Correlation coefficient

A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables.^[1] The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution.

Several types of correlation coefficient exist, each with their own definition and own range of usability and characteristics. They all assume values in the range from −1 to +1, where +1 indicates the strongest possible agreement and −1 the strongest possible disagreement. As tools of analysis, correlation coefficients present certain problems, including the propensity of some types to be distorted by outliers and the possibility of incorrectly being used to infer a causal relationship between the variables.^[2]

Types

Pearson

The Pearson product-moment correlation coefficient, also known as r, R, or Pearson's r, is a measure of the strength and direction of the linear relationship between two variables that is defined as the covariance of the variables divided by the product of their standard deviations. This is the best known and most commonly used type of correlation coefficient; when the term "correlation coefficient" is used without further qualification, it usually refers to the Pearson product-moment correlation coefficient

Intra-class

Intraclass correlation (ICC) is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups; it describes how strongly units in the same group resemble each other.

Rank

Rank correlation is a measure of the relationship between the rankings of two variables or two rankings of the same variable:

Spearman's rank correlation coefficient is a measure of how well the relationship between two variables can be described by a monotonic function.
The Kendall tau rank correlation coefficient is a measure of the portion of ranks that match between two data sets.
Goodman and Kruskal's gamma is a measure of the strength of association of the cross tabulated data when both variables are measured at the ordinal level.

References

↑ "correlation coefficient". NCME.org. National Council on Measurement in Education. Retrieved April 17, 2014. correlation coefficient: A statistic used to show how the scores from one measure relate to scores on a second measure for the same group of individuals. A high value (approaching +1.00) is a strong direct relationship, values near 0.50 are considered moderate and values below 0.30 are considered to show weak relationship. A low negative value (approaching -1.00) is similarly a strong inverse relationship, and values near 0.00 indicate little, if any, relationship.
↑ Boddy, Richard; Smith, Gordon (2009). Statistical methods in practice: for scientists and technologists. Chichester, U.K.: Wiley. pp. 95–96. ISBN 978-0-470-74664-6.

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[1] "correlation coefficient". NCME.org. National Council on Measurement in Education. Retrieved April 17, 2014. correlation coefficient: A statistic used to show how the scores from one measure relate to scores on a second measure for the same group of individuals. A high value (approaching +1.00) is a strong direct relationship, values near 0.50 are considered moderate and values below 0.30 are considered to show weak relationship. A low negative value (approaching -1.00) is similarly a strong inverse relationship, and values near 0.00 indicate little, if any, relationship.

[Boddy-2] Boddy, Richard; Smith, Gordon (2009). Statistical methods in practice: for scientists and technologists. Chichester, U.K.: Wiley. pp. 95–96. ISBN 978-0-470-74664-6.