Corrigendum to Power Approximations for Overall Average Effects in Meta-Analysis With Dependent Effect Sizes

Vembye, M. H., Pustejovsky, J. E., & Pigott, T. D. (2023). Power Approximations for Overall Average Effects in Meta-Analysis With Dependent Effect Sizes. Journal of Educational and Behavioral Statistics, 48(1), 70–102. DOI: 10.3102/10769986221127379

In this article, Equation 17 contained an error in the initial OnlineFirst and print versions. The correct formula is

$E\left({\ddot{\text{τ}}}^2\right)={\text{τ}}^2+{\text{ω}}^2\left(\frac{1-\frac{1}{a^2}{{\displaystyle \sum }}_{j=1}^J\frac{1}{k_j{\text{σ}}_j^4}}{1-\frac{1}{a^2}{{\displaystyle \sum }}_{j=1}^J\frac{1}{{\text{σ}}_j^4}}\right),$

where $a={\displaystyle \sum }_{j=1}^J\frac{1}{{\text{σ}}_j^2}$ . This formula is used in calculating weights used for the CE working model, which are then used for approximating the power of the test of average effect size. Equation 17 was also used in the calculations of several examples reported in the paper, and corrected values are as follows. This has also been corrected in the online article.

Table 1.

Hypothetical Studies in a Meta-Analysis

Study	Nj	${\text{σ}}_j^2$	kj	CHE Weight $\left(w_j\right)$	CE Weight $\left({\ddot{w}}_j\right)$	MLMA Weight $\left({\tilde{w}}_j\right)$	RE Weight $\left(w_j^{*}\right)$
A	28	1/7	1	6.02	5.99	5.77	6.40
B	32	1/8	3	10.00	6.70	13.87	7.23
C	40	1/10	2	10.71	8.05	12.43	8.82
D	48	1/12	3	13.85	9.30	17.17	10.34
E	56	1/14	4	16.54	10.46	20.70	11.80
F	64	1/16	2	15.34	11.54	16.21	13.19
G	80	1/20	2	17.91	13.49	18.03	15.79
H	96	1/24	1	15.38	15.19	13.87	18.18
I	128	1/32	2	23.94	18.05	21.70	22.43
J	180	1/45	3	31.76	21.57	26.41	28.12
K	192	1/48	5	35.93	22.23	28.88	29.27
L	256	1/64	2	33.28	25.14	26.13	34.53

In the example reported in the subsection “Using the Power Approximations: A Computational Example,” the correct weights for the CE working model are reported in the revised Table 1. Based on the corrected CE weights and assumed model parameters, we calculate ${\ddot{\text{τ}}}^2=0.0241$ , $\ddot{S}=0.004416$ , $\ddot{\text{λ}}=1.505$ , and $\ddot{\text{ξ}}=9.04$ . Using the correct weights, the power of the robust test based on the CE working model is 26.70% (cf. 26.90% reported in Table 2 of the original paper).

Figure 1 of the original paper depicted the approximate power of several different tests, including a test based on the CE working model, based on a realistic practical example. A revised version appears in Figure 1. Using the correct Equation 17 has a negligible effect on the power estimates for the CE working model with robust variance estimation (CE-RVE) depicted in the figure.

Figure 1.

Power for finding $\mathit{\text{μ}}=\mathit{0.1}$ with $\mathit{\text{τ}}=\mathit{0.115},\mathit{\omega }=\mathit{0.1}$ , and $\mathit{\text{ρ}}=\mathit{.5}$ across three different methods for obtaining n_j and ${\mathit{\text{σ}}}_{\text{j}}^{\mathit{2}}$ . For the stylized and pilot sample (empirical) methods, the average power is estimated across 100 iterations. Dashed lines indicate power of 80%.

Figure 2A of the original paper depicted the difference between approximate power and true (simulated) power for tests based on several different working models, including the test using the CE-RVE. A revised version appears in Figure 2A; only the panels in the fourth column (labeled CE-RVE) have any alterations. Using the correct Equation 17 leads to very small differences in approximate power for the CE-RVE, of no more than 2.1 percentage points in absolute value and usually much smaller (less than 0.1 percentage points). The substantive findings of the simulation study are not affected by these changes.

Figure 2A.

Power difference between approximated and true (simulated) power versus approximated power for the C(H)E working models, across different methods of sampling k _j and N _j . Solid lines indicate no discrepancy between approximated and simulated power. Dashed lines indicate five percentage points over- or underestimation of true power.

Supplemental Material

Supplemental Material, sj-docx-1-jeb-10.3102_10769986231207878 - Corrigendum to Power Approximations for Overall Average Effects in Meta-Analysis With Dependent Effect Sizes

Supplemental Material, sj-docx-1-jeb-10.3102_10769986231207878 for Corrigendum to Power Approximations for Overall Average Effects in Meta-Analysis With Dependent Effect Sizes by in Journal of Educational and Behavioral Statistics