[R-meta] Negative r value for effect sizes within studies

[Viechtbauer, Wolfgang (SP)]

Dear Stefanou,

Would be unusual, but in principle, yes, it can. However, only certain values of r are actually possible. Let's say there is a study with 5 estimates and you assume r = -0.6. Then the correlation matrix for this study would be:

v <- matrix(-.6, 5, 5); diag(v) <- 1

which is not positive definite, which we can see by examining its eigenvalues:

round(eigen(v)$values, 4)

(note the last eigenvalue is negative). The largest negative correlation possible when p=5 (to get at least a positive semi-definite matrix) is -0.25:

v <- matrix(-.25, 5, 5); diag(v) <- 1
round(eigen(v)$values, 4)

(the last eigenvalue is now essentially 0).

More generally, the lower bound is:

-1/(p-1)

So, you need to determine which study has the largest number of estimates and based on that, you know what the lower bound for r is (at least, when we assume that the correlation matrix is compound symmetric - as above).

Best,
Wolfgang

>-----Original Message-----
>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On
>Behalf Of Stefanou Revesz
>Sent: Friday, 03 September, 2021 18:47
>To: R meta
>Subject: [R-meta] Negative r value for effect sizes within studies
>
>Dear All,
>
>This may be too basic to ask. But can the r value chosen to represent
>the common correlation among effect sizes within studies be a negative
>one?
>
>For example, is there a situation where I would need to use the following?
>
>impute_covariance_matrix(..., r = -.6)
>
>Thank you,
>Stefanou