- Richard D Riley, senior lecturer in medical statistics1,
- Julian P T Higgins, senior statistician2,
- Jonathan J Deeks, professor of biostatistics1
- 1Department of Public Health, Epidemiology and Biostatistics, Public Health Building, University of Birmingham, Birmingham B15 2TT, UK
- 2MRC Biostatistics Unit, Institute of Public Health, Cambridge CB2 0SR, UK
- Correspondence to: R D Riley
- Accepted 11 November 2010
Meta-analysis is used to synthesise quantitative information from related studies and produce results that summarise a whole body of research.1 A typical systematic review uses meta-analytical methods to combine the study estimates of a particular effect of interest and obtain a summary estimate of effect.2 For example, in a meta-analysis of randomised trials comparing a new treatment with placebo, researchers will collect the estimates of treatment effect for each study, as measured by a relevant statistic such as a risk ratio, and then statistically synthesise them to obtain a summary estimate of the treatment effect.
Meta-analyses use either a fixed effect or a random effects statistical model. A fixed effect meta-analysis assumes all studies are estimating the same (fixed) treatment effect, whereas a random effects meta-analysis allows for differences in the treatment effect from study to study. This choice of method affects the interpretation of the summary estimates. We examine the differences and explain why a prediction interval can provide a more complete summary of a random effects meta-analysis than is usually provided.
Difference between fixed effect and random effects meta-analyses
Figure 1⇓ shows two hypothetical meta-analyses, in which estimates of treatment effect are computed and synthesised from 10 studies of the same antihypertensive drug. Each study provides an unbiased estimate of the standardised mean difference in change in systolic blood pressure between the treatment group and the control group. Negative estimates indicate a greater blood pressure reduction for patients in the treatment group than the control group.
The two meta-analyses give identical summary estimates of treatment effect of −0.33 with a 95% confidence interval of −0.48 to −0.18, but the first uses a fixed effect model and the second a random effects model. In the following two sections we explain why the summary result should be interpreted differently in these two examples because of the different meta-analysis models they use.
Fixed effect meta-analysis
Use of a fixed effect meta-analysis model assumes all studies are estimating the same (common) treatment effect. In other words, there is no between study heterogeneity in the true treatment effect. The implication of this model is that the observed treatment effect estimates vary only because of chance differences created from sampling patients. Hypothetically, if all studies had an infinite sample size, there would be no differences due to chance and the differences in study estimates would completely disappear.
I2 measures the percentage of variability in treatment effect estimates that is due to between study heterogeneity rather than chance.3 I2 is 0% in our fixed effect meta-analysis example, suggesting the variability in study estimates is entirely due to chance. This is visually evident by the narrow scatter of effect estimates with large overlap in their confidence intervals (fig 1, top⇑). The summary result of −0.33 (95% confidence interval of −0.48 to −0.18) in our example thus provides the best estimate of a common treatment effect, and the confidence interval depicts the uncertainty around this estimate. As the confidence interval does not contain zero, there is strong evidence that the treatment is effective.
Random effects meta-analysis
A random-effects meta-analysis model assumes the observed estimates of treatment effect can vary across studies because of real differences in the treatment effect in each study as well as sampling variability (chance). Thus, even if all studies had an infinitely large sample size, the observed study effects would still vary because of the real differences in treatment effects. Such heterogeneity in treatment effects is caused by differences in study populations (such as age of patients), interventions received (such as dose of drug), follow-up length, and other factors.
In the random effects example in figure 1⇑, I2 is 71%, suggesting 71% of the variability in treatment effect estimates is due to real study differences (heterogeneity) and only 29% due to chance.3 This is visually evident from the wide scatter of effect estimates with little overlap in their confidence intervals, in contrast to the fixed effect example (fig 1⇑). The random effects model summary result of −0.33 (95% confidence interval −0.48 to −0.18) provides an estimate of the average treatment effect, and the confidence interval depicts the uncertainty around this estimate. As the confidence interval does not contain zero, there is strong evidence that on average the treatment effect is beneficial.
Use and interpretation of meta-analysis in practice
Unfortunately, meta-analysis results are often interpreted in the same manner regardless of whether a fixed effect or random effects model is used. We reviewed 44 Cochrane reviews that each reported a random effects meta-analysis and found that none correctly interpreted the summary result as an estimate of the average effect rather than the common effect.4 Furthermore, only one indicated why the summary result from a random effects meta-analysis was clinically meaningful,5 arguing that, although real study differences (heterogeneity) in treatment effects existed (because of different doses), the studies were reasonably clinically comparable as the same drug was used and patient characteristics were similar.
Another problem is that a fixed effect meta-analysis model is often used even when heterogeneity is present. We examined 31 Cochrane reviews that did not use a random effects model and found that 26 had potentially moderate or large heterogeneity between studies (I2>25% as a guide3) yet still used a fixed effect model, without justifying why.4 Ignoring heterogeneity leads to an overly precise summary result (that is, the confidence interval is too narrow) and may wrongly imply that a common treatment effect exists when actually there are real differences in treatment effectiveness across studies.
Benefits of using prediction intervals
After a random effects meta-analysis, researchers usually focus on the average treatment effect estimate and its confidence interval. However, it is important also to consider the potential effect of treatment when it is applied within an individual study setting, as this may be different from the average effect. This can be achieved by calculating a prediction interval (fig 2⇓).6
Intervals akin to prediction intervals are commonly used in other areas of medicine. For example, when considering the blood pressure of an individual or the birthweight of an infant, we not only compare it with the average value but also with a reference range (prediction interval) for blood pressure or birthweight across the population. In the meta-analysis setting, our measures are treatment effects, and we work at the study level (rather than the individual level) with a population of study effects. We therefore can report the range of effects across study settings, providing a more complete picture for clinical practice. For instance, consider the random effects analysis in figure 1⇑ again, for which the 95% prediction interval is −0.76 to 0.09. Although most of this interval is below zero, indicating the treatment will be beneficial in most settings, the interval overlaps zero and so in some settings the treatment may actually be ineffective. This finding was masked when we focused only on the average effect and its confidence interval.
A prediction interval can be provided at the bottom of a forest plot (fig 3⇓). It is centred at the summary estimate, and its width accounts for the uncertainty of the summary estimate, the estimate of between study standard deviation in the true treatment effects (often denoted by the Greek letter τ), and the uncertainty in the between study standard deviation estimate itself.6 It can be calculated when the meta-analysis contains at least three studies, although the interval may be very wide with so few studies. A prediction interval will be most appropriate when the studies included in the meta-analysis have a low risk of bias.7 Otherwise, it will encompass heterogeneity in treatment effects caused by these biases, in addition to that caused by genuine clinical differences.
Antidepressants for reducing pain in fibromyalgia syndrome
Hauser and colleagues report a meta-analysis of randomised trials to determine the efficacy of antidepressants for fibromyalgia syndrome, a chronic pain disorder associated with multiple debilitating symptoms.8 Twenty two estimates of the standardised mean difference in pain (for the antidepressant group minus the control group) were available from the included trials (fig 3⇑), with negative values indicating a benefit for antidepressants. Studies used different classes of antidepressants, and other clinical and methodological differences also existed, resulting in large between study heterogeneity in treatment effect (I2=45%; between study standard deviation estimate=0.18). The authors therefore used a random effects meta-analysis and obtained a summary result of −0.43 (95% confidence interval −0.55 to −0.30), concluding that “antidepressant medications are associated with improvements in pain.”
The summary result here relates to the average effect of antidepressants across the trials. As the confidence interval is below zero, it provides strong evidence that on average antidepressants are beneficial; however, it does not indicate whether antidepressants are always beneficial. The authors acknowledge the heterogeneity of treatment effects but conclude that “although study effect sizes differed, results were mostly consistent.” This can be quantified more formally by a 95% prediction interval, which we calculated as −0.83 to −0.02 (fig 2⇑). This interval is entirely below zero and shows that antidepressants will be beneficial when applied in at least 95% of the individual study settings, an important finding for clinical practice.
Inpatient rehabilitation in geriatric patients
Bachmann and colleagues did a random effects meta-analysis of 12 randomised trials to summarise the effect of inpatient rehabilitation compared with usual care on functional outcome in geriatric patients (fig 4⇓).9 The summary odds ratio estimate is 1.36 (1.07 to 1.71), which indicates that the average effect of the intervention is to make the odds of functional improvement 1.36 times higher than usual care. As the confidence interval is above one, it provides strong evidence that the average intervention effect is beneficial.
However, there is large between study heterogeneity in intervention effect (I2=51%; between study standard deviation estimate=0.27), possibly because of differences in the type of intervention used (such as general or orthopaedic rehabilitation) and length of follow-up, among other factors. Responding to the heterogeneity, the authors state: “Pooled effects should be interpreted with caution because the true differences in effects between studies might be due to uncharacterised or unexplained underlying factors or the variability of outcome measures on functional status.”9 This cautionary note can be quantified by presenting a 95% prediction interval, which we calculate as 0.70 to 2.64. This interval contains values below 1 and so, although on average the intervention seems effective, it may not always be beneficial in an individual setting. Further research is needed to identify causes of the heterogeneity, in particular the subtypes of geriatric rehabilitation programmes that work best and the subgroups of patients that benefit most.
Between study heterogeneity in treatment effects is a common problem for meta-analysts. Although it is desirable to identify the causes of heterogeneity (by using meta-regression or subgroup analyses, for example),10 this is often not practically possible.11 12 For instance, there may be too few studies to examine heterogeneity reliably; no prespecified idea of what factors might cause heterogeneity; or a lack of necessary information (such as no individual participant data13). Even when factors causing heterogeneity are identified, unexplained heterogeneity may remain. Thus random effects meta-analysis, which accounts for unexplained heterogeneity, will continue to be prominent in the medical literature. Including a prediction interval, which indicates the possible treatment effect in an individual setting, will make these analyses more useful in clinical practice and decision making.14 Although our examples focused on the synthesis of randomised trials, prediction intervals can also be used in other meta-analysis settings such as studies of diagnostic test accuracy15 and prognostic biomarkers.16
Meta-analysis combines the study estimates of a particular effect of interest, such as a treatment effect
Fixed effect meta-analysis assumes a common treatment effect in each study and variation in observed study estimates is due only to chance
Random effects meta-analysis assumes the true treatment effect differs from study to study and provides an estimate of the average treatment effect
Interpretation of random effects meta-analysis is aided by a prediction interval, which provides a predicted range for the true treatment effect in an individual study
Cite this as: BMJ 2011;342:d549
We thank David Spiegelhalter and Simon Thompson for their comments and acknowledge their role in developing the concept of a prediction interval from random effects meta-analysis (see Higgins et al6). We thank the reviewers and editors for their helpful comments, which have greatly improved the content and clarity of the paper.
Contributors: All authors have undertaken applied and methodological research in the meta-analysis field over many years and work closely with the Cochrane Collaboration. JPTH and RDR conceived the paper. RDR performed the review of 44 Cochrane reviews that used random effects meta-analysis. JPTH (with colleagues Simon Thompson and David Spiegelhalter) identified the need for a prediction interval and subsequently derived how to calculate it. RDR and JJD performed the analyses for the two examples. RDR wrote the first draft and produced the figures and tables. All authors contributed to revising the paper accordingly. RDR is the guarantor.
Competing interests: All authors have completed the unified competing interest form at www.icmje.org/coi_disclosure.pdf (available on request from the corresponding author) and declare no support from any organisation for the submitted work; no financial relationships with any organisation that might have an interest in the submitted work in the previous three years; RDR and JJD are statistics editors for the BMJ.
Provenance and peer review: Not commissioned; externally peer reviewed.