Fail-safe N is a publication bias sensitivity statistic that answers one focused question: how many unpublished, null-result studies would need to exist in file drawers before your pooled effect became statistically non-significant (or practically trivial)? A large fail-safe N means your result is robust. A small one means a handful of suppressed studies could invalidate your finding.
Three methods dominate the literature: Rosenthal's classic approach, Orwin's target-effect variant, and Rosenberg's weighted refinement. Each asks a slightly different version of the same question, and choosing the wrong one can either overstate or understate your robustness. This guide walks through all three, compares them directly, and tells you when each is most appropriate.
Try our free Funnel Plot Generator to visualize asymmetry in your study pool before running fail-safe N calculations.
Why Fail-Safe N Matters More Than a Significant Funnel Test
Funnel plot asymmetry tests such as Egger's and Begg's have known limitations. They require at least 10 studies for reasonable power, and asymmetry can arise from heterogeneity rather than publication bias alone. Fail-safe N sidesteps these problems by framing the question as a tolerance calculation rather than a hypothesis test.
The practical threshold most reviewers use is: if the fail-safe N exceeds 5k + 10 (where k is the number of studies in your synthesis), the result is considered tolerant of publication bias.
Rosenthal's Method: The Classic 5k + 10 Rule
Rosenthal (1979) proposed the original fail-safe N as a straightforward way to address what he called the "file drawer problem." The calculation asks: how many studies averaging a null result (z = 0) would reduce the combined p-value to exactly the significance threshold, usually p = 0.05?
When to use Rosenthal's method: Use it when your outcome is binary (significant or not), when you want a widely recognized benchmark your reviewers will immediately understand, and when you have fewer than 10 studies where funnel tests lack power.
Limitation: Rosenthal's method assumes the null studies average exactly zero effect. Real file-drawer studies may have small but non-zero effects, making the estimate conservative.
Use our free Forest Plot Generator to visualize your k studies and their individual z-scores before computing the combined z-sum.
Orwin's Method: Setting a Practical Trivial Threshold
Orwin (1983) introduced a modification that many regard as more practically meaningful. Instead of asking how many null studies would push p above 0.05, Orwin asks: how many studies averaging a specified trivial effect would reduce the pooled effect size to a criterion threshold you define as negligible?
When to use Orwin's method: Use it when effect size magnitude matters more than p-value significance, when your field has established minimum clinically important differences, or when reviewers are likely to question practical relevance rather than statistical significance.
Limitation: The result depends entirely on your chosen criterion and trivial effect values. Two researchers using different thresholds will produce different fail-safe N values. Always report your chosen thresholds explicitly.
Rosenberg's Method: Incorporating Study Weights
Rosenberg (2005) observed that both Rosenthal's and Orwin's methods treat all studies as equally informative, which contradicts the standard practice in meta-analysis of weighting studies by their precision (inverse variance).
When to use Rosenberg's method: Use it when your studies vary substantially in sample size and precision, when your synthesis uses random-effects or inverse-variance weighting, and when you want the most methodologically rigorous of the three approaches.
See also our Sensitivity Analysis Tool to check whether excluding individual studies shifts your pooled estimate substantially before computing fail-safe N.
Side-by-Side Comparison
Rosenthal targets p-value significance, assumes equal-weighted null studies, uses the 5k + 10 benchmark. Most widely reported and easiest to explain.
Orwin targets a user-defined effect size threshold, allows customization of both the criterion and the trivial effect. Most useful when clinical significance drives interpretation.
Rosenberg incorporates inverse-variance weights, mirrors the actual weighting structure of the synthesis. Most internally consistent for precision-weighted analyses.
For most systematic reviews and meta-analyses, reporting Rosenthal's fail-safe N alongside a funnel plot provides sufficient evidence. For clinical trials where effect size magnitude matters clinically, adding Orwin's estimate strengthens the bias assessment.
Interpreting and Reporting Your Results
When you report fail-safe N in a manuscript, include three elements: the method used, the resulting number, and the tolerance threshold for comparison.
A complete Rosenthal report reads: "The fail-safe N (Rosenthal, 1979) was 483, exceeding the tolerance threshold of 85 (5k + 10 = 5 * 15 + 10), indicating the pooled effect is robust to the file-drawer problem."
Journals increasingly expect fail-safe N to be accompanied by funnel plot visualization and at least one asymmetry test. Fail-safe N answers "how many?" while funnel asymmetry addresses "is there a pattern suggesting suppression?"
Try our free Funnel Plot Generator to pair your fail-safe N calculation with visual evidence.
Key Takeaways
- Fail-safe N estimates how many unpublished null studies would overturn your meta-analysis finding, providing a tolerance measure for publication bias.
- Rosenthal's method targets statistical significance using the widely cited 5k + 10 benchmark; it is the most recognized and easiest to report.
- Orwin's method targets a user-defined practical threshold, making it more meaningful when clinical significance matters more than p-values.
- Rosenberg's method incorporates inverse-variance weights, aligning the bias calculation with the actual structure of your weighted synthesis.
- Always report the method used, the resulting N, and the comparison threshold so readers can evaluate robustness themselves.
- Fail-safe N is most informative when paired with a funnel plot and an asymmetry test rather than used in isolation.
- A fail-safe N below the tolerance threshold is a signal to run subgroup analyses and investigate whether unpublished studies exist through trial registries.
FAQ
What is a good fail-safe N value?
For Rosenthal's method, the standard benchmark is 5k + 10, where k is the number of studies in your meta-analysis. If your fail-safe N exceeds this threshold, your result is generally considered robust. There is no single universal "good" value; the threshold scales with the size of your synthesis.
Can fail-safe N replace Egger's test for publication bias?
No. Fail-safe N and Egger's test answer different questions. Fail-safe N asks how many suppressed studies would overturn your result. Egger's test asks whether the funnel plot shows the kind of asymmetry consistent with small-study effects or publication bias. Both have distinct limitations, and PRISMA guidelines recommend using multiple complementary methods.
Which software can calculate Rosenthal, Orwin, and Rosenberg fail-safe N?
The metafor package in R supports all three methods via the fsn() function with arguments type = "Rosenthal", type = "Orwin", and type = "Rosenberg". Comprehensive Meta-Analysis (CMA) software supports Rosenthal and Orwin.
Does fail-safe N work for random-effects meta-analyses?
Rosenthal's and Orwin's original formulations were derived for fixed-effects contexts. Rosenberg's weighted method is more compatible with random-effects syntheses because it uses the same weights as the random-effects model.
What should I do if my fail-safe N is below the tolerance threshold?
A below-threshold fail-safe N does not invalidate your meta-analysis, but it warrants additional investigation. Search trial registries such as ClinicalTrials.gov for registered but unpublished studies. Contact corresponding authors of included studies to ask about unpublished replications. Run a trim-and-fill analysis to estimate the adjusted effect after imputing missing studies.
Is fail-safe N still recommended in current reporting guidelines?
Fail-safe N remains acceptable and is still reported in many high-quality systematic reviews. However, the Cochrane Handbook and some methodologists suggest pairing it with contour-enhanced funnel plots and selection model approaches for a more complete assessment.
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