Biostatistics I: Descriptive Statistics

Correlation

Eleni-Rosalina Andrinopoulou
Department of Biostatistics, Erasmus Medical Center

e.andrinopoulou@erasmusmc.nl

7@erandrinopoulou
In this Section

▶ Correlation coefficients
▶ Examples

1
 Correlation

Correlation is a measure that describes the strength of the association

between two variables. Let’s assume that we have two continuous
variables, we can get the following relationships:

Positive correlation Negative correlation No correlation

2
2

1
1

1
Variable 2

Variable 2
Variable 2
0

0
0
−1

−1
−1
−2

−2
−2

−2 −1 0 1 2 −2 −1 0 1 2 −2 −1 0 1 2

Variable 1 Variable 1 Variable 1

2
Pearson Correlation

▶ magnitude of association
▶ linear association
▶ direction of the relationship

A relationship is linear when a change in one variable is associated with a

proportional change in the other variable
cov(X,Y)
Pearson Correlation: corr(X, Y) = sd(X)sd(Y) ,
where cov(X, Y) is the covariance and sd(X), sd(Y) are the standard
deviations

3
Spearman Correlation
▶ direction of the relationship
▶ monotonic relationship

In a monotonic relationship, the variables tend to change together, but

not always at a constant rate (as in the linear case)
The Spearman correlation coefficient is based on the ranked values:
cov(RX ,RY )
corrR (X, Y) = sd(R X )sd(RY )

What is rank?
Ranks are integers indicating the rank of some values. E.g. the rank of 3,
10, 16, 6, 2 is 2, 4, 5, 3, 1:

rank(c(3, 10, 16, 6, 2))

[1] 2 4 5 3 1
4
 Difference between Pearson and Spearman
Weak positive correlation: Strong positive correlation:
Pearson = 0.5 Spearman = 0.51 Pearson = 0.9 Spearman = 0.8
2

2
1

1
Variable 2

Variable 2
0

0
−1

−1
−2

−2
−2 −1 0 1 2 −2 −1 0 1 2

Variable 1 Variable 1

5
 Difference between Pearson and Spearman
What if there is a correlation but
Linear VS monotonic relationship:
this is not linear?
Pearson = −0.85 Spearman = −0.99
Pearson = 0.06 Spearman = 0.04
50

50
45
40

40
Variable 2

30

Variable 2

35
20

30
10

25
0

20
5 10 15 20
0 5 10 15 20 25 30
Variable 1
Variable 1
6
Other Correlation Measures

▶ Point-Biserial: It evaluates the association between a continuous

variable with a categorical (dichotomous) variable
▶ Intraclass: It evaluates the association between two continuous
variables that are structured in groups

Note
▶ Correlation must not be confused with causality
▶ If two variables are correlated, it does not imply that one variable
causes the changes in another variable