I am looking into computing a correlation between two variables, x and y, each having a general normal distribution and not i.i.d. normal distribution. In particular these two variables have a particular variance-covariance structure, i.e., Var(x)=A and Var(y)=B. Is it still OK to compute correlation between x and y as shown bellow or should I use any other approach?

Cor(x, y) = Cov(x, y) / sqrt(Var(x)*Vary)

In detail I have a set of simulated/true values stored in the variable x that has a particular structure due to the nature of values - there is some underlying data generation process giving rise to variance-covariance structure Var(x)=A. I then try to estimate these values based on some data which gives me variable y. Due to the nature of x, there is also an underlying structure in y that can be described by Var(y)=B. A and B are not the same due to conditioning on the observed data. I want to evaluate Cor(x, y) to validate the quality of estimation and could use a simple standard approach, but I wonder if I am making a mistake here by treating x and y as iid observations.


1 Answer 1


It is reasonable to calculate the correlation coefficient when the two variables are not i.i.d.

Indeed, if they were independent, the correlation between the two popopulations would be $0$ and you would expect the calculated sample correlation to be close to $0$. So getting a result significantly different from $0$ gives you some potentially useful information.

The problem, if there is one, is that there is a lot of information that the calculated correlation does not give you. The following charts (both from Wikipedia) might give you something to think about. (In the second with orange dots, called Anscombe's quartet, the correlation coefficient in each of the four charts is just over 0.81)




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