Standardized Mean Difference for Single-Case Designs
Also known as: Between-Case SMD, Single-Case d-Statistic, Hedges-Pustejovsky-Shadish d, SCD Standardized Mean Difference
The between-case standardized mean difference is an effect-size measure that puts the result of a single-case experiment on the same numerical scale as Cohen's d from a conventional between-groups study, so that single-case and group findings can be combined in the same meta-analysis. Developed by Larry Hedges, James Pustejovsky, and William Shadish in 2012, it solves a long-standing problem: the many ad hoc nonoverlap indices used in single-case research (PND, PAND, IRD, Tau-U) are not comparable in scale to the standardized mean differences that dominate the broader evidence-synthesis literature. Their estimator models the single-case data with a hierarchical model that separates within-case variation from between-case variation, then standardizes the estimated treatment effect by the total standard deviation — the same denominator a between-subjects d would use. A 2013 extension specialized the estimator to multiple-baseline designs across individuals. The result is a design-comparable effect size with a known variance, suitable for disability and special-education research where single-case studies are abundant.
Key highlights
- Produces an effect size on the same scale as Cohen's d, enabling single-case and between-subjects studies to be meta-analyzed together.
- Grounded in a hierarchical model that correctly separates within-case from between-case variance, unlike ad hoc nonoverlap indices.
- Supplies an estimated sampling variance, so effects can be weighted and pooled with standard inverse-variance methods.
- Includes a small-sample bias correction appropriate to the short series typical of single-case research.
Intuition
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How it works
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When to use it
Use the between-case standardized mean difference when you need an effect size from single-case data that is directly comparable to Cohen's d, typically because you are synthesizing single-case studies — or mixing them with group studies — in a meta-analysis. It is best suited to designs with several cases and reasonably long phases, especially multiple-baseline designs across individuals, where the between-case and within-case variance components can be estimated. It is less appropriate for a single AB case with very few data points, where the variance components are poorly identified, and it assumes the modeled level-shift structure adequately describes the data. When the research question concerns only whether an effect exists for one client rather than its comparable magnitude, simpler nonoverlap indices or a randomization test may suffice.
Strengths & limitations
- Produces an effect size on the same scale as Cohen's d, enabling single-case and between-subjects studies to be meta-analyzed together.
- Grounded in a hierarchical model that correctly separates within-case from between-case variance, unlike ad hoc nonoverlap indices.
- Supplies an estimated sampling variance, so effects can be weighted and pooled with standard inverse-variance methods.
- Includes a small-sample bias correction appropriate to the short series typical of single-case research.
- Requires several cases and adequate phase lengths to estimate the between- and within-case variance components reliably.
- Assumes a particular level-shift model and homogeneous variances that may not fit data with strong trends or changing variability.
- More technically demanding to compute and interpret than nonoverlap indices, requiring multilevel modeling.
- Estimates can be unstable when between-case variance is near zero or when the number of cases is very small.
Common pitfalls
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Applications
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Frequently asked
How does this effect size differ from nonoverlap indices like PND or Tau-U?
Nonoverlap indices summarize how much treatment-phase data exceed baseline-phase data on a percentage or rank scale that has no direct counterpart in between-subjects research. The between-case standardized mean difference instead divides the estimated treatment effect by the total standard deviation recovered from a hierarchical model, producing a number on the same scale as Cohen's d. This design comparability is its defining advantage: it lets single-case results be pooled with group-design results, which nonoverlap indices cannot do because they are not measured in standard-deviation units.
Why does the estimator need both within-case and between-case variance?
A between-subjects d is standardized by how much people differ from one another. In single-case data that population spread is hidden in two pieces: how much each individual's baseline level differs from other individuals (between-case variance) and how much an individual's sessions fluctuate around their own level (within-case variance). Hedges, Pustejovsky, and Shadish show that the sum of these two components reconstructs the cross-person standard deviation a group study would have observed, so standardizing by their sum is what makes the resulting d comparable across designs.
Can I compute this effect size from a single AB case?
Not reliably. The estimator depends on a between-case variance component, which describes how baseline levels vary across individuals; with only one case there is no information to estimate it. The method is designed for designs with several cases, such as multiple-baseline designs across individuals. For a single case, a randomization test or a nonoverlap index is more appropriate, though those quantities are not on the Cohen's d scale and cannot be combined directly with between-subjects effect sizes.
Sources
- 1.Hedges, L. V., Pustejovsky, J. E., & Shadish, W. R. (2012). A standardized mean difference effect size for single case designs. Research Synthesis Methods, 3(3), 224-239.
- 2.Hedges, L. V., Pustejovsky, J. E., & Shadish, W. R. (2013). A standardized mean difference effect size for multiple baseline designs across individuals. Research Synthesis Methods, 4(4), 324-341.
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Cite this page
ScholarGate. (2026, June 23). Standardized Mean Difference for Single-Case Designs. ScholarGate. https://scholargate.app/disability-studies/standardized-mean-difference-single-case