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.
- Scenario 1 depicts a strong positive association (r=0.9), similar to what we might see for the correlation between infant birth weight and birth length.
- Scenario 2 depicts a weaker association (r=0,2) that we might expect to see between age and body mass index (which tends to increase with age).
- Scenario 3 might depict the lack of association (r approximately = 0) between the extent of media exposure in adolescence and age at which adolescents initiate sexual activity.
- Scenario 4 might depict the strong negative association (r= -0.9) generally observed between the number of hours of aerobic exercise per week and percent body fat.
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.
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.
The correlation requires two scores from the same individuals. These scores are normally identified as X and Y. The pairs of scores can be listed in a table or presented in a scatterplot.
Example: We might be interested in the correlation between your SAT-M scores and your GPA at UNC.
Here are the Math SAT scores and the GPA scores of 13 of the students in this class, and the scatterplot for all 41 students:
The scatterplot has the X values (GPA) on the horizontal (X) axis, and the Y values (MathSAT) on the vertical (Y) axis. Each individual is identified by a single point (dot) on the graph which is located so that the coordinates of the point (the X and Y values) match the individual's X (GPA) and Y (MathSAT) scores.
For example, the student named "Obs5" (in the sixth row of the datasheet) has GPA=2.30 and MathSAT=710. This student is represented in the scatterplot by high-lighted and labled ("5") dot in the upper-left part of the scatterplot. Note that is to the right of MathSAT of 710 and above GPA of 2.30.
Note that the Pearson correlation (explained below) between these two variables is .32.
Correlations have three important characterstics. They can tell us about the direction of the relationship, the form (shape) of the relationship, and the degree (strength) of the relationship between two variables.- The Direction of a Relationship The correlation measure tells us about the direction of the relationship between the two variables. The direction can be positive or negative.
- Positive : In a positive relationship both variables tend to move in the same direction: If one variable increases, the other tends to also increase. If one decreases, the other tends to also.
- Negative : In a negative relationship the variables tend to move in the opposite directions: If one variable increases, the other tends to decrease, and vice-versa.
- The Form (Shape) of a Relationship : The form or shape of a relationship refers to whether the relationship is straight or curved.
- Linear : A straight relationship is called linear, because it approximates a straight line. The GPA, MathSAT example shows a relationship that is, roughly, a linear relationship.
- Curvilinear : A curved relationship is called curvilinear, because it approximates a curved line. An example of the relationship between the Miles-per-gallon and engine displacement of various automobiles sold in the USA in 1982 is shown below. This is curvilinear (and negative).
- The Degree (Strength) of a Relationship
- Perfect Relationship : When two variables are exactly (linearly) related the correlation coefficient is either +1.00 or -1.00. They are said to be perfectly linearly related, either positively or negatively.
- No relationship : When two variables have no relationship at all, their correlation is 0.00.
In the example above, GPA and MathSAT are positively related. As GPA (or MathSAT) increases, the other variable also tends to increase.
The direction of the relationship between two variables is identified by the sign of the correlation coefficient for the variables. Postive relationships have a "plus" sign, whereas negative relationships have a "minus" sign.
In this course we only deal with correlation coefficients that measure linear relationship. There are other correlation coefficients that measure curvilinear relationship, but they are beyond the introductory level.
Finally, a correlation coefficient measures the degree (strength) of the relationship between two variables. The mesures we discuss only measure the strength of the linear relationship between two variables. Two specific strengths are:
There are strengths in between -1.00, 0.00 and +1.00. Note, though. that +1.00 is the largest postive correlation and -1.00 is the largest negative correlation that is possible. Here are three examples:
Weight and Horsepower
The relationship between Weight and Horsepower is strong, linear, and positive, though not perfect. The Pearson correlation coefficient is +.92.
Drive Ratio and Horsepower
The relationship between drive ratio and Horsepower is weekly negative, though not zero. The Pearson correlation coefficient is -.59.
Drive Ratio and Miles-Per-Gallon
The relationship between drive ratio and MPG is weekly positive, though not zero. The Pearson correlation coefficient is .42.
- Prediction : Correlations can be used to help make predictions. If two variables have been known in the past to correlate, then we can assume they will continue to correlate in the future. We can use the value of one variable that is known now to predict the value that the other variable will take on in the future.
For example, we require high school students to take the SAT exam because we know that in the past SAT scores correlated well with the GPA scores that the students get when they are in college. Thus, we predict high SAT scores will lead to high GPA scores, and conversely.