Detect publication bias in small meta-analyses using the Doi plot and LFK index (Furuya-Kanamori, Barendregt & Doi, 2018), a superior alternative to funnel plots when your review includes fewer than 10 studies.
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| Study | Effect Size | Standard Error | |
|---|---|---|---|
Input the effect size (e.g., log odds ratio, standardized mean difference, or mean difference) and its standard error for each study in your meta-analysis. You can type values directly into the table, paste from a spreadsheet, or import a CSV or Excel file with automatic column detection.
Click Analyze to compute the normal quantile transformation and render the Doi plot. The tool plots effect sizes on the horizontal axis against their corresponding Z-scores (normal quantiles) on the vertical axis, producing a shape where asymmetry is visually conspicuous even with very few studies.
Review the LFK index value displayed alongside the plot. An absolute value less than 1 indicates no asymmetry (no evidence of publication bias), 1 to 2 indicates minor asymmetry warranting further investigation, and greater than 2 indicates major asymmetry suggesting substantial small-study effects.
Examine the D3.js rendered Doi plot for clustering of data points on one side of the centroid. Symmetric distribution around the center line suggests balanced reporting, while concentration on one side suggests that smaller or less precise studies systematically favor one direction of effect.
Use the LFK index alongside findings from a traditional funnel plot to triangulate your publication bias assessment. When fewer than 10 studies are included, the Doi plot provides more reliable asymmetry detection than Egger's regression test or visual inspection of funnel plots.
Download the Doi plot as a high-resolution PNG suitable for journal submission, copy the auto-generated R code for the metafor package to reproduce the analysis, copy the methods paragraph for your manuscript, or export the underlying data as CSV for further analysis.
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Get a Free QuoteThe LFK index uses three classification bands: absolute values below 1 indicate no asymmetry, 1 to 2 indicate minor asymmetry, and above 2 indicate major asymmetry. These thresholds were validated through simulation studies by Furuya-Kanamori et al. (2018) across a range of effect sizes and study counts. Unlike Egger's regression, the LFK index does not produce a p-value, so interpretation relies on the magnitude of the index rather than a binary significance threshold.
The Doi plot shape provides information that the LFK index alone cannot capture. Outlying studies that sit far from the main cluster may indicate genuine heterogeneity rather than publication bias. When the LFK index falls near a threshold boundary (e.g., 0.9 or 1.1), visual inspection of the plot distribution helps decide whether to report no asymmetry or minor asymmetry in your manuscript.
When your meta-analysis includes fewer than 10 studies, funnel plots lack the statistical power to detect asymmetry reliably, and Egger's test has unacceptable Type I and Type II error rates. The Doi plot maintains acceptable sensitivity with as few as 3 to 5 studies because its normal quantile transformation amplifies visual asymmetry patterns that would appear subtle or invisible in a funnel plot.
The Doi plot builds conceptually on the Galbraith radial plot (Galbraith, 1988) by using normal quantiles of study precision on the vertical axis. However, the Doi plot displays effect sizes on the horizontal axis rather than standardized effects, making asymmetry interpretation more intuitive. The centroid of the Doi plot corresponds to the pooled effect, with deviations on either side reflecting potential small-study effects.
Asymmetric Doi plots can arise from several mechanisms other than publication bias. Genuine heterogeneity (different true effects in different populations), differences in methodological quality between small and large studies, and time-lag bias (smaller studies published earlier with more extreme results) can all produce asymmetric plots. Always interpret asymmetry findings alongside clinical context and sensitivity analyses.
Publication bias is one of the five GRADE domains for rating certainty of evidence. When the LFK index shows major asymmetry (greater than 2) in your Doi plot, this provides quantitative justification for downgrading evidence certainty by one level for publication bias. Document the LFK index value and its classification in your GRADE evidence profile to support transparent decision-making.
The Doi plot was introduced by Furuya-Kanamori, Barendregt, and Doi (2018) as a refinement of the Galbraith radial plot for assessing publication bias in meta-analysis. While funnel plots have been the standard since Light and Pillemer (1984), they perform poorly with fewer than 10 studies because asymmetry in the funnel shape is difficult to detect visually or statistically when data points are sparse. The Doi plot addresses this fundamental limitation by applying a normal quantile transformation to study precision, producing a visualization where asymmetry patterns are more apparent across the full range of study sizes. Cochrane Handbook guidelines (Higgins et al., 2023) recommend against using Egger's test with fewer than 10 studies because of unacceptable rates of false positives and false negatives, making the Doi plot the preferred tool for small meta-analyses.
The LFK index quantifies plot symmetry by computing the difference in areas under the curve on each side of the centroid. The calculation involves integrating the area between each data point and the centroid line, then comparing left-side and right-side contributions. Absolute values below 1 suggest no asymmetry, 1 to 2 suggest minor asymmetry, and values exceeding 2 suggest major asymmetry. Simulation studies confirmed that the LFK index outperforms Egger's regression intercept and Begg's rank correlation when the number of studies is small (Furuya-Kanamori et al., 2018). Unlike these traditional tests, the LFK index does not depend on parametric assumptions about the relationship between effect size and precision.
The conceptual advantage of the Doi plot over a funnel plot lies in its axis transformation. A standard funnel plot places precision (or standard error) on the vertical axis and effect size on the horizontal axis, creating an inverted funnel shape when no bias is present. However, with 5 to 9 studies, the eye cannot reliably distinguish an asymmetric funnel from random scatter. The Doi plot's normal quantile axis stretches the distribution in a way that makes systematic departures from symmetry more visually salient. This property was demonstrated through Monte Carlo simulations across scenarios with 3, 5, 7, and 10 studies (Furuya-Kanamori et al., 2018), where the Doi plot consistently identified planted asymmetry that funnel plots missed.
Egger's test (Egger et al., 1997) remains the most cited statistical test for funnel plot asymmetry, but its statistical power is directly proportional to the number of studies. With fewer than 10 studies, the test has a Type II error rate exceeding 50% in many scenarios, meaning it will miss genuine asymmetry more often than it detects it. The Doi plot and LFK index fill this gap by providing both a visual and quantitative assessment that maintains reasonable performance even with very few studies. For meta-analyses with 10 or more studies, the Doi plot can still be reported alongside Egger's test as a complementary sensitivity analysis, offering a different perspective on the data distribution.
When interpreting asymmetry detection results, reviewers must distinguish between publication bias and other mechanisms that produce small-study effects. Genuine heterogeneity, where smaller studies are conducted in populations that truly respond differently, can create asymmetric plots without any selective reporting. Similarly, lower methodological quality in smaller studies (which often have fewer resources) can inflate effect estimates. The LFK index cannot differentiate these mechanisms from true publication bias, so always interpret asymmetry findings in the context of your risk of bias assessments, subgroup analyses, and clinical reasoning.
For a comprehensive publication bias assessment, pair this tool with our funnel plot and publication bias tool for contour-enhanced funnel plots with Egger's test and trim-and-fill analysis. The funnel plot alternative perspective provided by the Doi plot is most valuable when the two methods agree: if both show asymmetry, the evidence for bias is strengthened. Evaluate evidential value with our p-curve analysis tool, which assesses whether the distribution of significant p-values is consistent with a true underlying effect. Test the robustness of your pooled estimate with the leave-one-out sensitivity analysis tool, and visualize your primary meta-analysis results using the forest plot generator.
A Doi plot is a graphical method for detecting publication bias in meta-analysis, introduced by Furuya-Kanamori, Barendregt, and Doi (2018). Instead of plotting effect sizes against standard errors like a funnel plot, the Doi plot displays effect sizes on the horizontal axis against their corresponding normal quantiles (Z-scores) on the vertical axis. This transformation produces a plot where symmetry is easier to evaluate visually, especially when the number of included studies is small. The Doi plot builds on the concept of the Galbraith radial plot but uses normal quantiles to provide a more intuitive representation of the distribution of study results.
The LFK index (Luis Furuya-Kanamori index) is a quantitative measure of asymmetry in a Doi plot. It is calculated by comparing the areas under the curve on each side of the Doi plot centroid. An LFK index with an absolute value less than 1 indicates no asymmetry, suggesting no evidence of publication bias. Values between 1 and 2 indicate minor asymmetry, which may warrant further investigation. Values greater than 2 indicate major asymmetry, suggesting substantial publication bias or other small-study effects. Unlike Egger's regression test, the LFK index does not rely on a minimum number of studies to produce meaningful results.
The Doi plot is particularly useful when your meta-analysis includes fewer than 10 studies, where funnel plots and Egger's regression test have poor statistical power and are unreliable. Cochrane guidelines note that Egger's test should not be used with fewer than 10 studies because it cannot reliably distinguish true asymmetry from random variation. The Doi plot with its accompanying LFK index provides better detection of publication bias in these small meta-analyses. Even with 10 or more studies, the Doi plot can be used alongside funnel plots as a complementary visualization. Its normal quantile transformation makes asymmetry patterns more visually apparent than the traditional funnel shape.
In a symmetric Doi plot, the data points should be roughly evenly distributed around the center line, forming a shape that is approximately mirror-imaged on both sides. If the plot shows more data points or a larger area on one side, this indicates asymmetry. Left-sided asymmetry (more area on the left) suggests that smaller or less precise studies tend to report smaller effect sizes, while right-sided asymmetry suggests the opposite. The LFK index quantifies this visual impression numerically. Always interpret asymmetry in context, because factors other than publication bias (such as genuine heterogeneity or differences in study quality) can also produce asymmetric plots.
There is no strict minimum number of studies required for a Doi plot, and this is one of its key advantages over funnel plots and Egger's test. Simulation studies by Furuya-Kanamori et al. (2018) demonstrated that the Doi plot and LFK index maintain acceptable performance with as few as 3 to 5 studies. In contrast, Egger's regression test is generally considered unreliable with fewer than 10 studies due to insufficient statistical power. For very small meta-analyses (3 to 5 studies), interpret the LFK index with appropriate caution, as any single outlying study can substantially influence the result.
While the Doi plot addresses several shortcomings of funnel plots, it has its own limitations. First, like all asymmetry-based methods, it cannot distinguish publication bias from other causes of small-study effects, such as genuine differences in treatment effect across populations or differences in study quality. Second, the LFK index thresholds (less than 1, 1 to 2, greater than 2) are guidelines rather than formal statistical tests with defined Type I error rates. Third, the method assumes that the included studies estimate the same underlying effect, so substantial unexplained heterogeneity can affect interpretation. Finally, the Doi plot is relatively new compared to funnel plots, so some reviewers and journals may be less familiar with it. Including both a funnel plot and a Doi plot in your publication can help bridge this gap.
Detect publication bias with contour-enhanced funnel plots, Egger's test, Begg's test, and trim-and-fill analysis using our funnel plot and publication bias tool. Evaluate evidential value and detect p-hacking with our p-curve analysis tool. Visualize individual study estimates and pooled results with our forest plot generator for meta-analysis.
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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