What does a correlation coefficient of indicate about the relationship between two variables?

Correlation analysis measures how two variables are related. Thecorrelation coefficient (r) is a statistic that tells you the strengthand direction of that relationship. It is expressed as a positive ornegative number between -1 and 1. The value of the number indicates the strengthof the relationship:

• r = 0 means there is no correlation
• r = 1 means there is perfect positive correlation
• r = -1 means there is a perfect negative correlation

The sign of the correlation coefficient indicates whether the direction ofthe relationship is positive (direct) or negative (inverse).

Variables whichhave a direct relationship (a positive correlation) increase together and decrease together.

In aninverse relationship (a negative correlation), one variable increases while the other decreases.

While the sign indivates how one variable changes with respect to anothervariable, the magnitude of the number indicates the strength of a relationship.

It is important to remember that while correlation coefficients can be usedfor prediction (i.e. if we know the value for one variable, and thecorrelation, we can predict what the value of the second variable will be) theymay NOT be used for causation (i.e. we cannot say that one variable causesanother).

Example

Suppose you are reading a study of Regents exams. The investigator wantedto know if performance in grade school was related to scores on the Regentsexams. He did a correlation analysis on grade school performance and Regentsexam score, and found that r = .75 in his study. This tells you two things:

1. r is positive, so grade school performance and Regents exam score tendto increase and decrease together.
2. r is fairly close to 1, so the direct relationship is fairly strong.

If a correlation exists between two variables, this does NOT imply that onevariable causes another. Causation and correlation are two very differentthings.

The two correlation coefficients that appear most often in the literatureare the Pearson-product moment and the Spearmanrank sum.

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.

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.

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.

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