The Impact of Visualizing Nested Sets. An Empirical Study on Tree Diagrams and Unit Squares

Abstract

It is an ongoing debate, what properties of visualizations increase people’s performance when solving Bayesian reasoning tasks. In the discussion of the properties of two visualizations, i.e., the tree diagram and the unit square, we emphasize how both visualizations make relevant subset relations transparent. Actually, the unit square with natural frequencies reveals the subset relation that is essential for the Bayes’ rule in a numerical and geometrical way whereas the tree diagram with natural frequencies does it only in a numerical way. Accordingly, in a first experiment with 148 university students, the unit square outperformed the tree diagram when referring to the students’ ability to quantify the subset relation that must be applied in Bayes’ rule. As hypothesized, in a second experiment with 143 students, the unit square was significantly more effective when the students’ performance in tasks based on Bayes’ rule was regarded. Our results could inform the debate referring to Bayesian reasoning since we found that the graphical transparency of nested sets could explain these visualizations’ effect.

Visualizing the Bayesian 2-test case: The effect of tree diagrams on medical decision making

“…There is another strategy for improving Bayesian reasoning in the 1-test case, namely, visualizing the statistical information. Some prominent visualizations that have been developed are Euler diagrams (e.g., [ 29 – 31 ]), roulette-wheel diagrams (e.g., [ 32 , 33 ]), frequency grids (e.g., [ 23 , 34 , 35 ]), Eikosograms (sometimes also called unit squares or mosaic plots ; e.g., [ 36 – 39 ]), icon arrays (e.g., [ 32 , 40 , 41 ]), 2×2-tables (e.g., [ 14 , 42 ]), and tree diagrams (e.g., [ 14 , 33 , 42 – 44 ]). For an overview of these visualizations, see [ 14 ], and for corresponding visualizations regarding the 2-test case, see Fig 1 .…”

Why Can Only 24% Solve Bayesian Reasoning Problems in Natural Frequencies: Frequency Phobia in Spite of Probability Blindness

“…Additionally, various other factors are known to have an impact on performance in Bayesian reasoning tasks. Visualizations, for example tree diagrams (e.g., Yamagishi, 2003 ; Binder et al, 2018 ), unit squares (e.g., Böcherer-Linder and Eichler, 2017 ; Pfannkuch and Budgett, 2017 ), icon arrays (e.g., Brase, 2009 , 2014 ) or roulette wheel diagrams (e.g., Yamagishi, 2003 ; Brase, 2014 ), have been shown to improve accuracies in Bayesian situations (for an exception, see, e.g., Micallef et al, 2012 ). An overview and categorization of visualizations that were used to boost performance in Bayesian situations is provided by Khan et al ( 2015 ).…”

“…Why can still on average only a quarter of participants solve the problem correctly, although the task is presented in the beneficial natural frequency format? Many psychological theories explain, discuss, and specify in detail if and why natural frequencies facilitate Bayesian inferences (e.g., the nested sets-hypothesis or the ecological rationality framework, see Gigerenzer and Hoffrage, 1999 ; Lewis and Keren, 1999 ; Mellers and McGraw, 1999 ; Girotto and Gonzalez, 2001 , 2002 ; Hoffrage et al, 2002 ; Sloman et al, 2003 ; Barbey and Sloman, 2007 ; Pighin et al, 2016 ; McDowell et al, 2018 ) and how additional tools, such as visualizations, further increase their beneficial effect (e.g., Yamagishi, 2003 ; Brase, 2009 , 2014 ; Spiegelhalter et al, 2011 ; Micallef et al, 2012 ; Garcia-Retamero and Hoffrage, 2013 ; Micallef, 2013 ; Ottley et al, 2016 ; Böcherer-Linder and Eichler, 2017 ). However, a satisfying answer to the question why only 24% of participants solve Bayesian reasoning problems in natural frequency format correctly has not yet been found.…”

A visualization technique for Bayesian reasoning

“…The past literature suggests that visual supplements improve Bayesian reasoning performance in some, but not all, situations (Binder, Krauss, & Bruckmaier, ; Böcherer‐Linder & Eichler, ; Brase, , ; Cosmides & Tooby, ; Micallef, Dragicevic, & Fekete, ; Reani, Davies, Peek, & Jay, ; Sirota, Kostovičová, & Juanchich, ; Sloman, Over, Slovak, & Stibel, ; Wu, Meder, Filimon, & Nelson, ; Yamagishi, ). We will introduce a variation on past displays that represents probabilities with bar length and reveals the answer to Bayesian inference problems in terms of spatial relationships.…”

“…Another focus of past research has been the effect of presenting visual representations of the problem information (Binder et al, ; Böcherer‐Linder & Eichler, ; Brase, , ; Cosmides & Tooby, ; Micallef et al, ; Reani et al, ; Sirota et al, ; Sloman et al, ; Wu et al, ; Yamagishi, ). Although the pattern of results is complex, fairly consistent benefits have been observed for displays that represent information with bar graphs or icon arrays (Brase, , ; Sirota et al, , Exp.…”