Neth, H., Gradwohl, N., Streeb, D., Keim, D.A., & Gaissmaier, W. (2021).  Perspectives on the 2x2 matrix: Solving semantically distinct problems based on a shared structure of binary contingencies.  Frontiers in Psychology, 11, 567817.  doi: 10.3389/fpsyg.2020.567817

The 2x2 matrix lens model: Pipeline of perspectives

Hansjörg Neth, Nico Gradwohl, Dirk Streeb, Daniel A. Keim, Wolfgang Gaissmaier

Perspectives on the 2x2 Matrix: 
Solving semantically distinct problems based on a shared structure of binary contingencies



A partial cube model of the 2x2 matrixCognition is both empowered and limited by representations. The matrix lens model explicates tasks that are based on frequency counts, conditional probabilities, and binary contingencies in a general fashion. Based on a structural analysis of such perspective on representational accounts of cognition that recognizes representational isomorphs as opportunities, rather than as problems. The shared structural construct of a 2x2 matrix supports a set of generic tasks and semantic mappings that provide a tasks, the model links several problems and semantic domains and provides a new unifying framework for understanding problems and defining scientific measures. Our model’s key explanatory mechanism is the adoption of particular perspectives on a 2x2 matrix that categorizes the frequency counts of cases by some condition, treatment, risk, or outcome factor. By the selective steps of filtering, framing, and focusing on specific aspects, the measures used in various semantic domains negotiate distinct trade-offs between abstraction and specialization. As a consequence, the transparent communication of such measures must explicate the perspectives encapsulated in their derivation. To demonstrate the explanatory scope of our model, we use it to clarify theoretical debates on biases and facilitation effects in Bayesian reasoning and to integrate the scientific measures from various semantic domains within a unifying framework. A better understanding of problem structures, representational transparency, and the role of perspectives in the scientific process yields both theoretical insights and practical applications.

Why read this paper?

This paper is quite long and covers a wide array of concepts and topics.  So what can you expect to gain from reading it?

The article essentially promotes a notion of positive framing effects:  When starting from a shared representation, explicating its features and structure will illuminate a range of problems, rather than only provide curious puzzles that may evoke delight and surprise but ultimately remain obscure.  The representational structure studied here is the 2x2 matrix (aka. 2-by-2 table, contingency table, or confusion matrix).  We use a classification of the Titanic population (see Figure 2) and the notorious mammography problem (see Table 1) to introduce a family of problems that involve frequency counts, binary contingencies, and conditional probabilities.  

The article introduces two abstract frameworks for analyzing problems based on binary contingencies:  A general matrix lens model (see Section 2: The Matrix Lens Model and Figure 2) and a more specific partial cube model (see Figure 5).  Together, these models explain a wide range of phenomena on the basis of adopting particular perspectives on the simple representational construct of a 2x2 matrix.

In our opinion, this article makes three main contributions:

  1. Explain the scientific metrics used in several domains in a unifying framework: 
    Various scientific measures (e.g., common metrics for quantifying the performance of classification tasks and clinical diagnostics, but also the effects of risks and treatments) can be defined and explicated in a unifying framework. (See Section 4: Integration, especially Figure 6 and Table 3.)
  2. Clarify theoretical debates on facilitation effects in tasks of Bayesian reasoning: 
    Expressing probabilistic problems in terms of natural frequencies or the so-called short menu format are known to facilitate the correct solution to problems that ask for the inverse of a conditional probability (e.g., Gigerenzer & Hoffrage, 1995). We provide a genuinely representational explanation of these effects and explore its relation to related topics (e.g., nested set theory and the benefits of providing visualizations) (see Section 5.1: Perspectives on Natural Frequencies and Nested Sets).
  3. Provide a non-circular and non-trivial definition of representational transparency:
    A representation is transparent with respect to a specific task when it explicates the perspective required for solving the task.
    When applying this definition to measures derived from a 2×2 matrix, we obtain:
    A particular measure’s representation is transparent when it explicates the perspective adopted during the measure’s derivation.
    (See Section 2.4. Presenting for details.)

Apart from these main contributions, the paper makes some additional points:

  • The research traditions on problem solving and thinking and reasoning suffer from serious issues (see Section 1: Introduction).  In contrast to overly narrow or too vague accounts and a tendency to depict representational isomorphs as a source of biases or problems, we advocate a notion of positive framing effects: Starting from a shared representation has the potential of illuminating and linking a wide array of measures, problems, and semantic domains.
  • The matrix lens model (see Section 2: The Matrix Lens Model  and the pipeline of perspectives shown in Figure 2) describes how various scientific measures (e.g., a diagnostic test’s positive predictive value, PPV) are captured by
  1. filtering the world into a binary grid of frequency counts, 
  2. framing a 2x2 matrix (aka. binary contingency table), and 
  3. focusing on particular aspects (e.g., a diagnostic test's sensitivity vs. PPV).
  • The partial cube model (see Figure 5) is a special case of a binary grid (in which some cells are missing due to semantic constraints). It shows that so-called Bayesian situations are three-dimensional, and give rise to three distinct 2x2 matrices and six hierarchical trees. Section 1 of the Supplementary Material further shows how each level allows for the same 24 distinct perspectives.
  • Section 5.2 provides new perspectives on some notorious puzzles. We explicate the so-called cab problem (see Figure 9) and the Monty Hall Problem (see Figure 10).  Our structural analysis shows their similarities to other Bayesian tasks, but also illustrate that the typical responses can arise in multiple ways.
  • The initial steps of the matrix lens model are illustrated by the population of passengers on the Titanic.  Later (see Section 5.3. Perspectives on Surviving the Titanic), we address the question whether the ship's evacuation protocol successfully implemented the so-called Birkenhead drill (i.e., preferential treatment of women and children). Interestingly, the question “Were women and children more likely to survive the disaster than adult men?” can be answered in multiple and conflicting ways (see Figure 11). This shows that any conclusion drawn from data crucially depends on the particular perspective adopted and the corresponding measure that is being derived.

As you can see, the range of tasks and problems discussed in this paper really is quite wide. However, sweeping over all these concepts and topics makes sense when anchoring them in the shared representational construct of the 2x2 matrix.

Overall, we hope that this paper contributes to several theoretical debates and enables new insights into the structure of some well-known problems.  But rather than taking our word for it, just check out the article for yourself.

Keywords:  2x2 matrix, binary contingency table, 2-by-2 confusion matrix, framing effects, representational isomorphs, matrix lens model, partial cube model, Bayesian reasoning, conditional probability, natural frequencies, nested-set theory, problem solving, scientific measurement, clinical diagnostics, insight, transparency, visualization.


  • Neth, H., Gradwohl, N., Streeb, D., Keim, D.A., & Gaissmaier, W. (2021).  Perspectives on the 2x2 matrix: Solving semantically distinct problems based on a shared structure of binary contingencies.  Frontiers in Psychology, 11, 567817.  doi: 10.3389/fpsyg.2020.567817

2x2 matrix: by row vs. by column trees (Neth et al., 2021)

Projections of a 2x2 matrix (from 2D) into linear 1D sequences (from the Supplementary Material).

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