Correlation
Let’s say you want to test the
association between cortisol levels in
the blood and hours per week
studying statistics
Use Pearson’s correlation
Pearson correlation coefficient
Used to test for linear associations between two
continuous, (normally distributed) variables
Unitless
Values range from – 1 to + 1



0 indicates no linear correlation
+ 1 indicates perfect positive linear correlation
– 1 indicates perfect negative linear correlation
Negative association
Stronger
-1
Positive association
Weaker
Weaker
0
No association:
Value under H0
Stronger
+1
Same line, difference correlation
r = 0.985
r = 0.667
How Pearson correlation works
1.
2.
3.
Establish alpha (say, 0.05).
Start with a null hypothesis.
H0: There is no linear association between cortisol
levels and time spent in the wards. ρxy = 0
3. Compute a test statistic, called Pearson’s r.
Final steps for Pearson correlation
4. Compare rxy to a known distribution of Pearson
correlation coefficients to obtain a p-value.
5. Make a decision about rejecting H0.

As usual, if p > α, we do not reject H0; if p < α, we
reject H0.
Source: http://www.radford.edu/~jaspelme/statsbook/Chapter%20files/Table_of_Critical_Values_for_r.pdf
Stressed medical students example
Establish alpha: α = 0.05.
Write your null hypothesis:
2.
-
3.
There is no association between average number of hours per
week spent at the wards and cortisol levels. (ρxy = 0)
Compute rxy, the test statistic.
rxy = 0.736
22
8AM cortisol level (mcg/dL)
1.
20
18
16
14
12
45
50
55
60
65
Average hours per week in wards
70
75
Last steps
4. Compare rxy to a known distribution of r.
(degrees of freedom = n – 2)
rxy = 0.736
5. Make a decision about H0:
Since p > α, we do not reject H0.
Correlation coefficient interpretations
rxy
rxy
=1
=-1
≈ 0.8
≈ - 0.8
≈ 0.5
≈ - 0.5
≈0
≈ - 0.2
Caveat #1: Slope of the line
The slope of the best-fit line does not dictate the
strength of the association
Only the relative distance of the data points from the
best-fit determines the association
rxy = 1 for all
Caveat #2: Must be a linear association
Pearson’s r measures the strength of the linear
association between two continuous variables
Some variables may be related to each other, but not
linearly
rxy = 0
for all
Some associations may be positive or negative, but not
linearly related
Caveat #3: Outliers
Outliers often distort the linear association
rxy = 0.80
rxy = 0.88
rxy = 0.54