136:. It is difficult to establish the validity of any distributional assumption, and this is a common criticism of random effects meta-analyses. However, variations of the exact distributional form may not make much of a difference, and simulations have shown that methods are relatively robust even under extreme distributional assumptions, both in estimating heterogeneity, and calculating an overall effect size.
268:); with this normalisation however, it is not quite obvious what exactly would constitute "small" or "large" amounts of heterogeneity. For a constant heterogeneity (τ), the availability of smaller or larger studies (with correspondingly differing standard errors associated) would affect the I² measure; so the actual interpretation of an I² value is not straightforward.
132:. The model represents the lack of knowledge about why treatment effects may differ by treating the (potential) differences as unknowns. The centre of this symmetric distribution describes the average of the effects, while its width describes the degree of heterogeneity. The obvious and conventional choice of distribution is a
151:
Common meta-analysis models, however, should, of course, not be applied blindly or naively to collected sets of estimates. In case the results to be amalgamated differ substantially (in their contexts or in their estimated effects), a derived meta-analytic average may eventually not correspond to a
84:
Reasons for the additional variability are usually differences in the studies themselves, the investigated populations, treatment schedules, endpoint definitions, or other circumstances ("clinical diversity"), or the way data were analyzed, what models were employed, or whether estimates have been
56:
is a method used to combine the results of different trials in order to obtain a quantitative synthesis. The size of individual clinical trials is often too small to detect treatment effects reliably. Meta-analysis increases the power of statistical analyses by pooling the results of all available
143:
to the model has the effect of making the inferences (in a sense) more conservative or cautious, as a (non-zero) heterogeneity will lead to greater uncertainty (and avoid overconfidence) in the estimation of overall effects. In the special case of a zero heterogeneity variance, the random-effects
60:
As one tries to use meta-analysis to estimate a combined effect from a group of similar studies, the effects found in the individual studies need to be similar enough that one can be confident that a combined estimate will be a meaningful description of the set of studies. However, the individual
226:
While many of these estimators behave similarly in case of a large number of studies, differences in particular arise in their behaviour in the common case of only few estimates. An incorrect zero between-study variance estimate is frequently obtained, leading to a false homogeneity assumption.
598:
1011:
Veroniki, A.A.; Jackson, D.; Viechtbauer, W.; Bender, R.; Bowden, J.; Knapp, G.; Kuß, O.; Higgins, J.P.T.; Langan, D.; Salanti, G. (2016), "Methods to estimate the between-study variance and its uncertainty in meta-analysis",
1061:
Röver, C.; Bender, R.; Dias, S.; Schmid, C.H.; Schmidli, H.; Sturtz, S.; Weber, S.; Friede, T. (2021), "On weakly informative prior distributions for the heterogeneity parameter in
Bayesian random‐effects meta‐analysis",
31:. In a simplistic scenario, studies whose results are to be combined in the meta-analysis would all be undertaken in the same way and to the same experimental protocols. Differences between outcomes would only be due to
922:
Davey, J.; Turner, R.M.; Clarke, M.J.; Higgins, J.P.T. (2011), "Characteristics of meta-analyses and their component studies in the
Cochrane Database of Systematic Reviews: a cross-sectional, descriptive analysis",
275:
along with a confidence interval for the main effect may help getting a better sense of the contribution of heterogeneity to the uncertainty around the effect estimate.
156:. When individual studies exhibit conflicting results, there likely are some reasons why the results differ; for instance, two subpopulations may experience different
1450:
Borenstein, M.; Hedges, L. V.; Higgins, J. P. T.; Rothstein, H. R. (2010), "A basic introduction to fixed-effect and random-effects models for meta-analysis",
430:
685:
Cornell, John E.; Mulrow, Cynthia D.; Localio, Russell; Stack, Catharine B.; Meibohm, Anne R.; Guallar, Eliseo; Goodman, Steven N. (2014-02-18).
351:
Singh, A.; Hussain, S.; Najmi, A.N. (2017), "Number of studies, heterogeneity, generalisability, and the choice of method for meta-analysis",
518:
440:
901:
245:(its square root) by τ. Heterogeneity is probably most readily interpretable in terms of τ, as this is the heterogeneity distribution's
183:
of such tests especially in the very common case of only few estimates being combined in the analysis, as well as the specification of
1533:
414:
1212:
Rücker, G.; Schwarzer, G.; Carpenter, J.R.; Schumacher, M. (2008), "Undue reliance on I² in assessing heterogeneity may mislead",
121:. Unfortunately, literature-based meta-analysis may often not allow for gathering data on all (potentially) relevant moderators.
106:
501:
Bretthorst, G.L. (1999), "The near-irrelevance of sampling frequency distributions", in von der Linden, W.; et al. (eds.),
1380:
599:"Performance of statistical methods for meta-analysis when true study effects are non-normally distributed: A simulation study"
397:(2001), "Effect measures for meta-analysis of trials with binary outcomes", in Egger, M.; Davey Smith, G.; Altman, D. (eds.),
220:
1259:
742:"Is meta-analysis of RCTs assessing the efficacy of interventions a reliable source of evidence for therapeutic decisions?"
105:
In case the origin of heterogeneity can be identified and may be attributed to certain study features, the analysis may be
109:(by considering subgroups of studies, which would then hopefully be more homogeneous), or by extending the analysis to a
261:
1114:
Friede, T.; Röver, C.; Wandel, S.; Neuenschwander, B. (2017), "Meta-analysis of few small studies in orphan diseases",
1367:
1168:
Higgins, J. P. T.; Thompson, S. G.; Deeks, J. J.; Altman, D. G. (2003), "Measuring inconsistency in meta-analyses",
1609:
264:). I² relates the heterogeneity variance's magnitude to the size of the individual estimates' variances (squared
69:. The presence of some heterogeneity is not unusual, e.g., analogous effects are also commonly encountered even
284:
179:
or related test procedures. This common procedure however is questionable for several reasons, namely, the low
42:
37:
969:
Li, W.; Liu, F.; Snavely, D. (2020), "Revisit of test‐then‐pool methods and some practical considerations",
219:
is available. Bayesian estimation of the heterogeneity usually requires the specification of an appropriate
174:
45:
in outcomes that goes beyond what would be expected (or could be explained) due to measurement error alone.
1572:
Sutton, A. J.; Abrams, K. R.; Jones, D. R. (2001), "An illustrated guide to the methods of meta‐analysis",
1614:
331:
540:"A re-analysis of the Cochrane Library data: The dangers of unobserved heterogeneity in meta-analyses"
334:, in Higgins, J.P.T.; Thomas, J.; Chandler, J.; Cumpston, M.; Li, T.; Page, M.J.; Welch, V.A. (eds.),
1318:
1287:
551:
459:
Riley, R. D.; Higgins, J. P.; Deeks, J. J. (2011), "Interpretation of random-effects meta-analyses",
289:
125:
114:
250:
145:
133:
62:
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Higgins, J.P.T.; Thomas, J.; Chandler, J.; Cumpston, M.; Li, T.; Page, M.J.; Welch, V.A. (2019),
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484:
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242:
212:
129:
118:
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888:
Hardy, R.J.; Thompson, S.G. (1998), "Detecting and describing heterogeneity in meta-analysis",
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Overall, it appears that heterogeneity is being consistently underestimated in meta-analyses.
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32:
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468:
402:
360:
157:
797:
Borenstein, Michael; Hedges, Larry V.; Higgins, Julian P. T.; Rothstein, Hannah R. (2010).
1542:
Mosteller, F.; Colditz, G. A. (1996), "Understanding research synthesis (meta-analysis)",
799:
307:
298:
246:
188:
180:
110:
647:
Röver, C. (2020), "Bayesian random-effects meta-analysis using the bayesmeta R package",
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in a meta-analysis that is attributable to study heterogeneity (somewhat similarly to a
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539:
265:
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140:
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estimates of treatment effect will vary by chance; some variation is expected due to
53:
28:
1353:
488:
380:
1479:
1304:
840:
726:
800:"A basic introduction to fixed-effect and random-effects models for meta-analysis"
630:
854:
Cochran, W.G. (1954), "The combination of estimates from different experiments",
564:
172:
Statistical testing for a non-zero heterogeneity variance is often done based on
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758:
741:
510:
394:
208:
86:
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406:
191:
which is then only rejected in the presence of sufficient evidence against it.
65:. Any excess variation (whether it is apparent or detectable or not) is called
1337:
364:
256:
Another common measure of heterogeneity is I², a statistic that indicates the
90:
20:
16:
Research study variability considered during meta-analytic, systematic reviews
1181:
937:
824:
767:
710:
614:
902:
10.1002/(SICI)1097-0258(19980430)17:8<841::AID-SIM781>3.0.CO;2-D
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372:
1565:
1507:
909:
671:
1317:
Chiolero, A; Santschi, V.; Burnand, B.; Platt, R.W.; Paradis, G. (2012),
1260:"Basics of meta-analysis: I² is not an absolute measure of heterogeneity"
687:"Random-Effects Meta-analysis of Inconsistent Effects: A Time for Change"
335:
302:
237:
153:
1369:
Application of prediction intervals in meta-analyses with random effects
1258:
Borenstein, M.; Higgins, J.P.T.; Hedges, L.V.; Rothstein, H.R. (2017),
875:
199:
While the main purpose of a meta-analysis usually is estimation of the
1278:
1137:
1085:
1026:
429:
Cooper, Harris; Hedges, Larry V.; Valentine, Jeffrey C. (2019-06-14).
85:
adjusted in some way ("methodological diversity"). Different types of
1395:"Plea for routinely presenting prediction intervals in meta-analysis"
982:
702:
472:
1523:
1463:
867:
816:
27:
is a phenomenon that commonly occurs when attempting to undertake a
1128:
1076:
661:
1393:
IntHout, J; Ioannidis, J.P.A.; Rovers, M.M.; Goeman, J.J. (2016),
1376:
1366:
Bender, R.; Kuß, O.; Koch, A.; Schwenke, C.; Hauschke, D. (2014),
1486:
Fleiss, J. L. (1993), "The statistical basis of meta-analysis",
124:
In addition, heterogeneity is usually accommodated by using a
399:
Systematic reviews in health care: Meta-analysis in context
207:
is also crucial for its interpretation. A large number of (
1525:
Cochrane handbook for systematic reviews of interventions
1319:"Meta-analyses: with confidence or prediction intervals?"
337:
Cochrane
Handbook for Systematic Reviews of Interventions
160:. In such a scenario, it would be important to both know
97:) may also be more or less susceptible to heterogeneity.
538:
Kontopantelis, E.; Springate, D. A.; Reeves, D. (2013).
432:
The
Handbook of Research Synthesis and Meta-Analysis
330:Deeks, J.J.; Higgins, J.P.T.; Altman, D.G. (2021),
325:
323:
798:
401:(2nd ed.), BMJ Publishing, pp. 313–335,
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1054:
533:
531:
529:
128:, in which the heterogeneity then constitutes a
454:
452:
144:model again reduces to the special case of the
505:, Kluwer Academic Publishers, pp. 21–46,
164:consider relevant covariables in an analysis.
8:
746:Studies in History and Philosophy of Science
642:
640:
1574:Journal of Evaluation in Clinical Practice
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1413:
1288:1983/9cea2307-8e9b-4583-9403-3a37409ed1cb
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1145:
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946:
936:
757:
670:
660:
573:
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1488:Statistical Methods in Medical Research
603:Statistical Methods in Medical Research
319:
597:Kontopantelis, E.; Reeves, D. (2012).
7:
503:Maximum Entropy and Bayesian methods
353:Journal of the Neurological Sciences
1557:10.1146/annurev.pu.17.050196.000245
41:). Study heterogeneity denotes the
241:is commonly denoted by τ², or the
14:
1528:(2nd ed.), Wiley Blackwell,
1586:10.1046/j.1365-2753.2001.00281.x
1326:European Journal of Epidemiology
1214:BMC Medical Research Methodology
925:BMC Medical Research Methodology
249:, which is measured in the same
113:, accounting for (continuous or
740:Maziarz, Mariusz (2022-02-01).
649:Journal of Statistical Software
1544:Annual Review of Public Health
253:as the overall effect itself.
25:(between-) study heterogeneity
1:
271:The joint consideration of a
565:10.1371/journal.pone.0069930
262:coefficient of determination
35:(and studies would hence be
1415:10.1136/bmjopen-2015-010247
759:10.1016/j.shpsa.2021.11.007
691:Annals of Internal Medicine
511:10.1007/978-94-011-4710-1_3
435:. Russell Sage Foundation.
67:(statistical) heterogeneity
1631:
1500:10.1177/096228029300200202
1452:Research Synthesis Methods
1267:Research Synthesis Methods
1116:Research Synthesis Methods
1064:Research Synthesis Methods
1014:Research Synthesis Methods
805:Research Synthesis Methods
407:10.1002/9780470693926.ch16
1338:10.1007/s10654-012-9738-y
971:Pharmaceutical Statistics
365:10.1016/j.jns.2017.09.026
1182:10.1136/bmj.327.7414.557
938:10.1186/1471-2288-11-160
615:10.1177/0962280210392008
285:Homogeneity (statistics)
158:pharmacokinetic pathways
203:, investigation of the
1227:10.1186/1471-2288-8-79
890:Statistics in Medicine
258:percentage of variance
1375:, Joint statement of
672:10.18637/jss.v093.i06
332:"10.10 Heterogeneity"
290:Random effects model
126:random effects model
556:2013PLoSO...869930K
273:prediction interval
134:normal distribution
119:moderator variables
63:observational error
295:Standard deviation
243:standard deviation
235:The heterogeneity
221:prior distribution
130:variance component
75:multicenter trials
1610:Systematic review
1279:10.1002/jrsm.1230
1176:(7414): 557–560,
1138:10.1002/jrsm.1217
1086:10.1002/jrsm.1475
1027:10.1002/jrsm.1164
520:978-94-010-5982-4
442:978-1-61044-886-4
33:measurement error
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81:heterogeneity).
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1444:Further reading
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299:scale parameter
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247:scale parameter
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189:null hypothesis
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139:Inclusion of a
111:meta-regression
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87:effect measures
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17:
12:
11:
5:
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1580:(2): 135–148,
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1494:(2): 121–145,
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1408:(7): e010247,
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1070:(4): 448–474,
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862:(1): 101–129,
846:
789:
732:
697:(4): 267–270.
677:
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448:
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340:(6.2 ed.)
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231:Quantification
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1458:(2): 97–111,
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1332:(10): 823–5,
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811:(2): 97–111.
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122:
120:
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95:relative risk
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36:
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1379:, GMDS and
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752:: 159–167.
655:(6): 1–51,
209:frequentist
201:main effect
185:homogeneity
152:reasonable
115:categorical
43:variability
38:homogeneous
1604:Categories
1220:(79): 79,
1129:1601.06533
1077:2007.08352
931:(1): 160,
856:Biometrics
662:1711.08683
314:References
217:estimators
195:Estimation
107:stratified
91:odds ratio
21:statistics
1516:121128494
1102:220546288
999:212718520
825:1759-2887
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768:0039-3681
711:0003-4819
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544:PLOS ONE
489:32994689
481:21310794
467:: d549,
381:31073171
373:28967410
303:variance
279:See also
238:variance
213:Bayesian
154:estimand
101:Modeling
57:trials.
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187:as the
175:Cochran
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