I am comparing R packages including `metafor`

and `meta`

for meta-analysis. Both packages are excellent and most of the results are the same. However, I found that the confidence intervals of $tau^2$ and $I^2$ provided by `metafor`

are different from those of the `meta`

package (and also other meta-analysis software such as MedCalc). Am I missing any parameter in the `metafor`

package? Or how can I fix the inconsistency?

`> library(metafor) # Load metafor package > CCDES1=escalc(measure="ZCOR", ri=c(0.5, 0.6, 0.4, 0.2, 0.7, 0.45), ni=c(40, 90, 25, 400, 60, 50)) # Calculating individual effect size > AGGCCD=rma(yi=yi, vi=vi, data = CCDES1, method="DL") # Aggregating effect sizes > confint(AGGCCD) # Get confidence interval for indices of heterogeneity estimate ci.lb ci.ub tau^2 0.0819 0.0114 0.3023 tau 0.2861 0.1067 0.5498 I^2(%) 86.1663 46.4352 95.8332 H^2 7.2287 1.8669 23.9990 > library(meta) # Load meta package > m1=metacor(cor=c(0.5, 0.6, 0.4, 0.2, 0.7, 0.45), n=c(40, 90, 25, 400, 60, 50), sm="ZCOR", method.tau="DL",backtransf = TRUE) # Aggregating effect sizes > m1 COR 95%-CI %W(fixed) %W(random) 1 0.50 [0.2233; 0.7021] 5.72 15.47 2 0.60 [0.4487; 0.7179] 13.45 18.05 3 0.40 [0.0058; 0.6866] 3.40 13.23 4 0.20 [0.1040; 0.2923] 61.36 19.97 5 0.70 [0.5425; 0.8100] 8.81 16.95 6 0.45 [0.1962; 0.6473] 7.26 16.33 Number of studies combined: k=6 COR 95%-CI z p-value Fixed effect model 0.3584 [0.2895; 0.4236] 9.5396 < 0.0001 Random effects model 0.4875 [0.2714; 0.6568] 4.1045 < 0.0001 Quantifying heterogeneity: tau^2 = 0.0819; H = 2.69 [1.89; 3.82]; I^2 = 86.2% [72%; 93.2%] Test of heterogeneity: Q d.f. p-value 36.14 5 < 0.0001 Details on meta-analytical method: - Inverse variance method - DerSimonian-Laird estimator for tau^2 - Fisher's z transformation of correlations `

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#### Best Answer

The `metafor`

package computes the CI for $tau^2$ using the Q-profile method, which is described in:

Viechtbauer, W. (2007). Confidence intervals for the amount of heterogeneity in meta-analysis. *Statistics in Medicine, 26*(1), 37-52.

This is an exact CI, so it is guaranteed to have nominal coverage (under the assumptions of the model).

Since $I^2$ and $H^2$ are just functions of $tau^2$ (and the 'typical' sampling variance; for details, see here), one can just take the CI bounds for $tau^2$, plug them into the equation for obtaining $I^2$ and $H^2$, and thereby obtain exact CIs for $I^2$ and $H^2$.

The `meta`

package uses the method described in section A2 in:

Higgins, J. P. T., & Thompson, S. G. (2002). Quantifying heterogeneity in a meta-analysis. *Statistics in Medicine, 21*(11), 1539-1558.

This is an approximate CI that does not always have nominal coverage.

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