META-ANALYSIS RESEARCH TOOL

MetaSyn Statistical Converter for Meta-Analysis

Convert supported standard errors, sampling variances, Wald confidence intervals and ratio-scale results for meta-analysis with transparent assumptions, browser-local calculations and a predefined validation record.

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By Dr. Esmaeel Saeedy Robat, Founder of MetaSyn Academy Evidence checked: August 2026

A statistical conversion can be mathematically simple and still be methodologically wrong. Statistical Converter is designed for the narrow point in a review workflow where the effect measure has already been chosen, the relevant result has already been reported or extracted, and a supported representation needs to be converted without hiding the assumptions that make the conversion defensible.


That boundary matters. A standard error is not a standard deviation. A 95% confidence interval is not automatically a licence to divide its width by 3.92. An odds ratio, risk ratio, hazard ratio and rate ratio can all be represented on a natural-log scale, but they do not become the same effect measure merely because the algebra looks similar. Cochrane’s current guidance makes these distinctions explicit: ratio measures are usually analysed after natural-log transformation, while confidence-interval reconstruction depends on how the interval was formed and, for some continuous outcomes, may require a t distribution rather than the standard normal approximation.1


The practical reason for being strict is not theoretical. Published evidence has documented consequential extraction and calculation errors. In one sports-medicine investigation, potential SE/SD confusion changed reconstructed standardized effects; a separate umbrella review of depression and inflammatory biomarkers found specified extraction or calculation errors in 118 of 521 primary studies represented in the included meta-analyses.4,5 Those figures come from particular research contexts and should not be treated as a universal error rate. They do show why a converter should expose its assumptions instead of turning every reported number into another number on demand.


Methodological boundary
Statistical Converter starts after effect-measure selection.

If the question is whether a continuous outcome should be expressed as a mean difference or standardized mean difference, or whether a binary outcome should be represented by an odds ratio or risk ratio, that is an upstream methodological decision. Use the Effect Size Selection Worksheet for Meta-analysis and the companion effect-size selection guide. Statistical Converter does not make that scientific choice.


Where Statistical Converter belongs in the workflow

A meta-analysis moves through different kinds of judgment. Defining the estimand, selecting an effect measure, deciding which outcome and time point belong in the synthesis, and deciding whether studies are sufficiently compatible to pool are scientific tasks. Converting a known standard error to its sampling variance, or taking the natural logarithm of a positive ratio, is different: once the inputs and assumptions are fixed, the transformation is deterministic.


Where Statistical Converter begins in a meta-analysis workflow A sequence from research question and effect-measure selection through extraction, an assumption gate, Statistical Converter conversion, and synthesis software. Scientific judgment occurs before and after the deterministic conversion. Where Statistical Converter begins Scientific judgment is not delegated to the converter. Question &estimand Choose effectmeasure Extractreported result Supported &assumptions met? Statistical Converterconversion Synthesissoftware No / unknown → verify the source method; do not force the conversion methodological decisions deterministic operation model & interpretation
Statistical Converter occupies a narrow computational step between extraction and synthesis. The assumption gate is part of the researcher’s judgment, not a decision made by the software.

What the five supported conversions do

Version 1 is deliberately narrow. It supports five conversion families. The restriction is a feature of the methodology, not a missing menu. Each family has a defined input, analysis scale and output, and each has cases that Statistical Converter should refuse rather than estimate.


MC-01

Standard error and sampling variance

For the same estimator on the same scale, the sampling variance is the square of its standard error, and the standard error is the positive square root of the sampling variance.

v = SE²     and     SE = √v

This identity does not turn a study’s participant-level standard deviation into the sampling variance of an effect estimate. SD describes dispersion of observations; SE describes uncertainty in an estimate. Confusing those quantities can change weighting and standardized effects.1,4


MC-02

Supported additive-scale Wald CI to SE and variance

If a two-sided confidence interval is known to have been formed as an estimate plus or minus the matching standard-normal critical value multiplied by its SE, then its width can be inverted:

SE = (U − L) / (2 × z1−α/2)     ;     v = SE²

For a conventional large-sample 95% normal interval, the denominator is approximately 3.92. For 90% and 99% intervals it changes. The familiar 3.92 shortcut is therefore a consequence of a particular interval construction, not a property of the label “95% CI”. Cochrane specifically notes that small-sample mean-difference intervals may have been formed using a t distribution, in which case the relevant critical value depends on degrees of freedom.1


MC-03

Ratio measure plus supported Wald CI to log effect, SE and variance

Odds ratios, risk ratios, hazard ratios and rate ratios are positive quantities. In conventional meta-analysis they are usually analysed on the natural-log scale. Statistical Converter therefore transforms the point estimate and both interval bounds before reconstructing uncertainty.

y = ln(R),   l = ln(L),   u = ln(U),   SE = (u − l)/(2z),   v = SE²

This is the same scale logic used in Cochrane guidance and in the documented metafor::conv.wald() workflow for a Wald interval reported on a ratio scale.1,3 The common logarithmic transformation does not make OR, RR, HR and rate ratio substantively interchangeable. Each remains the effect measure that the study and review have defined.


MC-04

Ratio and natural-log ratio

For any positive ratio R, the transform is one-to-one:

y = ln(R)     and     R = exp(y)

The no-effect value 1 on the ratio scale becomes 0 on the log scale. Ratios of 0.5 and 2 are reciprocals; their natural logs have equal magnitude and opposite signs. That is one reason the log scale is useful for multiplicative effects.1


MC-05

Log ratio plus uncertainty to ratio and Wald CI

When the log effect and its uncertainty are already known, Statistical Converter accepts exactly one uncertainty input: SE or sampling variance. If variance is supplied, the software first takes its positive square root. It then forms a two-sided Wald interval on the log scale and exponentiates the point estimate and interval limits for display.

Llog = y − zSE;   Ulog = y + zSE;   R = exp(y);   CI = [exp(Llog), exp(Ulog)]

Here the Wald construction is explicit because Statistical Converter itself is constructing the interval. The scientific question remains whether a Wald interval is the interval the review needs for that effect and analysis.


Ratio scale and natural-log analysis scale A ratio scale with 0.5, 1 and 2 maps through the natural logarithm to negative 0.693, zero and positive 0.693. The transformation is reversible and does not make different ratio effect measures substantively equivalent. Original scale and analysis scale A multiplicative ratio becomes additive after the natural-log transform. Ratio / display scale 0.5 1 2 Natural-log analysis scale −0.693 0 +0.693 ln exp Shared transformation ≠ shared substantive meaning
OR, RR, HR and rate ratio can share this transformation pipeline while still representing different substantive quantities. Effect-measure choice remains upstream.

The confidence-interval gate: the most important assumption

The safest question is not “Is the interval 95%?” It is “Do I know how this interval was constructed, and is that construction compatible with the formula I am about to invert?”


The width formula is exact only for the model it is reversing: an estimate centred within a two-sided interval whose endpoints are the estimate plus or minus a known critical value times the same SE, on the scale used for reconstruction. A profile-likelihood interval is obtained from likelihood calculations, not by that generic geometry.6 Bootstrap intervals form another family of procedures.7 Exact intervals also use construction-specific methods.15 A visually symmetric interval can therefore still be non-Wald, while a ratio interval that looks asymmetric on its reported scale can be symmetric after logging.


Reported intervalGeneric Statistical Converter reconstruction?Reason
Known two-sided normal/Wald interval on the relevant scaleSupportedThe documented formula matches the interval construction.
t-based intervalNot supported in v1The critical value depends on degrees of freedom rather than a fixed standard-normal value.
Profile-likelihood intervalDo not forceEndpoints arise from a likelihood-based procedure, not generic estimate ± zSE geometry.
Bootstrap intervalDo not forcePercentile, BCa and related bootstrap intervals have their own construction.
Exact intervalDo not forceThe endpoint procedure is method-specific.
Construction unknownStop and verifyThe required assumption is not established by the confidence level alone.
Rounded reported boundsPossibly, if construction is known compatibleThe reconstruction is approximate and small off-centering can be caused by rounding.

When the right answer is not to convert

A converter should make stopping easy. If a paper reports a confidence interval but does not identify enough of the analysis to establish its construction, Statistical Converter cannot recover that missing methodological information from the endpoints. Apparent symmetry is not a substitute for documentation. The same is true when the point estimate and interval may have been reported on different scales, when a transformed interval has been back-reported without enough detail to reconstruct the original analysis, or when a source has rounded the estimate and bounds so heavily that the implied uncertainty becomes unstable. In these situations the next step is source verification, not a more aggressive formula.


Rounding deserves separate attention because it can create small discrepancies without making the underlying method wrong. The metafor documentation notes that rounded estimates and bounds can trigger its rough midpoint check and can make a back-calculated sampling variance slightly inaccurate.3 A097 therefore treats source precision as provenance. Record the values exactly as reported, retain the original number of decimal places, and label a reconstructed SE or variance as derived. Do not manufacture extra certainty by printing many more decimals simply because the calculator can display them.


This also explains why Statistical Converter asks the researcher to acknowledge the Wald assumption for CI-based reconstruction rather than silently inferring it. The acknowledgement is not a legal disclaimer or a box that makes the method valid. It records that the user has checked the methodological condition the software itself cannot establish from the numbers alone. If that condition remains unknown, an unresolved entry in the extraction record is scientifically preferable to a precise-looking but unjustified variance.


Decision gate for reconstructing a confidence interval A decision flow asks whether the analysis scale and interval construction are known, and whether the interval is a supported two-sided Wald interval. Only then does the flow reach SE reconstruction. Can this confidence interval be reconstructed? Reported estimate+ confidence interval Analysis scaleidentified? CI constructionknown? Compatible two-sidedWald/normal intervalon that scale? Supported:reconstruct SE / variance No or unknown:verify source method first Yes No No / unknown No
Symmetry by eye is not the gate. The researcher needs to know the interval’s construction and analysis scale. When that information is absent, verifying the source method is safer than forcing a back-calculation.

What the off-centering warning means

Statistical Converter includes a midpoint-compatibility warning for the CI-based methods. Its purpose is modest: flag a result that deserves another look. The logic is inspired by the rough check documented in metafor::conv.wald(), which examines whether an estimate lies approximately halfway between its CI bounds after any specified transformation.3 The current metafor documentation explicitly cautions that a warning may indicate a non-Wald interval or a scale mismatch, but can also arise from rounding.


That makes the warning a diagnostic prompt, not a statistical test. It cannot prove that an interval is Wald-type, and it cannot prove that an interval is invalid. A097 deliberately avoids language such as “CI validation” or “asymmetry test” for this feature. The correct response to a warning is to return to the source report and verify how the interval was calculated.


Worked examples

SE to sampling variance

If SE = 0.20, then v = 0.20² = 0.04. The calculation is exact for the same estimator and scale. It says nothing about the participant-level SD.

Known 95% Wald CI to SE

For a constructed teaching example with estimate 2.00 and a known compatible 95% normal/Wald CI of 1.02 to 2.98, the width is 1.96. Dividing by approximately 3.92 gives SE ≈ 0.50, so v ≈ 0.25.

Ratio CI to log-scale uncertainty

The metafor documentation gives OR = 1.37 with a 95% CI of 1.03 to 1.82. Its documented log-scale conversion gives ln(OR) = 0.3148 and v = 0.0211.3

Log ratio back to display scale

If ln(R) = 0.405465 and SE = 0.10, exponentiation gives R ≈ 1.50. A 95% Wald interval is first formed on the log scale and then exponentiated, yielding an asymmetric interval on the ratio scale.


Keep source truth

Never replace the reported value in your extraction record with the converted value. Keep the original statistic, its source location, the reported precision and the transformation record. The converted value is a derived quantity. This separation makes later checking possible and prevents a rounded or reconstructed value from being mistaken for the study’s original report.


Numerical precision and extreme values

Statistical Converter runs in JavaScript, whose ordinary Number values use finite double-precision floating-point representation. Very large finite calculations can exceed the representable range and become Infinity; exponentiating sufficiently negative values can approach the smallest positive representable values and eventually underflow to zero.11,12 Those are computing limits, not meaningful meta-analytic effect estimates.


The release specification therefore requires numerical-range guards. A back-transformation that cannot be represented as a finite, strictly positive ratio must return an explicit range state rather than display Infinity or a machine-zero result as if it were an ordinary estimate. The final public release also needs boundary tests tied to the exact calculation engine. Small differences in the last displayed decimal between statistical packages can arise from floating-point representation or rounding; they should be distinguished from differences caused by using a different statistical method.


How the MetaSyn Statistical Converter was verified

The development process was designed so that the expected results existed before implementation. The frozen v1 matrix contains 200 direct cases distributed across MC-01 to MC-05 and 20 cross-method or round-trip cases, for 220 predefined cases in total. It includes ordinary calculations, invalid states, warning paths and external reference cases. The validated v1 release record reports 220/220 passes.9,10


That is meaningful evidence about implementation behavior, but it is not a universal proof. A finite expected-output suite cannot establish that every possible input is appropriate, that every future browser environment will behave identically, or that a researcher’s confidence interval actually satisfies the assumption they acknowledged. Research-software guidance places value on explicit provenance and version-specific identification precisely because a verification result belongs to a particular software artifact rather than an abstract product name.9,10


Statistical Converter verification chain The process moves from scientific specification to frozen expected outputs, implementation, direct and boundary tests, reference and round-trip checks, and a version-specific release record. How the MetaSyn Statistical Converter was verified The tests check implementation against a predefined specification; they do not validate the scientific suitability of a user’s input. Scientificspecification Expected outputsfrozen beforeimplementation Calculationengine Direct · invalidwarning · boundaryreference checks+ tolerances Round-tripchecks Versionedreleaserecord Release claim = performance against the specified, versioned test set
The test result belongs to the tested engine and release record. Any calculation-engine change after testing requires a new parity check and, where relevant, a rerun of the affected test suite.

Published release record

The MetaSyn Statistical Converter is available in the Chrome Web Store. The released v1 calculation engine is recorded as passing 220/220 predefined expected-output cases.

This result reports performance against the specified, versioned test set. It is not a universal accuracy guarantee and does not determine whether a researcher’s inputs or scientific choices are appropriate. Release parity must be rechecked after any calculation-engine or distributed-package change.


Privacy by design and Chrome Web Store boundaries

Statistical Converter’s intended architecture is deliberately small: the calculation happens inside the extension rather than on a MetaSyn server. The validated development record reports no account, analytics, advertising, host permissions, browsing-history access, webpage-content access, saved calculation history, backend/API transmission or remote executable code. Those statements are release-specific implementation facts checked for the current public package and must be checked again before any Store update that changes permissions, networking, storage, analytics, account features or executable code.


This architecture also fits the Chrome Web Store’s current policy direction. Google requires an extension to have a narrow, easy-to-understand single purpose, to disclose its functionality accurately across the full user experience, and to limit user-data use to what is necessary for that disclosed purpose.13,14 The extension therefore performs the conversion itself; it is not a launcher for a website calculator. The Store listing, this page and the dedicated privacy policy must describe the same functionality and data behavior.


Extension data and website email are different

The statistical values entered into the Chrome extension are not the same thing as an email address voluntarily entered on the MetaSyn Academy website. The optional printable companion on this page uses the Academy’s website form and therefore involves website-side processing of the email address supplied for delivery. That website interaction is not part of Statistical Converter’s calculation engine and is disclosed separately in the Statistical Converter Privacy Policy.


Statistical Converter Conversion and Verification Record

A conversion is easier to audit when the original report and the derived quantity are kept together. The optional printable companion is therefore a transformation record, not another effect-size-selection worksheet. It begins after the effect measure has already been selected.


Printable implementation companion

Statistical Converter Conversion and Verification Record

Record the study and outcome, source location, original estimate and uncertainty, confidence level, interval construction if known, original and target scales, Statistical Converter method, exact inputs and outputs, formula/critical value, warning state, assumption acknowledgement, source rounding, software version, analyst/date and any unresolved concern.

Not included: estimand selection, choosing OR versus RR, deciding MD versus SMD, pooling compatibility, dependence strategy or model selection. Those decisions belong elsewhere in the review workflow.

Production note: the approved Academy Word master is required before the downloadable .docx is generated. A generic replacement is not used.


What Statistical Converter deliberately refuses to decide

The usefulness of a narrow tool depends partly on what it declines to automate. Statistical Converter does not infer the estimand from a paper, choose an effect measure, reconstruct raw group statistics, decide whether a reported confidence interval is scientifically appropriate, repair unit-of-analysis errors, choose sparse-data corrections, model dependence, select fixed-effect or random-effects synthesis, or decide whether a pooled result is meaningful. It also does not convert a mathematical relationship into substantive equivalence. An odds ratio remains an odds ratio after exponentiating a log odds ratio; it does not become a risk ratio.


The same principle applies to hazard ratios. The logarithm is a mathematical transformation; interpretation of a hazard ratio remains tied to time-to-event analysis and the assumptions of the underlying model. Methodological concerns about hazard ratios extend well beyond the algebra implemented by Statistical Converter.8


A transparent conversion is still a research decision

Statistical Converter is useful when the researcher’s question has already reached the point where a deterministic transformation is justified. It can remove repetitive arithmetic, reduce transcription steps and make the calculation path visible. It cannot supply missing methodological knowledge about how the source statistic was generated.


When the interval construction is known and compatible, conversion can be straightforward. When the construction is unknown, the defensible action is often to check the paper, supplement or analysis method rather than force a result. That is the boundary this page and the software are designed to preserve.


Meta-analysis methods, effect sizes and forest plotsReturn to the broader quantitative-synthesis hub for effect measures, models and interpretation. Effect Size Selection Worksheet for Meta-analysisUse this before Statistical Converter when the effect measure itself has not yet been selected and documented.

Chrome Web Store · Version 1.0.3MetaSyn Statistical Converter is now availableInstall the free extension from its official public listing. Calculations remain local in Chrome.
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Methodological scope. The formulas on this page apply only to the explicitly supported conversion families and their stated assumptions. A numerical result does not establish that the original statistic, interval construction, effect measure, study design or planned synthesis is scientifically appropriate.

References

  1. Higgins JPT, Li T, Deeks JJ, editors. Chapter 6: Choosing effect measures and computing estimates of effect. In: Cochrane Handbook for Systematic Reviews of Interventions. Current online version. Cochrane. Available from: https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-06
  2. Viechtbauer W. Conducting meta-analyses in R with the metafor package. J Stat Softw. 2010;36(3):1-48. doi: 10.18637/jss.v036.i03.
  3. Viechtbauer W. conv.wald(): Convert Wald-Type Confidence Intervals and Tests to Sampling Variances. metafor documentation. Accessed 28 Aug 2026. Available from: https://wviechtb.github.io/metafor/reference/conv.wald.html
  4. Sandercock G. The Standard Error/Standard Deviation Mix-Up: Potential Impacts on Meta-Analyses in Sports Medicine. Sports Med. 2024;54(6):1723-1732. doi: 10.1007/s40279-023-01989-9.
  5. Lee S, Lee KH, Park KM, et al. Impact of data extraction errors in meta-analyses on the association between depression and peripheral inflammatory biomarkers: an umbrella review. Psychol Med. 2023;53(5):2017-2030. doi: 10.1017/S0033291721003767.
  6. Venzon DJ, Moolgavkar SH. A method for computing profile-likelihood-based confidence intervals. Appl Stat. 1988;37(1):87-94. doi: 10.2307/2347496.
  7. Efron B, Tibshirani R. Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Stat Sci. 1986;1(1):54-75. doi: 10.1214/ss/1177013815.
  8. Hernán MA. The hazards of hazard ratios. Epidemiology. 2010;21(1):13-15. doi: 10.1097/EDE.0b013e3181c1ea43.
  9. Barker M, Chue Hong NP, Katz DS, et al. Introducing the FAIR Principles for research software. Sci Data. 2022;9:622. doi: 10.1038/s41597-022-01710-x.
  10. Smith AM, Katz DS, Niemeyer KE, FORCE11 Software Citation Working Group. Software citation principles. PeerJ Comput Sci. 2016;2:e86. doi: 10.7717/peerj-cs.86.
  11. MDN Web Docs. Number.MAX_VALUE. Accessed 28 Aug 2026. Available from: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Number/MAX_VALUE
  12. MDN Web Docs. Math.exp(). Accessed 28 Aug 2026. Available from: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Math/exp
  13. Google Chrome for Developers. Chrome Web Store Program Policies. Accessed 28 Aug 2026. Available from: https://developer.chrome.com/docs/webstore/program-policies
  14. Google Chrome for Developers. Limited Use. Chrome Web Store Program Policies. Accessed 28 Aug 2026. Available from: https://developer.chrome.com/docs/webstore/program-policies/limited-use
  15. R Core Team. binom.test: Exact Binomial Test. R stats documentation. Accessed 28 Aug 2026. Available from: https://stat.ethz.ch/R-manual/R-devel/library/stats/html/binom.test.html

Frequently asked questions

Can I convert any 95% confidence interval to a standard error?

No. The supported reconstruction requires a compatible two-sided Wald/normal interval on the relevant analysis scale. A 95% confidence level alone does not tell you how the interval was constructed. t-based, profile-likelihood, exact and bootstrap intervals require their own methods or additional information.

Does Statistical Converter choose the effect measure for my meta-analysis?

No. Effect-measure selection is an upstream methodological decision. Statistical Converter begins after the effect measure and relevant reported statistic are already known.

Why are odds ratios, risk ratios and hazard ratios converted to a log scale?

Positive ratio measures are usually analysed on a natural-log scale in conventional meta-analysis. The transform maps the no-effect ratio of 1 to 0 and converts multiplicative distances into additive distances. Sharing this transform does not make OR, RR, HR and rate ratio equivalent effect measures.

Does a midpoint warning mean my confidence interval is wrong?

No. It is a heuristic prompt to verify the source method. A warning can reflect a non-Wald interval or scale mismatch, but rounding can also cause off-centering. It is not a statistical validity test.

What does “220/220 tests passed” mean?

It means that the tested calculation engine returned the expected outcomes for the predefined cases in that version-specific test suite. It does not prove universal correctness and does not determine whether a researcher’s source statistic is methodologically appropriate.

Are the values I enter into the Chrome extension sent to MetaSyn Academy?

The validated v1 architecture is designed to perform calculations locally in the extension without sending calculation inputs to a MetaSyn backend. This implementation claim matches the current public architecture and must be rechecked after any Store release that changes data handling or permissions. The optional email form on the MetaSyn Academy website is separate and processes the email address voluntarily supplied for delivery of the printable companion.

Evidence and Store-listing review date: 29 August 2026. Software-specific privacy, version and test statements must be rechecked whenever the release package or calculation engine changes.