Visualize the full predicted distribution of true effect sizes in a random-effects meta-analysis. Instead of showing only the pooled mean and confidence interval, this tool plots the Normal(mu, tau-squared) density curve, the prediction interval, and the probability that the true effect exceeds any threshold you choose. The prediction interval made visual, following Higgins et al. (2009), Riley et al. (2011), and IntHout et al. (2016).
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| Study | Effect Size | Standard Error | |
|---|---|---|---|
Input the effect size and standard error for each study in your meta-analysis. You can type values manually, paste from a spreadsheet, import a CSV or Excel file, or receive data directly from the effect size calculator via the inter-tool pipeline.
The tool automatically fits a DerSimonian-Laird random-effects model and plots the Normal(mu, tau-squared) density curve. The curve shows the full predicted range of true effects across different populations and settings.
The darker shaded band shows the confidence interval of the pooled estimate. The dashed gold lines mark the prediction interval bounds. Circles along the x-axis indicate where each individual study effect falls relative to the predicted distribution.
Check the probability that the true effect is negative. Enter a custom threshold to compute the probability that the true effect exceeds that value in any given setting, moving beyond binary significance to probabilistic clinical reasoning.
Overlay a histogram of observed study effects to compare against the predicted Normal curve. Adjust the confidence level between 80%, 90%, 95%, and 99% to see how intervals change with different certainty requirements.
Download the plot as a high-resolution PNG or PDF for your manuscript. Copy the auto-generated methods paragraph, export reproducible R code using metafor and dnorm, or send data downstream to the forest plot or funnel plot via the pipeline.
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Get a Free QuoteThe confidence interval reflects uncertainty about the mean true effect. The prediction interval adds between-study heterogeneity, showing where a future true effect would fall. When I-squared is high, the prediction interval can be much wider than the confidence interval, sometimes crossing zero even when the CI is entirely positive.
Unlike I-squared (which is a percentage), tau is on the same scale as the effect sizes. If your pooled standardized mean difference is 0.40 with tau = 0.15, the true effects vary by about 0.15 units around the mean. This makes tau directly interpretable for clinical decision-making.
The Normal(mu, tau-squared) model is the standard assumption in random-effects meta-analysis. With small numbers of studies, the actual shape of the distribution cannot be verified. The predicted distribution should be treated as an approximation, not a definitive characterization.
Reporting that 'there is an 82% probability that the true effect exceeds the minimal clinically important difference of 0.30' is more informative than simply stating the pooled estimate is significant. Threshold probabilities bridge the gap between statistical significance and clinical relevance.
The DerSimonian-Laird estimator can return tau-squared = 0 when the true heterogeneity is small relative to sampling variability, especially with fewer than 10 studies. A zero estimate means the method could not detect heterogeneity, not that it is absent. Consider REML estimation for potentially less biased results.
Toggling the histogram overlay shows how the observed study effects compare to the predicted Normal distribution. If the observed effects cluster differently than the Normal curve suggests, it may indicate non-normal heterogeneity, outliers, or subgroup structure worth investigating.
When the prediction interval crosses zero, the probability of a negative true effect is non-trivial. This metric is critical for interventions where a negative effect means harm. Even a 15% probability of harm may be unacceptable in some clinical contexts.
After examining the distribution, send your studies to the forest plot generator for standard visualization, the funnel plot tool for publication bias assessment, or the sensitivity analysis tool to identify influential studies. The inter-tool pipeline passes data with one click.
The distribution of true effects is a concept at the heart of the random-effects model in meta-analysis. When researchers pool studies using a random-effects approach, they implicitly assume that each study estimates a different true effect, and that these true effects follow a Normal distribution centered on the pooled mean (mu) with variance equal to tau-squared (Higgins, Thompson, and Spiegelhalter, 2009). Most meta-analysis software displays only the pooled estimate and its confidence interval, but this captures only the uncertainty about the average. The full distribution, the bell curve of predicted true effects, reveals how much the treatment effect is expected to vary across different patient populations, clinical settings, and intervention protocols.
The prediction interval is the most practical summary of this distribution. Formally introduced by Higgins et al. (2009) and refined by Riley, Higgins, and Deeks (2011), the prediction interval estimates the range within which the true effect of a new study would be expected to fall. IntHout, Ioannidis, and Borm (2016) recommended using the t-distribution with k minus 2 degrees of freedom rather than the normal distribution, because with a small number of studies, the normal approximation underestimates the interval width. This tool implements the IntHout formula: mu plus or minus t(alpha/2, k-2) times the square root of (tau-squared plus the squared standard error of the pooled estimate). The prediction interval is always wider than the confidence interval and can cross zero even when the confidence interval does not, fundamentally changing the interpretation of the meta-analysis.
Tau and tau-squared are the key parameters that determine the shape of the distribution. Unlike I-squared, which is a relative measure of heterogeneity that depends on sample sizes, tau is expressed on the same scale as the effect sizes and has direct clinical meaning. If the pooled standardized mean difference is 0.50 with tau = 0.25, a researcher can immediately see that the true effects are expected to range roughly from about 0.00 to 1.00 (mu plus or minus 2 times tau), spanning from negligible to large effects by Cohen's conventions. The DerSimonian-Laird estimator used in this tool is the most widely cited method for estimating tau-squared, though it can underestimate the true between-study variance when the number of studies is small (Veroniki et al., 2016). For formal publications, consider comparing with REML estimates using the forest plot generator, which supports both estimators.
The threshold probability feature moves meta-analysis interpretation from binary significance testing to probabilistic clinical reasoning. Rather than asking whether the pooled effect is statistically significant, clinicians can ask: "What is the probability that the true effect exceeds the minimal clinically important difference?" This probability is computed from the Normal(mu, tau-squared) distribution using the standard normal cumulative distribution function. For example, if the minimal clinically important difference for a pain outcome is 0.3 standardized mean difference units, and the tool shows an 88% probability that the true effect exceeds 0.3, this communicates the practical importance of the intervention far more effectively than a p-value.
The histogram overlay provides a visual comparison between the observed study effects and the predicted Normal distribution. If the observed effects closely follow the Normal curve, the random-effects model assumption is supported. If the histogram shows bimodality (two peaks) or heavy tails, it may indicate that the Normal assumption is violated and that the true effects come from a mixture of subpopulations. In such cases, investigate potential moderators using our GOSH plot generator for all-subsets heterogeneity detection or the bubble plot generator for meta-regression visualization.
In practice, the distribution of true effects plot complements rather than replaces the standard forest plot. Present both in your systematic review: the forest plot for individual study results and the distribution plot for the predicted range of effects across settings. Combine with leave-one-out sensitivity analysis to assess whether removing any single study substantially changes the distribution shape. Use our heterogeneity calculator to obtain prediction intervals using alternative tau-squared estimators and compare them against the DerSimonian-Laird values from this tool.
The R code generated by this tool uses the metafor package (Viechtbauer, 2010) to fit the random-effects model and base R's dnorm function to plot the density curve. The code includes prediction interval computation via the predict() function, threshold probability calculation using pnorm, and publication-quality plot formatting. Reviewers and co-authors can reproduce the exact analysis in R, ensuring full computational transparency. For researchers reporting results in their manuscripts, the auto-generated methods paragraph follows PRISMA 2020 conventions and cites the foundational references for prediction intervals and heterogeneity estimation.
Before interpreting the distribution, verify that all effect estimates are on the same scale and direction. Mixing log odds ratios with standardized mean differences, or including studies where a higher effect means benefit alongside studies where a higher effect means harm, will produce a meaningless distribution. Use our effect size calculator to standardize all measures before entering data. For diagnostic test accuracy reviews, where bivariate models are more appropriate, see our SROC curve generator instead.
This plot visualizes the estimated distribution of true effect sizes across different settings in a random-effects meta-analysis. Instead of showing only the pooled mean and its confidence interval (which describes uncertainty about the average effect), the distribution plot shows the full Normal(mu, tau-squared) curve. This makes the prediction interval tangible: you can see the entire bell curve of expected true effects, where each point represents the probability density for a particular effect size value. Circles along the x-axis show where each observed study falls relative to the predicted distribution.
A forest plot displays individual study estimates with their confidence intervals and the pooled diamond, focusing on point estimates and precision. The distribution of true effects plot focuses on the between-study variability, showing the full probability density of where true effects are expected to lie. The forest plot answers 'what did each study find?' while the distribution plot answers 'across all possible settings, what range of true effects would we expect?' The prediction interval on a forest plot is a single line; here, it becomes a visible curve showing the probability of each possible true effect.
The prediction interval estimates the range within which the true effect of a future study conducted in a new setting would be expected to fall (Riley et al., 2011; IntHout et al., 2016). While the confidence interval of the pooled estimate reflects uncertainty about the mean true effect, the prediction interval incorporates both this uncertainty and the between-study heterogeneity (tau-squared). The prediction interval is always wider than the confidence interval and is computed as mu plus or minus t(alpha/2, k-2) times the square root of (tau-squared plus the squared standard error of the pooled estimate). When the prediction interval crosses zero, it means the true effect could plausibly be in either direction in some settings, even if the pooled confidence interval is entirely positive or negative.
Tau-squared is the estimated between-study variance in the true effects. It quantifies how much the true effect sizes vary across different study populations, interventions, or contexts. Tau is the square root of tau-squared and represents the between-study standard deviation, expressed on the same scale as the effect sizes. For example, if the pooled standardized mean difference is 0.50 with tau = 0.20, this means the true effects are expected to vary by about 0.20 units (one standard deviation) around the pooled mean. The distribution curve uses tau directly: it is a Normal distribution with mean equal to the pooled estimate and standard deviation equal to tau.
This probability is calculated from the estimated Normal(mu, tau-squared) distribution using the standard normal cumulative distribution function. If you set a threshold of 0.2, the tool computes the area under the distribution curve to the right of 0.2, giving you the estimated proportion of settings in which the true effect would exceed 0.2. This is clinically useful because it moves beyond a binary 'significant or not' conclusion to a probabilistic statement about effect magnitude. For instance, knowing that 85% of true effects are expected to exceed a clinically meaningful threshold of 0.2 is more informative than simply knowing the pooled estimate is 0.5.
When the estimated tau-squared is zero, there is no detectable between-study heterogeneity. The distribution of true effects collapses to a single point at the pooled estimate, and the density curve cannot be drawn (it would be infinitely narrow). In this case, the tool displays a message indicating that all studies appear to estimate the same underlying effect. The prediction interval equals the confidence interval when tau-squared is zero. A tau-squared of zero does not necessarily mean there is truly no heterogeneity, especially with a small number of studies, because the DerSimonian-Laird estimator can underestimate tau-squared when the number of studies is small.
Use this tool after conducting a standard random-effects meta-analysis, particularly when you observe moderate to substantial heterogeneity (I-squared above 30% to 50%). It is most informative alongside a forest plot. The forest plot shows individual study results and the pooled diamond, while this tool shows the full predicted distribution. It is especially valuable when you need to communicate to clinical decision-makers or policy audiences how much the true effect might vary across settings, or when you want to estimate the probability that the true effect in a new setting would exceed a clinically important threshold.
The tool uses the DerSimonian-Laird method to estimate the between-study variance (tau-squared). Study weights in the random-effects model are 1 / (within-study variance + tau-squared). The pooled estimate (mu) is the weighted mean of study effect sizes. The standard error of the pooled estimate is the inverse square root of the sum of the random-effects weights. Cochran Q is computed as the weighted sum of squared deviations from the fixed-effect pooled estimate, and I-squared is derived from Q as max(0, (Q - df) / Q). The prediction interval uses the t-distribution with k - 2 degrees of freedom (IntHout et al., 2016).
Visualize individual study results and the overall pooled estimate with our forest plot generator for meta-analysis. Identify which studies drive heterogeneity using our GOSH plot generator for all-subsets analysis. Test the robustness of the pooled estimate and distribution shape with leave-one-out sensitivity analysis. Quantify heterogeneity using multiple tau-squared estimators with the heterogeneity calculator.
Reviewed by
Dr. Sarah Mitchell holds a PhD in Biostatistics from Johns Hopkins Bloomberg School of Public Health and has over 15 years of experience in systematic review methodology and meta-analysis. She has authored or co-authored 40+ peer-reviewed publications in journals including the Journal of Clinical Epidemiology, BMC Medical Research Methodology, and Research Synthesis Methods. A former Cochrane Review Group statistician and current editorial board member of Systematic Reviews, Dr. Mitchell has supervised 200+ evidence synthesis projects across clinical medicine, public health, and social sciences. She reviews all Research Gold tools to ensure statistical accuracy and compliance with Cochrane Handbook and PRISMA 2020 standards.
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