How to get the asymptotic covariance matrix for the estimates of a correlation matrix?

[jeromyanglim]

This is a required step in Cheung and Chan (2005). However, they do not specify how to obtain such an estimate.

References:

• Cheung, M.W.L. & Chan, W. (2005). Meta-analytic structural equation modeling: a two-stage approach.. Psychological Methods; Psychological Methods, 10, 40. PDF

[jeromyanglim]

Equations 3, 4, and 5 in Olkin and Finn (1995) provide the formulas assuming rho. It sounds like it is common to replace rho with sample estimates of the correlations. Presumably this is more reasonable as the sample size used to estimate the correlations gets larger.

Let $R$ be a $p \times p$ correlation matrix, where there are $m=p(p-1)/2$ correlations. Thus, the covariance matrix $\Omega$ of the sample estimates of the correlation matrix is an $m \times m$ matrix.

There are several different logical ways of extracting correlations from a $p \times p$ matrix into a vector $\mathbf{r}$. The lower triangle or upper triangle can be extracted, and the extraction can either occur row-wise or column wise. A general representation in R is to have a data.frame where each row represents a correlation with four variables, matrix_row, matrix_col, vector_index, and r.

References

Olkin, I. & Finn, J.D. (1995). Correlations redux.. Psychological Bulletin, 118, 155.

[jeromyanglim]

Mike Cheung's R package metaSEM provides the asyCov function which estimates the asymptotic sampling covariance or a correlation/covariance matrix.

See the example. E.g.,

C1 <- matrix(c(1,0.5,0.4,0.5,1,0.2,0.4,0.2,1), ncol=3)  
asyCov(C1, n=100)

Which produces the following output:

             x2x1         x3x1        x3x2
x2x1 4.545457e-03 2.824226e-09 0.001818184
x3x1 2.824226e-09 6.144200e-03 0.003072097
x3x2 1.818184e-03 3.072097e-03 0.008626946