Research Methods & Design in Psychology

Lecture 4Correlation

Lecturer: James Neill

Readings

• Howell (Fundamentals)– Ch9 (Correlation)

• Howell (Methods)– Ch6 (Categorical Data and Chi-Square)

– Ch 9 (Correlation and Regression)

Overview

• Correlational analyses

• Types of answers

• Types of correlation

• Interpretation

• Assumptions / Limitations

The World is Made of Covariation

Covariations are the Building Block of Complex Models

1.000000E+00

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6.073440E-01

1.000000E+00

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3.684480E-01

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4.337140E-01

1.703010E-01

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1.000000E+00

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1.000000E+00

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1.000000E+00

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1.000000E+00

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3.690980E-01

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1.000000E+00

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1.000000E+00

6.657210E-01

2.406820E-01

3.872940E-01

4.337140E-01

4.594940E-01

2.620210E-01

2.794770E-01

3.535430E-01

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6.657210E-01

1.000000E+00

2.661950E-01

1.608430E-01

1.703010E-01

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1.718700E-01

1.982780E-01

2.188810E-01

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Correlational Research Questions

Are two variables related?Interesting questions tend to:

• test a novel relationship, e.g.– “is time spent studying for exams associated with

increased incidence of brain cancer?”

• avoid simply showing an expected relationship, e.g.– “is time spent studying studying for exams associated

with higher exam marks?”

Correlational analyses 1

Correlational analyses are used to examine the extent to which two variables have a simple linear relationship.

Correlations provide the building blocks for:– Factor analysis– Reliability– Regression– Etc.

Correlational analyses 2

Linear relationship between 2 variables:– direction and – strength– ranges from -1 to +1

• Sign indicates direction• Size indicates strength

Correlational analyses 3

Measures the extent to which:• differences in one variable can be

predicted from differences in the other variable

• one variable varies with another variable

A correlation is also an effect size.

Types of Answers

• No relationship (independence)

• Linear relationship:– As one variable increases, so does the other

(+ve)– As one variable increases, the other decreases

(-ve)

• Non-linear

• Restricted range

• Heterogeneous samples

Types of Correlation

• Phi / Cramer’s V

• Spearman’s rank / Kendall’s Tau b

• Point bi-serial rpb

• Product-moment or Pearson’s r

  Nominal Ordinal Int/Ratio

Nominal

Clustered bar-graphChi-squaredPhi (φ) or Cramer's V

Clustered bar-graphChi-squaredPhi (φ) or Cramer's V

Scatterplot, bar chart or error-bar chartPoint bi-serial correlation (rpb)

Ordinal  

Scatterplot or clustered bar chartSpearman's Rho or Kendall's Tau

RecodeScatterplotPoint bi-serial or Spearman/Kendall

Int/Ratio    

ScatterplotProduct-moment correlation (r)

tufte$X10 5 10 15 20

05

1015

tufte$X20 5 10 15 20

05

1015

tufte$X30 5 10 15 20

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1015

tufte$X40 5 10 15 20

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1015

Tufte: Graphics reveal data.

Nominal by Nominal

Contingency Tables 1

• Bivariate frequency tables

• Marginal totals

• Can include %

Contingency Tables 2

Example

Example

Example

Clustered Bar Graph

• Bar graph of frequencies or percentages with the category axis clustered by coloured bars to indicate the two variable’s categories

Example

Chi-square 1

Example

Example

Phi () & Cramer’s VPhi ()

• Two dichotomous variables (2x2, 2x3, 3x2)

• E.g., Gender & Pass/Fail

Cramer’s V

• Two dichotomous variables (3x3 or greater)

• E.g., Favourite Season x Favourite Sense

Example

Ordinal by Ordinal

Spearman’s rho (rs)

• For ranked (or recoded to ordinal) data

• Uses product-moment correlation, but interpretations must be adjusted to consider the underlying ranked scales– e.g. Olympic Placing vs. World Ranking

Kendall’s Tau-b

• Kendall’s tau-b

– for ordinal/ranked data– takes joint ranks into account– Ranges -1 to +1, but only for square tables

Dichotomous by Interval/Ratio

Point-biserial correlation

• Point-biserial correlation (rpb)

– one dichotomous & one continuous

variable– calculate as for Pearson’s r, but

interpretations must be adjusted to consider the underlying ranked scales

– e.g., gender and self-esteem

Example

Example

Product-moment correlation (r)• For two interval and/or ratio variables

• r = covxy

sxsy

Interval/Ratio by Interval/Ratio

Scatterplots

• Plot each pair of observations (X, Y)

• x = predictor variable (independent)

• y = criterion variable (dependent)

• Check for:– outliers– linearity

• ‘Line of best fit’

y = a + bx

The correlation between 2 variables is a measure of the degree to which:

• Pairs of numbers (points) cluster together around a best-fitting straight line

Scatterplot showing relationship between age & cholesterol

Figure 9.1

Infant Mortaility and Number of Physicians

Physicians per 100,000 Population

201816141210

Infa

nt

Mo

rta

lity

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8

6

4

2

0

-2

-4

-6

Strong positive (.81)

Figure 9.2

Life Expectancy and Health Care Costs

Health Care Expenditures

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Life

Exp

ect

an

cy (

Ma

les)

74

73

72

71

70

69

68

67

66

Weak positive (.14)

Figure 9.3

Cancer Rate and Solar Radiation

Solar Radiation

600500400300200

Bre

ast

Ca

nce

r R

ate

34

32

30

28

26

24

22

20

Moderately strong negative (-.76)

Stop global warming: Become a pirate

Correlation Estimation

Indicate level (high, med., or low) and sign of the correlation for:

a) number of guns in community and number firearm deaths

b) robberies and incidence of drug abuse

c) protected sex and incidence of AIDS

d) community education level and crime rate

e) solar flares and suicide

Covariance

• Variance shared by 2 variables

• Covariance reflects the direction of the relationship:

+ve cov indicates + relationship

-ve cov indicates - relationship.

Cross products

1))((

NYYXX

CovXY

Covariance – Cross-products-ve cross

products

X1

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Y1

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3

2

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1

1

0

-ve dev. products

-ve dev. products

+ve dev. products

+ve dev. products

Covariance

• Covariance is dependent on the scale of measurement used

• Can’t compare magnitude of cov across different scales of measurement (e.g., age by weight in kilos versus age by weight in grams).

• Therefore, standardise covariance-> correlation

Correlation formula

YX

XY

ssCov

r

The correlation between 2 variables is:• an effect size – i.e., standardised measure

of amount of covariation.

Correlation - SPSSCorrelations

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21

PearsonCorrelationSig.(2-tailed)NPearsonCorrelationSig.(2-tailed)N

CigaretteConsumption perAdult per Day

CHD Mortalityper 10,000

CigaretteConsumptionper Adult per

Day

CHDMortality per10,000

Correlation is significant at the 0.01 level(2-tailed).

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Hypothesis testing

• Almost all correlations are not 0, therefore “What is the likelihood that a relationship between variables is a ‘true’ relationship, or could it simply be a result of random sampling variability or ‘chance’”?

Significance of Correlation• Null hypothesis (H0): assumes that there is no

‘true’ relationship• Alternative hypothesis (H1): assumes that the

relationship is real • We initially assume the null hypothesis, and

evaluate whether the data support the alternative hypothesis

• Null hypothesis: H0: = 0 • Alternative hypothesis H0: 0

rho= population product-moment correlation coefficient

How do we test the null hypothesis?

• Use statistical tests of probability which produce a p value

• Convention specifies a criterion for statistical significance of 0.05 (alpha level)

• We generate a p value and compare it to 0.05.

• If p is less than 0.05, this indicates statistical significance and less than a 5% chance that the relationship being tested is due to random sampling variability

Imprecision in hypothesis testing

• Type I error: rejecting the null hypothesis when it is true

• Type II error: Accepting the null hypothesis when it is false

• Statistical significance is a function of effect size, sample size and alpha level

Significance of Correlation

• A 1- or 2-tailed significance test can be done in an effort to infer to a population

• Result will depend on the power of study (i.e., higher N, p, and r, more likely to be sig.)

• Alternatively look up r tables with df = N - 2

Scatterplot showing a confidence interval for a line of best fit

US States 4th Academic Achievement by SES

Significance of Correlationdf critical

(N-2) p = .055 .67

10 .50

15 .41

20 .36

25 .32

30 .30

50 .23

200 .11

500 .07

1000 .05

When we say that the correlation between Age and test Performance is significant, we mean that:

a. there is an important relationship between Age and test Performance

b. the true correlation between Age and Performance in the population is equal to 0

c. the true correlation between Age and Performance in the population is not equal to 0

d. getting older causes you to do poorly on tests

Descriptive Assumptions

• LOM >= interval• Linear relationships• No outliers

Inferential Assumptions

• Homoscedasticity• Similar, normal underlying

distributions• Correction for attenuation

(unreliability)• Minimal and normally distributed

measurement error

Homoscedasticity

Factors that affect correlation

• Restricted range• Heterogenous samples• Scale has no effect

Coefficient of Determination (r2)

• CoD = The proportion of variance or change in one variable that can be accounted for by another variable.

• e.g., r = .60, r2 = .36

Interpreting Correlation(Cohen, 1988)

• A correlation is an effect size, so guidelines re strength can be suggested.

Strength r r2

weak: .1 to .3 (1 to 10%)

moderate: .3 to .5 (10 to 25%)

strong: >.5 (> 25%)

Size of Correlation (Cohen, 1988)

WEAK (.1 - .3)

MODERATE (.3-.5)

STRONG (>.5)

Interpreting Correlation 1

Strength r r2

very weak 0 - .19 (0 to 4%)

weak .20 - .39 (4 to 16%)

moderate .40 - .59 (16 to 36%)

strong .60 - .79 (36% to 64%)

very strong .80 - 1.00 (64% to 100%)

Interpreting Correlation 2

Interpreting Correlation 3

Interpreting Correlation 4

Correlation of this scatterplot = .9

X1

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Y1

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1

1

0

Correlation of this scatterplot = .9

X1

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Y1

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What do you estimate the correlation of this scatterplot of height and weight to be?

a. -.5

b. -1

c. 0

d. .5

e. 1

WEIGHT

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HE

IGH

T

176

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166

What do you estimate the correlation of this scatterplot to be?

a. -.5

b. -1

c. 0

d. .5

e. 1

X

5.65.45.25.04.84.64.4

Y

14

12

10

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6

4

2

What do you estimate the correlation of this scatterplot to be?

a. -.5

b. -1

c. 0

d. .5

e. 1

X

1412108642

Y

6

5

5

5

5

5

4

Non-linear Relationships

Check scatterplot

Can a linear relationship ‘capture’ the lion’s share of the variance?

If so,use r.

Y2

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X2

40

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20

10

0

Non-linear relationships

• If non-linear, consider transforming variables to ‘create’ linear relationship = equivalent to finding a non-linear mathematical function to describe the relationship between the variables

Range restriction

• Range restriction is when sample contains restricted (or truncated) range of scores– e.g., fluid intelligence and age < 18 might

have linear relationship

• If range restriction, be cautious in generalising beyond the range for which data is available– E.g., fluid intelligence does not continue to

increase linearly with age after age 18

Range restriction

Heterogenous samples

• Sub-samples (e.g., males & females) may artificially increase or decrease overall r.

• Solution - calculate r separately for sub-samples & overall, look for differences

W1

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H1

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Scatterplot of Same-sex Relations & Opposite-sex Relations by Gender

boys r = .67

girls r = .52

Opp Sex Relations

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e S

ex R

elat

ions

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SEX

female

male

Scatterplot of Weight and Self-esteem by Gender

WEIGHT

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SE

10

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0

SEX

male

female

Males r = .50

Females r = -.48

Effect of Outliers• Outliers can disproportionately increase or

decrease r.• Options

– compute r with & without outliers– get more data for outlying values– recode outliers as having more conservative

scores– transformation– recode variable into lower level of measurement

Age & self-esteem ( r = .63)

AGE

8070605040302010

SE

10

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2

0

Age & self-esteem (outliers removed) r = .23

AGE

40302010

SE

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1

Checklist1. Graphs & Scatterplots

– Outliers?– Linear?– Does each variable have a reasonable range?– Are there subsamples to consider?

2. Choose appropriate measure of Association

3. Conduct inferential test (if needed)

4. Interpret/Discuss

Dealing with several correlations

• Scatterplot matrices and correlation matrices organised correlations amongst several variables at once.

• Example (Kliewer et al, 1998)

– 99 young children

– Measured level of• Witnessed violence, Intrusive thoughts,

Social support, and Internalizing symptoms

Correlation matrix

Scatterplot matrix

Reporting

• Relate back to research hypothesis• Describe & interpret correlation

– direction of relationship– size/strength– significance

• If many correlations, report in a table• Acknowledge limitations e.g.,

– Heterogeneity (sub-samples)– Range restriction– Causality?

Writing“Number of children and marital

satisfaction were inversely related to each other, r (48) = -.35, p<.05, indicating that contentment in marriage declines as couples elect to have more children. Overall, number of children explained approximately 10% of the variance in marital satisfaction, a small-moderate effect.”

Also see end of Howell (Fundamentals) Ch9 for an example write-up

Key Points• Covariations are the building blocks of

reliability analysis, factor analysis, multiple regression

• Correlation does not prove causation – may be in opposite direction, co-causal, or due to other variables

• Check scatterplots to see whether a correlation makes sense

• Choose appropriate measure of association based on levels of measurement

• Use r, r2 and statistical significance

References• Rank order correlation

http://faculty.vassar.edu/lowry/ch3b.html• Correlation coefficient

http://www.sportsci.org/resource/stats/correl.html

• Correlationhttp://www2.chass.ncsu.edu/garson/pa765/correl.htm

• Correlationhttp://www.uwsp.edu/psych/stat/7/correlat.htm