ABA Kendall Tau-B Tied-Pair Rank Association Calculator for Clinicians

Clinicians & ABA Professionals / Data, Outcomes and Clinical Decision-Making.

Confirm that tau-b matches the question

Kendall's treatment of ties and its publisher record describe a rank-correlation approach that accommodates ties. The current SciPy kendalltau documentation gives the tau-b counting definition used in this worksheet. These sources do not make tau-b an ABA standard, establish a universal cutoff, or approve a clinical decision.

Use the worksheet for two complete ordinal variables recorded on the same independent units. The variables may use different ordered scales, such as a severity rating and a priority rank, but each scale's direction must be stable and interpretable before calculation. Tau-b asks whether a higher value on one variable tends to accompany a higher value on the other. Numerical equality is not the question.

Do not use this calculator for nominal labels, incomplete pairs, repeated observations from the same person, a multi-rater ranking panel, or a treatment-effect estimate. Kendall W addresses concordance among multiple complete rankings. Weighted kappa addresses agreement between ordered categories under a chosen weighting rule. Lin CCC and Bland-Altman methods address paired continuous measurements under different assumptions.

The current BACB Ethics Codes and BCBA Test Content Outline provide professional context for competent measurement and interpretation. Neither requires tau-b. The Standards for Educational and Psychological Testing support keeping interpretation tied to the intended use and available evidence.

Register the paired-rank plan

Complete this record before viewing the coefficient.

Design fieldPrespecified entryObservable questionIndependent unit of analysisVariable X and scale directionVariable Y and scale directionPairing keyTie meaning and handlingPreserve equal values as tiesObserved range and possible restrictionInclusion and exclusion rulesMissing-pair ruleStop; complete pairs requiredData version and extraction timePrimary statisticKendall tau-bIntended consequenceQualified reviewers

Reversing one scale after seeing the result reverses the sign. Removing tied or inconvenient observations changes the pair counts. Preserve the original version, document any justified sensitivity plan prospectively, and label each result separately.

Enter one row per independent unit

Pairing keyX ordinal valueY ordinal valueSource/version noteUnit 1Unit 2Unit 3...

Use the original ordered values. Dense ranks, competition ranks, category codes, and raw ordinal scores can yield the same pair ordering when they preserve ties and direction, but undocumented recoding can change the estimand. Stop if a unit appears more than once, a pair is missing, or the scales do not have defensible order.

Classify every unordered pair of units

With n units, inspect n(n-1)/2 unordered pairs. For units i and j, calculate the signs of xi-xj and yi-yj.

• Concordant P: both differences have the same nonzero sign.
• Discordant Q: the differences have opposite nonzero signs.
• Tied only in X T: the X difference is zero and the Y difference is nonzero.
• Tied only in Y U: the Y difference is zero and the X difference is nonzero.
• Tied in both: both differences are zero. Record the pair, but do not add it to T or U.

These five classifications are mutually exclusive and exhaustive. Check that P+Q+T+U+B = n(n-1)/2, where B is the number tied in both. The identity must hold before calculating tau-b; otherwise a comparison was omitted or counted twice.

PairSign of X differenceSign of Y differenceClass P, Q, T, U or BReview note

Calculate Kendall tau-b

Use unrounded counts:

tau_b = (P-Q) / sqrt((P+Q+T)(P+Q+U))

The numerator is the balance of concordant and discordant pairs. The two denominator factors adjust for ties unique to each variable. If either factor is zero, tau-b is undefined. For a valid positive denominator, tau-b ranges from minus one to one. Round only the displayed result.

Work a fictional tied example

Five fictional units have X=[1,2,2,4,5] and Y=[1,3,2,4,4]. No person or clinical record is represented.

PairClassificationReason1 with 2, 3, 4 or 54 concordantBoth variables increase2 with 3TX is tied; Y differs2 with 4 or 52 concordantBoth variables increase3 with 4 or 52 concordantBoth variables increase4 with 5UY is tied; X differs

Thus P=8, Q=0, T=1, U=1, and B=0. The ten classified pairs equal 5(4)/2. Both tie-adjusted denominator factors equal 8+0+1=9.

tau_b = 8 / sqrt((8+1)(8+1)) = 8/9 = 0.8888888889

The number describes directional ordinal association in this fabricated table. It cannot show absolute agreement, validate either scale, or establish that the association is clinically important.

Check sign and denominator boundaries

Two identical strict rankings produce taub=1; reversing one strict ranking produces taub=-1. If every X value is tied while Y varies, then P+Q+T may be positive but P+Q+U=0, so the denominator is zero and tau-b is undefined. If every Y value is tied while X varies, the other factor is zero. If both variables are constant, both denominator factors are zero.

Pairs tied in both variables affect the total pair audit but not P, Q, T, or U. Do not silently assign them to concordance. A result near one can still arise from a narrow observed range, dependent units, unreliable observations, or a question with little clinical consequence. It should not be compared mechanically with a coefficient from a sample having a materially different range or tie pattern.

Interpret the coefficient with the pair audit

Use the ABA Kendall tau-b calculator as an inspectable calculation record. Report the scale definitions and direction, unit count, total unordered pairs, P, Q, T, U, both-tied count, denominator, coefficient, missingness decision, and any prespecified sensitivity results. Review the discordant pairs directly rather than treating them as noise.

Tau-b is symmetric: swapping X and Y does not change the value. It measures monotone ordinal association, not agreement on the same category and not equality on a continuous scale. The coefficient cannot identify the correct variable, prove reliability, demonstrate treatment integrity, establish experimental control, or support causal claims.

The location of discordance can matter more than its total. Two data sets can share the same tau-b while differing on which units are ordered differently. A qualified reviewer should connect those visible patterns to definitions, observation conditions, consequences, and the perspectives of affected people.

Stop when the design exceeds this worksheet

Pause for unmatched or missing pairs, repeated or clustered observations, non-independent units, survey weights, partial orders, more than two variables, or a plan to select the direction after inspecting results. Obtain statistical review for confidence intervals, exact or asymptotic hypothesis tests, permutation procedures, covariate adjustment, sample-size planning, or generalization beyond the observed units.

Do not use tau-b as a replacement for a direct graph. A scatter or jittered ordinal plot can reveal tied bands, nonlinear patterns, range restriction and influential observations. Preserve the source data and the review narrative next to the coefficient.

Protect observations and provenance

Use the minimum information needed for review. Keep identifiable unit-level material only in approved systems, restrict access, and retain an audit trail. The HHS Privacy Rule summary and HHS Security Rule summary describe federal requirements for regulated entities. State and organizational rules may add duties. This page is not legal advice.

Copyable result record

Result fieldValueData versionX and Y definitions/directionsIndependent unit and pairing keyComplete pairs nTotal unordered pairsConcordant PDiscordant QX-only ties TY-only ties UBoth-tied pairs BDenominator factors and denominatorKendall tau-bMissing or excluded entriesDiscordant pairs reviewedIntended interpretation and limitsReviewer and review date

Related resources

• ABA Kendall W Multi-Rater Rank Concordance Calculator for Clinicians
• ABA Weighted Kappa Ordinal Agreement Calculator for Clinicians
• ABA Krippendorff Alpha Missing-Data Agreement Calculator for Clinicians
• ABA Lin Concordance Correlation Precision-Bias Calculator for Clinicians

Sources

• Kendall, The Treatment of Ties in Ranking Problems.
• Publisher DOI for Kendall's treatment-of-ties paper.
• SciPy kendalltau documentation.
• Behavior Analyst Certification Board Ethics Codes.
• BCBA Test Content Outline, Sixth Edition.
• Standards for Educational and Psychological Testing.
• HHS Summary of the HIPAA Privacy Rule.
• HHS Summary of the HIPAA Security Rule.