Correlation Analysis


In correlation analysis, we estimate a sample correlation coefficient, more specifically the Pearson Product Moment correlation coefficient. The sample correlation coefficient, denoted r,

ranges between -1 and +1 and quantifies the direction and strength of the linear association between the two variables. The correlation between two variables can be positive (i.e., higher levels of one variable are associated with higher levels of the other) or negative (i.e., higher levels of one variable are associated with lower levels of the other).

The sign of the correlation coefficient indicates the direction of the association. The magnitude of the correlation coefficient indicates the strength of the association.

For example, a correlation of r = 0.9 suggests a strong, positive association between two variables, whereas a correlation of r = -0.2 suggest a weak, negative association. A correlation close to zero suggests no linear association between two continuous variables.

It is important to note that there may be a non-linear association between two continuous variables, but computation of a correlation coefficient does not detect this. Therefore, it is always important to evaluate the data carefully before computing a correlation coefficient. Graphical displays are particularly useful to explore associations between variables.

The figure below shows four hypothetical scenarios in which one continuous variable is plotted along the X-axis and the other along the Y-axis.

Four Correlation Graphs.png

 

Thinking man icon signifying a problem for the student to solve

 

A study of a random sample of 100 Americans summarizes the relationship between alcohol consumption and age with a correlation coefficient r= 0.03. The value of r tells us:

 
 
 
 

 

Example - Correlation of Gestational Age and Birth Weight

A small study is conducted involving 17 infants to investigate the association between gestational age at birth, measured in weeks, and birth weight, measured in grams.

Infant ID #

Gestational Age (weeks)

Birth Weight (grams)

1

34.7

1895

2

36.0

2030

3

29.3

1440

4

40.1

2835

5

35.7

3090

6

42.4

3827

7

40.3

3260

8

37.3

2690

9

40.9

3285

10

38.3

2920

11

38.5

3430

12

41.4

3657

13

39.7

3685

14

39.7

3345

15

41.1

3260

16

38.0

2680

17

38.7

2005

We wish to estimate the association between gestational age and infant birth weight. In this example, birth weight is the dependent variable and gestational age is the independent variable. Thus y=birth weight and x=gestational age. The data are displayed in a scatter diagram in the figure below.

Scatter graph of birth weight as a function of gestationa age

Each point represents an (x,y) pair (in this case the gestational age, measured in weeks, and the birth weight, measured in grams). Note that the independent variable, gestational age) is on the horizontal axis (or X-axis), and the dependent variable (birth weight) is on the vertical axis (or Y-axis). The scatter plot shows a positive or direct association between gestational age and birth weight. Infants with shorter gestational ages are more likely to be born with lower weights and infants with longer gestational ages are more likely to be born with higher weights.