Linearity Assumption: the correlation coefficient requires that the underlying relationship between the two variables under consideration is linear. If the relationship is known to be linear, or the observed pattern between the two variables appears to be linear, then the correlation coefficient provides a reliable measure of the strength of the linear relationship. If the relationship is known to be non-linear, or the observed pattern appears to be non-linear, then the correlation coefficient is not useful, or at least questionable.
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The calculation of the correlation coefficient for two variables, say X and Y, is simple to understand. Let zX and zY be the standardised versions of X and Y, respectively, that is, zX and zY are both re-expressed to have means equal to 0 and standard deviations (s.d.) equal to 1. The re-expressions used to obtain the standardised scores are in equations (1) and (2):
The adjusted correlation coefficient is obtained by dividing the original correlation coefficient by the rematched correlation coefficient, whose sign is that of the sign of original correlation coefficient. The sign of adjusted correlation coefficient is the sign of original correlation coefficient. If the sign of the original r is negative, then the sign of the adjusted r is negative, even though the arithmetic of dividing two negative numbers yields a positive number. The expression in (4) provides only the numerical value of the adjusted correlation coefficient. In this example, the adjusted correlation coefficient between X and Y is defined in expression (4): the original correlation coefficient with a positive sign is divided by the positive-rematched original correlation.
(a) Scatter plots of associated (but not correlated), non-associated and correlated variables. In the lower association example, variance in y is increasing with x. (b) The Pearson correlation coefficient (r, black) measures linear trends, and the Spearman correlation coefficient (s, red) measures increasing or decreasing trends. (c) Very different data sets may have similar r values. Descriptors such as curvature or the presence of outliers can be more specific.
For quantitative and ordinal data, there are two primary measures of correlation: Pearson's correlation (r), which measures linear trends, and Spearman's (rank) correlation (s), which measures increasing and decreasing trends that are not necessarily linear (Fig. 1b). Like other statistics, these have population values, usually referred to as ρ. There are other measures of association that are also referred to as correlation coefficients, but which might not measure trends.
(a) Distribution (left) and 95% confidence intervals (right) of correlation coefficients of 10,000 n = 10 samples of two independent normally distributed variables. Statistically significant coefficients (α = 0.05) and corresponding intervals that do not include r = 0 are highlighted in blue. (b) Samples with the three largest and smallest correlation coefficients (statistically significant) from a.
Because P depends on both r and the sample size, it should never be used as a measure of the strength of the association. It is possible for a smaller r, whose magnitude can be interpreted as the estimated effect size, to be associated with a smaller P merely because of a large sample size3. Statistical significance of a correlation coefficient does not imply substantive and biologically relevant significance.
The Pearson correlation coefficient can also be used to quantify how much fluctuation in one variable can be explained by its correlation with another variable. A previous discussion about analysis of variance4 showed that the effect of a factor on the response variable can be described as explaining the variation in the response; the response varied, and once the factor was accounted for, the variation decreased. The squared Pearson correlation coefficient r2 has a similar role: it is the proportion of variation in Y explained by X (and vice versa). For example, r = 0.05 means that only 0.25% of the variance of Y is explained by X (and vice versa), and r = 0.9 means that 81% of the variance of Y is explained by X. This interpretation is helpful in assessments of the biological importance of the magnitude of r when it is statistically significant.
The polychoric correlation coefficient is a measure of association for ordinal variables which rests upon an assumption of an underlying joint continuous distribution. More specifically, in Karl Pearson\u2019s original definition an underlying joint normal distribution is assumed. In this article, the definition of the polychoric correlation coefficient is generalized so that it allows for other distributional assumptions than the joint normal distribution. The generalized definition is analogous to Pearson\u2019s definition, and the two definitions agree under bivariate normal distributions. Moreover, the polychoric correlation coefficient is put into a framework of copulas which is both mathematically and practically convenient. The theory is illustrated with examples which, among other things, show that the measure of association suffers from lack of statistical robustness. 2ff7e9595c
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