If you work with rare clinical populations, applied behavior analysis, or special education, it's common to accumulate several single-case experimental design studies (N=1) on the same intervention long before there's ever a group trial. The question that comes up sooner or later is: can these studies be combined into a meta-analysis, the way you would with randomized controlled trials? The answer is yes, but with an important caveat this article covers in full: the classic meta-analysis toolkit (the same one you'd use to combine ten group trials) can't be applied to single-case data without adaptation, because the starting point itself — each study's effect size — isn't computed the same way.
Why this isn't a meta-analysis like any other
In a group design, the effect size (a Cohen's d, say) comes from comparing the mean of an intervention group to a control group, with a known variance estimable from sample size. In a single-case design there's no control group in that sense: each participant is their own control, and what you have are time series of repeated measurements, usually few (5-15 points per phase) and autocorrelated — today's value resembles yesterday's more than chance alone would predict. Computing a Cohen's d as if these were two independent groups ignores that autocorrelation and produces a biased effect size, almost always inflated.
This doesn't mean single-case designs can't be meta-analyzed. It means you need an effect size built specifically for single-case time series, with a sampling variance that actually accounts for the autocorrelation. There are two broad families of solution, and it's worth understanding both before choosing.
Nonoverlap indices: useful, but with a real limit
The oldest and most widespread family are nonoverlap indices, which compare intervention-phase data directly against baseline data without modeling anything:
- PND (Percentage of Non-overlapping Data): the percentage of intervention-phase points that exceed the most extreme baseline value. The simplest to compute and easiest to explain, but also the most criticized: it has no known sampling distribution, so you can't compute a standard error or confidence interval the usual way.
- PEM (Percentage Exceeding the Median): the same idea as PND but using the baseline median instead of the most extreme value, making it somewhat more robust to a single outlying baseline point.
- NAP (Nonoverlap of All Pairs): compares every baseline point against every intervention point and computes the proportion of pairs where the intervention "wins." It's directly related to the area under the ROC curve, and does admit a reasonable variance approximation.
- Tau-U (Parker et al., 2011): the most modern and most widely used today, because on top of quantifying nonoverlap it corrects for baseline trend (if the behavior was already improving before the intervention, Tau-U discounts that) and has a published standard error formula.
The practical problem is this: of the four, only Tau-U has a reasonably accepted standard error that lets you plug it directly into the classic meta-analysis machinery (weighting by precision, computing heterogeneity, pooling in a random-effects model). PND, in particular, still gets reported a lot in individual studies because it's intuitive, but you shouldn't try to meta-analyze it as if it were a Cohen's d with its own standard error: it doesn't have one.
The modern approach: between-case standardized mean difference (BC-SMD)
The approach that has become the reference standard over the last decade is the between-case standardized mean difference (BC-SMD), developed by Hedges, Pustejovsky and Shadish (2012, 2013). The core idea: instead of comparing data points directly, you fit a hierarchical linear model to the raw data of each case (level 1: repeated measurements over time within each phase; level 2: cases), estimate the level difference between phases while controlling for trend and autocorrelation, and standardize it using the between-case variance. The result is a d index on the same scale as a group-design Cohen's d, with a real sampling variance.
That last part is what matters for a meta-analysis: having a genuine sampling variance means the BC-SMD can be combined using the standard random-effects toolkit, its heterogeneity computed, and — this is the most striking part of the method — even pooled in the same meta-analysis with group-design studies reporting a conventional Cohen's d, because both live on the same scale. The R package scdhlm (Pustejovsky) implements the computation. The real practical limitation is that you need the raw data for each case, not just the summary effect usually reported in publications — often you have to request it from the authors or extract it from the original article's figures with a tool like WebPlotDigitizer, which adds a data-extraction step group-design meta-analyses don't have.
The alternative when you have raw data from several studies: multilevel modeling
When you have raw data from multiple cases across multiple studies (not just a summary effect per study), another option is the multilevel approach of Van den Noortgate and Onghena (2003, 2008): a three-level model where repeated measurements (level 1) are nested within cases (level 2), which are in turn nested within studies (level 3). It's, in spirit, very similar to an individual participant data meta-analysis: instead of meta-analyzing already-summarized effects, you model the raw data from all studies at once and get a direct estimate of the mean effect and the variance at each level (between cases within a study, and between studies). It's more demanding in terms of data and model complexity, but it makes better use of the information when you have it available.
From the effect-size table to the forest plot
Once you have, study by study, an effect on a scale with known sampling variance (BC-SMD, or Tau-U with its standard error), the rest of the process is ordinary meta-analysis: compute heterogeneity (I², Cochran's Q, τ²), decide between a fixed- or random-effects model (random almost always, given the expected variability across cases, interventions and contexts), and plot the result in a forest plot with the pooled effect. You can use the forest plot and funnel plot generator and the heterogeneity calculator directly for that part: once you have an effect and its 95% CI per study, it doesn't matter that the original design was single-case.
One warning that avoids a common mistake: don't mix effects computed with different indices in the same forest plot (say, a Tau-U from one study next to a BC-SMD from another) as if they were on the same scale, because they aren't. Pick a single index and apply it consistently across all included studies, or transform everything to a common scale before pooling.
A worked example
Imagine five single-case studies on the same behavioral intervention, each with its BC-SMD already computed from the raw data:
Study 1: d = 0.81, 95% CI [0.32, 1.30]
Study 2: d = 0.54, 95% CI [0.10, 0.98]
Study 3: d = 1.12, 95% CI [0.55, 1.69]
Study 4: d = 0.38, 95% CI [−0.05, 0.81]
Study 5: d = 0.67, 95% CI [0.21, 1.13]
With those five rows you already have exactly what the forest plot generator asks for: an effect and its 95% confidence interval per study. The pooled result, the heterogeneity and the recommended model come out the same as in any other meta-analysis — the difference was entirely in how you got to those five numbers, not in what you do with them afterward.
PRISMA doesn't change (much)
Even though the effect-size calculation is different, the study-selection process is reported exactly as in any systematic review: with a complete PRISMA flow diagram, inclusion criteria decided before the search, and exclusion reasons broken down one by one. The one thing usually added at the eligibility stage is a specific criterion: that the study reports (or allows reconstructing) the raw data needed to compute a comparable effect size, which is precisely what most often falls short in this literature.
When this is worth doing, and when it isn't
With two or three single-case studies, the formal meta-analysis apparatus probably doesn't add much over a careful narrative synthesis describing each case. Formal meta-analysis starts to be justified from roughly 8-10 studies onward, a threshold similar to the informal one used for any meta-analysis, though there's no strict rule. And if the available studies are very heterogeneous in design (some AB, some ABAB, some multiple baseline) or population, it's worth asking whether the research question is specific enough that combining them makes clinical sense, beyond being statistically possible.
Frequently asked questions
Can I combine single-case designs with group studies in the same meta-analysis?
Yes, specifically with the BC-SMD approach: being on the same scale as a group-design Cohen's d, both types of study can be meta-analyzed together. It's one of the reasons this method has become the reference standard over nonoverlap indices.
What software is used for this?
Mainly R: the scdhlm package for BC-SMD, SingleCaseES for nonoverlap indices (PND, PEM, NAP, Tau-U), and nlme or lme4 for the Van den Noortgate and Onghena multilevel model. There isn't yet an option as accessible as JASP for this specific type of analysis.
I only have the PND reported in the original studies — is that usable?
You can report it as descriptive data for the individual study, but a formal meta-analysis needs an index with known sampling variance. If you can't recompute the BC-SMD or Tau-U for lack of raw data, be transparent about the limitation rather than forcing a quantitative synthesis with what you have.
Fixed or random effects — which is expected?
Random, almost always. Single-case studies combine very different participants, interventions and contexts even when they nominally address the "same" intervention, so assuming a single true population effect (the fixed-effect assumption) is rarely defensible.
Where do I get raw data if the original study didn't publish it?
The usual route is contacting the authors directly, or extracting values from the published figures with a digitizing tool like WebPlotDigitizer. Always document the extraction method in your manuscript, since it introduces a source of error a reviewer will want to know about.
From per-study effect to the full meta-analysis
The free kit covers study selection with PRISMA, forest plot, heterogeneity and the APA 7 write-up — the same process, whether the effect comes from a group design or a single case.
See the meta-analysis kit →If you need the statistical analysis of a single individual single-case design before considering combining it with others, the article on single-case designs (N=1) covers visual analysis, effect-size indices, and when to use each design (AB, ABAB, multiple baseline). And if you already have the full manuscript written, the paper reviewer is free and tells you what a Q1 reviewer would object to before the journal does.