What is the uncertainty on the (Pearson) correlation coefficient?

Question

Do you know what is the uncertainty on the Pearson correlation coefficient as a function of the uncertainty on the measurement in the data set.

I know of an expression giving the uncertainty related to the limited size of the data set, but I'm looking for the uncertainty related to the measurement of the data itself, which I think must dominate.

Many thanks!

Answer 1

Suppose that the variables $X,X'$ are fixed with means $\mu, \mu'$, standard deviations $\sigma, \sigma'$, and correlation $\rho$. Suppose they are observed with errors $Z,Z'$ that are normally distributed with mean 0 and standard deviations $\epsilon, \epsilon'$. Then to a first approximation:

\begin{align} Var[\text{observed }\rho] &=Var\left[\frac{n \sum (X+Z)(X'+Z') - \sum(X+Z)\sum(X'+Z')}{n^2 SD[X+Z] SD[X+Z']}\right] \\ \\ &\simeq Var\left[\frac{n \sum (X+Z)(X'+Z') - n^2 \mu \mu'}{n^2 \sqrt{(\sigma^2+\epsilon^2)(\sigma'^2+\epsilon'^2)}}\right]\\ \\ &=\frac{Var\left[\Sigma (X+Z)(X'+Z')\right]}{n^2 (\sigma^2+\epsilon^2)(\sigma'^2+\epsilon'^2)}\\ \\ &=\frac{E\left[\big(\Sigma (XX'+XZ'+ZX'+ZZ')\big)^2\right]-n^2\mu^2 \mu'^2}{n^2 (\sigma^2+\epsilon^2)(\sigma'^2+\epsilon'^2)}\\ \\ &=\frac{E\left[\Sigma (X^2X'^2+X^2Z'^2+Z^2X'^2+Z^2Z'^2)\right]-n^2\mu^2 \mu'^2}{n^2 (\sigma^2+\epsilon^2)(\sigma'^2+\epsilon'^2)}\\ \\ &=\frac{(\mu^2+\sigma^2+\epsilon^2)(\mu'^2+\sigma'^2+\epsilon'^2)+2\rho\sigma\sigma'(2\mu\mu'+\rho\sigma\sigma')-\mu^2 \mu'^2}{(\sigma^2+\epsilon^2)(\sigma'^2+\epsilon'^2)}\\ \end{align}

Answer 2

$\newcommand\tsi{\tilde\sigma} \newcommand\tY{\tilde Y} \newcommand\tZ{\tilde Z}$ Let $(Y_1,Z_1),\dots,(Y_n,Z_n)$ be iid copies of a pair $(Y,Z)$ of real-valued random variables (r.v.'s) with finite fourth moments and correlation $\rho\in(-1,1)$. Let $R=R_n$ be the Pearson correlation coefficient for the "random sample" $(Y_1,Z_1),\dots,(Y_n,Z_n)$. Then, by the multivariate delta method, the r.v. $$\frac{R_n-\rho}{\tsi/\sqrt n}$$ converges (as $n\to\infty$) in distribution to a standard normal r.v., where $\tsi:=\sqrt{\tsi^2}$, $$\tsi^2:=Var\big(\tY\tZ-\tfrac\rho2(\tY^2+\tZ^2)\big),$$ $$\tY:=\frac{Y-EY}{\sqrt{Var\,Y}},\quad \tZ:=\frac{Z-EZ}{\sqrt{Var\,Z}}.$$ Here it is assumed that $Var\,Y$, $Var\,Z$, and $\tsi$ are nonzero.

So, the asymptotic standard deviation $\tsi/\sqrt n$ of $R_n$ can serve as a natural measure of uncertainty of the values of $R_n$ due to the variation in the values of the random points $(Y_1,Z_1),\dots,(Y_n,Z_n)$ and their failure to lie on a straight line.

Details on all this can be found on page 1016 of this paper; see, in particular, Theorem 3.4 (and the paragraph preceding it) and Remark 3.5 on that page.

Theorems 3.4 and Corollary 3.8 in the linked paper also provide (optimal) $O(1/\sqrt n)$ bounds on the rate of the mentioned convergence of the distribution of $R_n$ to normality. See also Remark 3.2 in the linked paper concerning general results on asymptotic expansions of nonlinear functions of the multivariate sample mean due to Bhattacharya and Ghosh.