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I'm conducting a meta-analysis on estimating a gold standard measure from other simpler measures. To determine the error of estimation, I derived the standard deviations of the differences between the two measures. Therefore, as effect sizes, I'm using the natural logarithm of the standard deviations of the differences $(lnSD)$ and their sampling variance $(1/2[n-1])$, as described on Nakagawa et al.

However, many studies included in my analysis, the same subject produces multiple results (multiple estimates [stages in the data.frame below]). Thus, I should determine the variance-covariance matrix of the effects. However, I am not sure if it is appropriate to determine the variance-covariance matrix using any correlation value for this type of effect, because effects variances are based only on the sample size $(1/2[n-1])$.

Is there any alternative to determine the variance-covariance matrix for this type of data, or I should just include the study as a random effect?

    study    yi     vi exercise stage
1      A 2.351 0.0250      row     3
2      A 2.351 0.0250      row     4
3      A 2.261 0.0250      row     5
4      A 1.922 0.0250      row     7
5      A 1.887 0.0250      row    10
6      B 1.884 0.0313    cycle     3
7      B 1.851 0.0313    cycle     4
8      B 1.774 0.0313    cycle     7
9      B 1.740 0.0313    cycle    10
10     C 1.831 0.0278    cycle     3
11     D 1.732 0.0294      run     5
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Professor Wolfgang Viechtbauer answered this question on the R help listserv for meta analysis. The covariance between two $\ln(SD)$ is calculated as $r^2/(2(n-1))$; where $r$ is the correlation between the two variables.

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