Correlation

Brian J. Piper, Ph.D., M.S.

Objectives

• Scatterplots• Correlation

– r– r2

• Regression

Sir Francis Galton

• Polymath: meteorology, geography• Behavioral Genetics

– Twin studies– Eugenics

1822-1911

Scatterplot

Scatterplots Aren’t Perfect

Big-Picture

Karl Pearson’s Product-Moment Correlation (r)

• Range: -1.0 to 0 to +1.0• Sign: + or –• Strength: +1.0/-1.0 or 0.0

1857-1936

Assumptions of Correlation

• 1) interval or ratio data• 2) X & Y normally distributed• 3) linear relationship

Also, degrees of freedom = N - 2

Age Differences in Tower of London Behavior (N=325)

Piper et al. (2012) Behavior Research Methods, 44, 110-123. http://pebl.sourceforge.net/

Tower of LondonExample #2

•Developed by Tim Shallice in 1982 as a simplified version of the Tower of Hanoi

•Sensitive to brain damage

Dose-Response (Example #3)• Threshold Model: This is the

standard model in pharmacology. Increasing the dose beyond a certain point will produce a linear response.

• Caffeine example: Increasing the dose of caffeine will cause an increase response (e.g. heart rate).

Threshold Model

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6

Dose

Re

spo

nse

a

a= No Observable Effect Level

Restriction of Range

• Optimal test of your hypothesis of a correlation requires as much variability as possible

Restriction of Range

• Optimal test of your hypothesis of a correlation requires as much variability as possible

Coefficient of Determination

• r2 = proportion of variance accounted for

r r2

.80 .64

.40 .16

Tower of London by Age

Best Performance

Trail Making Test: 19.5

Tower of London: 40.9

Piper et al. (2012) Behavior Research Methods, 44, 110-123. . http://pebl.sourceforge.net/

Terminology

• Correlational Design: several variables measured simultaneously

• Correlation (r): statistic• Amphetamine dose and locomotor activity

example

Directionality

• # of Churches (A) & # violent crimes (B) have r = +0.30

• Does A cause B?• Does B cause A?

Directionality

• # of Churches (A) & # violent crimes (B) have r = +0.30

• Does A cause B?• Does B cause A?• Does a 3rd variable (population size, C)

independently cause A & B?

Exercise

• Correlation among measures?• Paired t-test?