Editing Meta-analysis (section)

==== 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" />