1. Bivariate Correlation
Lesson 10

2. Measuring Relationships
Correlation
degree relationship b/n 2 variables
linear predictive relationship
Covariance
If X changes, does Y change also?
e.g., height (X) and weight (Y) ~

3. Covariance Variance How much do scores (Xi) vary from mean?
(standard deviation)2
Covariance
How much do scores (Xi, Yi) from their means

4. Covariance: Problem How to interpret size
Different scales of measurement
Standardization
like in z scores
Divide by standard deviation
Gets rid of units
Correlation coefficient (r)

5. Pearson Correlation Coefficient
Both variables quantitative (interval/ratio)
Values of r
between -1 and +1
0 = no relationship
Parameter = ρ (rho)
Types of correlations
Positive: change in same direction
X then Y; or X then Y
Negative: change in opposite direction
X then Y; or X then Y ~

6. Correlation & Graphs Scatter Diagrams Also called scatter plots
1 variable: Y axis; other X axis
plot point at intersection of values
look for trends
e.g., height vs shoe size ~

7. Scatter Diagrams 84 78
Height 72 66 60 6 7 8 9 10 11 12
Shoe size

8. Slope & value of r Determines sign positive or negative
From lower left to upper right
positive ~

9. Slope & value of r
From upper left to lower right
negative ~

10. Width & value of r Magnitude of r
draw imaginary ellipse around most points
Narrow: r near -1 or +1
strong relationship between variables
straight line: perfect relationship (1 or -1)
Wide: r near 0
weak relationship between variables ~

11. Width & value of r Strong negative relationship Weak relationship
Weight
Chin ups 3 6 9 12 15 18 21
100
150
200
250
300
Strong negative relationship
r near -1
Weight
Chin ups 3 6 9 12 15 18 21
100
150
200
250
300
Weak relationship
r near 0

12. Strength of Correlation
Coefficient of Determination
Proportion of variance in X explained by relationship with Y
Example: IQ and gray matter volume
r = (statisically significant)
R2 = .0625
Approximately 6% of differences in IQ explained by relationship to gray matter volume ~

13. Guidelines for interpreting strength of correlation
Table 5.2 Interpreting a correlation coefficient
Size of Correlation (r)
General coefficient interpretation
.8 to 1.0
Very strong relationship
.6 to .8
Strong relationship
.4 to .6
Moderate relationship
.2 to .4
Weak relationship
.0 to .2
Weak to no relationship
*The same guidelines apply for negative values of r
*from Statistics for People Who (Think They) Hate Statistics: Excel 2007 Edition By Neil J. Salkind

14. Factors that affect size of r
Nonlinear relationships
Pearson’s r does not detect more complex relationships
r near 0 ~ Y X

15. Factors that affect size of r
Range restriction
eliminate values from 1 or both variable
r is reduced
e.g. eliminate people under 72 inches ~

16. Hypothesis Test for r H0: ρ = 0 rho = parameter H1: ρ ≠ 0 ρCV
df = n – 2
Table: Critical values of ρ
PASW output gives sig.
Example: n = 30; df=28; nondirectional
ρCV =
decision: r = .285 ? r = -.38 ? ~

17. Using Pearson r Reliability Inter-rater reliability
Validity of a measure
ACT scores and college success?
Also GPA, dean’s list, graduation rate, dropout rate
Effect size
Alternative to Cohen’s d ~

18. Evaluating Effect Size
Pearson’s r
r = ± .1
r = ± .3
r = ±.5 ~
Cohen’s d
Small: d = 0.2
Medium: d = 0.5
Large: d = 0.8
Note: Why no zero before decimal for r ?

19. Correlation and Causation
Causation requires correlation, but...
Correlation does not imply causation!
The 3d variable problem
Some unkown variable affects both
e.g. # of household appliances negatively correlated with family size
Direction of causality
Like psychology  get good grades
Or vice versa ~

20. Point-biserial Correlation
One variable dichotomous
Only two values
e.g., Sex: male & female
PASW/SPSS
Same as for Pearson’s r ~

21. Correlation: NonParametric
Spearman’s rs
Ordinal
Non-normal interval/ratio
Kendall’s Tau
Large # tied ranks
Or small data sets
Maybe better choice than Spearman’s ~

22. Correlation: PASW Data entry 1 column per variable Menus
Analyze  Correlate  Bivariate
Dialog box
Select variables
Choose correlation type
1- or 2-tailed test of significance ~

23. Correlation: PASW Output
Figure 6.1 – Pearson’s Correlation Output

24. Reporting Correlation Coefficients
Guidelines
No zero before decimal point
Round to 2 decimal places
significance: 1- or 2-tailed test
Use correct symbol for correlation type
Report significance level
There was a significant relationship between the number of commercials watch and the amount of candy purchased, r = +.87, p (one-tailed) < .05.
Creativity was negatively correlated with how well people did in the World’s Biggest Liar Contest, rS = -.37, p (two-tailed) = .001.