Editing Meta-analysis (section)

===Statistical models for aggregate data===

==== Fixed effect model ====
[[File:Generic forest plot.png|thumb|Forest Plot of Effect Sizes]]
The fixed effect model provides a weighted average of a series of study estimates.<ref>{{Cite journal |last1=Nikolakopoulou |first1=Adriani |last2=Mavridis |first2=Dimitris |last3=Salanti |first3=Georgia |date=2014 |title=How to interpret meta-analysis models: fixed effect and random effects meta-analyses |url=https://ebmh.bmj.com/lookup/doi/10.1136/eb-2014-101794 |journal=Evidence Based Mental Health |language=en |volume=17 |issue=2 |pages=64 |doi=10.1136/eb-2014-101794 |pmid=24778439 |issn=1362-0347}}</ref> The inverse of the estimates' variance is commonly used as study weight, so that larger studies tend to contribute more than smaller studies to the weighted average.<ref>{{Cite journal |last=Dekkers |first=Olaf M. |date=2018 |title=Meta-analysis: Key features, potentials and misunderstandings |journal=Research and Practice in Thrombosis and Haemostasis |language=en |volume=2 |issue=4 |pages=658–663 |doi=10.1002/rth2.12153 |pmc=6178740 |pmid=30349883}}</ref> Consequently, when studies within a meta-analysis are dominated by a very large study, the findings from smaller studies are practically ignored.<ref name="pmid11884693">{{cite journal | vauthors = Helfenstein U | title = Data and models determine treatment proposals--an illustration from meta-analysis | journal = Postgraduate Medical Journal | volume = 78 | issue = 917 | pages = 131–134 | date = March 2002 | pmid = 11884693 | pmc = 1742301 | doi = 10.1136/pmj.78.917.131 }}</ref> Most importantly, the fixed effects model assumes that all included studies investigate the same population, use the same variable and outcome definitions, etc.<ref>{{Cite journal |last1=Dettori |first1=Joseph R. |last2=Norvell |first2=Daniel C. |last3=Chapman |first3=Jens R. |date=2022 |title=Fixed-Effect vs Random-Effects Models for Meta-Analysis: 3 Points to Consider |journal=Global Spine Journal |language=en |volume=12 |issue=7 |pages=1624–1626 |doi=10.1177/21925682221110527 |issn=2192-5682 |pmc=9393987 |pmid=35723546}}</ref> This assumption is typically unrealistic as research is often prone to several sources of [[study heterogeneity|heterogeneity]].<ref>{{Cite journal |last1=Hedges |first1=Larry V. |last2=Vevea |first2=Jack L. |date=1998 |title=Fixed- and random-effects models in meta-analysis. |url=http://doi.apa.org/getdoi.cfm?doi=10.1037/1082-989X.3.4.486 |journal=Psychological Methods |language=en |volume=3 |issue=4 |pages=486–504 |doi=10.1037/1082-989X.3.4.486 |s2cid=119814256 |issn=1939-1463}}</ref><ref>{{Cite journal |last1=Rice |first1=Kenneth |last2=Higgins |first2=Julian P. T. |last3=Lumley |first3=Thomas |date=2018 |title=A re-evaluation of fixed effect(s) meta-analysis |url=https://www.jstor.org/stable/44682165 |journal=Journal of the Royal Statistical Society. Series A (Statistics in Society) |volume=181 |issue=1 |pages=205–227 |doi=10.1111/rssa.12275 |jstor=44682165 |issn=0964-1998}}</ref>

If we start with a collection of independent effect size estimates, each estimate a corresponding effect size  <math>i = 1,\ldots,k</math>  we can assume that <math display="inline">y_i = \theta_i + e_i</math> where <math>y_i</math> denotes the observed effect in the <math>i</math>-th study, <math>\theta_i</math> the corresponding (unknown) true effect, <math>e_i</math> is the sampling error, and <math>e_i \thicksim N(0, v_i)</math>.  Therefore, the <math>y_i</math>’s are assumed to be unbiased and [[Normal distribution|normally distributed]] estimates of their corresponding true effects. The sampling variances (i.e., <math>v_i</math> values) are assumed to be known.<ref name=":5" />

==== Random effects model ====

Most meta-analyses are based on sets of studies that are not exactly identical in their methods and/or the characteristics of the included samples.<ref name=":5" /> Differences in the methods and sample characteristics may introduce variability (“heterogeneity”) among the true effects.<ref name=":5" /><ref>{{Cite journal |last1=Holzmeister |first1=Felix |last2=Johannesson |first2=Magnus |last3=Böhm |first3=Robert |last4=Dreber |first4=Anna |last5=Huber |first5=Jürgen |last6=Kirchler |first6=Michael |date=2024-08-06 |title=Heterogeneity in effect size estimates |journal=Proceedings of the National Academy of Sciences |language=en |volume=121 |issue=32 |pages=e2403490121 |doi=10.1073/pnas.2403490121 |issn=0027-8424 |pmc=11317577 |pmid=39078672|bibcode=2024PNAS..12103490H }}</ref> One way to model the heterogeneity is to treat it as purely random. The weight that is applied in this process of weighted averaging with a random effects meta-analysis is achieved in two steps:<ref>{{cite journal | vauthors = Senn S | title = Trying to be precise about vagueness | journal = Statistics in Medicine | volume = 26 | issue = 7 | pages = 1417–1430 | date = March 2007 | pmid = 16906552 | doi = 10.1002/sim.2639 | s2cid = 17764847 | doi-access = free }}</ref>

# Step 1: Inverse variance weighting
# Step 2: Un-weighting of this inverse variance weighting by applying a random effects variance component (REVC) that is simply derived from the extent of variability of the effect sizes of the underlying studies.

This means that the greater this variability in effect sizes (otherwise known as [[study heterogeneity|heterogeneity]]), the greater the un-weighting and this can reach a point when the random effects meta-analysis result becomes simply the un-weighted average effect size across the studies. At the other extreme, when all effect sizes are similar (or variability does not exceed sampling error), no REVC is applied and the random effects meta-analysis defaults to simply a fixed effect meta-analysis (only inverse variance weighting).

The extent of this reversal is solely dependent on two factors:<ref name="ReferenceA">{{cite journal | vauthors = Al Khalaf MM, Thalib L, Doi SA | title = Combining heterogenous studies using the random-effects model is a mistake and leads to inconclusive meta-analyses | journal = Journal of Clinical Epidemiology | volume = 64 | issue = 2 | pages = 119–123 | date = February 2011 | pmid = 20409685 | doi = 10.1016/j.jclinepi.2010.01.009 }}</ref>

# Heterogeneity of precision
# Heterogeneity of effect size

Since neither of these factors automatically indicates a faulty larger study or more reliable smaller studies, the re-distribution of weights under this model will not bear a relationship to what these studies actually might offer. Indeed, it has been demonstrated that redistribution of weights is simply in one direction from larger to smaller studies as heterogeneity increases until eventually all studies have equal weight and no more redistribution is possible.<ref name="ReferenceA"/>
Another issue with the random effects model is that the most commonly used confidence intervals generally do not retain their coverage probability above the specified nominal level and thus substantially underestimate the statistical error and are potentially
overconfident in their conclusions.<ref name="Brockwell2001">{{cite journal | vauthors = Brockwell SE, Gordon IR | title = A comparison of statistical methods for meta-analysis | journal = Statistics in Medicine | volume = 20 | issue = 6 | pages = 825–840 | date = March 2001 | pmid = 11252006 | doi = 10.1002/sim.650 | s2cid = 16932514 }}</ref><ref name="Noma2011">{{cite journal | vauthors = Noma H | title = Confidence intervals for a random-effects meta-analysis based on Bartlett-type corrections | journal = Statistics in Medicine | volume = 30 | issue = 28 | pages = 3304–3312 | date = December 2011 | pmid = 21964669 | doi = 10.1002/sim.4350 | hdl-access = free | hdl = 2433/152046 | s2cid = 6556986 }}</ref> Several fixes have been suggested<ref>{{cite journal | vauthors = Brockwell SE, Gordon IR | title = A simple method for inference on an overall effect in meta-analysis | journal = Statistics in Medicine | volume = 26 | issue = 25 | pages = 4531–4543 | date = November 2007 | pmid = 17397112 | doi = 10.1002/sim.2883 | s2cid = 887098 }}</ref><ref>{{cite journal | vauthors = Sidik K, Jonkman JN | title = A simple confidence interval for meta-analysis | journal = Statistics in Medicine | volume = 21 | issue = 21 | pages = 3153–3159 | date = November 2002 | pmid = 12375296 | doi = 10.1002/sim.1262 | s2cid = 21384942 }}</ref> but the debate continues on.<ref name="Noma2011" /><ref name="pmid19016302">{{cite journal | vauthors = Jackson D, Bowden J | title = A re-evaluation of the 'quantile approximation method' for random effects meta-analysis | journal = Statistics in Medicine | volume = 28 | issue = 2 | pages = 338–348 | date = January 2009 | pmid = 19016302 | pmc = 2991773 | doi = 10.1002/sim.3487 }}</ref> A further concern is that the average treatment effect can sometimes be even less conservative compared to the fixed effect model<ref>{{cite journal | vauthors = Poole C, Greenland S | title = Random-effects meta-analyses are not always conservative | journal = American Journal of Epidemiology | volume = 150 | issue = 5 | pages = 469–475 | date = September 1999 | pmid = 10472946 | doi = 10.1093/oxfordjournals.aje.a010035 | doi-access = free }}</ref> and therefore misleading in practice. One interpretational fix that has been suggested is to create a prediction interval around the random effects estimate to portray the range of possible effects in practice.<ref>{{cite journal | vauthors = Riley RD, Higgins JP, Deeks JJ | title = Interpretation of random effects meta-analyses | journal = BMJ | volume = 342 | pages = d549 | date = February 2011 | pmid = 21310794 | doi = 10.1136/bmj.d549 | s2cid = 32994689 }}</ref> However, an assumption behind the calculation of such a prediction interval is that trials are considered more or less homogeneous entities and that included patient populations and comparator treatments should be considered exchangeable<ref name="pmid23494781">{{cite journal | vauthors = Kriston L | title = Dealing with clinical heterogeneity in meta-analysis. Assumptions, methods, interpretation | journal = International Journal of Methods in Psychiatric Research | volume = 22 | issue = 1 | pages = 1–15 | date = March 2013 | pmid = 23494781 | pmc = 6878481 | doi = 10.1002/mpr.1377 }}</ref> and this is usually unattainable in practice.

There are many methods used to estimate between studies variance with restricted maximum likelihood estimator being the least prone to bias and one of the most commonly used.<ref>{{Cite journal |last1=Langan |first1=Dean |last2=Higgins |first2=Julian P.T. |last3=Jackson |first3=Dan |last4=Bowden |first4=Jack |last5=Veroniki |first5=Areti Angeliki |last6=Kontopantelis |first6=Evangelos |last7=Viechtbauer |first7=Wolfgang |last8=Simmonds |first8=Mark |date=2019 |title=A comparison of heterogeneity variance estimators in simulated random-effects meta-analyses |journal=Research Synthesis Methods |language=en |volume=10 |issue=1 |pages=83–98 |doi=10.1002/jrsm.1316 |pmid=30067315 |s2cid=51890354 |issn=1759-2879|doi-access=free |hdl=1983/c911791c-c687-4f12-bc0b-ffdbe42ca874 |hdl-access=free }}</ref> Several advanced iterative techniques for computing the between studies variance exist including both maximum likelihood and restricted maximum likelihood methods and random effects models using these methods can be run with multiple software platforms including Excel,<ref name="Manual">{{cite web |title=MetaXL User Guide |url=http://www.epigear.com/index_files/MetaXL%20User%20Guide.pdf |access-date=2018-09-18}}</ref> Stata,<ref name="metaan">{{cite journal|url=https://www.researchgate.net/publication/227629391|title=Metaan: Random-effects meta-analysis| vauthors = Kontopantelis E, Reeves D |date=1 August 2010|journal=Stata Journal|volume=10|issue=3|pages=395–407|via=ResearchGate |doi= 10.1177/1536867X1001000307 |doi-access=free}}</ref> SPSS,<ref>{{Cite journal |last1=Field |first1=Andy P. |last2=Gillett |first2=Raphael |date=2010 |title=How to do a meta-analysis |url=http://doi.wiley.com/10.1348/000711010X502733 |journal=British Journal of Mathematical and Statistical Psychology |language=en |volume=63 |issue=3 |pages=665–694 |doi=10.1348/000711010X502733|pmid=20497626 |s2cid=22688261 }}</ref> and R.<ref name=":5">{{Cite journal |last=Viechtbauer |first=Wolfgang |date=2010 |title=Conducting Meta-Analyses in R with the metafor Package |url=http://www.jstatsoft.org/v36/i03/ |journal=Journal of Statistical Software |language=en |volume=36 |issue=3 |doi=10.18637/jss.v036.i03 |s2cid=15798713 |issn=1548-7660|doi-access=free }}</ref>

Most meta-analyses include between 2 and 4 studies and such a sample is more often than not inadequate to accurately estimate [[study heterogeneity|heterogeneity]]. Thus it appears that in small meta-analyses, an incorrect zero between study variance estimate is obtained, leading to a false homogeneity assumption. Overall, it appears that heterogeneity is being consistently underestimated in meta-analyses and sensitivity analyses in which high heterogeneity levels are assumed could be informative.<ref name=KontopantelisP1>{{cite journal | vauthors = Kontopantelis E, Springate DA, Reeves D | title = A re-analysis of the Cochrane Library data: the dangers of unobserved heterogeneity in meta-analyses | journal = PLOS ONE | volume = 8 | issue = 7 | pages = e69930 | year = 2013 | pmid = 23922860 | pmc = 3724681 | doi = 10.1371/journal.pone.0069930 | veditors = Friede T | doi-access = free | bibcode = 2013PLoSO...869930K }}</ref> These random effects models and software packages mentioned above relate to study-aggregate meta-analyses and researchers wishing to conduct individual patient data (IPD) meta-analyses need to consider mixed-effects modelling approaches.<ref name="ipdforest">{{cite journal |url= https://www.researchgate.net/publication/257316967 |title=A short guide and a forest plot command (ipdforest) for one-stage meta-analysis| vauthors = Kontopantelis E, Reeves D |date=27 September 2013|journal=Stata Journal|volume=13|issue=3|pages=574–587 |via= ResearchGate |doi=10.1177/1536867X1301300308 |doi-access=free}}</ref>/

==== Quality effects model ====

Doi and Thalib originally introduced the quality effects model.<ref name="Doi_2008">{{cite journal | vauthors = Doi SA, Thalib L | title = A quality-effects model for meta-analysis | journal = Epidemiology | volume = 19 | issue = 1 | pages = 94–100 | date = January 2008 | pmid = 18090860 | doi = 10.1097/EDE.0b013e31815c24e7 | s2cid = 29723291 | doi-access = free }}</ref> They<ref>{{cite journal | vauthors = Doi SA, Barendregt JJ, Mozurkewich EL | title = Meta-analysis of heterogeneous clinical trials: an empirical example | journal = Contemporary Clinical Trials | volume = 32 | issue = 2 | pages = 288–298 | date = March 2011 | pmid = 21147265 | doi = 10.1016/j.cct.2010.12.006 }}</ref> introduced a new approach to adjustment for inter-study variability by incorporating the contribution of variance due to a relevant component (quality) in addition to the contribution of variance due to random error that is used in any fixed effects meta-analysis model to generate weights for each study. The strength of the quality effects meta-analysis is that it allows available methodological evidence to be used over subjective random effects, and thereby helps to close the damaging gap which has opened up between methodology and statistics in clinical research. To do this a synthetic bias variance is computed based on quality information to adjust inverse variance weights and the quality adjusted weight of the ''i''th study is introduced.<ref name="Doi_2008"/> These adjusted weights are then used in meta-analysis. In other words, if study ''i'' is of good quality and other studies are of poor quality, a proportion of their quality adjusted weights is mathematically redistributed to study ''i'' giving it more weight towards the overall effect size. As studies become increasingly similar in terms of quality, re-distribution becomes progressively less and ceases when all studies are of equal quality (in the case of equal quality, the quality effects model defaults to the IVhet model – see previous section). A recent evaluation of the quality effects model (with some updates) demonstrates that despite the subjectivity of quality assessment, the performance (MSE and true variance under simulation) is superior to that achievable with the random effects model.<ref>{{cite journal | vauthors = Doi SA, Barendregt JJ, Khan S, Thalib L, Williams GM | title = Simulation Comparison of the Quality Effects and Random Effects Methods of Meta-analysis | journal = Epidemiology | volume = 26 | issue = 4 | pages = e42–e44 | date = July 2015 | pmid = 25872162 | doi = 10.1097/EDE.0000000000000289 | doi-access = free }}</ref><ref>{{cite journal | vauthors = Doi SA, Barendregt JJ, Khan S, Thalib L, Williams GM | title = Advances in the meta-analysis of heterogeneous clinical trials II: The quality effects model | journal = Contemporary Clinical Trials | volume = 45 | issue = Pt A | pages = 123–129 | date = November 2015 | pmid = 26003432 | doi = 10.1016/j.cct.2015.05.010 }}</ref> This model thus replaces the untenable interpretations that abound in the literature and a software is available to explore this method further.<ref name="Epigear">{{cite web|url=http://www.epigear.com/ |title=MetaXL software page |publisher=Epigear.com |date=2017-06-03 |access-date=2018-09-18}}</ref>

==== Network meta-analysis methods ====
[[File:Indirekt jämförelse.jpg|thumb|300px|right|A network meta-analysis looks at indirect comparisons. In the image, A has been analyzed in relation to C and C has been analyzed in relation to B. However the relation between A and B is only known indirectly, and a network meta-analysis looks at such indirect evidence of differences between methods and interventions using statistical method.]]

Indirect comparison meta-analysis methods (also called network meta-analyses, in particular when multiple treatments are assessed simultaneously) generally use two main methodologies.<ref>{{Cite journal |last1=Rouse |first1=Benjamin |last2=Chaimani |first2=Anna |last3=Li |first3=Tianjing |date=2017 |title=Network meta-analysis: an introduction for clinicians |journal=Internal and Emergency Medicine |language=en |volume=12 |issue=1 |pages=103–111 |doi=10.1007/s11739-016-1583-7 |issn=1828-0447 |pmc=5247317 |pmid=27913917}}</ref><ref>{{Cite journal |last1=Phillips |first1=Mark R. |last2=Steel |first2=David H. |last3=Wykoff |first3=Charles C. |last4=Busse |first4=Jason W. |last5=Bannuru |first5=Raveendhara R. |last6=Thabane |first6=Lehana |last7=Bhandari |first7=Mohit |last8=Chaudhary |first8=Varun |last9=for the Retina Evidence Trials InterNational Alliance (R.E.T.I.N.A.) Study Group |last10=Sivaprasad |first10=Sobha |last11=Kaiser |first11=Peter |last12=Sarraf |first12=David |last13=Bakri |first13=Sophie J. |last14=Garg |first14=Sunir J. |last15=Singh |first15=Rishi P. |date=2022 |title=A clinician's guide to network meta-analysis |journal=Eye |language=en |volume=36 |issue=8 |pages=1523–1526 |doi=10.1038/s41433-022-01943-5 |issn=0950-222X |pmc=9307840 |pmid=35145277}}</ref> First, is the Bucher method<ref>{{cite journal | vauthors = Bucher HC, Guyatt GH, Griffith LE, Walter SD | title = The results of direct and indirect treatment comparisons in meta-analysis of randomized controlled trials | journal = Journal of Clinical Epidemiology | volume = 50 | issue = 6 | pages = 683–691 | date = June 1997 | pmid = 9250266 | doi = 10.1016/s0895-4356(97)00049-8 }}</ref> which is a single or repeated comparison of a closed loop of three-treatments such that one of them is common to the two studies and forms the node where the loop begins and ends. Therefore, multiple two-by-two comparisons (3-treatment loops) are needed to compare multiple treatments. This methodology requires that trials with more than two arms have two arms only selected as independent pair-wise comparisons are required. The alternative methodology uses complex [[statistical model]]ling to include the multiple arm trials and comparisons simultaneously between all competing treatments. These have been executed using Bayesian methods, mixed linear models and meta-regression approaches.{{citation needed|date=June 2018}}

=====Bayesian framework=====
Specifying a Bayesian network meta-analysis model involves writing a directed acyclic graph (DAG) model for general-purpose [[Markov chain Monte Carlo]] (MCMC) software such as WinBUGS.<ref name="Valkenhoef, G. 2012">{{cite journal | vauthors = van Valkenhoef G, Lu G, de Brock B, Hillege H, Ades AE, Welton NJ | title = Automating network meta-analysis | journal = Research Synthesis Methods | volume = 3 | issue = 4 | pages = 285–299 | date = December 2012 | pmid = 26053422 | doi = 10.1002/jrsm.1054 | s2cid = 33613631 }}</ref> In addition, prior distributions have to be specified for a number of the parameters, and the data have to be supplied in a specific format.<ref name="Valkenhoef, G. 2012"/> Together, the DAG, priors, and data form a Bayesian hierarchical model. To complicate matters further, because of the nature of MCMC estimation, overdispersed starting values have to be chosen for a number of independent chains so that convergence can be assessed.<ref>{{cite journal |vauthors=Brooks SP, Gelman A | year = 1998 | title = General methods for monitoring convergence of iterative simulations | url = http://www.stat.columbia.edu/~gelman/research/published/brooksgelman2.pdf| journal = Journal of Computational and Graphical Statistics | volume = 7 | issue = 4| pages = 434–455 | doi=10.1080/10618600.1998.10474787| s2cid = 7300890 }}</ref> Recently, multiple [[R (programming language)|R]] software packages were developed to simplify the model fitting (e.g., ''metaBMA''<ref>{{Cite web | vauthors = Heck DW, Gronau QF, Wagenmakers EJ, Patil I |title=metaBMA: Bayesian model averaging for random and fixed effects meta-analysis |url=https://CRAN.R-project.org/package=metaBMA |access-date=9 May 2022 |website=CRAN|date=17 March 2021 }}</ref> and ''RoBMA''<ref>{{Cite web | vauthors =  Bartoš F, Maier M, Wagenmakers EJ, Goosen J, Denwood M, Plummer M |title=RoBMA: An R Package for Robust Bayesian Meta-Analyses |date=20 April 2022 |url=https://CRAN.R-project.org/package=RoBMA |access-date=9 May 2022}}</ref>) and even implemented in statistical software with graphical user interface ([[Graphical user interface|GUI]]): [[JASP]]. Although the complexity of the Bayesian approach limits usage of this methodology, recent tutorial papers are trying to increase accessibility of the methods.<ref>{{Cite journal | vauthors = Gronau QF, Heck DW, Berkhout SW, Haaf JM, Wagenmakers EJ |date=July 2021 |title=A Primer on Bayesian Model-Averaged Meta-Analysis |journal=Advances in Methods and Practices in Psychological Science |language=en |volume=4 |issue=3 |pages= |doi=10.1177/25152459211031256 |s2cid=237699937 |issn=2515-2459|doi-access=free |hdl=11245.1/ec2c07d1-5ff0-431b-b53a-10f9c5d9541d |hdl-access=free }}</ref><ref>{{Cite journal | vauthors = Bartoš F, Maier M, Quintana D, Wagenmakers EJ |date=2020-10-16 |title=Adjusting for Publication Bias in JASP & R - Selection Models, PET-PEESE, and Robust Bayesian Meta-Analysis | journal = Advances in Methods and Practices in Psychological Science |url=https://osf.io/75bqn |doi=10.31234/osf.io/75bqn |s2cid=236826939 |doi-access=free |hdl=11245.1/5540e87c-0883-45e6-87de-48d2bf4c1e1d |hdl-access=free }}</ref> Methodology for automation of this method has been suggested<ref name="Valkenhoef, G. 2012" /> but requires that arm-level outcome data are available, and this is usually unavailable. Great claims are sometimes made for the inherent ability of the Bayesian framework to handle network meta-analysis and its greater flexibility. However, this choice of implementation of framework for inference, Bayesian or frequentist, may be less important than other choices regarding the modeling of effects<ref name="ReferenceC">{{cite journal | vauthors = Senn S, Gavini F, Magrez D, Scheen A | title = Issues in performing a network meta-analysis | journal = Statistical Methods in Medical Research | volume = 22 | issue = 2 | pages = 169–189 | date = April 2013 | pmid = 22218368 | doi = 10.1177/0962280211432220 | s2cid = 10860031 }}</ref> (see discussion on models above).

=====Frequentist multivariate framework=====
On the other hand, the frequentist multivariate methods involve approximations and assumptions that are not stated explicitly or verified when the methods are applied (see discussion on meta-analysis models above). For example, the mvmeta package for Stata enables network meta-analysis in a frequentist framework.<ref>{{cite journal | vauthors = White IR | year = 2011 | title = Multivariate random-effects meta-regression: updates to mvmeta | journal = The Stata Journal | volume = 11 | issue = 2| pages = 255–270| doi = 10.1177/1536867X1101100206 | doi-access = free }}</ref> However, if there is no common comparator in the network, then this has to be handled by augmenting the dataset with fictional arms with high variance, which is not very objective and requires a decision as to what constitutes a sufficiently high variance.<ref name="Valkenhoef, G. 2012" /> The other issue is use of the random effects model in both this frequentist framework and the Bayesian framework. Senn advises analysts to be cautious about interpreting the 'random effects' analysis since only one random effect is allowed for but one could envisage many.<ref name="ReferenceC"/> Senn goes on to say that it is rather naıve, even in the case where only two treatments are being compared to assume that random-effects analysis accounts for all uncertainty about the way effects can vary from trial to trial. Newer models of meta-analysis such as those discussed above would certainly help alleviate this situation and have been implemented in the next framework.

=====Generalized pairwise modelling framework=====
An approach that has been tried since the late 1990s is the implementation of the multiple three-treatment closed-loop analysis. This has not been popular because the process rapidly becomes overwhelming as network complexity increases. Development in this area was then abandoned in favor of the Bayesian and multivariate frequentist methods which emerged as alternatives. Very recently, automation of the three-treatment closed loop method has been developed for complex networks by some researchers<ref name="Manual" /> as a way to make this methodology available to the mainstream research community. This proposal does restrict each trial to two interventions, but also introduces a workaround for multiple arm trials: a different fixed control node can be selected in different runs. It also utilizes robust meta-analysis methods so that many of the problems highlighted above are avoided. Further research around this framework is required to determine if this is indeed superior to the Bayesian or multivariate frequentist frameworks. Researchers willing to try this out have access to this framework through a free software.<ref name="Epigear"/>

====Diagnostic test accuracy meta-analysis====

Diagnostic test accuracy (DTA) meta-analyses differ methodologically from those assessing intervention effects, as they aim to jointly synthesize pairs of sensitivity and specificity values. These parameters are typically analyzed using hierarchical models that account for the correlation between them and between-study heterogeneity. Two commonly used models are the '''bivariate random-effects model''' and the '''hierarchical summary receiver operating characteristic (HSROC) model'''. These approaches are recommended by the ''Cochrane Handbook for Systematic Reviews of Diagnostic Test Accuracy'' and are widely used in reviews of screening tests, imaging tools, and laboratory diagnostics.<ref name="Reitsma2005">Reitsma JB, Glas AS, Rutjes AWS, Scholten RJPM, Bossuyt PMM, Zwinderman AH (2005). Bivariate analysis of sensitivity and specificity produces informative summary measures in diagnostic reviews. ''J Clin Epidemiol''. 58(10):982–990. doi:[https://doi.org/10.1016/j.jclinepi.2005.02.022 10.1016/j.jclinepi.2005.02.022]</ref><ref name="Rutter2001">Rutter CM, Gatsonis CA (2001). A hierarchical regression approach to meta-analysis of diagnostic test accuracy evaluations. ''Stat Med''. 20(19):2865–84. doi:[https://doi.org/10.1002/sim.942 10.1002/sim.942]</ref><ref name="CochraneDTA">McInnes MDF, Moher D, Thombs BD, et al. Preferred Reporting Items for a Systematic Review and Meta-analysis of Diagnostic Test Accuracy Studies: The PRISMA-DTA Statement. ''JAMA''. 2018;319(4):388–396. doi:[https://doi.org/10.1001/jama.2017.19163 10.1001/jama.2017.19163]</ref>

Beyond the standard hierarchical models, other approaches have been developed to address various complexities in diagnostic accuracy synthesis. These include methods that incorporate differences in threshold effects, account for covariates through meta-regression, or improve applicability by considering test setting and clinical variation. Some frameworks aim to adapt the synthesis to reflect intended use conditions more directly. These extensions are part of an evolving body of methodology that reflects growing experience in the field and increasing demands from clinical and policy decision-makers.<ref name="Deeks2020">Deeks JJ, Takwoingi Y. Two decades of progress in test accuracy systematic reviews: Managing meta-analytical complexity. ''J Clin Epidemiol''. 2020;122:92–102. doi:[https://doi.org/10.1016/j.jclinepi.2020.03.003 10.1016/j.jclinepi.2020.03.003]</ref>

==== Aggregating IPD and AD ====
Meta-analysis can also be applied to combine IPD and AD. This is convenient when the researchers who conduct the analysis have their own raw data while collecting aggregate or summary data from the literature. The generalized integration model (GIM)<ref name=Zhang2020/> is a generalization of the meta-analysis. It allows that the model fitted on the individual participant data (IPD) is different from the ones used to compute the aggregate data (AD). GIM can be viewed as a model calibration method for integrating information with more flexibility.