance SD and


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;or a given set of ata thecovariance et+eenX an Y is1.20. $he SDofX is 2 an the SD

of Y is !. $he resulting correlationis

a. .20

. .!0

c. .#0

. 1.20

Covariance: SD: andcorrelation' ;uiui< >uestion'ignificance of correlation


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#3

9nterpreting correlation


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#4

Coefficient of &eter,ination (r2)

5 oD $he proportion ofvariance or change in one

variale that can e accountefor y another variale.

5 e.g., r .'0, r2

.!'

9nterpreting correlation


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9nterpreting correlation(Co!en: 1/--)

( correlation is an effect sie of correlation


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Si>e of correlation 9ohen, 1MM:

4F(E 9.1 .!:


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#%

9nterpreting correlation(?vans: 1//#)

trengt! r r2

very +ea3 0 .1 90 to #G:

+ea3 .20 .! 9# to 1'G:moerate .#0 .% 91' to !'G:

strong .'0 .* 9!'G to '#G:

very strong .M0 1.00 9'#G to 100G:

orrelation of this scatterplot


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X1

403020100

Y1

3

3

2

2

1

1

0

orrelation of this scatterplot .

Scale has no effect

on correlation.

orrelation of this scatterplot


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X1

100+0,070-0.0403020100

Y1

2

222222

222

111

11111

11

00000

orrelation of this scatterplot .

Scale has no effect

on correlation.

4h i h l i f hi


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4hat o you estimate the correlation of thisscatterplot of height an +eight to e6

a. .%. 1

c. 0

. .%

e. 1

/%

73727170-+-,-7---.

%02%3

17-

174

172

170

1-,

1--

4h t ti t th l ti f thi


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4hat o you estimate the correlation of thisscatterplot to e6

a. .%

. 1

c. 0

. .%

e. 1

X

.-.4.2.04.,4.-4.4

Y

14

12

10

,

-

4

2

4h t ti t th l ti f thi


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4hat o you estimate the correlation of thisscatterplot to e6

a. .%

. 1

c. 0

. .%

e. 1

X

141210,-42

Y

-

.

.

.

.

.

4

rite up' ?@a,ple


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rite8up' ?@a,ple

UNumer of chilren an maritalsatisfaction +ere inversely relate9r 9#M: .!%,p Z .0%:, such that

contentment in marriage teneto e lo+er for couples +ith morechilren. Numer of chilren

e8plaine appro8imately 10G ofthe variance in maritalsatisfaction, a smallmoerateeffect 9see ;igure 1:.W


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=ssu,ptions and

li,itations(Pearson product8,o,entlinear correlation)

=ssu,ptions and li,itations


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1. "evels of measurement [ interval

2. orrelation is not causation

!. "inearity

1. Fffects of outliers2. Nonlinearity

#. Normality

%. Bomosceasticity

'. Range restriction

* Beterogenous samples

=ssu,ptions and li,itations

orrelation is not causation e.g.,


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g

correlation et+een ice cream consumption an crime,ut actual cause is temperature

orrelation is not causation e.g.,


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gStop gloal +arming /ecome a pirate

ausation may e


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yin the eye of the

eholertHs a rather interestingphenomenon. Fvery time

press this lever, thatgrauate stuent reathesa sigh of relief.

?ffect of outliers


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?ffect of outliers

5 7utliers can isproportionatelyincrease or ecrease r.

5 7ptions

? compute r+ith & +ithout outliers? get more ata for outlying values

? recoe outliers as having more

conservative scores? transformation

? recoe variale into lo+er level of

measurement

(ge & selfesteem


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(ge & self esteem9r .'!:

(NF

M0*0'0%0#0!02010

SF

10

M

'

#

2

0

(ge & selfesteem


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(ge & self esteem9outliers remove: r .2!

(NF

#0!02010

SF

.

M

*

'

%

#

!

2

1


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on8linear relations!ips

f nonlinear, consier5 Does a linear relation help6

5 $ransforming variales to Vcreatelinear relationship

5 Kse a nonlinear mathematical

function to escrie therelationship et+een the variales


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or,alit$

5 $he an Q ata shoul e samplefrom populations +ith normal istriutions

5 Do not overly rely on a single inicator ofnormalityY use histograms, s3e+ness an3urtosis, an inferential tests 9e.g.,

Shapiro4il3s:5 Note that inferential tests of normality are

overly sensitive +hen sample is large


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Bomosceasticity

Range restriction


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Range restriction

5 Range restriction is +hen thesample contains restricte 9ortruncate: range of scores? e.g., cognitive capacity an age Z 1M

might have linear relationship5 f range restriction, e cautious in

generalisingeyon the range for

+hich ata is availale? F.g., cognitive capacity oes not

continue to increase linearly +ith ageafter age 1M

Range restriction


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--

g

B l


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Beterogenous samples

Susamples 9e.g.,males & females:may artificiallyincrease or

ecrease overall r. Solution calculate

r separately for su

samples & overall,loo3 for ifferences

/1

,070-00

%1

1+0

1,0

170

1-0

10

140

130

Scatterplot of Samese8 &


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p7ppositese8 Relations y ener

A r .'*B r .%2

pp Se5 6elations

7-.43210

Sa!eSe56elatio

ns

7

-

.

4

3

2

SX

(e!ale

!ale

A r .'*B r .%2

Scatterplot of 4eight an Self


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p gesteem y ener

4FB$

1201101000M0*0'0%0#0

SF

10

M

'

#

2

0

SF

male

female

A r .%0Br .#M


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&ealing wit! several

correlations

Dealing +ith several correlations


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Scatterplot matricesorganisescatterplots ancorrelations

amongst severalvariales at once.

Bo+ever, they arenot etaile over formore than aout fivevariales at a time.

g

orrelation matri8


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F8ample of an (P( Style

orrelation $ale


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Scatterplotmatr

i8


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/#

u,,ar$

6e$ points


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/%

$ p

1. ovariations are the uilingloc3sof more comple8 analyses,e.g., reliaility analysis, factor analysis,

multiple regression2. orrelation oes not prove

causation? may e in opposite

irection, cocausal, or ue to othervariales.


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6e$ points


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//

$ p

%. onsier effect si>e9e.g., C,ramerHs V, r, r2: an irection ofrelationship

'. onuct inferential test9if neee:.

6e$ points


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$ p

*. nterpret)Discuss Relate ac3 to research

hypothesis Descrie & interpret correlation

(direction: si


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Fvans, J. D. 91':. Straightfr!ard statistics

fr the behaviral sciences. Pacific rove, (/roo3s)ole Pulishing.

Bo+ell, D. . 9200*:. "#ndamental statistics

fr the behaviral sciences. /elmont, (4as+orth.

Bo+ell, D. . 92010:. Statistical methds fr

psychlgy9*th e.:. /elmont, (4as+orth.

Open Office 9,press


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Open Office 9,press $his presentation +as mae using

7pen 7ffice mpress. ;ree an open source soft+are.

http))+++.openoffice.org)prouct)impress.html
http://www.openoffice.org/product/impress.htmlhttp://www.openoffice.org/product/impress.html

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