Pools the effect sizes of several comparable studies into one estimate — inverse-variance fixed-effect and DerSimonian–Laird random-effects — and reports how much the studies disagree (Cochran's Q, I², tau²) and whether small studies skew the pool (Egger's test), with a forest table. Use when a user asks to "run a meta-analysis", "pool these trial results", "what's the overall effect size across the studies?", or "how heterogeneous are the findings?". Not for a single study's significance (use `test-significance`) or for assembling the study set (use `systematic-review`); this skill is the statistical endpoint once comparable studies are in hand.
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Pools the effect sizes of several comparable studies into one estimate — inverse-variance fixed-effect and DerSimonian–Laird random-effects — and reports how much the studies disagree (Cochran's Q, I², tau²) and whether small studies skew the pool (Egger's test), with a forest table. Use when a user asks to "run a meta-analysis", "pool these trial results", "what's the overall effect size across the studies?", or "how heterogeneous are the findings?". Not for a single study's significance (use `test-significance`) or for assembling the study set (use `systematic-review`); this skill is the statistical endpoint once comparable studies are in hand.
license
MIT
metadata
{"category":"evidence-verification","method":"Meta-analysis (fixed-effect and DerSimonian–Laird random-effects pooling)","origin":"G. V. Glass, 1976 (term); R. DerSimonian & N. Laird, 1986 (random-effects estimator)","version":"2.0.0"}
Meta-Analysis
Meta-analysis — named by Glass (1976); standard random-effects estimator by DerSimonian and Laird (1986) — pools effect estimates from studies measuring the same thing, precision-weighted, into one estimate with a confidence interval. Its core principle: report pooling and heterogeneity together — a pooled number is only as meaningful as the agreement behind it (Higgins & Thompson, 2002).
When to invoke
Invoke when:
Several studies report the same effect on a common scale and a pooled answer is wanted: "pool these trials", "overall effect size", "meta-analysis of {intervention}".
How consistent a literature is: "how heterogeneous are the findings?".
A systematic-review has produced a comparable included-study table.
Do NOT invoke when:
Only one study — use test-significance; nothing to pool.
The studies still need finding and screening — use systematic-review.
How much certainty the pooled body deserves — use evidence-appraisal (GRADE).
No common effect scale exists — write a narrative synthesis; pooling unlike measures gives a precise, meaningless number.
Procedure
1 — Fix the question and the inclusion rule
State the PICO and which studies qualify before touching numbers. If the pooling ends a systematic-review, the protocol already answers this — never renegotiate inclusion after seeing results. Fewer than three comparable studies: report them individually; pooling adds false authority.
2 — Extract per-study effect sizes on one scale
Record each study's name, its effect on a common scale (log odds ratio, standardized mean difference, or mean difference — pick one and convert), and its standard error or 95 % CI (SE = CI width ÷ 3.92). Record every conversion for the audit trail. Carry each study's assess-study-bias rating alongside its effect.
3 — Pool both ways
Compute the inverse-variance fixed-effect and DerSimonian–Laird random-effects estimates. They answer different questions: fixed-effect assumes one true effect; random-effects estimates the mean of a distribution of true effects. Report both; when they diverge materially, heterogeneity is the story, not the pooled point.
4 — Quantify heterogeneity
Read Cochran's Q with its p-value, I², and tau². I² is the share of observed variance beyond sampling error (Higgins & Thompson, 2002). The familiar adjectives come from a different paper — Higgins, Thompson, Deeks & Altman (BMJ, 2003) — which assigns low, moderate and high to I² values of 25 %, 50 % and 75 %, and says a naive categorisation "would not be appropriate for all circumstances" and that the adjectives are tentative. They are anchor points, not bands, and reading them as cut-offs is this skill's simplification. Moderate or high I² calls for random-effects plus subgroup explanation (by design, population, dose); at high I² no single number should headline — report the range and its drivers.
5 — Probe small-study effects
Run Egger's regression test for funnel-plot asymmetry (Egger et al., 1997). Interpret it only with ten or more studies; below that it is underpowered — a null result proves nothing, a significant one merely suggests. Asymmetry may reflect publication bias or genuine small-study effects (different populations, lower quality); say which is more plausible and why.
6 — Report with sensitivity checks
Fill the output template: both pooled estimates with intervals, which headlines and why; heterogeneity with interpretation; the Egger result or why it is not interpreted; the forest table so every study's contribution is visible; and leave-one-out on the heaviest study — whether one trial drives the pool.
Output template
## Meta-analysis — {question / PICO}
**Studies pooled:** {k} ({designs}); scale: {SMD | log OR | MD}; source: {protocol/search}
**Fixed-effect:** {estimate} (95 % CI {low} to {high})
**Random-effects (DL):** {estimate} (95 % CI {low} to {high}); tau² = {tau2}
**Headline estimate:** {FE | RE} — {reason tied to heterogeneity}
**Heterogeneity:** Q = {Q} (df {df}, p = {p}); I² = {i2} % — {low | moderate | high}; drivers: {subgroups or "unexplained"}
**Small-study effects:** Egger t({df}) = {t}, p = {p} — {interpretation | not interpreted, k < 10}
**Forest table:** {study | effect | SE | 95 % CI | weight %}
**Sensitivity:** without {heaviest study}: {estimate} ({unchanged | changed} conclusion)
**Caveats:** {risk-of-bias profile; conversions applied}
Mandatory fields: studies pooled, both pooled estimates, headline choice with reason, heterogeneity with interpretation, the Egger line, and the sensitivity line. A pooled estimate without its heterogeneity line is not a result.
Worked example
Question (illustrative, invented studies): does a workplace mindfulness programme reduce burnout versus waitlist — standardized mean difference on an emotional-exhaustion scale, negative = less burnout? Six RCTs, SMDs and standard errors in examples/mindfulness-burnout.json. python3 scripts/metapool.py --file examples/mindfulness-burnout.json prints:
Forest table (fixed-effect weights)
study effect SE 95% CI weight %
------------------------------------------------------------------
Andersen 2017 -0.420 0.150 [-0.714, -0.126] 16.3
Bianchi 2018 -0.250 0.120 [-0.485, -0.015] 25.4
Chen 2019 -0.610 0.200 [-1.002, -0.218] 9.1
Dubois 2020 -0.100 0.110 [-0.316, 0.116] 30.2
Ekstrom 2021 -0.380 0.170 [-0.713, -0.047] 12.7
Farah 2022 -0.550 0.240 [-1.020, -0.080] 6.3
------------------------------------------------------------------
Fixed-effect pooled: -0.301 SE 0.0605 95% CI [-0.419, -0.182]
Random-effects (DL): -0.330 SE 0.0788 95% CI [-0.485, -0.176]
Heterogeneity: Q = 7.828 (df 5, p = 0.1660); I^2 = 36.1%; tau^2 = 0.0131
Egger's test: intercept = -4.013 (slope 0.274), t(4) = -4.273, p = 0.0129
note: k = 6 < 10 -- Egger's test has low power; interpret with care.
Reading: I² = 36 % is moderate, so the random-effects estimate (SMD −0.33, 95 % CI −0.49 to −0.18) headlines, fixed-effect −0.30 beside it; both exclude zero. Dubois 2020 carries 30 % of the weight and is near null; without it the pool is −0.39 (95 % CI −0.53 to −0.25) with I² = 0 % — the conclusion (small-to-moderate reduction) stands, though its size hinges on one large near-null trial, and the report says so. The Egger intercept is negative and nominally significant — the two smallest studies (Chen 2019, Farah 2022) show the largest effects, a small-study pattern worth naming — but with k = 6 it is no proof of publication bias.
Verification
Every effect and SE traces to its source table or figure; conversions recorded.
Recompute the pool with scripts/metapool.py (by hand for k ≤ 4): weights sum to 100 %, estimate = Σ(w·y)/Σw.
The headline choice (fixed vs random) follows from the heterogeneity line, not the better-looking interval.
Egger's test is marked "not interpreted" when k < 10; otherwise the plausible cause of asymmetry is named.
The leave-one-out check on the heaviest study is reported; exclusions are named, with both pools shown.
Companion tool
scripts/metapool.py does the arithmetic of steps 3–5: from a JSON or CSV file it prints the forest table, both pooled estimates with 95 % CIs, Cochran's Q with p, I², tau², and Egger's test (low-power note when k < 10). Stdlib only; deterministic. Sample output: the worked example.
JSON: [{"study": "A", "effect": 0.5, "se": 0.2}, ...] or {"studies": [...]}; CSV: study,effect,se or study,effect,ci_low,ci_high. By hand this works for three or four studies; beyond that, use the tool.
Pair with adjacent skills
systematic-review — the upstream pipeline; its included-study table is the input.
assess-study-bias — rate each study (RoB 2) before pooling; biased studies pool into a precise biased estimate.
evidence-appraisal — GRADE the pooled body; I² feeds its inconsistency factor, Egger its publication-bias factor.
test-significance — the single-study case.
quantitative-sanity-check — recompute pooled figures from published meta-analyses before repeating them.
Do not report a pooled estimate without heterogeneity; I² = 80 % is a cover-up, not a summary.
Do not choose fixed versus random by which gives the wanted answer; decide from what the studies are and the heterogeneity statistics.
Do not interpret Egger's test below about ten studies; underpowered asymmetry tests are decoration.
Do not mix effect scales; convert to one scale and record the conversion.
Do not let pooling launder quality; ten high-risk-of-bias studies pool into one high-risk number.
Do not drop outliers silently to lower I²; name exclusions and show the pool both ways.
Reference
G. V. Glass, "Primary, Secondary, and Meta-Analysis of Research," Educational Researcher, vol. 5, no. 10, pp. 3–8, 1976. doi:10.3102/0013189X005010003
R. DerSimonian and N. Laird, "Meta-analysis in clinical trials," Controlled Clinical Trials, vol. 7, no. 3, pp. 177–188, 1986. doi:10.1016/0197-2456(86)90046-2
J. P. T. Higgins and S. G. Thompson, "Quantifying heterogeneity in a meta-analysis," Statistics in Medicine, vol. 21, no. 11, pp. 1539–1558, 2002. doi:10.1002/sim.1186 — I² and its estimation; it contains no low/moderate/high categorisation.
J. P. T. Higgins, S. G. Thompson, J. J. Deeks and D. G. Altman, "Measuring inconsistency in meta-analyses," BMJ, vol. 327, no. 7414, pp. 557–560, 2003. doi:10.1136/bmj.327.7414.557 — the tentative low/moderate/high adjectives at I² = 25 %, 50 % and 75 %.
M. Egger, G. Davey Smith, M. Schneider, and C. Minder, "Bias in meta-analysis detected by a simple, graphical test," BMJ, vol. 315, pp. 629–634, 1997. doi:10.1136/bmj.315.7109.629
M. Borenstein, L. V. Hedges, J. P. T. Higgins, and H. R. Rothstein, Introduction to Meta-Analysis. Wiley, 2009. doi:10.1002/9780470743386
