How the method works

This page explains how a triad answer is analysed, in plain language, and then in more detail for anyone who wants the geometry. It does not claim that the geometry makes a self-report more truthful.

The book reference for the wider design is Reinhard and Reddy (2026), Designing Sensemaking Studies, Palgrave Macmillan.

What a person actually does

A participant shares a short story about something that happened. They then place that story on one or more signifiers.

A triad is a triangle. Each corner is a phrase. The participant puts a marker somewhere inside the triangle, closer to the phrase that fits the story better. A slider is the same idea with two poles instead of three.

The marker is their interpretation of their own story. An analyst does not code the story in order to place it.

A triad answer is a set of weights

The marker is recorded as three positive weights, one for each corner, that add up to one. If the marker sits in the middle, the three weights are equal. If it sits near one corner, that corner's weight is large and the other two are small.

A slider is the same kind of answer with two weights instead of three.

Only the relative size of the parts carries information. The weights are compositional data: they live on a simplex. For three parts, that simplex is the triangle itself.

Why a plain average can mislead

Because the parts must sum to one, they cannot vary independently. If one weight goes up, one or both of the others must come down, even when that trade-off is not what the participant was trying to say.

That constraint has been known for a long time. Pearson (1897) showed that a correlation between ratios that share a part can be an artefact. Chayes (1960) showed that a constant sum pushes raw covariances negative: for each part, its covariances with the other parts add up to the negative of that part's variance. Aitchison (1982, 1986) treated the simplex as a space with its own geometry, in which the useful operations are on ratios of parts.

So an ordinary average of the raw marker positions, or a Pearson correlation between raw weights, or an unmodified straight-line model, can report the shape of the triangle rather than a feature of the stories.

What "the average" means here

The average that stays inside the geometry is the closed geometric mean. In words: average the logarithms of each corner's weight, turn those averages back into weights, and rescale them so they sum to one.

That point can be drawn on the triangle. It is not the arithmetic mean of the raw coordinates. The same idea is the back-transform of the arithmetic mean of the log-ratio coordinates described below (Aitchison, 1986; Pawlowsky-Glahn and Egozcue, 2001).

Heat maps have the same problem

A heat map that smooths the markers as if the triangle were an ordinary square lets probability leak off the edges. The distortion is worst near the corners and the sides, which is where people place a story when one phrase fits much better than the others.

A simplex-aware heat map smooths in a way that respects the triangle, either with a kernel defined on the simplex (Aitchison and Lauder, 1985) or by smoothing in log-ratio coordinates and mapping the result back (Chacón, Mateu-Figueras and Martín-Fernández, 2011).

What self-placement does, and what it does not

The participant, not a coder, positions the story. That moves interpretation from the analyst to the narrator, and it removes the need for two coders to agree on what a story "means" before a pattern can be seen.

It does not guarantee that the signifier was a good question, or that the participant understood the corners the way the designer hoped. That is the job of co-design, below.

It also does not make the story more truthful. The geometry only keeps the analysis from distorting what people said, once they have said it.

Participatory co-design

The geometry is the same in every study. The corners of the triangle are not. A triad is informative when its three phrases name distinctions that matter to the people in that setting, and that they can use without feeling judged.

Realist evaluation (Pawson and Tilley, 1997), grounded in critical realism (Bhaskar, 1975), is the logic used here. A method does not work in the abstract. It produces outcomes through mechanisms that fire only in particular contexts. The shorthand is context + mechanism = outcome (C + M = O).

For this method, the mechanism is narration, self-interpretation, and making sense of the patterns together. The context is who is invited, how, where, and with what protection. Co-design fits the instrument to that context before data are collected, so a case can be read as a context–mechanism–outcome configuration rather than a generic success story.

Four traditions supply the practical questions:

  • Dervin's Sense-Making (Dervin, 1998; Dervin and Foreman-Wernet, 2003). People make sense in time and place: a situation, a gap, a bridge, an outcome. Meaning is made in the moment. It is not retrieved from a stored opinion. That is why the prompt asks for a specific recent experience before it asks for any placement, and why prompts and signifiers are phrased around what happened and how it felt.
  • Liberating Structures (Lipmanowicz and McCandless, 2013). Short group formats, such as 1-2-4-All and Troika Consulting, that give every participant a voice. They are used before collection, to surface the situations worth asking about and the distinctions that should become signifiers, and after collection, to interpret the patterns together. McCandless, interviewed for the book on 26 June 2025, describes the purpose this way: with a group he is "not doing a study" but collecting their observations, and he wants to know whether "this way we're relating [is] generating a deeper relationship or more trust", and whether "the context in which we're operating [is] becoming clear enough that there's a path" (interview, as quoted in Reinhard and Reddy, 2026). He also advises convening "people as diverse as we can get … that represent the people who care about that particular challenge".
  • Human Systems Dynamics and the CDE model (Eoyang and Holladay, 2013; HSD Institute). Three conditions shape patterns in a human system: the Container (what holds the system together and defines what is in and out), the Differences (the distinctions that hold tension), and the Exchanges (the connections through which information and resources move). As a design checklist, those become three questions for each instrument: what is the container, which differences matter, and what exchanges will carry the patterns back to people who can act.
  • Realist evaluation, as above. C + M = O is how a finished case is read, including the limits of that reading.

Case stories that use this structure, including their limits, are on Case stories. Terms are collected in the glossary.

Optional: a closer look at the geometry

You can stop at the previous section. What follows is the same argument with the coordinates written out. The symbols are defined in place. No special software is required to read them.

The sample space

A triad response is a vector of three positive weights that sum to one. Formally it is a point on the two-dimensional simplex:

S² = { w = (w₁, w₂, w₃) : wᵢ > 0, w₁ + w₂ + w₃ = 1 }

A slider with two poles is the one-dimensional case: two parts that sum to one.

The closure problem

Pearson (1897) showed that correlations between indices that share a component can be spurious. Chayes (1960) showed that the constant-sum constraint forces a negative bias into raw covariances. If the parts sum to a constant, then for each part i the covariances with all the other parts add up to −var(wᵢ), which is negative. A rise in one part is offset by a fall in the others, whatever the participants actually think.

Aitchison (1982, 1986) turned that observation into a statistical programme: the Aitchison geometry, in which the natural operations are on ratios of parts. Applying ordinary Euclidean statistics (arithmetic means, Pearson correlations, standard linear models, unmodified kernel smoothing) to raw triad coordinates therefore risks results that are artefacts of the geometry.

Log-ratio coordinates

The centred log-ratio maps a composition into a space where standard tools apply, at the cost of a zero-sum constraint. With g the geometric mean of the three parts:

clr(w) = (ln(w₁/g), ln(w₂/g), ln(w₃/g))

The isometric log-ratio (Egozcue et al., 2003) goes one step further and yields unconstrained orthonormal coordinates. For three parts, one convenient choice is:

z₁ = (1/√2) · ln(w₁/w₂)

z₂ = (1/√6) · ln(w₁ w₂ / w₃²)

Distances in the Aitchison geometry equal ordinary Euclidean distances between these (z₁, z₂) coordinates. Means, covariances, clustering, and hypothesis tests can be carried out in that plane and mapped back to the triangle for display. For a two-part slider, the isometric log-ratio reduces to a rescaled logit.

Placements exactly on a vertex or an edge contain zeros, and logarithms of zero are undefined. Those zeros are handled by a multiplicative replacement (Martín-Fernández, Barceló-Vidal and Pawlowsky-Glahn, 2003) that preserves the ratios among the non-zero parts.

The closed geometric mean, written out

The centre of a set of compositions in this geometry is the closed geometric mean: the component-wise geometric mean, rescaled to sum to one.

ḡ = C[ exp( (1/N) · Σₖ ln w⁽ᵏ⁾ ) ]

C rescales a vector so its parts sum to one. This is the same point as the back-transform of the arithmetic mean of the isometric log-ratio coordinates (Aitchison, 1986; Pawlowsky-Glahn and Egozcue, 2001).

References

  • Aitchison, J. (1982). The statistical analysis of compositional data. Journal of the Royal Statistical Society: Series B, 44(2), 139–177. https://doi.org/10.1111/j.2517-6161.1982.tb01195.x
  • Aitchison, J. (1986). The Statistical Analysis of Compositional Data. Chapman and Hall.
  • Aitchison, J., and Lauder, I. J. (1985). Kernel density estimation for compositional data. Journal of the Royal Statistical Society: Series C (Applied Statistics), 34(2), 129–137. https://doi.org/10.2307/2347365
  • Bhaskar, R. (1975). A Realist Theory of Science. Leeds Books.
  • Chacón, J. E., Mateu-Figueras, G., and Martín-Fernández, J. A. (2011). Gaussian kernels for density estimation with compositional data. Computers & Geosciences, 37(5), 702–711. https://doi.org/10.1016/j.cageo.2009.12.011
  • Chayes, F. (1960). On correlation between variables of constant sum. Journal of Geophysical Research, 65(12), 4185–4193. https://doi.org/10.1029/JZ065i012p04185
  • Dervin, B. (1998). Sense-making theory and practice: An overview of user interests in knowledge seeking and use. Journal of Knowledge Management, 2(2), 36–46. https://doi.org/10.1108/13673279810249369
  • Dervin, B., and Foreman-Wernet, L. (2003). Sense-Making Methodology Reader. Hampton Press.
  • Egozcue, J. J., Pawlowsky-Glahn, V., Mateu-Figueras, G., and Barceló-Vidal, C. (2003). Isometric logratio transformations for compositional data analysis. Mathematical Geology, 35(3), 279–300. https://doi.org/10.1023/A:1023818214614
  • Eoyang, G. H., and Holladay, R. J. (2013). Adaptive Action: Leveraging Uncertainty in Your Organization. Stanford University Press.
  • HSD Institute. CDE model. https://www.hsdinstitute.org/
  • Lipmanowicz, H., and McCandless, K. (2013). The Surprising Power of Liberating Structures. Liberating Structures Press. https://www.liberatingstructures.com/
  • Martín-Fernández, J. A., Barceló-Vidal, C., and Pawlowsky-Glahn, V. (2003). Dealing with zeros and missing values in compositional data sets using nonparametric imputation. Mathematical Geology, 35(3), 253–278. https://doi.org/10.1023/A:1023866030544
  • Pawlowsky-Glahn, V., and Egozcue, J. J. (2001). Geometric approach to statistical analysis on the simplex. Stochastic Environmental Research and Risk Assessment, 15(5), 384–398. https://doi.org/10.1007/s004770100077
  • Pawson, R., and Tilley, N. (1997). Realistic Evaluation. Sage.
  • Pearson, K. (1897). Mathematical contributions to the theory of evolution. On a form of spurious correlation which may arise when indices are used in the measurement of organs. Proceedings of the Royal Society of London, 60, 489–498. https://doi.org/10.1098/rspl.1896.0076
  • Reinhard, C. D., and Reddy, A. (2026). Designing Sensemaking Studies. Palgrave Macmillan. https://link.springer.com/book/10.1007/978-3-032-25166-4