Chapter 11

Correlation

Pt 1: Nov. 12, 2013

Correlation

• Association between scores on two variables– e.g., age and coordination skills in children, price

and quality

– Use scatterplots to see the relationship– Rule of thumb – if 1 var is a “predictor”, put it on the x

axis

Patterns of Correlation

• Linear correlation – straight line relationship (appropriate to compute corr)

• Curvilinear correlation – U or S shaped curves• No correlation – no trend to points in scatterplot

• Positive correlation – points move from lower left to upper right (pos slope)

• Negative correlation – points move from upper left to lower right (neg slope)

Degree of Linear CorrelationThe Correlation Coefficient

• Figure correlation using products of deviation scores• Multiply pos x pos get positive results• Multiply negative x negative get positive results, which we want• Multiply pos x neg get negative results

1) Find means of x variable (Mx) and y variable (My)2) Find deviation scores for each person for x variable (x-Mx)

and y variable (y-My)3) Sum these up across the sample4) divide by sqrt of (SSx)(SSy)

– where SSx=sum of squared deviations for x variable and SSy=sum of squared deviations for y variable

• Formula for the correlation coefficient:

r = Σ [(x – Mx)(y – My)]sqrt [(SSx)(SSy)]

where SSx = Σ (x-Mx)2

where SSy = Σ (y-My)2

• Positive perfect correlation: r = +1• No correlation: r = 0• Negative perfect correlation: r = –1

• Example on board…

Correlation and Causality

• Three possible directions of causality:

1. X Y

2. X Y

3. Z

X Y

Can only determine causality w/longitudinal study or a true experiment (w/random assignment) to rule out 3rd

variables (z)

Examples of 3rd variable explainingthe correlation between x & y?

Issues in Interpreting the Correlation Coefficient

• Statistical significance – for correlation, test is whether true corr in pop = 0.– If corr is statistically signif, means it is highly unlikely

that we’d get this corr if true pop corr = 0.

• Proportionate reduction in error– r2 = proportion of variance (in y) accounted for (by x)– Used to compare correlations

• r = .3, r2 = .09; • r = .6, r2 = .36 (so 4x as big)

Issues in Interpreting the Correlation Coefficient

• Restriction in range– With limited range, corr is different than what it

would be with full range (more variability)– Correlate job perf with hiring test score

• But only hire people w/high test scores, so limited range (selective of good performers)

Size of r: Cohen’s Guidelines

• What is a large corr?– Cohen’s guidelines:

• > .5 or -.5 = large, .3 or -.3 = moderate, .1 or -.1 = small

• Unusual to have corr above .5 or -.5• Consider average r = .19 for job satisfaction & job

perf…– Interpretation?

Correlation in Research Articles• Scatter diagrams occasionally shown• Correlation matrix presented in table. Notice only lower triangle completed &

corr of variable w/itself represented with dash.– In text: “The correlation between acculturation and assimilation was significant (r = .56, p

< .05).