Correlation Coefficient

The correlation coefficient, or coefficient of correlation, typically denoted by the letter r, evaluates the similarity of two sets of measurements (i.e., two dependent variables) obtained from the same sample. To do so, the coefficient of correlation quantifies the amount of information, or shared variance, common to the two variables.

The idea of correlation is rather old, but the modern approach and definition of the coefficient of correlation was initiated by Francis Galton (in an evolutionary context) in the end of the 19th century and formalized by Karl Pearson in the early 20th century. The sampling distribution of this coefficient was mostly derived a few years later by Ronald Fisher, and this was the source of a lifelong enmity between these two giants of statistics.

The correlation coefficient takes values between −1 and +1 (inclusive). A value of +1 shows that the two series of measurements are measuring the same thing, whereas a value of −1 indicates that the two measurements are still measuring the same thing, but one measurement varies inversely to the other. A value of 0 indicates that the two series of measurements have nothing in common. It is important to note that the coefficient of correlation measures only the linear relationship between two variables and that its value is very sensitive to outliers.

The squared correlation gives the proportion of common variance between two variables and is also called the coefficient of determination. Subtracting the coefficient of determination from unity gives the proportion of variance not shared between two variables. This quantity is called the coefficient of alienation.

The significance of the coefficient of correlation can be tested with an F or a t test. This entry presents three different approaches that can be used to obtain p values: (1) the classical approach, which relies on Fisher’s F distributions; (2) the Monte Carlo approach, which relies on computer simulations to derive empirical approximations of sampling distributions; and (3) the nonparametric permutation (also known as randomization) test, which evaluates the likelihood of the actual data against the set of all possible configurations of these data. In addition to p values, confidence intervals can be computed using Fisher’s Z transform or the more modern, computationally based nonparametric Efron’s bootstrap.

The coefficient of correlation always overestimates the intensity of the correlation in the population and needs to be corrected in order to provide a better estimation. The corrected value is called shrunken or adjusted. This entry also presents variations of the [Page 317]correlation coefficients (often called nonparametric measures of correlation) that can be used with ordinal or nominal data.