The short answer
A fixed-effect model asks what the studies show if they are all estimating the same thing. A random-effects model asks what the average effect is across a population of studies that differ, and how much the effect varies. In applied research the studies nearly always differ in participants, settings, interventions and methods, so random effects is usually the more realistic assumption, and it is the more common default. A fixed-effect analysis is defensible in a narrower set of circumstances, described below. In either case the choice should be reasoned and stated, and the other analysis is commonly shown as a check.
The choice should not be made by testing for heterogeneity and then picking the model that the test favors. The Q test has low power with few studies and high power with many, so it cannot reliably decide. The choice is about what question to answer and what assumption about the studies is credible, which is a matter of subject-matter judgment. The methods are described in detail on the pages for fixed-effect and random-effects meta-analysis.
The assumptions side by side
| Fixed effect | Random effects | |
|---|---|---|
| Assumption | One true effect shared by all studies | True effects vary between studies around a mean |
| Source of differences | Sampling error only | Sampling error and real differences |
| Weights | Inverse of each study's variance | Inverse of the sum of the study's variance and tau-squared |
| Large studies | Dominate the pooled estimate | Lose some of their advantage; weights are more even |
| Pooled result describes | The common effect, or a weighted average of the included studies | The mean of a distribution of effects |
| Confidence interval | Narrower | Wider when heterogeneity is present |
| Needs | Nothing beyond the study estimates | An estimate of tau-squared, which is imprecise with few studies |
A worked comparison
The six simulated studies used throughout this site show the arithmetic. They are invented for illustration. Under the fixed-effect model, with weights equal to the inverse variances, 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. Under random effects the estimate is 0.22 and the interval runs from 0.10 to 0.35. The point estimates are almost the same, because the heterogeneity is small, but the interval is wider, which reflects the uncertainty about the variation between studies.
The difference becomes important when the studies disagree and differ in size. Suppose a collection has one large trial with an effect of 0.05 and standard error 0.05, and three small trials with effects of 0.60, 0.75 and 0.55. The fixed-effect estimate is 0.10, essentially the large trial's result, because it carries nearly all of the weight. With the heterogeneity estimated at 0.124, the random-effects estimate is 0.43, because the small trials count for more. Neither number is the truth. They answer different questions, and the gap between them is itself information: it says that the conclusion depends on how the studies are weighted, which should prompt an examination of why the large trial differs from the small ones, which may be real heterogeneity or small-study bias.
How the weights move
The practical difference is in the weights. The table shows, for the six simulated studies, the share of the total weight each receives under the two models. Because the estimated between-study variance is added to every study's own variance under random effects, the weights move toward equality: the most precise study loses share and the least precise gains it. With larger heterogeneity the movement is greater, and in the limit of very large between-study variance every study would receive nearly the same weight regardless of its size, so that the pooled estimate approaches a simple average.
| Study | Standard error | Fixed effect | Random effects |
|---|---|---|---|
| Study A | 0.15 | 13.9% | 14.8% |
| Study B | 0.12 | 21.6% | 21.2% |
| Study C | 0.20 | 7.8% | 9.0% |
| Study D | 0.10 | 31.2% | 27.6% |
| Study E | 0.18 | 9.6% | 10.9% |
| Study F | 0.14 | 15.9% | 16.6% |
This is why a result can look different under the two models even when the point estimates are close: the studies contributing to it are weighted differently, and the confidence intervals are computed from different variances.
The details that matter under random effects
Choosing random effects is only the first step. The analysis also needs an estimator for the between-study variance and a method for the confidence interval. The older DerSimonian-Laird estimator is simple and common, but tends to underestimate the variance with few studies. Restricted maximum likelihood and the Paule-Mandel estimator are generally preferred in methodological comparisons. For the interval, the conventional normal-theory approach ignores the uncertainty in the estimated variance and is too narrow with few studies, and the Hartung-Knapp-Sidik-Jonkman adjustment, which uses a t distribution, gives better coverage. A prediction interval should accompany the confidence interval to show the range of effects expected in a new setting. These details are set in the protocol and reported, since they can change the result as much as the choice between fixed and random effects. See random-effects meta-analysis.
Small-study effects and the choice of model
The two models respond differently to bias in small studies. Because a random-effects analysis gives small studies more weight than a fixed-effect analysis does, it is more affected when small studies show larger effects for reasons such as publication bias or poorer methods. A noticeable difference between the fixed-effect and random-effects estimates, with the random-effects estimate showing the larger effect, is a signal worth investigating, with a funnel plot and sensitivity analyses. It does not show that either model is wrong. It suggests that the conclusion depends on how much influence the small studies are given, and the report should say so. When funnel plot asymmetry is present, neither model corrects for it, and more specialized methods are used only as sensitivity analyses. See publication bias.
Conventions in different fields
Practice varies. In clinical medicine, random effects is the usual primary model in reviews of trials, with a fixed-effect analysis shown as a sensitivity analysis, particularly where studies are few. In psychology, education and management, where studies of the same construct differ widely in populations and measures, random effects is standard and is often described as the model that allows generalization beyond the sample of studies. In ecology and some other fields, models that include random effects for species, sites or laboratories, in addition to study, are used to deal with non-independence. Some older literature, and some commercial software defaults, used fixed effects when a heterogeneity test was not significant, a practice now discouraged. Authors should follow the conventions of their field and the instructions of the journal, and explain their choice in a way that reviewers from the field will accept.
A short decision guide
State the question
Is the aim to describe the included studies, or to estimate an average effect across a wider population of studies and settings?
Judge the similarity of the studies
Compare populations, interventions, comparators, outcomes and designs. Would you expect the effect to be the same in all of them?
Choose in the protocol
Write down the model and its reason. In most reviews the primary analysis is random effects with a justified estimator and interval method.
Run the other model as a check
Show how much the estimate and interval change, and explain differences.
Report the variation
Give tau-squared and a prediction interval under random effects, and interpret them.
When a fixed-effect model is defensible
- The studies were designed to be as alike as possible, for example trials in one development program with the same protocol, so that a common effect is plausible.
- The question concerns the studies actually included, and no generalization is intended.
- There are very few studies, and the random-effects estimate would depend on an unstable estimate of the between-study variance, provided the limits are acknowledged.
- It is a sensitivity analysis alongside a random-effects primary analysis.
Even then, the fixed-effect confidence interval applies only to the common effect or the weighted average of the included studies, and it should not be read as an interval for the effect in other settings.
When random effects is the better choice
- The studies differ in populations, settings, interventions, outcome measures or designs, as in almost any review that draws on the published literature.
- The aim is to generalize beyond the included studies, for example to inform a decision in a new setting.
- There is evidence of heterogeneity, or good reason to expect it.
- The audience will want to know how much the effect varies, which a random-effects analysis estimates and a prediction interval conveys.
The model has costs. The between-study variance is estimated poorly with few studies, so the interval may be too narrow unless an adjustment is used, and small studies receive more weight than under fixed effects, which can matter if small studies are biased. These are discussed on the method page.
Common mistakes
- Choosing by the Q test or I squared. Neither decides which assumption is credible.
- Reporting only the model that gave the preferred answer. The choice should be fixed in the protocol, and the other result shown.
- Calling a fixed-effect result "the effect." It is the common effect only if the assumption holds.
- Ignoring tau-squared and the prediction interval under random effects. The mean alone hides the variation.
- Using the default estimator and interval without thought. DerSimonian-Laird with a normal interval is inadequate with few studies.
- Mixing up the term. Fixed effects in a regression is a different idea from the fixed-effect model in meta-analysis.
Reporting the choice
The report should name the model, give the reason for it, and state the estimator of the between-study variance and the method for the confidence interval under random effects. It should give heterogeneity statistics and, under random effects, a prediction interval. The result of the other model is shown in a sensitivity analysis if it could change the conclusion. The figure and the table should state which model produced them, since weights and intervals differ. Reviewers will ask why one model was chosen, and a clear answer, based on the clinical and methodological features of the studies, is the best protection. The review is reported according to PRISMA 2020.
Support
The choice of model, the estimator and the sensitivity analyses are decided in the analysis plan, as part of the meta-analysis service.
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Frequently asked questions
Which model should I use if I only have a few studies?
Random effects is still the more credible assumption if the studies differ, but the between-study variance is imprecise, so use an adjusted interval, show the fixed-effect result too, and be open about the limits.
Do fixed and random effects give the same answer?
When there is little heterogeneity they give nearly the same estimate. When studies differ, particularly in size, the estimates and intervals can diverge.
Is random effects always more conservative?
Its interval is wider when heterogeneity is present, but it gives more weight to small studies, which can make the estimate more affected by small-study bias.
Should I decide the model before looking at the data?
Yes. Specify it in the protocol, with a reason, and show the alternative as a sensitivity analysis.
What is a common-effect model?
Another name for the fixed-effect model, which some authors prefer because it states the assumption of one common effect.
How do I interpret a random-effects result?
As the average of a distribution of true effects, reported with the variation between studies and, ideally, a prediction interval.
References
- 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.
- Higgins JPT, Thompson SG, Spiegelhalter DJ. A re-evaluation of random-effects meta-analysis. J R Stat Soc Ser A Stat Soc. 2009;172(1):137-159.
- 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.
- IntHout J, Ioannidis JPA, Borm GF. The Hartung-Knapp-Sidik-Jonkman method for random effects meta-analysis is straightforward and considerably outperforms the standard DerSimonian-Laird method. BMC Med Res Methodol. 2014;14:25.
- 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.