# Solved – The confidence interval of \$tau^2\$ and \$I^2\$ produced by the meta and metafor packages

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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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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