### Correlation

```CORRELATION
Overview of Correlation
 What
is a Correlation?
 Correlation Coefficients
 Coefficient of Determination
 Test for Significance
 Correlation and Causality
 Partial and Part Correlations
What is a Correlation?
Degree of linear relationship between
variables
 Each individual is measured on both
variables

What is a Correlation?
 Comparison
of the way scores deviate
from their means on the two variables
 Standardized covariance
Cross-Product Deviation
Find the difference between each
person’s scores and the mean of the
variable (deviation).
 For each person, multiply the two
deviations together.
 Do the deviations tend to go in the
same direction?

Covariance
deviations and average them.
 The more covariance, the more the two
variables go together, or co-vary.
 Covariance is not standardized, so it’s
hard to interpret.

Pearson r
 Standardized
covariance
 Used for two interval/ratio variables
 Varies from -1 to +1
Pearson r

Absolute value indicates strength of
relationship
 .1
- small
 .3 - medium
 .5 - large
Pearson r

 Positive:
increases on one variable
correspond to increases on the other
variable
 Negative: increases on one variable
correspond to decreases on the other
variable
Other Correlation Coefficients
 Ordinal

variables
Spearman rho or Kendall’s tau
 Dichotomous
variable with
interval/ratio variable
 Point
biserial r (discrete dichotomy)
 Biserial r (continuous dichotomy)
Other Correlation Coefficients
 Two
dichotomous variables
 Phi
coefficient
Dichotomous variables are usually at
the nominal level.
 Numbers are assigned to the two
categories in an arbitrary way.
 The way the numbers are assigned
determines the sign of the correlation
coefficient.

Review Question!
How is covariance related to the
correlation coefficient?
Coefficient of Determination
Measures proportion of explained
variance in Y based on X
 r2

Testing r for Significance
Null hypothesis is usually that r is zero
in the population.
 One tailed vs. two-tailed

Assumptions
 Appropriate
types of data
 Independent observations
 Normal distributions
 Linear relationship
Example APA format
The Pearson r was computed between
rated enjoyment of frog legs and level
of neuroticism. The correlation was
statistically significant, r(58) = .28, p =
.03.
Review Question
If r = .28, then r2 = .08. What does the .08
represent?
Review Question!
If p = .03, what probability does the .03
represent? There is a 3% chance of…..?
Correlation and Causality
A correlation by itself does not show
that one variable causes the other.
 A correlation may be consistent with a
causal relationship.

The Third Variable Problem

A correlation between X and Y could
be caused by a third variable
influencing both X and Y.
The Directionality Problem

A correlation between X and Y could
be a result of X causing Y or Y causing
X.
Partial Correlation
 Used
to “partial out” the effects of a third
variable (X2) on the relationship between X1
and Y
 Correlation between X1 and Y with the
influence of X2 removed from both X1 and Y
Partial r2
X1
Y
X2
Interpreting Partial
Correlations
 Compare
the simple bivariate
correlation to the partial correlation.
 If the partial correlation is lower, it
suggests that X2 is mediating the
relationship between X1 and Y.
Part Correlation
 Also
called: semi-partial correlation
 Correlation between X1 and Y with the
influence of X2 (and other predictor
variables) removed from just X1
 Indicates amount of unique variance in
Y explained by X1
 Used in Multiple Regression Analysis
Part r2
X1
Y
X2
Partial r2
X1
Y
X2
Choosing Stats
Patrons at a bar are randomly assigned to one
of three information conditions. In one
condition, they taste a beer without being given
any information about it. In a second condition,
they are told that it is an inexpensive brand of
beer. In a third condition, they are told that it is
an expensive brand of beer. Their ratings of the
taste quality are compared across the three
conditions.
```