Confidence Intervals for the Overall Effect Size in Random-Effects Meta-Analysis

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1 Psychologcal Methods 008, Vol. 13, No. 1, Copyrght 008 by the Amercan Psychologcal Assocaton X/08/$1.00 DOI: / X Confdence Intervals for the Overall Effect Sze n Random-Effects Meta-Analyss Julo Sánchez-Meca and Fulgenco Marín-Martínez Unversty of Murca One of the man objectves n meta-analyss s to estmate the overall effect sze by calculatng a confdence nterval (CI). The usual procedure conssts of assumng a standard normal dstrbuton and a samplng varance defned as the nverse of the sum of the estmated weghts of the effect szes. But ths procedure does not take nto account the uncertanty due to the fact that the heterogenety varance ( ) and the wthn-study varances have to be estmated, leadng to CIs that are too narrow wth the consequence that the actual coverage probablty s smaller than the nomnal confdence level. In ths artcle, the performances of 3 alternatves to the standard CI procedure are examned under a random-effects model and 8 dfferent estmators to estmate the weghts: the t dstrbuton CI, the weghted varance CI (wth an mproved varance), and the quantle approxmaton method (recently proposed). The results of a Monte Carlo smulaton showed that the weghted varance CI outperformed the other methods regardless of the estmator, the value of, the number of studes, and the sample sze. Keywords: meta-analyss, random-effects model, confdence ntervals, heterogenety varance, standardzed mean dfference Meta-analyss s a research methodology that ams to ntegrate, by applyng statstcal methods, the results of a set of emprcal studes about a gven topc. To accomplsh ts purpose, a meta-analyss requres a thorough search of the relevant studes, and the results of each ndvdual study have to be translated nto the same metrc (Cooper, 1998; Lpsey & Wlson, 001). Dependng on such study characterstcs as the desgn type and how the varables mpled were measured, the meta-analyst has to select one of the dfferent effect-sze ndces and apply t to all of the studes of the meta-analyss (Grssom & Km, 005). So, when the dependent varable s contnuous and the purpose of each study s to compare the performance between two groups, the standardzed mean dfference s the most usual effectsze ndex (Cooper, 1998; Hedges & Olkn, 1985). If the Julo Sánchez-Meca and Fulgenco Marín-Martínez, Department of Basc Psychology and Methodology, Faculty of Psychology, Espnardo Campus, Unversty of Murca, Murca, Span. Ths artcle was supported by a grant from the Mnstero de Educacón y Cenca of the Spansh Government and by Fondo Europeo de Desarrollo Regonal funds for Project No. SEJ /PSIC. Correspondence concernng ths artcle should be addressed to Julo Sánchez-Meca, Department of Basc Psychology and Methodology, Faculty of Psychology, Espnardo Campus, Unversty of Murca, Murca, Span. E-mal: jsmeca@um.es dependent varable s dchotomous or has been dchotomzed, then effect-sze ndces such as an odds rato (or ts log transformaton), a rsk rato (or ts log transformaton), or a rsk dfference can be appled (Egger, Smth, & Altman, 001; Haddock, Rndskopf, & Shadsh, 1998; Sánchez- Meca, Marín-Martínez, & Chacón-Moscoso, 003). If all of the varables are contnuous, then an effect-sze ndex from the r famly can be appled, such as the Pearson correlaton coeffcent or ts Fsher s Z transformaton (Hunter & Schmdt, 004; Rosenthal, 1991; Rosenthal, Rosnow, & Rubn, 000). In general, the statstcal analyss usually appled n metaanalyss has three man objectves: (a) to estmate the overall effect sze of the populaton to whch the studes pertan; (b) to assess f the heterogenety found among the effect estmates can be explaned by chance alone or f, on the contrary, the ndvdual studes exhbted true heterogenety, that s, varablty produced by real dfferences among the populaton effect szes; and, (c) f heterogenety cannot be explaned by samplng error alone, to search for study characterstcs that could operate as moderator varables of the effect estmates. Our focus n ths artcle was the frst objectve, that s, to estmate the populaton effect sze. To estmate the populaton effect sze from a set of ndvdual studes, an average of the effect estmates s calculated by weghtng each one of them by ts nverse varance, and a confdence nterval (CI) s thus obtaned 31

2 3 SÁNCZ-MECA AND MARÍN-MARTÍNEZ around t. Most of the effect-sze ndces usually appled n meta-analyss are approxmately normally dstrbuted and ther samplng varances can be easly estmated by smple algebrac formulas (Fless, 1994; Rosenthal, 1994; Shadsh & Haddock, 1994). As a consequence, meta-analyses typcally calculate a CI for the overall effect sze assumng a standard normal dstrbuton to estmate the populaton effect sze, wth the samplng varance estmated as the nverse of the sum of the estmated weghts. Ths procedure performs well when the effect estmates obtaned n the studes dffer among themselves only by samplng error, that s, when the effect estmates assume a fxed-effects model or the heterogenety varance s small. However, when the underlyng statstcal model n the meta-analyss s a random-effects model, the emprcal coverage probablty of ths CI for the average effect sze systematcally underestmates the nomnal confdence level (Brockwell & Gordon, 001, 007; Sdk & Jonkman, 00). In recent years, the random-effects model has been consdered the most realstc statstcal model n meta-analyss (Feld, 001, 003; Hedges & Vevea, 1998; Overton, 1998; Raudenbush, 1994). Therefore, to obtan CIs for the overall effect sze wth a good coverage probablty s an mportant ssue. Our purpose n wrtng ths artcle was to compare the performances of three alternatve CI procedures wth that based on the standard normal dstrbuton to estmate the overall effect sze when the underlyng statstcal model s a random-effects model. Moreover, we also examned whether dfferent heterogenety varance estmators affect the coverage probablty of the CIs for the overall effect sze. Thus, we started from the dea that a good CI procedure to estmate an overall effect sze should offer good coverage, that s, close to nomnal, and the coverage should not be affected by the value of the heterogenety varance, by the heterogenety varance estmator used n the metaanalyss, or by the number of studes. The four CI procedures analyzed here are very smple to calculate, not requrng teratve numercal computaton. Other methods of obtanng CIs that are computatonally more complex and are not addressed here are those of Bggerstaff and Tweede (1997) or the profle lkelhood method of Hardy and Thompson (1996). The Random-Effects Model Let k be a set of ndependent emprcal studes about a gven topc and ˆ be the effect-sze estmate obtaned n the th study. The underlyng statstcal model can be represented as ˆ e, (1) where e s the samplng error of ˆ. Usually e s assumed to be normally dstrbuted, e N(0, ), wth beng the wthn-study varance. The random-effects model assumes that each sngle study estmates ts own parametrc effect sze and, as a consequence, consttutes a random varable wth mean and between-studes varance. The between-studes varance, also named heterogenety varance, represents the varablty between the estmated effect szes due not to wthn-study samplng error but to true heterogenety among the studes. In other words, the heterogenety varance represents the varablty produced by the nfluence of the dfferental characterstcs of the studes, such as the desgn qualty, the characterstcs of the subjects n the samples, or dfferences n the program mplementaton. Ths mples that each parametrc effect sze,, can be decomposed as ε, () wth ε representng the dfference between the parametrc effect sze of the th study,, and the parametrc mean,. The errors ε are usually assumed to be normally dstrbuted, wth heterogenety varance, ε N(0, ). It s also assumed that the errors e and ε are ndependent. So, combnng Equatons 1 and enables us to formulate the random-effects model as ˆ e ε, (3) and, as a consequence, the estmated effect szes ˆ are assumed to be normally dstrbuted wth mean and varance, ˆ N(, ). When there s not true heterogenety, then the betweenstudes varance s zero, 0, and the random-effects model becomes a fxed-effects model, that s, all of the ndvdual studes estmate the same parametrc effect sze 1... k. In ths case, Equaton 3 smplfes to ˆ e, and the effect estmates ˆ are assumed to be normally dstrbuted wth mean and varance, ˆ N(, ). Thus, the fxed-effects model can be consdered a partcular case of the random-effects model when dfferences among the effect estmates are only due to samplng error. Both models, those of random and fxed effects, can be extended to nclude moderator varables. They are not presented here, however, as our purpose s to compare the performance of dfferent procedures to calculate a CI around the overall effect sze. CIs for the Overall Effect Sze One of the man objectves n meta-analyss s to obtan an average effect-sze estmate from a set of ndependent effect-sze estmates and to calculate a CI around t to estmate the parametrc effect sze,. In practce, the studes ncluded n a meta-analyss have dfferent sample szes and, as a consequence, the precson of the effect-sze estmates vares among them. A good estmator of the mean

3 CONFIDENCE INTERVALS IN META-ANALYSIS 33 parametrc effect sze should take nto account the precson of the effect estmates. The most usual procedure to acheve ths objectve conssts of weghtng each effect-sze estmate by ts nverse varance. In a random-effects model, the unformly mnmum varance unbased estmator (UMVU) of s gven by ˆ UMVU w ˆ (4) w (Vechtbauer, 005), wth w beng the optmal or true weghts w 1/. The samplng varance of ˆ UMVU s gven by Vˆ UMVU 1. (5) w If, n a meta-analyss, the populaton samplng varance of each study,, and the populaton heterogenety varance,, are known, then ˆ UMVU can be calculated and, as t s asymptotcally normally dstrbuted, a 100(1 )% CI assumng a standard normal dstrbuton can be calculated by ˆ UMVU z 1 / Vˆ UMVU ), (6) where z 1/ s the 100(1 /) percentle of the standard normal dstrbuton, beng the sgnfcance level. The z Dstrbuton CI In practce, nether the parametrc heterogenety varance,, nor the parametrc samplng varances of the sngle studes,, are known. Therefore, they have to be estmated from the data reported n the studes. Ths means that Equaton 6 cannot ever be appled. For most of the effectsze ndces usually appled n meta-analyss, unbased estmators of the samplng varance, ˆ, have been derved, and several estmators can be found n the lterature to estmate the heterogenety varance n a meta-analyss, ˆ (Sdk & Jonkman, 007; Vechtbauer, 005). Once we have an unbased samplng varance estmator, ˆ, to be appled n each study and a heterogenety varance estmator, ˆ, the optmal weghts, w, can be estmated by ŵ 1/ˆ ˆ. Therefore, the formula for estmatng the parametrc mean effect sze,, n meta-analyss s gven by ˆ ŵ ˆ ŵ, (7) and ts samplng varance s usually estmated as Vˆ ˆ 1. (8) ŵ The typcal procedure to calculate a CI around an overall effect sze assumes a standard normal dstrbuton and estmates the samplng varance of ˆ by Equaton 8. Here we refer to ths procedure as the z dstrbuton CI, whch s obtaned by ˆ z 1 / Vˆ ˆ. (9) However, ths procedure does not take nto account the uncertanty produced by the fact that the wthn-study and the between-studes varances have to be estmated (Bggerstaff & Tweede, 1997). As Sdk and Jonkman (003) have contended, The normalty assumpton for ˆ s not strctly true n practce (nor s Vˆ (ˆ ) the true varance), because the ŵ values are estmates. Nonetheless, ths s the commonly used practce for constructng CIs (p. 1196). The man consequence of assumng a standard normal dstrbuton to obtan a CI for ˆ wth Equaton 9 s that ts actual coverage probablty s smaller than the nomnal confdence level, the wdth of the CI beng too narrow. As Vechtbauer (005) has shown, estmatng the optmal weghts, w, usng unbased estmates of and, results n an estmate of the samplng varance of ˆ that s negatvely based. As a consequence of ths negatve bas, the samplng varance of ˆ wll be underestmated on average, and researchers wll attrbute unwarranted precson to ther estmate of. (p. 63) Moreover, several Monte Carlo studes have shown that the underestmaton of the nomnal confdence level wth the z dstrbuton CI s more severe as the between-studes varance ncreases and as the number of studes decreases. The z dstrbuton CI only presents good coverage probablty n meta-analyses wth a large number of studes and very lttle or zero heterogenety varance (Brockwell & Gordon, 001, 007; Follmann & Proschan, 1999; Hartung & Makamb, 003; Makamb, 004; Sdk & Jonkman, 00, 003, 005, 006). The t Dstrbuton CI To solve the problems of coverage probablty wth the z dstrbuton CI, t has been proposed n the lterature (Follmann & Proschan, 1999; Hartung & Makamb, 00) to assume a Student t reference dstrbuton wth k 1 degrees of freedom, nstead of the standard normal dstrbuton, and to estmate the samplng varance of ˆ n Equaton 8 wth ˆ t k 1,1 / Vˆ ˆ, (10)

4 34 SÁNCZ-MECA AND MARÍN-MARTÍNEZ wth t k1, 1/ beng the 100(1 /) percentle of the t dstrbuton wth k 1 degrees of freedom. Here we refer to ths procedure as the t dstrbuton CI. Usng a t dstrbuton produces CIs that are wder than those of the standard normal dstrbuton, n partcular for meta-analyses wth a small number of studes, and, consequently, ths should mprove the coverage probablty, as Follmann and Proschan (1999) have found. The Weghted Varance CI One procedure that has not yet wdely been used n meta-analyss s that proposed by Hartung (1999), whch conssts of calculatng a CI for the overall effect sze assumng a Student t dstrbuton wth k 1 degrees of freedom and estmatng the samplng varance of ˆ wth a weghted extenson of the usual formula, Vˆ w(ˆ): ŵ ˆ ˆ Vˆ wˆ, (11) k 1 ŵ where ŵ 1/ˆ ˆ and ˆ s the overall effect sze defned n Equaton 7 assumng a random-effects model. It can be shown that the statstc (ˆ )/Vˆ w(ˆ)s approxmately dstrbuted as a t dstrbuton wth k 1 degrees of freedom (Hartung, 1999; Sdk & Jonkman, 00). Therefore, a CI around the overall effect sze can be computed by ˆ t k 1,1 / Vˆ w(ˆ ). (1) Followng Sdk and Jonkman (003, 006), here we refer to ths procedure as the weghted varance CI. Prevous smulatons seem to offer good coverage of ths procedure when the effect-sze ndex s the log odds rato (Makamb, 004; Sdk & Jonkman, 00, 006), the standardzed mean dfference (Sdk & Jonkman, 003), and the unstandardzed mean dfference and the rsk dfference (Hartung & Makamb, 003). In partcular, the weghted varance CI offers a better coverage probablty than the z dstrbuton CI except when the between-studes varance s zero, 0 (Hartung, 1999; Hartung & Makamb, 003; Sdk & Jonkman, 00, 003). The Quantle Approxmaton (QA) Method The fourth method of calculatng a CI for the overall effect sze that s ncluded n ths study has been recently proposed by Brockwell and Gordon (007). The method conssts of approxmatng, by means of ntensve computaton, the quantles of the dstrbuton of the statstc M ˆ /Vˆ ˆ and then usng the 100(1 /)% percentle of the M dstrbuton to calculate a CI for the overall effect sze by ˆ b 1 / Vˆ ˆ (13) (Brockwell & Gordon, 007, p. 4538), where Vˆ (ˆ ) s the usual formula to estmate the samplng varance of ˆ, defned n Equaton 8, and b 1/ s the 100(1 /)% percentle of the dstrbuton of M emprcally approached by Monte Carlo smulaton. Unlke the other three procedures for calculatng a CI for the overall effect sze n a random-effects meta-analyss, the crtcal values n the Brockwell and Gordon (007) method are obtaned by smulatng thousands of meta-analyses from a random-effects model and varyng the number of studes between and 30 and the heterogenety varance between 0 and 0.5. The effect-sze ndex that they used n the smulatons was the log odds rato, as t s a very common effect estmator n the medcal lterature. Once Brockwell and Gordon (007) obtaned the observed values for the quantles 100(/)% and 100(1 /)% of the M statstc, they adjusted a regresson equaton for the quantles as a functon of the number of studes, k: b 1/ k k lnk (14) (Brockwell & Gordon, 007, p. 4538). Thus, the crtcal values, b 1/, to be used n the CI formula (Equaton 13) of Brockwell and Gordon (007) are estmated from Equaton 14. For example, f a meta-analyss has k 10 studes, then the correspondng crtcal value for a 95% nomnal confdence level s b Here we refer to ths procedure as the QA method. Brockwell and Gordon (007) have found a better performance of ths procedure than those of the z and t dstrbuton CIs, usng the DerSmonan and Lard (1986) estmator of the heterogenety varance, but they dd not compare the QA method wth the weghted varance CI. Heterogenety Varance Estmators To calculate a CI around the overall effect sze n a meta-analyss where a random-effects model s assumed, an estmate of the heterogenety varance s needed. Although meta-analyses typcally use the heterogenety varance estmator proposed by DerSmonan and Lard (1986), alternatve estmators have been proposed that seem to offer better propertes than the usual estmator. Some of the alternatves are based on nonteratve estmaton procedures, whereas others requre teratve computatons. Dfferent heterogenety varance estmators dffer n respect to such statstcal propertes as bas and mean square error (Sdk & Jonkman, 007; Vechtbauer, 005, 007), and an ssue that has not yet been wdely studed s whether the selecton of the heterogenety varance estmator has an effect on the performance of dfferent CIs for the overall

5 CONFIDENCE INTERVALS IN META-ANALYSIS 35 effect sze. Next, we present formulas to calculate eght dfferent heterogenety varance estmators that could be used to obtan CIs for the overall effect sze under a random-effects model. Hunter and Schmdt () Estmator Hunter and Schmdt (1990, pp ; see also Hunter & Schmdt, 004, pp ) proposed to estmate the heterogenety varance by calculatng the dfference between the total varance of the effect estmates and an average of the estmated wthn-study varances, ˆ. A smplfed formula of ths estmator s gven by ˆ Q k ŵ, (15) FE where ŵ FE 1/ˆ s the nverse varance of the th study assumng a fxed-effects model, wth ˆ beng the wthnstudy varance estmate for the th study. Q s the heterogenety statstc usually appled to test the homogenety hypothess (Hedges & Olkn, 1985): Q ŵ FE ˆ ˆ FE, (16) wth ˆ FE beng the mean effect sze, assumng a fxedeffects model; that s, ˆ FE ŵ FE ˆ FE ŵ. (17) If Q k, then ˆ s negatve and, as a consequence, t has to be truncated to zero. Hedges () Estmator The estmator of the populaton heterogenety varance conssts of calculatng the dfference between an unweghted estmate of the total varance of the effect szes and an unweghted estmate of the average wthn-study varance (Hedges, 1983, p. 391; see also Hedges & Olkn, 1985, p. 194): ˆ ˆ ˆ UW k 1 1 ˆ, (18) k where ˆ UW s an unweghted mean of the effect szes ˆ ˆ UW. (19) k As ˆ s not a nonnegatve heterogenety varance estmator, t has to be truncated to zero when ˆ 0. DerSmonan and Lard () Estmator The heterogenety varance estmator usually appled n the meta-analytc lterature s that proposed by DerSmonan and Lard s (1986) estmator, whch s based on the moments method, conssts of estmatng the populaton heterogenety varance by Q k 1 ˆ, (0) c where Q s the heterogenety statstc defned n Equaton 16 and c s gven by c ŵ FE ŵ FE FE ŵ. (1) When Q (k 1), then ˆ s negatve and, lke ˆ, t has to be truncated to zero. ˆ Malzahn, Böhnng, and Hollng () Estmator and Malzahn, Böhnng, and Hollng (000) proposed a momentbased nonparametrc estmator of the populaton heterogenety varance specfcally desgned to be used only wth the standardzed mean dfference, d. It s also based on the dfference of an estmate of the total varance of the d ndces and an estmate of the average wthn-study varance of the d ndces. It s obtaned by ˆ 1 ˆ ˆ FE k 1 1 k N n E n C 1 k ˆ () (Malzahn et al., 000, p. 6; see also Malzahn, 003), wth N n E n C beng the total sample sze of the th study; ˆ FE was defned n Equaton 17, ˆ s the d ndex for the th study, and s gven by 1 N 4 cm N, (3) wth c(m ) beng the correcton factor of the d ndex for small sample szes, defned n Equaton 33. Applcatons of ths estmator are lmted to meta-analyses where the effectsze ndex s the d ndex. When ˆ has a negatve value, t s truncated to zero.

6 36 SÁNCZ-MECA AND MARÍN-MARTÍNEZ Hartung and Makamb () Estmator Hartung and Makamb (003; see also Makamb, 004) proposed a postve heterogenety varance estmator that attempts to mprove the performance of the usual estmator. A smplfed formula of ths estmator s gven by ˆ Q k 1 Qc, (4) wth Q and c defned n Equatons 16 and 1, respectvely. An advantage of ths estmator s that t cannot yeld negatve values. Sdk and Jonkman () Estmator Another estmator of the heterogenety varance n metaanalyss, recently proposed by Sdk and Jonkman (005), also yelds nonnegatve values. The estmator s a smple nonteratve estmator of the heterogenety varance that s based on a reparametrzaton of the total varance n the effect estmates, ˆ. It s obtaned by ˆ vˆ1 ˆ ˆ vˆ k 1, (5) (Sdk & Jonkman, 005, p. 371; see also Sdk & Jonkman, 007), where vˆ r 1, r ˆ /ˆ0, and ˆ0 s an ntal estmate of the heterogenety varance, whch can be defned, for example, as ˆ 0 ˆ ˆ UW k, (6) ˆ UW beng the unweghted mean of the effect estmates, defned n Equaton 19, and ˆ vˆ s gven by ˆ vˆ vˆ1 ˆ. (7) Thus, n the estmator, the weghts are a functon not only of the wthn-study varances but also of a crude estmate of the heterogenety varance. Although other ntal ˆ0 estmates can be proposed, we used the one orgnally recommended by Sdk and Jonkman (005). Maxmum Lkelhood () Estmator The sx heterogenety varance estmators presented above are nonteratve. Two teratve estmators proposed n the meta-analytc lterature to estmate the heterogenety varance are based on maxmum lkelhood and restrcted maxmum lkelhood estmaton (Brockwell & Gordon, 001; DerSmonan & Lard, 1986; Hardy & Thompson, 1996; Raudenbush & Bryk, 1985). For a specfed convergence crteron, the formula that enables us to estmate the populaton heterogenety varance by maxmum lkelhood under a random-effects model s gven by ˆ ŵ ˆ ˆ ˆ (8) ŵ (Sdk & Jonkman, 007; Vechtbauer, 005, p. 68), wth ŵ 1/ˆ ˆ, where ˆ s ntally estmated by any of the nonteratve estmators of the heterogenety varance or settng ˆ 0 and ˆ s gven by ˆ ŵ ˆ ŵ. (9) In each teraton of Equatons 8 and 9, the estmate of has to be checked to avod havng negatve values truncatng t to zero. Convergence s usually acheved wthn fewer than 10 teratons. Restrcted Maxmum Lkelhood Estmator The second teratve estmator of the heterogenety varance n a random-effects model s based on restrcted maxmum lkelhood estmaton (). The estmator of compensates for the negatve bas of the estmator by applyng a lnear combnaton of the effect szes. The estmator of the heterogenety varance s gven by ˆ ŵ ˆ ˆ ˆ 1 (30) ŵ ŵ (Vechtbauer, 005, p. 69). The teratve procedure s smlar to that of the estmator. When ˆ 0, t s truncated to zero to avod negatve values. An Example To llustrate the calculatons and the extent to whch dfferent CI procedures and heterogenety varance estmators can yeld dfferences n the nterval estmatons of the overall effect sze, we have selected the example cted n Hedges and Olkn (1985, p. 5), composed of the results of 10 studes on the effectveness of open versus tradtonal educaton on student creatvty. The effect-sze ndex appled was the standardzed mean dfference, d. Table 1 presents the d value, d ; the sample sze; and the estmated

7 CONFIDENCE INTERVALS IN META-ANALYSIS 37 Table 1 Effect Estmates, Sample Szes, and Estmated Wthn-Study Varances for the Example Data Study d n E n C ˆ wthn-study varance, ˆ, for each study. In total, we calculated 3 CIs assumng a random-effects model, resultng from the combnaton of four CI procedures (z dstrbuton CI wth Equaton 9, t dstrbuton CI wth Equaton 10, weghted varance CI wth Equaton 1, and QA method wth Equaton 13) wth eght heterogenety varance estmators (Equatons 15, 18, 0,, 4, 5, 8, and 30). Table shows the results obtaned wth the dfferent CI procedures. A frst nterestng result s that the standard errors of ˆ obtaned wth the usual formula, Vˆ (ˆ ), were more dependent on the estmator than were those calculated wth the weghted samplng varance, Vˆ w(ˆ). In partcular, standard errors obtaned from the weghted varance, Vˆ w(ˆ), vared from to 0.169, whereas standard errors from the usual varance, Vˆ (ˆ ), ranged from to 0.19; that s, the range for Vˆ (ˆ ) was about 1 tmes larger than that of Vˆ w(ˆ). Moreover, a postve relatonshp was found between the standard errors from the usual formula and the estmates. However, the overall effect estmates, ˆ, vared as a functon of the estmator from 0.30 to As expected, for common values of ˆ, the wdth of the CIs based on the z dstrbuton was narrower than the wdths obtaned wth the t dstrbuton CI, the weghted varance CI, and the QA method (except for the heterogenety varance estmator). On average, the wdths of the CIs based on the z dstrbuton, t dstrbuton, weghted varance, and QA method were 0.680, 0.784, 0.758, and 0.83, respectvely. The wdth of the CI obtaned wth the QA method was slghtly larger than that of the t dstrbuton CI because of the dfferent crtcal value used:.374 for b 1/ versus.6 for t k1,1/. Furthermore, the wdth of the CIs obtaned wth the weghted varance procedure through the dfferent heterogenety varance estmators vared about 1 tmes less (SD 0.004) than those obtaned wth z dstrbuton, t dstrbuton, and QA method CIs (SDs 0.046, 0.053, and 0.055, respectvely). Therefore, the weghted varance CI seems to be less dependent on the estmator than are the other three CI procedures. Ths example llustrates how the selecton of the estmator and the procedure for calculatng a CI for the overall effect sze can affect the results. Monte Carlo Study Although several Monte Carlo studes have compared the coverage probablty of the usual CI and that proposed by Hartung (1999) under a random-effects model, the extent to whch dfferent heterogenety varance estmators can affect ther performance has not yet been wdely examned. In prevous studes, the usual estmator was used (Sdk & Jonkman, 00, 006), and ts nfluence on the coverage probablty has been compared wth one (Hartung & Makamb, 003; Makamb, 004) or two (Sdk & Jonkman, 003) alternatve estmators of the heterogenety varance only n some cases. Moreover, a comparson of the performance of the four CI procedures has not yet been carred out. However, only one of these smulaton studes focused on the standardzed mean dfference as the effect-sze ndex (Sdk & Jonkman, 003). Fnally, prevous smulaton stud- Table Results Based on Dfferent Heterogenety Varance Estmators and Confdence Interval (CI) Procedures for the Example Data Wdth of the 95% CI for estmator ˆ ˆ Vˆ (ˆ) Vˆ w (ˆ) z dstrbuton t dstrbuton Weghted varance QA method Note. CI confdence nterval; Hunter and Schmdt estmator; Hedges estmator; DerSmonan and Lard estmator; Malzahn, Böhnng, and Hollng estmator; Hartung and Makamb estmator; Sdk and Jonkman estmator; maxmum lkelhood estmator; restrcted maxmum lkelhood estmator.

8 38 SÁNCZ-MECA AND MARÍN-MARTÍNEZ es have not manpulated the parametrc mean effect sze,, because t s expected that CIs calculated from z and t dstrbutons should be nvarant to a locaton shft (Brockwell & Gordon, 001; Sdk & Jonkman, 005). However, as the weghts used n the calculatons of the overall effect sze, ˆ, and of the varances for ˆ are estmated weghts, ŵ, and not the true or correct weghts, w, changes n could affect the coverage probablty of the CIs. Ths s of partcular nterest when the effect-sze ndex s the standardzed mean dfference, as the wthn-study varance of each study s a functon of the parametrc effect sze,. Therefore, we carred out Monte Carlo smulatons to determne whether (a) dfferent assumptons about the underlyng samplng dstrbuton of the overall effect sze (standard normal dstrbuton, Student t dstrbuton, and quantle approxmaton); (b) dfferent estmators of the samplng varance of ˆ (the usual or the weghted samplng varance); (c) dfferent heterogenety varance estmators; and (d) changes n the parametrc mean effect sze,, can affect the coverage probablty when constructng a CI around an overall standardzed mean dfference n a random-effects meta-analyss. Moreover, the performance of the dfferent CI procedures was examned as a functon of such factors as the number of studes, the value of the heterogenety varance, and the average sample sze. Fnally, for comparson purposes, the four CI procedures were also calculated from the optmal or correct weghts, w. From the results of prevous research, we had several expectatons. Frst, n respect to the z dstrbuton CI, we expected to fnd (a) a good adjustment of the emprcal coverage probablty to the nomnal confdence level only when 0 and (b) an emprcal coverage probablty that becomes ncreasngly less than the nomnal coverage probablty as ncreases and the number of studes, k, decreases. Second, the t dstrbuton CI should offer better coverage than that based on the standard normal dstrbuton (Follmann & Proschan, 1999). In respect to the weghted varance CI, we expected to obtan a closer approxmaton to the nomnal coverage probablty by the actual coverage probablty regardless of the values of and k (Makamb, 004; Sdk & Jonkman, 003). The QA method should offer better coverage as and the number of studes ncrease. Furthermore, Sdk and Jonkman (003) found that the coverage probablty of the weghted varance CI s less affected by the estmator than s that based on the standard normal dstrbuton. In partcular, they showed ths fndng wth three estmators (,, and estmators). We expected to generalze ths fndng to the eght estmators examned here and, thus, to show the hgher robustness to changes n the estmator of the weghted varance CI, n comparson wth that of the z dstrbuton, t dstrbuton, and QA method CIs. Fnally, as expected from the statstcal theory, the z dstrbuton CI appled on the optmal or correct weghts should offer the best adjustment to the nomnal level. In our smulaton study, the effect-sze ndex was the standardzed mean dfference. To smulate each ndvdual study, we defned a two-group desgn (e.g., expermental vs. control) and a contnuous outcome. Defne w as the wthn-study varance of observatons for study. Under a random-effects model, the populaton standardzed mean dfference for each study,, was defned as E C w, (31) where E and C were the populaton means for the expermental and the control groups n the th study and w was the common populaton standard devaton of the th study. For each study, normal dstrbutons n the expermental and the control groups were assumed for the contnuous outcome. The populaton standardzed mean dfferences,, were normally dstrbuted wth mean and varance, that s, N(, ). Here, and correspond wth and, respectvely, n prevous equatons. From the normal dstrbuton of values, collectons of k ndependent studes were randomly generated to smulate a meta-analyss. Once a value was randomly selected, the th study was smulated by generatng two normal dstrbutons (for the expermental and control groups) wth means of E and C 0 and common standard devaton w 1. Then, pars of ndependent samples (expermental and control) were randomly selected from the two dstrbutons of the contnuous outcome wth sample szes n E n C, and the means, y E and y C, and the standard devatons, S E and S C, were calculated. Thus, for the th study, was estmated by the d ndex d cm y E y C S, (3) where c(m ) s a correcton factor for small sample szes that s approached by cm 1 3 4n E n C 9 (33) (Hedges & Olkn, 1985), and S s the pooled wthn-study standard devaton, gven by S n E 1S E n C 1S C. (34) n E n C In ths context, the d values match the ˆ estmates defned n the equatons n prevous sectons of ths artcle. The parametrc wthn-study varance of d s gven by

9 CONFIDENCE INTERVALS IN META-ANALYSIS 39 n E n C n E n C n E n C (35) (Hedges & Olkn, 1985). As s unknown n practce, d s substtuted n Equaton 35 for. So the samplng varance of d s estmated by ˆ n E n C n E n C d n E n C. (36) For each one of the k studes n a meta-analyss, the d ndex and both the populaton ( ) and the estmated (ˆ ) wthn-study varances were calculated by applyng the Equatons 3, 35, and 36, respectvely. Then, wth the data of each smulated meta-analyss, we performed the followng calculatons: 1. The eght heterogenety varance estmators presented above were computed (ˆ, Equaton 15; ˆ, Equaton 18; ˆ, Equaton 0; ˆ, Equaton ; ˆ, Equaton 4; ˆ, Equaton 5; ˆ, Equaton 8; and ˆ, Equaton 30).. For each study, eght estmated weghts, ŵ, under a random-effects model were calculated by applyng ŵ 1/ˆ ˆ, wth ˆ gven n Equaton 36 and by substtutng the eght heterogenety varance estmators for ˆ. 3. Also, for each study, the optmal weghts, w, defned as w 1/, were computed for comparson purposes. 4. For each of the eght estmated weghts, a weghted average effect sze, ˆ, was calculated wth Equaton 7, as were both the correspondng standard varance, Vˆ (ˆ ), and weghted varance, Vˆ w(ˆ ), by usng Equatons 8 and 11, respectvely. 5. Wth the optmal weghts, the UMVU mean effect sze, ˆ UMVU ; ts varance, V(ˆ UMVU ); and the weghted varance adapted to the w weghts, V w (ˆ UMVU ), were also calculated by usng Equatons 4, 5, and 11, respectvely. 6. For each of the nne average effect szes (the eght ˆ versons and ˆ UMVU ) and ther correspondng standard and weghted varances, four CI procedures were calculated: z dstrbuton CI (wth Equaton 9 for the eght ˆ versons and Equaton 6 for ˆ UMVU ), t dstrbuton CI (wth Equaton 10), weghted varance CI (wth Equaton 1), and the QA method (wth Equaton 13). In all cases, the nomnal confdence level was fxed at 100(1 ) 95%. To examne the performance of the dfferent CI procedures, we manpulated the followng factors n the smulatons. Frst, the heterogenety varance,, was manpulated wth values 0, 0.04, 0.08, 0.16, and 0.3. Note that for 0, the assumed model s not a random- but a fxed-effects model. The values for were selected n an attempt to reflect those usually found n real meta-analyses wth the d ndex. Second, the average parametrc standardzed mean dfference,, was manpulated wth values 0.5 and 0.8, whch can be consdered to be effects of medum and hgh magntude, respectvely (Cohen, 1988). Thrd, the number of studes, k, n each meta-analyss was manpulated, wth values 5, 10, 0, 40, and 100. Fnally, the average sample sze of the studes ncluded n the meta-analyses was manpulated wth values 30, 50, 80, and 100. The sample sze dstrbuton used n our smulatons was obtaned from a revew of the meta-analyses publshed n 18 nternatonal psychologcal journals, wth a Pearson skewness ndex of (for more detals, see Sánchez-Meca & Marín- Martínez, 1998). Thus, four vectors of fve sample szes each were selected, averagng 30, 50, 80, or 100, usng the skewness ndex gven above to approxmate real data, wth the followng values for N : (1, 16, 18, 0, 84), (3, 36, 38, 40, 104), (6, 66, 68, 70, 134), and (8, 86, 88, 90, 154). Each vector of fve samples was then replcated ether, 4, 8, or 0 tmes to generate meta-analyses of k 5, 10, 0, 40, and 100 studes, respectvely. For each smulated study, the sample szes for expermental and control groups were equal (n E n C ), wth N n E n C. For example, the sample sze vector (1, 16, 18, 0, 84) meant that the expermental and control groups had sample szes of n E n C 6, 8, 9, 10, and 4, respectvely. The smulaton study was programmed n GAUSS (Aptech Systems, 001). In total, 00 condtons were manpulated [5 ( values) ( values) 5(k values) 4 (N values)] and, for each of them, 10,000 replcates (metaanalyses) were performed. From the 10,000 replcates for each condton, the emprcal coverage probablty was calculated for the 36 CIs by computng the proporton of nterval estmates that ncluded the parametrc effect sze,. Results and Dscusson Table 3 presents the average emprcal coverage probabltes and ther standard devatons through the 100 smulated condtons for each of the 36 CIs calculated when 0.5: 3 CIs resultng from the applcaton of eght heterogenety varance estmators and four CI procedures (normal dstrbuton wth the usual varance of ˆ, t dstrbuton wth the usual varance of ˆ, t dstrbuton wth the weghted varance of ˆ, and the QA method) and 4 CIs obtaned for the UMVU average effect sze, ˆ UMVU, that s, wth the optmal or correct weghts, w. Although n practce the w weghts are unknown, these 4 CIs were ncluded n

10 40 SÁNCZ-MECA AND MARÍN-MARTÍNEZ Table 3 Average Emprcal Coverage Probabltes Wth Standard Devatons Through the 100 Smulated Condtons for Each Confdence Interval Procedure and Estmator ( 0.5) z dstrbuton Confdence nterval procedure t dstrbuton Weghted varance QA method Weghts ˆ M SD M SD M SD M SD Optmal estmator Note. QA quantle approxmaton; Hunter and Schmdt estmator; Hedges estmator; DerSmonan and Lard estmator; Malzahn, Böhnng, and Hollng estmator; Hartung and Makamb estmator; Sdk and Jonkman estmator; maxmum lkelhood estmator; restrcted maxmum lkelhood estmator. our smulatons purely for comparson purposes wth the CIs obtaned from the estmated between-studes and wthn-study varances. In the table, the mean effect sze, ˆ, s also presented. Ths was obtaned when usng the optmal weghts and when the optmal weghts were estmated by applyng eght dfferent estmators. Table 3 shows that the estmated mean effect sze usng the optmal weghts was practcally unbased through the 100 smulated condtons ˆ UMVU 0.496, whereas the mean effect szes obtaned wth the estmated weghts showed a slght negatve bas, wth average values rangng from to We can therefore consder that the mean effect estmates were very smlar among themselves as well as beng practcally unbased and, as a consequence, dfferences found among the estmated coverage probabltes of the dfferent CIs cannot be due to a dfferental bas n the mean effect estmates but must be due rather to the dfferent CI procedures and heterogenety varance estmators. Wth respect to the coverage probabltes of the CIs, the frst result from Table 3 that should be noted s that, as expected from statstcal theory, the z dstrbuton CI calculated wth the optmal weghts was very close to the nomnal confdence level of 0.95 (mean observed coverage.950), as well as that obtaned wth the weghted varance CI (mean observed coverage.950). However, the CI calculated assumng an approxmate t dstrbuton wth the usual varance of ˆ UMVU overstated the nomnal confdence level (mean observed coverage.968), as dd the CI obtaned by the QA method (mean observed coverage.971). Therefore, good coverage may be obtaned when usng the optmal weghts by assumng an approxmate normal dstrbuton wth the usual varance, V(ˆ UMVU ), or from an approxmate t dstrbuton wth the weghted varance, V w (ˆ UMVU ). However, n real meta-analyses, the only weghts that can be obtaned are the estmated weghts, ŵ, whch have been calculated here for eght dfferent heterogenety varance estmators. As Table 3 shows, CIs based on the normal dstrbuton and the usual varance (Vˆ ˆ 1/ ŵ ) presented emprcal coverage probabltes clearly under the nomnal confdence level (mean estmated coverage probablty through the eght estmators:.935), whereas CIs based on the t dstrbuton and the usual varance obtaned emprcal coverages slghtly over the nomnal confdence level (mean estmated coverage probablty.957). On the one hand, the understatement of the nomnal confdence level found for the z dstrbuton CI concded wth the results of prevous smulaton studes (Brockwell & Gordon, 007; Follmann & Proschan, 1999; Hartung & Makamb, 003; Makamb, 004; Sdk & Jonkman, 00, 003, 005, 006). On the other hand, the slght overstatement of the nomnal confdence level found wth the t dstrbuton CI was smlar to that obtaned by Follmann and Proschan (1999) but dd not concde wth the slght understatement found by Brockwell and Gordon (007). The CIs obtaned by the quantle approxmaton method showed, n general, coverage probabltes slghtly over the nomnal confdence level (mean estmated coverage probablty:.961). In partcular, the mean actual coverage probablty obtaned wth the QA method when was estmated by the estmator was.959, a result slghtly over that reported by Brockwell and Gordon (007, p. 4540, Table III) of.951. The CIs based on the t dstrbuton and the

11 CONFIDENCE INTERVALS IN META-ANALYSIS 41 weghted varance of ˆ, Vˆ w(ˆ ), showed better coverage than that of the other CI procedures, wth a mean estmated coverage probablty through the eght estmators of.947. The weghted varance CI however, showed a slght but systematc understatement of the nomnal level for all of the estmators (wth the excepton of the estmator). The good coverage acheved by the weghted varance CI was coherent wth the results obtaned n prevous studes (Sdk & Jonkman, 00, 003, 006). Furthermore, the varablty n the coverage probabltes of the weghted varance CIs through the eght estmators was clearly smaller (SD.00) than those found wth the z dstrbuton, the t dstrbuton, and the QA method CIs (SDs.010,.008, and.007, respectvely). Ths fndng means that the weghted varance CI was less affected by changes n the estmator used to calculate the weghts than the z dstrbuton, t dstrbuton, and QA method CIs. In fact, the mean coverage probabltes for the eght estmators wth the weghted varance CI only ranged from.945 to.951, whereas z dstrbuton, t dstrbuton, and QA method CIs obtaned mean coverage probabltes that ranged from.94 ( and estmators) to.957 ( estmator), from.950 ( and estmators) to.974 ( estmator), and from.954 ( and estmators) to.977 ( estmator), respectvely. In the same way, Table 3 shows how the coverage probabltes of the weghted varance CIs obtaned wth a gven estmator through the 100 condtons were also less varable among them (wth standard devatons between.004 and.008) than were those obtaned for the z dstrbuton, the t dstrbuton, and the QA method CIs (wth standard devatons between.00 and.09, between.014 and.00, and between and.00, respectvely). These results confrm and extend those obtaned by Sdk and Jonkman (003), who used only three estmators (,, and estmators) versus the eght estmators examned here. Therefore, on average, the weghted varance CI yelded coverage probabltes closer to the nomnal confdence level wth a lower varablty and was less dependent on the estmators as compared wth the z dstrbuton, t dstrbuton, and QA method CIs. Whereas Table 3 shows the results for 0.5, Table 4 presents the same results for 0.8. Thus, by comparng the emprcal coverage probabltes n both tables, t s possble to assess whether changes n the locaton parameter have an effect on the performance of the dfferent CI procedures. For the four CI procedures, the emprcal coverage probabltes found for 0.8 were slghtly lower than were those obtaned for 0.5. The mean estmated coverage probabltes through the eght estmators for 0.8 were.931,.954,.94, and.957 for the z dstrbuton, t dstrbuton, weghted varance, and QA method CIs, respectvely. Thus, for 0.8, the best coverage was acheved by the t dstrbuton CI, followed by the QA method and the weghted varance CI. The systematc decrease n the coverage probabltes for 0.8 wth respect to those for 0.5 may be due to the slght negatve bas exhbted by the estmated mean effect szes, ˆ. The ncrease n the negatve bas of ˆ as ncreases s consstent wth the results found by Vechtbauer (005) and could be the reason for the decrease n the coverage probabltes. Apart from ths result, the weghted varance CI showed coverage probabltes that were less varable and less dependent on the estmator (standard devaton through the eght estmators.00) than were those for the z Table 4 Average Emprcal Coverage Probabltes Wth Standard Devatons Through the 100 Smulated Condtons for Each Confdence Interval Procedure and Estmator ( 0.8) z dstrbuton Confdence nterval procedure t dstrbuton Weghted varance QA method Weghts ˆ M SD M SD M SD M SD Optmal estmator Note. QA quantle approxmaton; Hunter and Schmdt estmator; Hedges estmator; DerSmonan and Lard estmator; Malzahn, Böhnng, and Hollng estmator; Hartung and Makamb estmator; Sdk and Jonkman estmator; maxmum lkelhood estmator; restrcted maxmum lkelhood estmator.

12 4 SÁNCZ-MECA AND MARÍN-MARTÍNEZ dstrbuton, t dstrbuton, and QA method CIs (SDs.01,.009, and.008, respectvely). In our smulatons, we manpulated the heterogenety varance,, wth values 0, 0.04, 0.08, 0.16, and 0.3. One of the man problems of the z dstrbuton CI s that ts coverage probablty decreases under the nomnal level as the heterogenety varance ncreases. Fgure 1 shows the emprcal coverage probabltes of the 36 CI procedures as a functon of for 0.5, and Table 5 presents the emprcal coverage probabltes for the two most extreme values tested here: 0 and 0.3. As expected from prevous studes, Fgure 1A shows how the emprcal coverage probablty for CIs calculated assumng a normal dstrbuton and estmated weghts systematcally decreased as ncreased, regardless of the heterogenety varance estmator used n calculatng the weghts. As Table 5 shows, the mean coverage probablty through the eght estmators for 0.3 was.90. Only when 0 dd ths CI procedure yeld good coverage for all of the estmators (mean estmated coverage.963), wth the excepton of the and estmators, whch overstated the nomnal confdence level (Ms.977 and.984, respectvely). The overstatement of the nomnal confdence level obtaned wth the and estmators was due to the fact that both of them are nonnegatve estmators of and, as a consequence, when 0, they are postvely based, leadng to CIs that are too wde. For 0.04 and wth the excepton of the estmator, the coverage probabltes for ths CI procedure were nadmssbly under.94. Therefore, as ncreases, the wdth of the CIs for the z dstrbuton method becomes too narrow, wth the consequence that the actual coverage s under the nomnal level. The t dstrbuton CI yelded coverage probabltes that also decreased as ncreased for all the heterogenety varance estmators (Fgure 1B and Table 5). In contrast to the z dstrbuton CI, however, as ncreased, the actual coverage probablty got closer to the nomnal level. The unadjustment of the emprcal coverage to the nomnal level was more pronounced for small values. Thus, for 0, the t dstrbuton CI obtaned coverage probabltes nadmssbly larger than the nomnal level (mean estmated coverage through the eght estmators.977). For 0.3, the t dstrbuton CI showed good coverage (mean estmated coverage probablty.947). Therefore, assumng a t dstrbuton and the usual formula for the samplng varance seems to offer good coverage for large values. The results found for the QA method were very smlar to those of the t dstrbuton CI: a better adjustment of the emprcal coverage to the nomnal level as ncreased (Fgure 1C and Table 5). In partcular, for 0.3, the QA method acheved very good coverage (mean emprcal coverage through the eght estmators.950), even slghtly better than that of the t dstrbuton CI. For 0, the mean coverage was clearly over the nomnal level (mean est- A Emprcal coverage probablty B Emprcal coverage probablty C Emprcal coverage probablty D Emprcal coverage probablty Heterogenety varance Heterogenety varance Heterogenety varance Heterogenety varance Fgure 1. Average emprcal coverage probabltes as a functon of the parameter, for the four confdence nterval (CI) procedures wth the eght heterogenety varance estmators and the optmal weghts ( 0.5). A: z dstrbuton CI. B: t dstrbuton CI. C: quantle approxmaton method. D: weghted varance CI. Hunter and Schmdt estmator; Hedges estmator; DerSmonan and Lard estmator; Malzahn, Böhnng, and Hollng estmator; Hartung and Makamb estmator; Sdk and Jonkman estmator; maxmum lkelhood estmator; restrcted maxmum lkelhood estmator.

13 CONFIDENCE INTERVALS IN META-ANALYSIS 43 Table 5 Average Emprcal Coverage Probabltes Wth Standard Devatons Through the 100 Smulated Condtons for Each Confdence Interval Procedure and Two Values of (0.5) Confdence nterval procedure z dstrbuton t dstrbuton Weghted varance QA method Weghts M SD M SD M SD M SD 0 Optmal estmator Optmal estmator Note. QA quantle approxmaton; Hunter and Schmdt estmator; Hedges estmator; DerSmonan and Lard estmator; Malzahn, Böhnng, and Hollng estmator; Hartung and Makamb estmator; Sdk and Jonkman estmator; maxmum lkelhood estmator; restrcted maxmum lkelhood estmator. mated coverage.979) and slghtly worse than that of the t dstrbuton CI. The results for the QA method are smlar to those found by Brockwell and Gordon (007). The smlarty of the results found for the t dstrbuton and the QA method CIs s due to the fact that they only dffer n the crtcal value used to calculate the CI: a crtcal value from a Student t dstrbuton wth k 1 degrees of freedom and a quantle estmated from Equaton 14 that s a functon of k, respectvely. For example, for k 10, the respectve crtcal values are.6 and.374. As both procedures propose the same samplng varance for the overall effect sze, the wdth of the correspondng CIs s very smlar and, as a consequence, the estmated coverage probabltes are also very close to each other. In any case, as ncreases, the QA method seems to offer slghtly better coverage than that of the t dstrbuton CI. Whereas the z dstrbuton, t dstrbuton, and QA method CIs exhbted emprcal coverages that were affected by the value of, the weghted varance CI acheved good coverage regardless of the value of and the estmator (Fgure 1D), although always wth a coverage probablty slghtly under the nomnal confdence level. Even for 0, the weghted varance CI outperformed the z dstrbuton, t dstrbuton, and QA method CIs. In fact, as Table 5 shows, for 0, the mean estmated coverage probablty of weghted varance CIs through the eght estmators was.949, whereas those of z dstrbuton, t dstrbuton, and QA method CIs were.963,.977, and.979, respectvely. As a consequence, the weghted varance CI may be appled even for small values of. For 0.3, the mean emprcal coverage for the weghted varance CI was.947, smlar to that of the t dstrbuton CI and slghtly under that of the QA method. As Table 5 shows, the observed coverage probablty wth the weghted varance CI was less varable for each estmator and through the eght estmators than were those of the other three CI procedures. It seems that the mproved formula for estmatng the samplng varance proposed by Hartung (1999), together wth the use of crtcal values from a Student t dstrbuton, enables one to approprately accommodate the uncertanty due to estmatng the between-studes and wthn-study varances. Wth the optmal weghts, the coverage probablty of the four CI procedures was not affected by the value of but,

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