(Dis)Advantages of correlation vs. $R^2$ vs. p-value of linear regression for two variables? I would like to know what are advantages and disadvantages of $R^2$ vs. correlation (e.g. cor() in R) vs. p-value of linear regression for two variables/features? What other ways exist to measure whether two variables/features correlate?
 A: 1) Note: R-squared is simply the square of Pearson's correlation coefficient
Disadvantages: 
1) R^2 and r are only appropriate for linear relationships, so if there is a nonlinear relationship then, generally speaking although not always, r will fail to detect the relationship (you can test this by generating fake data for two variables and calculating the correlation)
2) The use of r also assumes that there is a bivariate normal distribution between the two variables (this is condition is often violated)
P-values vs. R^2:
--> there is a distinction between whether a variable(s) is important versus whether it is statistically significant
An "important" variable means that a variable explains a (relatively large) proportion of the variance in Y, and you can measure importance by looking at the change in the R^2 value for each variable that you add to the model
A statistically significant variable is a variable that has a p-value that is below a certain threshold (commonly this threshold is .05).
The problem with p-values are that they are sensitive to the sample size (please review the concept of "power"). Generally speaking, as the sample size increases, the p-value will decrease (in order to see this, please review the formula for the t-test statistic that is used in order to generate the p-value)
Alternatives:
Spearman's Rank Correlation Coefficient (Advantage: non-parametric, less
sensitive to outliers)
Hoeffding's D (Advantage: non-parametric, can detect some nonlinear relationships) 
Kendall's Tau (Advantage: its distribution will converge to a normal distribution faster than other correlation metrics, so this is useful when the sample size is small)
