1. Use Pearson’s 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

2. 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
Positive association -1 +1
Stronger Weaker
Weaker Stronger
No association:
Value under H0

3. Same line, difference correlation

4. How Pearson correlation works
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.

5. 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:

6. Stressed medical students example
Establish alpha: α = 0.05.
Write your null hypothesis:
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

7. (degrees of freedom = n – 2)
Last steps
4. Compare rxy to a known distribution of r.
(degrees of freedom = n – 2)
5. Make a decision about H0:
Since p > α, we do not reject H0.
rxy =

8. Correlation coefficient interpretations
rxy
rxy
= 1
= - 1
≈ 0.8
≈ - 0.8
≈ 0.5
≈ - 0.5
≈ 0
≈ - 0.2

9. 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

10. 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
Some associations may be positive or negative, but not linearly related
rxy = 0 for all

11. Caveat #3: Outliers rxy = 0.80 rxy = 0.88 rxy = 0.54
Outliers often distort the linear association
rxy = 0.80
rxy = 0.88
rxy = 0.54