Meta-Analysis Calculator

How to use the meta-analysis calculator

Calculate a meta-analysis online with numiqo. Enter the summary results of at least two independent studies in the table above, one study per row. The calculator combines the studies and shows the pooled effect with its 95% confidence interval, a forest plot, heterogeneity statistics, and a funnel plot. You can choose fixed-effect or random-effects pooling.

What data can I enter?

Under Effect sizes, choose Standard error or 95% CI. For odds, risk, or hazard ratios, enter the natural logarithm of the ratio and of its confidence limits, not the ratio itself; results are shown as ratios again. From a 95% confidence interval, the standard error is (upper - lower) / (2 * 1.96). This only works for a normal-approximation 95% interval that is symmetric on the analysis scale, so use the reported standard error when you have it.

Enter studies and view results

1. Choose your data format and effect measure. Keep the same group order and outcome direction in every study.
2. Import a spreadsheet, paste cells from Excel, or type into the table above. Then select the column for each required input. Study labels are optional.
3. Choose the model. For Random effects, you can change the Estimator and the Hartung–Knapp adjustment. Results update automatically.
4. Review the pooled results, forest plot, study weights, model comparison, heterogeneity, and funnel plot below the table.

To exclude a study without deleting it, use the filter above the table. Empty rows are ignored; missing or invalid values in an included study must be corrected first. Use export / import above the table to save your data as Excel or .numiqo files.

For example, a study with 12 events among 100 participants in Group 1 and 20 among 100 in Group 2 has a risk ratio of 0.60. Enter those four counts in one row and add at least one more study to get a pooled result.

Effect size converter

Before pooling, every study is converted to a common effect measure. Events and totals become an odds ratio, risk ratio, or risk difference; means and SDs become a mean difference or Hedges' g; correlations are converted to Fisher's z and shown as r. The Study results table lists the converted effect and its 95% confidence interval for each study. Results appear once at least two studies are included.

How are the estimates calculated?

The pooled effect is a weighted mean of the study effects. Fixed-effect pooling weights each study by the inverse of its variance, w = 1 / v. Random-effects pooling uses w = 1 / (v + tau-squared), where tau-squared is the between-study variance, estimated by REML (default) or DerSimonian-Laird.

The Hartung-Knapp adjustment is on by default for random effects. It rescales the standard error of the pooled effect and uses a t distribution with k - 1 degrees of freedom, where k is the number of studies. Without it, confidence intervals use the normal approximation. All intervals use a 95% level.

Ratios are pooled on the log scale, correlations on Fisher's z scale. Mean differences are Group 1 minus Group 2, and Hedges' g includes the small-sample correction. If a binary study contains a zero cell, 0.5 is added to all four cells, and the calculator reports this; for the risk difference, the correction is used only for the variance. For odds and risk ratios, studies with no events, or only events, in both groups carry no information and are not pooled. The calculations use the R package metafor, which runs in your browser, so your data are not sent to a server.

How do I interpret the results?

A fixed-effect model assumes one common true effect. A random-effects model allows the true effect to differ between studies. Choose the model based on your research question and study designs, not on the results. The model comparison table shows both estimates.

In the forest plot, each study is shown with its 95% confidence interval, and the diamond shows the pooled effect. For random effects, a dotted line shows the 95% prediction interval: the range in which the true effect of a new, comparable study is expected to lie. With few studies, it can be unreliable. The no-effect line is 1 for ratios and 0 for differences and correlations. See the confidence interval tutorial for more on intervals.

Heterogeneity describes how much the true effects differ between studies. The calculator reports Cochran's Q with its df and p-value, I-squared, and tau-squared. For Q greater than zero, I-squared is max(0, (Q - df) / Q) * 100; it is zero when Q is zero. An I-squared of zero does not prove that all studies share one true effect, especially with few studies.

The funnel plot shows each study's effect against its standard error. Egger's test appears with at least ten included studies. An asymmetric funnel can have several causes and does not by itself prove publication bias.

Scope and limitations

The calculator pools independent study estimates. It does not account for multiple outcomes from the same study or shared control groups. Subgroup analysis, meta-regression, network meta-analysis, trim-and-fill, and Mantel-Haenszel or Peto pooling are not available.

When reporting results, state the effect measure, model, tau-squared estimator, whether the Hartung-Knapp adjustment was used, and any continuity correction.

Methods and further reading

• Cochrane Handbook, Chapter 10: Analysing data and undertaking meta-analyses covers model choice, inverse-variance pooling, and heterogeneity.
• metafor effect-size documentation describes effect measures and alternative sampling-variance estimators. Different software settings can produce different results.