Correlation coefficient
A correlation coefficient is a numerical measure of some type of linear correlation, meaning a statistical relationship between two variables.[a] 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.[citation needed]
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 correlation and 0 indicates no correlation.[2] 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 (for more, see Correlation does not imply causation).[3]
Types
[edit]There are several different measures for the degree of correlation in data, depending on the kind of data: principally whether the data is a measurement, ordinal, or categorical.
Pearson
[edit]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.[4] 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
[edit]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
[edit]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.
Tetrachoric and polychoric
[edit]The polychoric correlation coefficient measures association between two ordered-categorical variables. It's technically defined as the estimate of the Pearson correlation coefficient one would obtain if:
- The two variables were measured on a continuous scale, instead of as ordered-category variables.
- The two continuous variables followed a bivariate normal distribution.
When both variables are dichotomous instead of ordered-categorical, the polychoric correlation coefficient is called the tetrachoric correlation coefficient.
Interpreting correlation coefficient values
[edit]Correlation between two variables have different associations that are measured in values such as r or R. Correlation values range from −1 to +1, where ±1 indicates the strongest possible correlation and 0 indicates no correlation between variables.[5]
| r or R | r or R | Strength or weakness of association between variables[6] |
|---|---|---|
| +1.0 to +0.8 | -1.0 to -0.8 | Perfect or very strong association |
| +0.8 to +0.6 | -0.8 to -0.6 | Strong association |
| +0.6 to +0.4 | -0.6 to -0.4 | Moderate association |
| +0.4 to +0.2 | -0.4 to -0.2 | Weak association |
| +0.2 to 0.0 | -0.2 to 0.0 | Very weak or no association |
See also
[edit]- Correlation disattenuation
- Coefficient of determination
- Correlation and dependence
- Correlation ratio
- Distance correlation
- Goodness of fit, any of several measures that measure how well a statistical model fits observations by summarizing the discrepancy between observed values and the values expected under the model
- Multiple correlation
- Partial correlation
Notes
[edit]- ^ 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.[1]
References
[edit]- ^ "correlation coefficient". NCME.org. National Council on Measurement in Education. Archived from the original on July 22, 2017. Retrieved April 17, 2014.
- ^ Taylor, John R. (1997). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements (PDF) (2nd ed.). Sausalito, CA: University Science Books. p. 217. ISBN 0-935702-75-X. Archived from the original (PDF) on 15 February 2019. Retrieved 14 February 2019.
- ^ 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.
- ^ Weisstein, Eric W. "Statistical Correlation". mathworld.wolfram.com. Retrieved 2020-08-22.
- ^ Taylor, John R. (1997). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements (PDF) (2nd ed.). Sausalito, CA: University Science Books. p. 217. ISBN 0-935702-75-X. Archived from the original (PDF) on 15 February 2019. Retrieved 14 February 2019.
- ^ "The Correlation Coefficient (r)". Boston University.
