Pearsons Product Moment Correlation coefficient r

Francis Galton, in his empirical analysis of the dependence of sweetpea seed weights and plant heights on the properties of the parents, laid down the 

The r value is commonly employed to quantify the degree of association between predicted values (from either a physics-based or empirical model) and observed values from eqn (9.1). The endpoints could be as diverse as estimates of affinity from 3-dimensional protein-ligand complexes, to estimates of solubility from a quantitative structure-activity model. The coefficient of variation (r ) expresses the fraction of the variation in the observed values that is explained by the predicted values, or more generally the fraction of the variation in the y-data that is explained by the x-data. 

The value of r or r is often used to compare the performance of different models or modelling methods. However, it is not widely recognised that the 

Even though the two sub-datasets are drawn from the same overall population, the r values for the sub-correlations are lower than r value for the overall correlation, as the standard deviations for the sub-datasets are lower than the population standard deviation. The r value can be a useful comparative statistic if comparing the performance of dilferent modelling approaches on a common dataset. 

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For continuous data, the Pearson product moment correlation coefficient, r, is calculated. Since continuous data are used here, certain fundamental assumptions... 

Correlation coefficients are used to look for relationships between two variables, and the most common correlation coefficient used is the Pearson product-moment correlation coefficient (r). When calculating correlation coefficients, the two variables must be at the interval or ratio level (2), which means that correlation coefficients cannot be used with category data that are dichotomous (mutually exclusive) and non-numerical (like animation/non-animation group, male/female, single/married/divorced, etc.). Values for the Pearson r vary from -1 to +1. Negative r-values imply negative correlations (as one variable increases, the other decreases) while positive r-values imply positive correlations (as one variable increases, so does the other and vice versa) r-values of 0 imply no relationship between the two variables. It is important to note that Pearson r-values assume linear relationships between the two variables if non-linear relationships are expected or observed, correlation ratios rj) that recognize non-linear relationships can be calculated (10). 

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