I would like to obtain the expectation and variance of the squared Pearson sample correlation ($\operatorname{E}(R_{lk}^2)$ and $V(R_{lk}^2)$) between two random variables $l$ and $k$ following a bivariate normal distribution.
I have been trying to trace back the results of a paper, stating that:
\begin{align}\operatorname{E}(\tilde{R}_{lk}^2)\approx R_{lk}^2 + \frac{(1 - \tilde{R}_{lk}^2)}{N}&\end{align} $\tag{Eq. 1a}$
and thus that:
\begin{align}\operatorname{E}(R_{lk}^2)\approx \tilde{R}_{lk}^2 - \frac{(1 - \tilde{R}_{lk}^2)}{N}&\end{align} $\tag{Eq. 1b}$
where $R_{lk}^2$ is the true squared correlation, $\tilde{R}_{lk}^2$ is the observed squared sample correlation and $N$ is the sample size. They claim that this result can be obtained with the $delta-method$. My understanding of the delta-method is that one can obtain an approximate expression for the Variance of a function of a random variable, such that:
\begin{align}Var(f(X))={({f'(\mu)})^2}Var(X)&\end{align} $\tag{Eq. 2a}$
Any idea how to get these results?
Derive Expected Squared Correlation Using Delta Method
Using the delta-method, I came out with this solution, but it is different from the first two equations I showed. In applying the delta-method, consider that $X=\tilde{R}$, $f(X)=\tilde{R}^2$, $f(\mu)=R_{lk}^2$ and $f'(\mu)=2R_{lk}$, where $R_{lk}$ is the true Pearson correlation.
\begin{align}Var(\tilde{R}_{lk}^2) = {({f'(R_{lk})})^2}Var(\tilde{R}_{lk})&\end{align} $\tag{Eq. 2b}$
Since $Var(\tilde{R}_{lk})$ can be interpreted as the the squared standard error of the sample correlation (Derivation of the standard error for Pearson's correlation coefficient):
\begin{align}Var(\tilde{R}_{lk}) = \frac{(1 - \tilde{R}_{lk}^2)}{N - 2}&\end{align} $\tag{Eq. 3}$
we do get:
\begin{align}Var(\tilde{R}_{lk}^2) = 4R_{lk}^2\frac{(1 - \tilde{R}_{lk}^2)}{N - 2}&\end{align} $\tag{Eq. 4}$
Noting that $Var(\tilde{R}_{lk}^2) = \operatorname{E}(\tilde{R}_{lk}^2)-\operatorname{E}(\tilde{R}_{lk})^2$, and the $\operatorname{E}(\tilde{R}_{lk})=0$, then we obtain:
\begin{align}\operatorname{E}(\tilde{R}_{lk}^2) = 4R_{lk}^2\frac{(1 - \tilde{R}_{lk}^2)}{N - 2}&\end{align} $\tag{Eq. 5a}$
Derive Expected Squared Correlation Using Taylor Expansion
Instead, if I use the Taylor expansion I do get the same results as the paper. The general formula for the Taylor expansion applied to this case is (as in Expected value of inverse?):
\begin{align}\operatorname{E}(\tilde{R}_{lk}^2) = R_{lk}^2+2 \frac{1}{2} Var(\tilde{R_{lk}})&\end{align}$\tag{Eq. 6}$
and again, using the standard error $Var(\tilde{R}_{lk})$ as expressed above, we get:
\begin{align}\operatorname{E}(\tilde{R}_{lk}^2) = R_{lk}^2+2 \frac{1}{2} \frac{(1 - \tilde{R}_{lk}^2)}{N - 2}&\end{align} $\tag{Eq. 6a}$
which by rearranging we get:
\begin{align}\operatorname{E}(R_{lk}^2) = \tilde{R}_{lk}^2-\frac{(1 - \tilde{R}_{lk}^2)}{N - 2}&\end{align} $\tag{Eq. 6b}$
Does it seem correct to you? Note that I do get the same result if I use the law of total variance (not shown here).
Derive the Variance of the Squared Correlation Using Delta Method
Regarding the variance of the sample squared correlation $Var(\tilde{R}_{lk}^2)$, I can apply the delta-method:
\begin{align}Var(\tilde{R}_{lk}^2) = {({f'(R_{lk})})^2}Var(\tilde{R}_{lk})&\end{align}$\tag{Eq. 7}$
with the variance of the true correlation coefficient $R_{lk}$:
\begin{align}Var(\tilde{R}_{lk}) = \frac{(1 - {R}_{lk}^2)^2}{N}&\end{align}$\tag{Eq. 8}$
and get
\begin{align}Var(\tilde{R}_{lk}^2) = {4R_{lk}^2} \frac{(1 - {R}_{lk}^2)^2}{N}&\end{align} $\tag{Eq. 9}$
Ultimately we could replace $R_{lk}^2$ in Eq. 9 with the expression in Eq. 6b, and obtain a formula for the variance in sample squared correlation in terms of squared sample correlation ($\tilde{R}_{lk}^2$), instead of the true squared correlation ($R_{lk}^2$).
