The model
The fixed-effect model, also called the common-effect model, assumes that there is a single true effect, theta, and that each study's estimate y_i differs from it only through sampling error, with variance v_i. Written compactly, each estimate is the true effect plus an error term, and the errors are independent with known variances. If the model is true, the best estimate of theta is obtained by weighting each study by the inverse of its variance. The estimate is more precise than any single study, because it uses all the information.
Two interpretations of the model coexist and should be kept apart. In the first, the assumption of a common effect is taken literally: the studies are essentially replicates, and the pooled result is the effect that all of them were estimating. In the second, no common effect is assumed, and the analysis is described as a weighted average of the effects in the particular studies included, answering the question of what they show on average, with weights proportional to precision. The second view has been argued to be a legitimate and sometimes useful reading of the fixed-effect analysis, but it limits the conclusion to the studies at hand and makes no claim about other settings.
Inverse-variance weighting
With the weight of study i defined as w_i = 1/v_i, the pooled estimate is the sum of the products w_i times y_i divided by the sum of the weights. Its variance is the reciprocal of the sum of the weights, and the 95 percent confidence interval is the pooled estimate plus and minus 1.96 times the square root of that variance. A z test of the null hypothesis of no effect compares the estimate with its standard error. This is the generic inverse-variance method, and because it works on any effect estimate with a standard error, it applies to mean differences, standardized mean differences, log ratios, correlations after transformation, and adjusted estimates from observational studies.
The weights make clear what the method does. A study with half the standard error has four times the weight, so large precise studies dominate. In a collection with one very large trial and several small ones, the pooled estimate will be close to the large trial's, and the confidence interval only slightly narrower than its own. This is correct if the common effect assumption holds and is a problem if it does not.
Mantel-Haenszel and Peto methods
For binary outcomes, two alternatives to the inverse-variance method are in routine use with a fixed-effect model. The Mantel-Haenszel method computes the pooled risk ratio, odds ratio or risk difference by weighting the study-specific counts directly, without first estimating each study's log ratio and variance. It is reliable when events are infrequent or studies are small, because it does not require the log ratio to be estimable in every study, and it handles studies with a zero in one cell without a continuity correction in the pooled estimate, although a correction may be used in computing its variance. It has been shown to perform better than inverse variance in sparse data.
The Peto method estimates the odds ratio from the difference between the observed and the expected number of events in the treatment group, with the expectation computed under the null hypothesis of no effect. It performs well when events are rare, effects are small to moderate and the groups in each study are of similar size. It is biased when there are large effects or marked imbalance between the groups, and it should not be used then. The choice of method is made for the data at hand, and the sensitivity of the result to it is examined where events are few. See the guide on zero-event studies.
Testing the assumption
Whether the common-effect assumption is reasonable is partly a matter of judgment about the studies and partly of statistics. The Q statistic tests the null hypothesis that all studies share one effect by comparing the weighted squared deviations from the pooled estimate with a chi-squared distribution on k minus 1 degrees of freedom, where k is the number of studies. A significant Q suggests that the studies differ more than chance allows. The test has low power when studies are few or small, so a non-significant result does not establish homogeneity, and high power when studies are many or large, so a significant result can reflect trivial differences. For this reason, deciding between fixed and random effects on the basis of a Q test is not recommended.
It is better to decide from the subject matter before looking at the results, and to use the heterogeneity statistics to describe the variation. If the studies are clinically and methodologically similar, and the heterogeneity is small, a fixed-effect analysis may be defensible. If not, random effects, or no pooling at all, is more appropriate. See heterogeneity.
When a fixed-effect analysis is defensible
There are circumstances in which a fixed-effect analysis is a reasonable primary choice. When a small number of studies were designed together to be as similar as possible, for example the trials of a single drug development program with the same protocol, a common effect is plausible. When the question concerns the studies actually included and not a wider population, the fixed-effect weighted average answers it directly. When there are very few studies and the random-effects estimate would depend on an unstable estimate of tau squared, the fixed-effect result is more stable, provided the limits are acknowledged. And when it is used as a sensitivity analysis alongside a random-effects primary analysis, it shows how much the conclusion depends on the weighting. In each case, the reason is stated and the other analysis is shown.
Where it misleads
The main danger is false precision. When true effects differ, the fixed-effect confidence interval ignores that variation and is too narrow, so that the claim of statistical significance may be overstated. The pooled estimate also gives most of its weight to the largest studies, which may differ from the others in important ways and may be the ones most affected by bias. A conclusion based on the fixed-effect model then reflects those studies in particular, and not the evidence as a whole. In the presence of small-study effects, where smaller studies show larger effects, the fixed-effect estimate is less affected than the random-effects estimate, since it gives less weight to small studies, which can be an advantage, but it does not cure the underlying problem.
These issues are the reason many methodologists regard random effects as the safer default for studies that differ, and the reason an analysis that reports only the fixed-effect result needs justification. See random-effects meta-analysis.
A comparison on the same data
The six simulated studies used in the introductory guide show how the two weightings differ. The studies are simulated for illustration and are not real research. Under the fixed-effect model the weights are the inverse variances, and the pooled standardized mean difference is 0.22 with a 95 percent confidence interval from 0.11 to 0.33. The Q statistic is 6.16 on 5 degrees of freedom, and the estimated between-study variance is 0.0045. Re-weighting under random effects with that variance gives a pooled value of 0.22 with an interval from 0.10 to 0.35.
| Study | Effect | Standard error | Fixed-effect weight | Random-effects weight |
|---|---|---|---|---|
| Study A | 0.30 | 0.15 | 13.9% | 14.8% |
| Study B | 0.10 | 0.12 | 21.6% | 21.2% |
| Study C | 0.45 | 0.20 | 7.8% | 9.0% |
| Study D | 0.22 | 0.10 | 31.2% | 27.6% |
| Study E | -0.05 | 0.18 | 9.6% | 10.9% |
| Study F | 0.38 | 0.14 | 15.9% | 16.6% |
Study D, the most precise, carries 31.2 percent of the weight under the fixed-effect model and 27.6 percent under random effects, while the least precise, Study C, rises from 7.8 to 9.0 percent. The point estimates are almost identical here because the heterogeneity is small, but the random-effects interval is wider. With larger heterogeneity both the estimates and the intervals would move further apart, and the choice would matter more.
Software and practical notes
Every standard meta-analysis program can fit a fixed-effect model, and several label it differently, as common-effect, fixed-effect or inverse-variance, so the settings and not the label should be checked. In R the metafor and meta packages offer inverse-variance, Mantel-Haenszel and Peto options, in Stata the meta suite and user-written commands do, and RevMan provides them as options for binary data. For binary outcomes with sparse data, check which continuity correction, if any, the program applies by default, since the default can differ between methods and between programs. Whichever program is used, the analysis is best run from a script, and the version and options recorded, so that it can be repeated.
Reporting
A fixed-effect analysis is reported with the method of weighting, the pooled estimate with its confidence interval, the Q statistic and the other heterogeneity measures, and a statement of why a fixed-effect model was chosen. The software, the version and the settings are given. Because many readers will assume random effects, the model should be named clearly in the abstract, the methods, the figure and the table. The analysis is reported according to PRISMA 2020, with the forest plot showing weights from the model used.
How we can help
Support for fixed-effect meta-analysis
The method can be supported at different depths. Choose what you need, and the scope is agreed in writing before work begins.
Feasibility check
A review of your studies and data to confirm that the method is suitable and which approach fits.
Analysis and figures
The analysis run to a prespecified plan, with forest plots and the other figures.
Methods and results text
Written for the manuscript and aligned with PRISMA 2020 or the relevant extension.
Manuscript and submission
Optional: the full paper, the reporting checklist and the submission materials.
Frequently asked questions
What is the difference between a fixed-effect and a common-effect model?
The terms are used for the same model by most authors. Common-effect is preferred by some because it describes the assumption directly, while fixed-effect can be confused with fixed effects in regression.
When should I use a fixed-effect model?
When studies are very similar and a single true effect is plausible, when the question concerns only the included studies, or as a sensitivity analysis alongside random effects. The reason should be stated.
Should I choose between fixed and random effects using the Q test?
No. The test has low power with few studies and too much power with many. The decision should rest on the similarity of the studies, judged before looking at the results.
Which method is best for rare events with a fixed-effect model?
Mantel-Haenszel is generally reliable, and the Peto method suits rare events with small effects and balanced groups. Check the result against an alternative approach.
Why does one large study dominate my pooled estimate?
Inverse-variance weights give much more weight to precise studies, so a very large study can dominate. Under random effects the weights are more even when heterogeneity is present.
Is the fixed-effect model wrong if the studies differ?
It is not wrong as a description of a weighted average of the included studies, but its confidence interval ignores differences between them and is too narrow for any wider inference.
References
- Deeks JJ, Higgins JPT, Altman DG, editors. Chapter 10: Analysing data and undertaking meta-analyses. In: Higgins JPT, Thomas J, Chandler J, et al., editors. Cochrane Handbook for Systematic Reviews of Interventions. Cochrane; current version available at training.cochrane.org/handbook.
- Borenstein M, Hedges LV, Higgins JPT, Rothstein HR. A basic introduction to fixed-effect and random-effects models for meta-analysis. Res Synth Methods. 2010;1(2):97-111.
- Rice K, Higgins JPT, Lumley T. A re-evaluation of fixed effect(s) meta-analysis. J R Stat Soc Ser A Stat Soc. 2018;181(1):205-227.
- Hedges LV, Vevea JL. Fixed- and random-effects models in meta-analysis. Psychol Methods. 1998;3(4):486-504.
- Hunter JE, Schmidt FL. Fixed effects vs. random effects meta-analysis models: implications for cumulative research knowledge. Int J Sel Assess. 2000;8(4):275-292.
- Cochran WG. The combination of estimates from different experiments. Biometrics. 1954;10(1):101-129.
- Hoaglin DC. Misunderstandings about Q and 'Cochran's Q test' in meta-analysis. Stat Med. 2016;35(4):485-495.
- Mantel N, Haenszel W. Statistical aspects of the analysis of data from retrospective studies of disease. J Natl Cancer Inst. 1959;22(4):719-748.
- Bradburn MJ, Deeks JJ, Berlin JA, Russell Localio A. Much ado about nothing: a comparison of the performance of meta-analytical methods with rare events. Stat Med. 2007;26(1):53-77.
- Page MJ, McKenzie JE, Bossuyt PM, et al. The PRISMA 2020 statement: an updated guideline for reporting systematic reviews. BMJ. 2021;372:n71. doi:10.1136/bmj.n71