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The difference between the Pearson correlation and the Spearman correlation is that the Pearson is most appropriate for measurements taken from an interval scale, while the Spearman is more appropriate for measurements taken from ordinal scales. Examples of interval scales include "temperature in Fahrenheit" and "length in inches", in which the ...
Loosely speaking, cross-correlation is a generalization of the Pearson's correlation. Specifically, when comparing two time series, cross-correlation seeks to obtain a relationship between lags of each series.
The correlation gives you a bounded measurement that can be interpreted independently of the scale of the two variables. The closer the estimated correlation is to $\pm 1$, the closer the two are to a perfect linear relationship. The regression slope, in isolation, does not tell you that piece of information.
For the correlation on the first dataset I get a Pearson of -0.74 and for the second I get -0.885. Now I want to find out whether these coefficients are significantly different from each other. Now I want to find out whether these coefficients are significantly different from each other.
Pearson's correlation is a measure of the linear relationship between two continuous random variables. It does not assume normality although it does assume finite variances and finite covariance. When the variables are bivariate normal, Pearson's correlation provides a complete description of the association.
The complete proof of how to derive the coefficient of determination R2 from the Squared Pearson Correlation Coefficient between the observed values yi and the fitted values y^i can be found under the following link:
While correlation typically refers to Pearson's correlation coefficient, there are other types of correlation, such as Spearman's. The correlation between X and Y is the same as the correlation between Y and X. In contrast, the unstandardized coefficient typically changes when moving from a model predicting Y from X to a model predicting X from Y.
Specifically, suppose that you think the two dichotomous variables (X,Y) are generated by underlying latent continuous variables (X*,Y*). Then it is possible to construct a sequence of examples where the underlying variables (X*,Y*) have the same Pearson correlation in each case, but the Pearson correlation between (X,Y) changes.
Binary & Continuous: Point-biserial correlation coefficient -- a special case of Pearson's correlation coefficient, which measures the linear relationship's strength and direction. Library: SciPy (pointbiserialr)
In this case, Pearson correlation will underestimate the true linear relationship between the two latent traits, especially in the mid-range of the correlation metric. On the other hand, when the cutoffs are clearly asymmetrical on both continuous variables, the tetrachoric correlation will generally overestimate the true relationship.