BIVARIATE CORRELATION - University of Richmond
Correlation
Research Problem:
What is the relationship between two variables?
Relationship between hours studying (X)
and grades on a midterm (Y)?
Relationship between stressful life events (X)
and number of illness symptoms (Y)?
Correlation = Direction and strength of (linear) relationship between two variables
I. The Scatterplot
What is the relationship between hours studying (X) and scores on a quiz (Y)?
|Student |Hours |Score |
|A |1 |1 |
|B |1 |3 |
|C |3 |4 |
|D |4 |5 |
|E |6 |4 |
|F |7 |6 |
II. Pearson Correlation Coefficeint
Symbol: r
r can range from -1.0 to +1.0
Sign (+/-) indicates “direction”
Value indicates “strength”
Measures a “linear” relationship only
(a) Direction of relationship between X, Y
Positive (+r) = As X goes up, Y goes up
Negative (-r) = As X goes up, Y goes down
(b) Strength of a relationship between X, Y
Closer to ( 1.0, stronger
Closer to 0, weaker
when r = 0 ( X,Y relationship not defined
by a straight line
Pearson Correlation Coefficient
-1.0 0 +1.0
1. Closer to 0 = weaker
2. Closer to (1.0 = stronger
3. r close to (1.0 very rare in social science
4. r ( ( .30 considered important
5. r ( 0, no linear relationship between X & Y
What does r represent?
r = degree to which X & Y vary together
degree to which X & Y vary separately
r = covariance of X & Y
variance of X & Y
Definitional Formula for Pearson r:
r = [pic]
SP = “Sum of Products”
SS = Sum of Squared Deviations
SP = ((X-[pic])(Y-[pic])
SSX=((X-[pic])2
SSy=((Y-[pic])2
Variance interpretation of r :
r 2 = % of variance in Y explained by its linear
relationship with X (and vice versa)
r 2 = “Coefficient of determination”
% of shared variance between X & Y
% of variance in Y predicted by X
III. Factors that affect the size of r
6. r ( 0 could mean many things:
• No relationship at all between X & Y
Non-linear relationship between X & Y
• Restricted range on X and/or Y
• Outlier may be causing problem
• Non-linear relationships
Curvilinear relationship
• Restricted range
Low variability on X and/or Y
• Outliers
Extreme value on X and/or Y
Examples of how restricted range can distort a correlation
Example of how an outlier can distort a correlation
IV. Correlation vs. Causality:
• Correlation tells you two variables are related
• Does NOT tell you why!!
• Do not draw causal inferences from a correlation
X ( Y
Y ( X
example:
r = -.30 #friends, depression
r = +.40 hours studying, grades
• Causal inferences require an “experiment”
V. Other Correlation Coefficients
Pearson r used when X & Y are at least interval level
Many types of correlation coefficients for other data
Spearman ( ordinal (rank) data
Point-biserial ( nominal X, interval/ratio Y
Phi ( nominal X & Y
-----------------------
Y
(Score)
X (Hours Studying)
No Linear Relationship
Perfect Positive Relationship
Perfect Negative Relationship
X
Y
Z
Third variable problem
Figure 16-3 (p. 524)
Examples of positive and negative relationships. (a) Beer sales are positively related to temperature. (b) Coffee sales are negatively related to temperature.
Figure 16-5 (p. 525)
Examples of different values for linear correlations: (a) shows a strong positive relationship, approx +.90; (b) shows a relatively weak negative correlation, approx –.40; (c) shows a perfect negative correlation, correlation = –1.0; (d) shows no linear trend, correlation = 0.0.
r = .90
r = -.40
r = -1.0
r = .00
(a) In this example, the full range of X and Y values shows a strong, positive correlation, but the restricted range of scores produces a correlation near zero.
(b) An example in which the full range of X and Y values shows a correlation near zero, but the scores in the restricted range produce a strong, positive correlation.
A demonstration of how one extreme data point (an outlier) can influence the value of a correlation.
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