How to use the meta-analysis calculator
- Paste one study per line: a name, the effect estimate and its standard error, separated by commas. Use the same kind of effect for every study.
- Choose the confidence level. If your effects are natural logs of ratios (log odds ratios or log risk ratios), tick the box to see results as ratios.
- Read the pooled effect from both models, the heterogeneity measures Q, I squared and tau squared, and the table of studies with their weights.
- Compare the fixed-effect and random-effects results. If they differ much, or if I squared is large, the studies do not agree closely.
Formula
weight w = 1 / SE^2
fixed-effect estimate = sum(w x effect) / sum(w)
Q = sum(w x (effect - fixed estimate)^2), compared with k - 1 degrees of freedom
I squared = max(0, (Q - (k - 1)) / Q)
tau squared = max(0, (Q - (k - 1)) / (sum(w) - sum(w^2) / sum(w)))
random-effects weight = 1 / (SE^2 + tau squared)
k is the number of studies. The random-effects estimate is the weighted mean using the new weights, and its standard error is the square root of 1 / the sum of those weights. This is the DerSimonian-Laird method.
Four studies
Four studies report effects of 0.30, 0.10, 0.55 and 0.20, with standard errors of 0.10, 0.15, 0.20 and 0.12. The fixed-effect estimate is 0.2599 with a 95 percent interval of 0.1331 to 0.3867. Q is 3.650 on 3 degrees of freedom (p = 0.3018), I squared is 17.8 percent and tau squared is 0.00392.
The random-effects estimate is 0.2608, with an interval of 0.1175 to 0.4041. The first study carries 41.9 percent of the fixed-effect weight and 38.4 percent of the random-effects weight, because tau squared adds the same amount to every variance and so evens the weights out a little.
Fixed effect or random effects
The fixed-effect model assumes that every study estimates the same true effect and that differences are due to chance alone. The random-effects model allows each study to have its own true effect drawn from a distribution, and tau squared measures the spread of that distribution. When tau squared is 0 the two models agree.
Heterogeneity is not a reason to throw studies away, but it is a reason to ask why they differ. The interval from the random-effects model is the one to use when the studies differ, and it is wider.
Limits to keep in mind
With only a few studies, the estimate of tau squared is imprecise and the normal-based interval can be too narrow. The tool pools numbers that you provide. It cannot judge the quality of the studies, publication bias or whether the studies are similar enough to combine. It is not clinical decision support.
To build the interval for a single study, see the confidence interval calculator and the standard error calculator. To turn two groups into an effect size, use the Cohen's d calculator or the odds ratio and relative risk calculator.
Assumptions and limits
- Each study supplies an effect and a standard error on the same scale, and the studies are independent.
- Intervals use the normal distribution, not a t distribution, and no small-sample adjustment is applied.
- The DerSimonian-Laird estimate of tau squared is used. Other estimators give somewhat different values.
- Up to 200 studies are accepted, with at least 2.
Frequently asked questions
What does I squared tell me?
It is the share of the variation among studies that comes from real differences rather than chance, as an estimate. A value of 0 percent means the spread is what chance alone would give. Higher values mean the studies disagree more than chance explains. It is imprecise when there are few studies.
How do I enter a ratio such as an odds ratio?
Take the natural log of each ratio and of its confidence limits to get an effect and a standard error, enter those, and tick the box to see results as ratios. The standard error of a log ratio is the width of the log interval divided by 3.92 for a 95 percent interval.
Why is the random-effects interval wider?
It adds the between-study variance tau squared to every study, which increases the uncertainty about the pooled value. If tau squared is 0, the two intervals are the same.
Is a forest plot drawn?
No plot is drawn. The table lists each study with its effect, interval and weight so you can copy the numbers into the tool you use for figures.