Tau-U
Also known as: Tau-U Single-Case, Parker Tau-U, Kendall Tau Nonoverlap, Tau-U Effect Size
Tau-U is a rank-based effect-size index for single-case research that combines the degree of nonoverlap between baseline and treatment phases with the trend within phases, and that can optionally subtract out any improving trend already present in the baseline. Developed by Richard Parker, Kimberly Vannest, and colleagues in 2011, it extends the Nonoverlap of All Pairs (NAP) statistic by adding a Kendall-style trend component, giving practitioners a single index that is robust to outliers, has a known sampling distribution for significance testing, and does not unfairly credit a treatment for change that the baseline was already heading toward.
Key highlights
- Uses all pairwise comparisons, so no single baseline outlier can dominate the result, unlike PND.
- Can explicitly correct for an existing baseline trend, separating treatment effect from pre-existing improvement.
- Built on Kendall's S, so it comes with a known sampling distribution, significance test, and confidence interval.
- Reported on an interpretable -1 to +1 scale and readily combined across cases for meta-analysis.
Intuition
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How it works
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When to use it
Use Tau-U when you want an outlier-robust, inferentially grounded effect size for a single-case AB contrast, and especially when the baseline shows a trend you need to control for. It suits short phases typical of practice evaluation and produces confidence intervals for reporting and synthesis. It is less appropriate when phases are extremely short (few pairs make the variance estimate unstable), when the baseline trend is so steep that correction overwhelms the contrast, or when a fully model-based analysis (e.g., multilevel or Bayesian single-case models) is warranted for autocorrelated, multi-tier data.
Strengths & limitations
- Uses all pairwise comparisons, so no single baseline outlier can dominate the result, unlike PND.
- Can explicitly correct for an existing baseline trend, separating treatment effect from pre-existing improvement.
- Built on Kendall's S, so it comes with a known sampling distribution, significance test, and confidence interval.
- Reported on an interpretable -1 to +1 scale and readily combined across cases for meta-analysis.
- With very short phases the number of pairs is small, making the variance estimate and resulting p-values unstable.
- Baseline-trend correction can over-adjust when baselines are short and noisy, occasionally producing values outside the expected range that must be capped.
- As a nonparametric rank statistic it ignores the magnitude of differences, only their direction, so two effects of very different size can score similarly.
- Does not by itself model autocorrelation, so for strongly serially dependent data a model-based approach may be preferable.
Common pitfalls
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Applications
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Frequently asked
When should I apply the baseline-trend correction?
Apply it when visual inspection or a trend test shows the baseline is already moving in the direction of the desired change, so that an uncorrected index would credit the treatment for improvement that was already underway. If the baseline is flat or trending away from improvement, correction is unnecessary and can add variance; many practitioners test the baseline trend first and only correct when it is non-trivial. Always report which variant you used.
How does Tau-U relate to NAP and Kendall's tau?
Tau-U is built on the same all-pairs comparison logic as NAP and on Kendall's rank correlation. The between-phase contrast component of Tau-U is essentially a rescaled NAP. Tau-U's addition is the within-baseline trend count, which lets it both incorporate phase trend and optionally subtract a baseline trend. So NAP is the pure nonoverlap special case, and Tau-U is the trend-aware generalization.
Can Tau-U values exceed +1 or fall below -1?
The pure nonoverlap component is bounded in [-1, +1], but when baseline-trend correction is combined across multiple cases or when corrections are large relative to short phases, computed Tau-U values can occasionally fall slightly outside that range. Software typically reports these and they are usually truncated to the bound for interpretation; values far outside the range signal that phases are too short or the correction is over-adjusting.
Sources
- 1.Parker, R. I., Vannest, K. J., Davis, J. L., & Sauber, S. B. (2011). Combining nonoverlap and trend for single-case research: Tau-U. Behavior Therapy, 42(2), 284–299.
- 2.Parker, R. I., & Vannest, K. J. (2009). An improved effect size for single-case research: Nonoverlap of all pairs. Behavior Therapy, 40(4), 357–367.
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Cite this page
ScholarGate. (2026, June 22). Tau-U. ScholarGate. https://scholargate.app/social-work/tau-u-single-case