Exact Inference on the Random-Effects Model for. Meta-Analyses with Few Studies

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1 Bimetria (2017), xx, x, pp Prited i Great Britai Exact Iferece the Radm-Effects Mdel fr Meta-Aalyses with Few Studies BY H. MICHAEL Departmet f Statistics, Stafrd Uiversity habe.michael@gmail.cm S. THORNTON, M. XIE Departmet f Statistics, Rutgers Uiversity AND L. TIAN Departmet f Bimedical Data Sciece, Stafrd Uiversity 1. INTRODUCTION The radm effects mdel is fte used t accut fr betwee-study hetergeeity whe cductig a meta-aalysis. Whe the distributi f the primary study treatmet effect estimates is apprximately rmal, the simple rmal-rmal mdel is cmmly used, ad the DerSimia-Laird ( DL ) methd ad its variatis are the mst ppular apprach t estimatig the mdel s parameters ad perfrmig statistical iferece (DerSimia & Laird, 1986). Hwever, the DL methd is based a asympttic apprximati ad its use is ly justified whe the umber f studies is large. I may fields, the umber f studies used i a meta-aalysis r sub-meta-aalysis rarely exceeds 20 ad is typically fewer tha 7 (Davey et al., 2011), leavig ifereces based the DL estimatr questiable. Ideed, extesive simulati studies have C 2016 Bimetria Trust

2 2 fud that the cverage prbability f the DL-based cfidece iterval (CI) ca be substatially lwer tha the mial level i varius settigs (Ktpatelis et al., 2010; ItHut et al., 2014), leadig t false psitives. Oe reas fr the failure f the DL methd is that the asympttic apprximati igres the variability i estimatig the hetergeeus variace, which ca be substatial whe the umber f studies is small (Higgis et al., 2009). Varius remedies have bee prpsed t crrect the uder-cverage f DL-based cfidece itervals. Hartug & Kapp (2001) prpsed a ubiased estimatr f the variace f the DL pit estimatr explicitly accutig fr the variability i estimatig the hetergeus variace. Sidi & Jma (2006) used the heavy-tailed t-distributi t apprximate the distributi f a mdified Wald-type test statistic based the DL estimatr. Usig the mre rbust t- rather tha rmal distributi has als bee prpsed (Berey et al., 1995; Raghuatha, 1993; Fllma & Prscha, 1999). Hardy & Thmps (1996); Vagel & Ruhi (1999); Viechtbauer (2005); ad Raudebush (2009) prpsed prcedures based maximum-lielihd estimati. Nma (2011) further imprved the perfrmace f the lielihd-based iferece prcedure whe the umber f study is small by usig a Bartlett-type crrecti. Bayesia appraches icrpratig exteral ifrmati have bee develped by may authrs (Smith et al., 1995; Higgis & Whitehead, 1996; Bdar et al., 2017). Hwever, with few exceptis, mst f these methds still deped a asympttic apprximati ad their perfrmace with very few studies has ly bee examied by specific simulati studies. T vercme these difficulties, ptetially cservative but exact iferece prcedures fr the radm effects mdel have bee prpsed (Fllma & Prscha, 1999; Wag et al., 2010; Liu et al., 2017) ad Wag & Tia (2017). A permutati rather tha the asympttic limitig distributi is used t apprximate the distributi f the relevat test statistics ad thus the validity f the assciated iferece is guarateed fr ay umber f studies. Hwever, due t the discreteess f the permutati distributi, the

3 3 highest sigificace level that may be achieved withut radmizati depeds the umber f studies. Fr example, a 95% cfidece iterval ca ly be cstructed with mre tha 5 studies. The mai ctributi f this paper is t prpse a set f ew methds fr cstructig exact, ucditial, -radmized CIs fr the lcati parameter f the rmal-rmal mdel by ivertig exact tests. The cverage level f the resultig CI is guarateed t be abve the mial level, up t Mte Carl errr, as lg as the meta-aalysis ctais mre tha 1 study. After emplyig several techiques t accelerate cmputati, the ew CI ca be easily cstructed a persal cmputer. Simulatis suggest that the prpsed CI typically is t verly cservative. I Secti 2, we preset ur prcedure fr cstructig exact CIs fr the ppulati mea; i Secti 3, we reprt results frm cmprehesive simulati studies; i Secti 4, we illustrate the prpsed methd with a real data example; ad i Secti 5 we cclude the paper with additial discussi. 2. METHOD The bserved data csist f Y 0 = {Y, = 1,, K}, where Y fllws a radm effects mdel, id. Y θ N(θ, σ 2 ), θ id. N(µ 0, τ 2 0 ), = 1,, K, with the variaces σ > 0, = 1,..., K, assumed w.the radm effects mdel implies The simple parametric mdel id. Y N(µ 0, σ 2 + τ2 0 ), = 1,, K. (1) I the ctext f a meta-aalysis, the pairs (Y, σ 2 ), = 1,..., K, are iterpreted as bserved effects ad w withi-study variaces draw frm K studies, respectively. The ubserved

4 4 ppulati effect ad betwee-study variace are µ 0 ad τ 2 0, respectively. The gal is iferece the lcati parameter µ 0, viewig τ 2 0 as a uisace parameter. The typical umber f studies depeds the area f research ad ca be small, e.g., K 10. by With τ 2 0 w, the uifrmly miimum variace ubiased estimatr f µ 0 uder (1) is give K=1 Y (τ σ2 ) 1 K=1 (τ σ2 ) 1. As τ 2 0 is uw, DerSimia & Laird (1986) prpse substitutig a simplified methd f mmets estimatr, where K=1 ˆτ 2 DL = max (Y ˆµ F ) 2 /σ 2 (K 1) 0,, K=1 K=1 σ 4 K=1 ˆµ F = Ki=1 Y Ki=1 is the miimum variace ubiased estimatr f µ 0 uder a fixed effects mdel, i.e., whe τ 2 0 = 0. The resultig estimatr is w as the DerSimia-Laird estimatr f µ 0 : ˆµ DL = K=1 Y (ˆτ 2 DL + σ2 ) 1 K=1 (ˆτ 2 DL + σ2 ) 1. By a aalgus substituti, a level 1 α cfidece iterval fr µ 0 is give by 1/2 1/2 K K ˆµ DL z 1 α/2 (ˆτ 2 DL + σ2 ) 1, ˆµ DL + z 1 α/2 (ˆτ 2 DL + σ2 ) 1. (2) =1 The justificati f the CI give i (2) relies the asympttic apprximati T 0 (µ 0 ; Y) = (ˆµ DL µ 0 ) 2 =1 K (ˆτ 2 DL + σ2 ) 1 χ 2 1 (3) as the umber f studies, K, grws t ifiity ad max{σ }/ mi{σ } is uifrmly buded. Hwever, whe K is mderate r small, the distributi f T 0 (µ 0 ; Y) depeds τ 2 0 ad may be very differet frm a χ 2 1 distributi. Csequetly, the fiite-sample perfrmace f the CI give =1

5 5 by (2) is fte usatisfactry. We prpse cstructig a exact CI fr µ0 by first cstructig a exact cfidece regi fr (µ0, τ20 ). T this ed, let T (µ, τ2 ); Y0 dete a test statistic, which may deped the ull parameter (µ, τ2 ), fr the simple hypthesis (µ0, τ20 ) = (µ, τ2 ). The specific chice f T (µ, τ2 ); Y0 will be discussed later ad here we ly assume that a high value f T (µ, τ2 ); Y0 represets gruds fr rejecti. Fr a give chice f T (µ, τ2 ); Y0, a 1 α level CI fr µ0 ca be cstructed as fllws: 1. Obtai buds [µmi, µmax ] ad [τ2mi, τ2max ] fr µ0 ad τ Fr each pair f µ ad τ2 i a R R grid f pits [µmi, µmax ] [τ2mi, τ2max ], a. Cmpute the ull distributi f T (µ, τ2 ); Y0, i.e., the distributi f T (µ, τ2 ); Y(µ, τ2 ), where e, = 1,, K Y(µ, τ2 ) = Y id. e N(µ, σ2 + τ2 ), = 1,, K. with Y i h b. Cmpute the p-value pµ,τ2 (Y0 ) := P T (µ, τ2 ); Y0 > T (µ, τ2 ); Y(µ, τ2 ). 3. Obtai a cfidece regi fr (µ0, τ20 ) as Ω1 α (Y0 ) := {(µ, τ2 ) : pµ,τ2 (Y0 ) > α}. 4. Prject Ω1 α (Y0 ) t the µ axis t btai a CI fr µ0 : {µ : (µ, τ2 ) Ω1 α (Y0 )}. This methd geerates the exact CI fr µ0 i the sese that pr µ0 {µ : (µ, τ2 ) Ω1 α (Y0 )} 1 α.

6 6 This is due t the fact that pr µ0 {µ : (µ, τ2 ) Ω1 α (Y0 )} pr (µ0, τ20 ) Ω1 α (Y0 ) =pr pµ0,τ2 (Y0 ) α 0 =pr(u α) = 1 α, where the radm variable U fllws the uit uifrm distributi. Here, we assume that τ0 [τ2mi, τ2max ]. If τ2mi ad τ2max are chse depedig the data i such a way that pr(τ2mi < τ2 < τ2max ) 1 β, the the guarateed cverage prbability f the prpsed CI is 1 α β 1 α fr very small β. The cumulative distributi fucti (CDF) f T (µ, τ2 ); Y(µ, τ2 ) may t be aalytically tractable, but it is well defied fr ay give grid pit (µ, τ2 ) ad ca always be apprximated by a Mte Carl simulati. T be specific, give (µ, τ2 ), we may apprximate the distributi f T (µ, τ2 ); Y(µ, τ2 ) i 2a as fllws: 2a Fr b = 1,, B, id. a. Geerate e 1b,, e Kb N(0, 1). = µ + (σ2 + τ2 )1/2 e, b. Let Yb = 1,, K, ad let Yb = b, = 1,, K. Yb c. Let T b = T (µ, τ2 ); Yb be the crrespdig test statistic based the ge- erated data Yb. The empirical distributi f {T 1,, T B } ca be used t apprximate the distributi f T (µ, τ2 ); Y(µ, τ2 ). Sice the estimati f the ull distributi i 2a des t deped ay asympttic apprximati, bth the p-value, pµ,τ2 (Y0 ), ad the cfidece regi, Ω1 α (Y0 ), are exact if we ca

7 7 safely igre the errrs f the grid apprximati ad the Mte Carl simulati abve, which ca be ctrlled by icreasig the grid desity ad B i step 2a, respectively. Because the data Y, = 1,, K, are distributed as N(µ, σ2 + τ20 ), = 1,..., K, wheever the shifted data Y µ, = 1,..., K, are distributed as N(0, σ + τ20 ), = 1..., K, we restrict ur fcus t equivariat statistics (Lehma & Rma, 2006), that is, T satisfyig T (µ, τ2 ); Y0 = T (0, τ2 ), Y0 µ, where Y0 µ = {Y µ, = 1,..., K}. I this situati, testig the ull H0 : (µ0, τ20 ) = (µ, τ2 ) based the data Y0 is the same as testig the ull H0 : (µ0, τ20 ) = (0, τ2 ) based the shifted data Y0 µ. Whe the test statistic is equivariat, the cmputatis i step 2a eed ly be perfrmed ce fr each τ2 i the grid rather tha each pair (µ, τ2 ). Thus, althugh a 2-dimesial grid is used i the algrithm, the cmputatial cmplexity remais liear i the grid size, R. Mre specifically, steps 2 3 becme: 20. Fr each τ2 f a R-sized grid [τ2mi, τ2max ], a. Cmpute the distributi f T (0, τ2 ); Y(0, τ2 ). b. Cmpute q1 α;τ2, the 1 α quatile f T (0, τ2 ); Y(0, τ2 ). c. Cmpute Ω1 α (τ2 ; Y0 ) = {(µ, τ2 ) T (µ, τ2 ); Y0 = T (0, τ2 ); Y0 µ q1 α;τ2 }. 30. Cmpute a (1 α)-level cfidece regi fr (µ0, τ20 ) as [ Ω1 α (τ2 ; Y0 ). τ2 [τ2mi,τ2max ] I this paper, we prpse usig the test statistics T (µ, τ2 ); Y0 = T 0 (µ; Y) + c0 T li (µ, τ2 ); Y, (4)

8 8 where T 0 (µ; Y) is the same Wald-type test statistic used i the Dersimia-Laird prcedure, T li { (µ, τ 2 ); Y } = 1 2 K (Y ˆµ DL ) 2 ˆτ 2 DL + σ2 =1 + lg { 2π(ˆτ 2 DL + σ2 )} + K =1 1 2 (Y µ) 2 τ 2 + σ 2 + lg { 2π(τ 2 + σ 2 )}, ad c 0 is a tuig parameter ctrllig the relative ctributis f these tw statistics. While T 0 (µ; Y) directly fcuses the lcati parameter µ 0, T li { (µ, τ 2 ); Y }, similar t the lielihd rati test statistic, targets the cmbiati f µ 0 ad τ 2 0 ad helps t cstruct a arrwer CI f µ 0 whe the umber f studies is small. The prpsed test statistics satisfy the equivariace cditi, esurig speedy cmputati whe carryig ut the prcedure a typical persal cmputer. A further simplificati affrded by this chice f test statistics is that step 2 c may be carried ut by slvig a simple quadratic iequality: A(τ)µ B(τ)µ 0 + C(τ) < 0, where K 1 c 0 A(τ) = ˆτ 2 =1 DL + + σ2 2(τ 2 + σ 2 ) > 0, K 2µ ˆ 0 DL B(τ) = + c 0Y 2 =1 τˆ 0DL + σ 2 τ 2 + σ 2, K c 0 Y 2 C(τ) = 2 τ 2 + σ 2 + lg τ2 + σ 2 ˆτ 2 DL + (Y ˆµ DL ) 2 σ2 ˆτ 2 DL + + ˆµ2 σ2 DL =1 K =1 1 ˆτ 2 DL + σ2 q 1 α;τ 2. (5) As a result, the cfidece iterval f µ 0 whe τ 0 = τ Ω 1 α (τ 2 ; Y 0 ), is simply the segmet with edpits ( B(τ) ± 1/2 2A(τ), τ 2 ), whe (τ) = B(τ) 2 4A(τ)C(τ) 0, ad a empty set, therwise.

9 T chse τ 2 mi ad τ2 max i step 1 f the algrithm, we may use the edpits f a 100(1 β)%, e.g., 99.9%, cfidece iterval f τ 2 0. This CI ca be cstructed by ivertig the pivtal statistic 9 T 3 (τ 2 ) = (WY) { WΣ(τ)W } 1 (WY), where Y = (Y 1,, Y K ), Σ(τ) = diag { σ τ2,, σ 2 K + τ2}, ad 1 / K i=1 i 1 2 / K i=1 i K / K i=1 i W = 1 / K i=1 i 2 / K i=1 i 1 K / K i=1 i. 1 / K i=1 i 2 / K i=1 i K / K i=1 i 1 The pivt fllws a χ 2 K 1 distributi whe τ2 = τ 2 0. Sice ur gal is a CI fr µ 0, the shape f the cfidece regi is crucial t its perfrmace: the prjecti f Ω 1 α (Y 0 ) t the µ axis shuld be as small as pssible, relative t the area f the cfidece regi. Figure 1 plts tw cfidece regis with the same cfidece cefficiet, but substatially differet prjected legths. T avid a verly cservative CI, we prefer a cfidece regi with budaries parallel t the τ axis, r early s. The shape f { Ω 1 α (X 0 ) is determied by the way we cmbie T 0 (µ; Y) ad T li (µ, τ 2 ); Y } r, mre geerally, by the chice f T { (µ, τ 2 ); Y }. Because the prpsed statistics (4) are quadratic i µ, the resultig cfidece regis are a ui f itervals with similar ceters ad ted t t prduce verly cservative CIs whe the tuig parameter c 0 is chse apprpriately. The prpsed test statistic was chse t balace perfrmace ad cmputati csts. Fr example, the true lielihd rati test statistic uder mdel (1) may be mre ifrmative tha { T li (µ, τ 2 ); Y }, but its evaluati ivlves cmputig the maximum lielihd estimate ad is substatially slwer. The prpsed algrithm is easily parallelized, s further gais i cmputig speed are available.

10 10 Fig. 1: The prjecti f the cfidece regi; the slid ad dashed thic lies are budaries f tw cfidece regis. Remar 1. Prjectig the cfidece regi parallel t the directi f the uisace parameter is a gemetric iterpretati f a well-w apprach t cstructig -radmized, ucditial, exact tests i the presece f uisace parameters. I geeral, give a parameter f iterest, θ, ad uisace parameter, η, let p θ,η (Y 0 ) dete the p-value fr testig the ull hypthesis H 0 : (θ 0, η 0 ) = (θ, η) cditial the bserved data, Y 0. A exact level α test fr

11 11 the cmpsite ull hypthesis H 0 : θ 0 = θ rejects the ull if sup η p θ,η (Y 0 ) < α. This test is cservative by cstructi. A crrespdigly cservative CI may be btaied by iversi as {θ : sup η p θ,η (Y 0 ) > α} = {θ : p θ,η (Y 0 ) > α fr sme η}, i.e., the prjecti described i (2). See, e.g., Suissa & Shuster (1985) fr a applicati t cmparig prprtis frm tw idepedet bimial distributis. 3. NUMERICAL STUDY I this secti, we study the small-sample perfrmace f the prpsed methd thrugh a cmprehesive simulati study. Observed data are simulated uder the radm effects mdel Y N(µ, τ σ2 ), = 1,, K, where σ 1,, σ K, are K equally spaced pits i the iterval [1, 5], that is, σ = 1 + 4( 1)/(K 1), = 1,, K. The ppulati variace τ 2 0 taes values 0, 12.5, ad 25 t mimic settigs with lw, mderate, ad high study hetergeeity, respectively. The crrespdig I 2 measures f hetergeeity are apprximately 0, 50%, ad 70%, respectively. I the first set f simulatis, we examie the effect f the tuig parameter c 0 the perfrmace f the prpsed methd. Fr each set f simulated data, we cstruct a series f CIs usig the prpsed methd with c 0 ragig frm 0 t 2.5 i icremets f 0.1, ad the umber f studies K rages frm 3 t 20. Based results frm simulated datasets uder each cmbiati f settigs, we calculate the empirical cverage levels ad average legths f the resultig 95% CIs. I all settigs, the empirical cverage levels f the prpsed CIs are abve the mial level ad therefre we ptimize pwer by selectig the value f c 0 with the shrtest CI legths. Whe K 10, the chice f c 0 des t have a pruced effect CI legth. Whe K is betwee 3 ad 6, the settig f primary iterest, assigig mre weight t the lielihd ratitype statistic typically reduces the legth f the CIs. We summarize the value f c 0 achievig

12 12 the miimum mea 95% CI legth i Figure 2. Based these results, we suggest fr a tuig parameter c 0 = 1.2 fr meta-aalyses with fewer tha 6 studies, c 0 = 0.6 fr meta-aalysess with 6 10 studies, c 0 = 0.2 fr meta-aalysis with studies, ad c 0 = 0 fr aalysis with mre tha 20 studies. I the secd set f simulatis, we cmpare the perfrmace f the prpsed CIs with existig alteratives. Fr replicates at each data-geerati settig described abve, we cstruct CIs usig the DerSimia-Laird, Sidi-Jma, ad restricted maximum lielihd asympttic variace estimates, as well as the prpsed CI with the recmmeded tuig parameter. I Figure 3 we summarize the average cverage ad legths f these CIs. I the presece f mderate hetergeeity, I 2 = 0.5, the empirical cverage level f the DL methd is belw 90% whe K 10, with the lwest cverage 75% whe the umber f studies is 3. The CIs based the Sidi-Jma estimatr have better cverage, but still drp belw 90% whe K 5. I ctrast, the prpsed exact CIs usig the recmmeded tuig parameter settigs d t fall belw the mial 95% cverage level. Mrver, the cverage level is t verly cservative eve fr small Ks. The legth f the 95% CI is cmparable t the legths f the asympttic CIs, whe these match the mial cverage level, e.g., K = 20. Whe I 2 = 0, i.e., the radm effects mdel degeerates t the fixed effects mdel, all methds, icludig the asympttic estimatrs, ctrl the Type 1 errr. Sidi-Jma s CI is verly cservative eve fr mderate K values, while the prpsed CIs, als verly cservative at lwer values f K, imprve steadily as K icreases. Whe I 2 = 0.70, ly the prpsed CIs maitai the prper cverage level, while all methds fall belw the mial level fr K as large as Several ther estimatrs, icludig Hedges-Oli, Huter-Schmidt, ad maximum lielihd, were als tested, with perfrmace fud t be geerally itermediate betwee the perfrmace f the DerSimia-Laird ad Sidi-Jma estimatrs.

13 13 Fig. 2: The chice f c 0 achievig the miimum mea 95% CI legth is pltted agaist the umber K f studies, at 3 levels f betwee-study hetergeeity. 4. EXAMPLE Tai et al. (2015) cduct a radm effects meta-aalysis f 59 radmized ctrlled trials t determie if icreased calcium itae affects be mieral desity ( BMD ). Altgether, these trials measured the chages i BMD at five seletal sites ver three time pits ad measured

14 14 Fig. 3: Cmparis by 95% CI cverage ad legth f the prpsed estimatr with 3 cmmly used estimatrs based asympttic apprximatis. Data was geerated accrdig t mdel (1) with the umber f studies K varyig betwee 3 ad 20 ad the rati f betwee- t average withi-variace adjusted t give 3 levels f betwee-study hetergeeity. The prpsed estimatr achieves the mial size at all cfiguratis, with vercverage evidet where the hetergeeity is lw r the studies is very few (3 4). the effect f calcium itae BMD frm dietary surces ad frm calcium supplemets. We illustrate the prpsed methd usig fur meta-aalyses. The first meta-aalysis ivestigates chages i BMD f the lumbar spie ad is based the fidigs f 27 trials that lasted fewer tha 18 mths. As shw i Table 1, the 95% CI prduced by the prpsed exact methd des t differ very much frm the 95% CI based the DL methd. The tw itervals have a similar

15 15 legth ad are cetered arud a BMD differece f abut 1.2. We als cstruct the exact CI by permutig a Hdge-Lehma type estimatr (Liu et al., 2017). The resultig iterval is very similar t the iterval prduced by the prpsed methd. These similarities are t be expected sice the rmality assumptis f the DL estimatr may t be t ureasable fr a metaaalysis based such a large umber f primary studies. Tw f the ther radm effects meta-aalyses ivestigate chages i BMD i the hip ad frearm fr trials f size six ad five, respectively, that lasted fr mre tha tw years. The furth aalysis we csider here is the meta-aalysis f three trials that lasted fewer tha 18 mths ad measured chages i BMD fr the ttal bdy f subjects. Fr these three metaaalyses, hwever, the umber f studies is small, ad the DL methd may be expected t fall shrt f the mial level. I the hip study, the prpsed exact methd ad the DL methd bth yield the same cclusi, prducig 95% cfidece itervals rejectig the ull f chage i BMD, althugh the exact methd prduces cfidece itervals that are wider tha their DL cuterparts. I ctrast, the DL 95% cfidece itervals fr the frearm ad ttal bdy studies fid a sigificat chage i BMD whereas the exact methd des t, suggestig that the DL methd may be givig a false psitive i these tw cases. The itervals ad their legths are give i Table 1. Nte that the exact 95% CI based the permutati methd is t available fr the last tw meta aalyses, sice the umber f studies is fewer tha DISCUSSION We have prpsed a methd t cstruct a exact CI fr the ppulati mea uder the rmalrmal mdel cmmly used i meta-aalysis. Apprpriate cverage is guarateed, up t Mte Carl errr, eve whe the umber f studies used i the meta-aalysis is as small as 2. While cveiet, the rmal assumpti fr the study-specific treatmet effect estimate may t be

16 16 Study K DerSimia-Laird Permutati Prpsal lumbar spie (0.841) (0.970) (0.958) ttal hip (1.345) (2.298) (2.087) frearm (3.169) (4.583) ttal bdy (1.511) (3.536) Table 1: Radm effects meta-aalyses f the effect f calcium supplemets percetage chage i be mieral desity (Tai et al. (2015), Figs. 1, 3, ad 7). The meta-aalyses were carried ut usig the DerSimia-Laird variace estimatr (as i Tai et al. (2015)), the permutati test f Wag & Tia (2017), applicable t meta-aalyses with 6 r mre studies, ad the prpsed exact methd. O the tw smaller meta-aalyses (K = 3, 5) the prpsed exact methd fails t reject the ull f chage, whereas the asympttic DL methd des reject. valid i ther settigs. Fr example, the treatmet effect estimate may be a dds rati frm a 2x2 ctigecy table. If the ttal sample sizes are small r if cell etries are clse t 0, the rmal assumpti fr the dds rati may be iapprpriate. Mre geerally, Y may be a quatity relevat t a treatmet effect with Y θ fllwig a -rmal, e.g., hypergeemtric, distributi depedig the study-specific parameter θ. I such a case, the mdel fr θ ad the crrespdig iferece prcedure warrat further research. Mre recetly, there have bee several ew develpmets cfidece distributi ad related geeralized fiducial iferece that have facilitated ew iferece prcedures fr meta-aalysis (Xie & Sigh, 2013; Claggett et al., 2014). These develpmets may als be prmisig directis fr develpig exact iferece prcedures fr meta-aalysis.

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