Meta-Analysis Calculator
Pool the results of several studies in one step. Paste effect sizes with their confidence intervals or standard errors, events and totals from each trial, or group means and SDs, and get fixed-effect and DerSimonian–Laird random-effects estimates with 95% confidence intervals, Q, I², τ², a prediction interval and a forest plot you can download as SVG or PNG. Free, no sign-up, and it runs entirely in your browser.
Paste from a spreadsheet or separate values with tabs, commas or semicolons; the name is optional and a header row is skipped. Use tabs or semicolons if your numbers have decimal commas. Group 1 is the treatment or exposed group.
Paste two or more studies to see the pooled estimates, heterogeneity statistics and a forest plot.
How to Use the Meta-Analysis Calculator
- 1. Choose what your studies report. Events and totals for binary outcomes (pooled as odds ratios), Means, SDs and group sizes for continuous outcomes (pooled as Hedges' g), or Effect sizes with a CI or SE when papers give a ready-made estimate.
- 2. For effect sizes, set the format and scale. Say whether each row gives a 95% interval or a standard error, and whether the measure is additive (a mean difference, Hedges' g) or a ratio (odds, risk or hazard ratio), which is pooled on the log scale.
- 3. Paste one study per line. Start with the study name, then the numbers in the order the label shows. A header row is skipped, and any line that cannot be used is reported with the reason.
- 4. Read the results and download the plot. You get both models' estimates with intervals, Q, I², τ², a prediction interval and each study's weight, plus a forest plot to save as SVG or PNG.
The Fixed-Effect and Random-Effects Models
Each study i contributes an effect yᵢ and its sampling variance vᵢ. Ratios are analysed as natural logs, so yᵢ is ln OR, ln RR or ln HR, and results are converted back with exp() for display.
Formulas
Fixed effect: wᵢ = 1/vᵢ, θ = Σwᵢyᵢ / Σwᵢ, SE(θ) = 1/√Σwᵢ
Heterogeneity: Q = Σwᵢ(yᵢ − θ)², df = k − 1, I² = max(0, (Q − df)/Q)
DerSimonian–Laird: τ² = max(0, (Q − df)/C), C = Σwᵢ − Σwᵢ²/Σwᵢ
Random effects: wᵢ* = 1/(vᵢ + τ²), θ* = Σwᵢ*yᵢ / Σwᵢ*, SE(θ*) = 1/√Σwᵢ*
Prediction interval: θ* ± tk−2 × √(τ² + SE(θ*)²), with t the 97.5th percentile of the t distribution on k − 2 degrees of freedom
- The fixed-effect model assumes every study estimates one true effect, so differences between studies are sampling error alone and large studies dominate.
- The random-effects model assumes the true effects themselves vary, with variance τ², and estimates their mean. Adding τ² to every variance evens out the weights and widens the interval.
- Effect sizes from raw data: log odds ratios from events and totals, adding 0.5 to every cell of a table that has a zero (a study with no events in either group is left out); Hedges' g from means and SDs, using the pooled SD and Hedges' exact small-sample correction; and a standard error from a 95% interval as (upper − lower) / (2 × 1.96), on the log scale for ratios.
Worked Example: Five Trials of Appointment Reminders
Five trials compared text-message reminders with no reminder, counting patients who missed an appointment. Press Load example in the events-and-totals mode to enter them; an odds ratio below 1 favours reminders.
| Trial | Reminder | Control | OR [95% CI] | Weight (random) |
|---|---|---|---|---|
| A | 12/100 | 25/100 | 0.409 [0.192, 0.870] | 19.6% |
| B | 8/60 | 18/62 | 0.376 [0.149, 0.948] | 15.0% |
| C | 38/250 | 41/245 | 0.892 [0.551, 1.443] | 30.7% |
| D | 4/40 | 11/38 | 0.273 [0.078, 0.950] | 9.5% |
| E | 20/150 | 38/148 | 0.445 [0.245, 0.810] | 25.3% |
The fixed-effect model gives a pooled OR of 0.554 (95% CI [0.408, 0.753], z = −3.78, p < .001). Heterogeneity is moderate: Q = 6.80 on 4 degrees of freedom (p = 0.147) and I² = 41.2%. On the log scale Σwᵢ = 41.0323 and C = 29.7711, so τ² = (6.8035 − 4) / 29.7711 = 0.0942. The random-effects model then gives OR = 0.504 (95% CI [0.329, 0.773], z = −3.14, p = .002).
Trial C is the largest and the only one close to no effect. Under fixed effects it carries 40.4% of the weight; under random effects, 30.7%, which is why the random-effects estimate sits further from 1. The prediction interval, 0.152 to 1.670, crosses 1: on average reminders work, but a new setting could see no benefit.
Interpreting Heterogeneity: Q, I² and τ²
- Q tests whether the studies share one true effect. With few studies it has little power, so a non-significant Q is weak evidence of homogeneity.
- I² is the share of the variation in estimates that reflects real differences rather than chance. Higgins and colleagues suggested 25%, 50% and 75% as low, moderate and high; the Cochrane Handbook uses overlapping bands (0–40% might not be important, 30–60% moderate, 50–90% substantial, 75–100% considerable) because I² also depends on how precise the studies are.
- τ² is the variance of the true effects, in the units of the analysis (log units for ratios), and the prediction interval turns it into a range for the effect in a new study — often more useful than the interval for the average.
Reading and Reporting a Forest Plot
Each row of a forest plot is a study: the square marks its estimate, with an area proportional to its weight, and the line its 95% confidence interval. The diamonds are the pooled estimates, their width spanning the interval, and the dashed vertical line marks no effect (1 for ratios, 0 for differences). Ratio axes are logarithmic, so 0.5 and 2 sit equally far from 1.
Report the model, the estimator, the pooled estimate with its interval, and the heterogeneity statistics: "In a random-effects meta-analysis (DerSimonian–Laird) of five trials, reminders reduced the odds of a missed appointment, OR = 0.50, 95% CI [0.33, 0.77], p = .002; I² = 41.2%, τ² = 0.094." PRISMA 2020 asks for each study's estimate and interval, ideally in a plot — which is what a forest plot is — and for the summary estimate with its precision and measures of heterogeneity.
Limits and Common Mistakes
- Trusting DerSimonian–Laird intervals with few studies. With fewer than about ten studies, τ² is imprecise and DL intervals can be too narrow; REML with the Hartung–Knapp adjustment, in dedicated software, is often preferred.
- Pooling studies that measure different things. Combine the same outcome in comparable populations, and do not mix odds ratios with risk ratios, or change scores with final values, without converting them.
- Ignoring small-study effects. Pooling cannot correct for unpublished null results; inspect a funnel plot when you have enough studies.
- Choosing the model from the Q test. Decide in advance; random effects is the usual default when studies differ in design or population.
Frequently Asked Questions
- Should I use a fixed-effect or random-effects model?
- Use fixed effect only if you believe every study estimates the same true effect — for example, near-identical trials of one intervention. When studies differ in populations, doses or settings, the random-effects model is more realistic, because it estimates the average of a distribution of true effects. Decide before looking at the results, and report both if reviewers expect it.
- How do I meta-analyse odds ratios reported with confidence intervals?
- Choose Effect sizes with a CI or SE, keep the format at effect and 95% limits, and set the scale to Ratio. The calculator takes logs, derives each standard error as (ln upper − ln lower) / (2 × 1.96), pools on the log scale, and converts the results back to odds ratios.
- What happens to studies with zero events?
- If one group has no events, the odds ratio is undefined, so the calculator adds 0.5 to all four cells of that study's table and marks it with a dagger. A study with no events in either group (or only events) tells you nothing about the odds ratio and is left out, with a note. For many zero-event studies, methods such as Peto's odds ratio or exact approaches are better.
- How do I know the calculator's results are correct?
- Its formulas are tested against the worked examples of Borenstein, Hedges, Higgins and Rothstein's Introduction to Meta-Analysis (2009), as reproduced by the metafor package. For their six-study example of means, it returns a random-effects g of 0.358 (95% CI [0.152, 0.565]) with τ² = 0.0373 and I² = 58.3%, the published values.
- Can I use this for a systematic review I plan to publish?
- It suits learning, checking a published analysis, or a quick pooled estimate. A publishable review usually also needs subgroup and sensitivity analyses, funnel plots and alternative τ² estimators, which dedicated software such as R's metafor or RevMan provides — use this calculator to sanity-check those results.