Measures of Association

Pearson's Correlation Coefficient

A typical example for measuring the association between two variables measured on an interval/ratio scale is the analysis of relationship between a person's height and weight. Each of these two characteristic variables is measured on a continuous scale. The appropriate measure of association for this situation is the Pearson's correlation coefficient.

The Pearson's correlation coefficient, r (rho), measures the strength of the linear relationship between the two variables measured on a continuous scale. The coefficient r takes on the values of −1through +1. Values of −1or +1 indicate a perfect linear relationship between the two variables, whereas a value of 0 indicates no linear relationship. Correlation coefficients that differ from 0 but are not +1or −1 indicate a linear relationship, although not a perfect linear relationship. Negative values simply indicate the direction of the association: As one variable increases, the other decreases. In practice, r (the population correlation coefficient) is estimated by r, the correlation coefficient derived from sample data.

Although the Pearson's correlation coefficient is a measure of the strength of an association (specifically the linear relationship), it is not a measure of the significance of the association. The significance of the association is a separate analysis of the sample correlation coefficient, r, using a t test to measure the difference between the observed r and the expected r under the null hypothesis.

Spearman Rank-Order Correlation Coefficient

The Spearman rank-order correlation coefficient (Spearman rho) is designed to measure the strength of a monotonic (in a constant direction) association between two variables measured on an ordinal or ranked scale. Examples that indicate the Spearman rho should be used to include data obtained on preferences where the data result from ranking. It is also appropriate for data collected on a scale that is not truly interval in nature, such as data obtained from Likert-scale administration. Any interval data may be transformed to ranks and analyzed with the Spearman rho, although this results in a loss of information; for instance, this may be done if one variable of interest is measured on an interval scale and the other is measured on an ordinal scale. Like the Pearson's correlation coefficient, the Spearman rho may be tested for its significance. A similar measure of strength of association is the Kendall tau, which may also be applied to measure the strength of a monotonic association between two variables measured on an ordinal or rank scale.

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