Fixed & random-effects meta-analysis
Combine independent study effect estimates while inspecting their precision and between-study differences.
Concepts and reasoning
Synthesize estimates of the same meaningful question
A meta-analysis combines study estimates, not simply all participants as though they came from one large undifferentiated study. Each study supplies an effect estimate and a measure of its precision. Before combining them, establish that they address sufficiently comparable populations, interventions, comparators, outcomes, and times.
The effect scale must be coherent. A mean difference in milligrams cannot be combined directly with one in grams. Ratios such as risk ratios and odds ratios are usually analyzed on their logarithmic scale. An odds ratio is not a risk ratio, and a standardized mean difference is not an ordinary difference in the original outcome units.
Two models, two targets
A common-effect model assumes all studies estimate one underlying effect; their observed differences arise from sampling error. A random-effects model allows the underlying effects themselves to vary across studies and estimates their mean under a distributional model. The latter adds between-study variance, often written
Choosing random effects does not make unlike studies scientifically comparable or remove bias from poor study design. It changes the weighting and the estimand. The choice should follow the evidence question and assumptions, not merely whether a heterogeneity test crosses a significance threshold. Cochrane’s synthesis chapter discusses these interpretation choices.
Mathematical foundations
Precision determines the weights
For study estimate
Halving a standard error quadruples its weight. Sample size is related to precision, but it is not itself the weight when outcome variances or event frequencies differ.
Under a simple random-effects model, use
A confidence interval for
Worked example
Pool two hypothetical log risk ratios
Suppose Study A reports log risk ratio −0.20 with standard error 0.10 and Study B reports −0.40 with standard error 0.20. Their common-effect weights are 100 and 25.
The pooled log ratio is
| Study | Log risk ratio | Standard error | Common-effect weight |
|---|---|---|---|
| A | −0.200 | 0.100 | 100.000 |
| B | −0.400 | 0.200 | 25.000 |
The more precise study receives four times the weight. Averaging the two ratios, or averaging the log estimates without weights, would answer a different calculation.
For illustration only, stipulate
Interpreting the results
Distinguish the average effect from the range of effects
The common-effect estimate assumes studies target one underlying effect; the random-effects estimate summarizes a distribution of underlying effects. Their difference is not a contest in which the more favorable answer should be chosen. Match the model to the research question and between-study differences.
A confidence interval for the pooled mean describes uncertainty in that average. A prediction interval additionally reflects estimated between-study heterogeneity and is intended to describe plausible underlying effects in comparable settings. It is not an interval for an individual patient's response. With few studies, both heterogeneity and prediction limits can be unstable.
Tau squared is heterogeneity on the analysis scale. I squared describes the relative contribution of heterogeneity to observed variation and depends on study precision; it is not the percentage of studies that are unreliable. On a log-ratio scale, exponentiate estimates and both interval limits before describing ratios. Cochrane's handbook discusses these distinctions.
Assumptions and limitations
Comparable estimates are a prerequisite
Studies must address sufficiently related populations, interventions, comparators, outcomes, and follow-up periods for pooling to be meaningful. Statistical weighting cannot resolve an incoherent target question. Risk ratios, odds ratios, and hazard ratios should not be combined merely because each is a ratio.
Inverse-variance methods require effect estimates and their standard errors on the same scale. For ratio measures this is usually the natural-log scale. Study estimates should be independent; multiple outcomes, time points, or comparisons sharing participants violate that assumption unless their dependence is modeled.
A random-effects distribution is a model for genuine effect variation, not a correction for bias. Selective publication, biased study designs, and inconsistent outcome measurement can distort both the pooled effect and heterogeneity. A small standard error does not make a biased study more credible.
Diagnostics and sensitivity analysis
Investigate disagreement and influence
Inspect the forest plot for effect direction, interval width, and influential studies. Recheck extracted scales, standard errors, units, and reference groups before attributing an outlier to clinical heterogeneity. A reversed comparison can manufacture disagreement.
Leave-one-out results show whether a single study strongly changes the pooled estimate or heterogeneity. They support investigation, not automatic exclusion. Examine prespecified sensitivity analyses based on study quality, design, or outcome definition and explain why each exclusion is relevant.
Funnel asymmetry has several possible causes, including heterogeneity and selective reporting. With few studies, visual or formal assessments have limited information. Compare statistical findings with the search strategy, registered protocols, and unavailable results. Report uncertainty in heterogeneity and avoid presenting a precise average as evidence of a uniform effect across settings.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- Cochrane Handbook, chapter 10: analysing data and undertaking meta-analyses — Free guidance on synthesis, heterogeneity and interpretation.