Visualizing Bayesian situations about the use of the unit square Andreas Eichler 1, Markus Vogel 2 and Katharina Böcherer-Linder 3

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1 Visualizing Bayesian situations about the use of the unit square Andreas Eichler 1, Markus Vogel 2 and Katharina Böcherer-Linder 3 1 University of Kassel, Germany; eichler@mathematik.uni-kassel.de 2 Heidelberg, University of Education, Germany; vogel@ph-heidelberg.de 3 Freiburg, University of Education, Germany; katharina.boechererlinder@ph-freiburg.de In this paper we focus on the unit square as a visualization of Bayesian Situations, i.e. situations that are based on conditional probabilities and Bayes rule from a theoretical point of view. We refer to systematically comparing the unit square and more common tools for visualizing Bayesian situations, i.e. the tree diagram and the 2x2-table. In this comparison, we focus on computing probabilities by applying Bayes rule, on problems of changing the base rate in Bayesian situations and on visualizing independency. Based on our theoretical considerations which are additionally underlined with some empirical findings from our previous research we found the unit square as being a useful visualization for supporting students understanding of Bayesian situations. Keywords: Visualizations, Bayes situations, unit square. Introduction Representation and visualization are at the core of understanding in mathematics (Duval, 2002, p. 312). Following Biehler and Burrill (2011), Duval s proposition is also true regarding statistics and probability, since the ability to read and to interpret graphical representations of data is a fundamental statistical idea for teaching statistics in school. Research in recent decades gained a lot of knowledge that people have great difficulties in adequately interpreting Bayesian situations (e.g. Kahneman & Tversky, 1982). But there is also empirical evidence that visualizations could potentially improve the ability of interpretation (e.g. Binder, Krauss, & Bruckmaier, 2015). A wellknown problem in this field that is often cited in textbooks (Veaux, Velleman, & Bock, 2012), educational research (Sturm & Eichler, 2014) or psychological research (Binder et al., 2015) is the interpretation of a positive test result in a diagnosis of a disease. We outline such a diagnosis problem below without taking a specific disease like tuberculosis or cancer into account: The probability of a person getting a specific disease is 5%. If a person has this disease the probability that he or she will have a positive test result is 80%. If a person does not have the disease the probability that he or she will have a positive test result is 10%. What is the probability that a person, who received a positive test result, actually has the disease? We will refer repeatedly to this task in the following sections of the paper, where we aim to argue for the unit square as a didactically useful visualization concerning Bayesian situations. Theoretical background of visualizing Bayesian situations Visualizations in Bayesian situations could be used for different aims. One aim could be to communicate risks by visualizing Bayesian situations in a way that facilitates a specific situation s interpretation and allows us to estimate a risk adequately (e.g. Spiegelhalter & Gage, 2014). A further aim of visualizing Bayesian situations is to improve students learning in terms of acquiring

2 and using conceptual and cognitive structures (Greeno, Collins, & Resnick, 1996, p. 22) which implies a more global and conceptual understanding of Bayesian situations, and which goes beyond the ability to solve purely specific tasks in a Bayesian situation. The difference between a more local aim of communicating and a more global aim of learning potentially impacts on the selection of a visualization of Bayesian situations. Whereas there is a considerable amount of different visualizations for communicating Bayesian situations (Khan, Breslav, Glueck, & Hornbæk, 2015), visualizations of Bayesian situations for students learning in the sense of the more global aim mentioned above are mainly restricted to two main types of visualization, namely the tree diagram and the 2x2-table (e.g. Veaux et al., 2012). Although the 2x2-table and also the tree diagram seem to improve students reasoning in Bayesian situations (Binder et al., 2015), we propose a further visualization of Bayesian situations, i.e. the unit square. The unit square The unit square is a square with side length 1. The unit square is a statistical graph (Tufte, 2013), which means, that the sizes of the partitioned areas are proportional to the sizes of the represented data. Therefore the proportion of incidences in a population, i.e. the base rate, is represented in a numerical and geometrical sense. Referring to the diagnosis task mentioned above, the unit square is partitioned into four areas concerning the events having the disease (D), not having the disease ( ), getting a positive test result ( ) or a negative test result ( ) (fig. 1). The vertical partitioning is determined by the probabilities of having the disease ( ) or not having the disease ( 95%). The horizontal partitioning depends on the vertical partitioning and, thus, represent conditional probabilities, i.e. the probability that an ill person gets a positive test result ( left side above) and the probability that a healthy person gets a positive test result ( ; right side above). The areas represent joint probabilities, i.e. and. The natural frequencies shown in figure 1 represent from the perspective of probability theory expected values for the compounded events (8), (2), (19) and (171). Figure 1 provides a unit square that represents the diagnosis task by using these natural frequencies. Of course, there are several alternatives possible, e.g. by indicating probabilities, by replacing absolute frequencies with probabilities, or by showing sums at the bottom of the square (e.g. Böcherer-Linder, Eichler, & Vogel, 2015). Figure 1: The unit square representing the diagnosis task An extensive approach to using the unit square is discussed by Oldford (2003). Calling the diagram eikosogram, he examined different well known Bayesian situations like diagnosis tests, the

3 Monty Hall problem, the Prisoner s Dilemma or the problem of green and blue cabs (Kahneman & Tversky, 1982, p. 156). Based on this approach Politzer (2014) used the metaphor of a tank and its subdivisions to visualize Bayes theorem with an eikosogram and, particularly, to discuss deductive arguments like modus ponens or contraposition. The first approach that we are aware of that empirically investigated the efficiency of the unit square was provided by Bea (1995). Bea reported that the unit square outperformed the tree diagram concerning the long-term success regarding the ability to solve tasks including conditional probabilities. As with Oldford, Bea also used probabilities to indicate the proportions shown in the unit square. Figure 2 Transformation process from the 2x2-table (left side) and the weighted 2x2-table (in the middle) to the unit square (right side). On the basis of Bea s results, Eichler and Vogel (2013, 2015) used the unit square as a form of a weighted 2x2-table to visualize two categorical variables that are also the basis of Bayesian situations like the diagnosis task. The transformation process from a 2x2-table and a weighted 2x2- table to the unit square is illustrated according to the diagnosis task in figure 2. In a further step of development natural frequencies were used not only in the weighted 2x2-table, but also in the unit square (Böcherer-Linder et al., 2015; Eichler & Vogel, 2013) since representing statistical information with natural frequencies was found to improve Bayesian reasoning (e.g. Brase, 2014). Unit square, tree diagram and 2x2-table: a threefold comparison Comparing the visualizations of the unit square, the tree diagram and the 2x2-table we focus on three crucial aspects of conceptual understanding of Bayesian situations, i.e. applying Bayes rule, estimating the influence of the base rate, and recognizing independency of involved variables. Computing probabilities by applying and visualizing Bayes rule The main question in a Bayesian situation represented by the diagnosis task concerns the conditional probability of having a disease given a positive test result. The main challenge for calculating this probability is to identify relationships of subsets and sets. Thus, the intersection of disease and test positive ( ) has to be identified as a subset of test positive ( ), i.e.. We regard the visualization of this relation of sets in figure 3: The relation of the relevant subsets and sets is apparent in the 2x2-table: For calculating, the relation from an interior field representing the subset and the sum of two neighboring fields representing the whole set has to be considered (dotted line). This sum is also represented in the row. Similarly, in the unit square the relation of an area representing the subset and the sum of

4 two neighboring areas representing the two subsets and has to be considered. In contrast to the 2x2-table, the set must be identified as the compound set. Differently to the 2x2-table and the unit square, the tree diagram has a hierarchical structure. Relations of subsets and sets like are transparently visualized in this hierarchy. However, the relation that is necessary for calculating is not as transparently visualized in the hierarchical order of branches in the tree diagram, since two branches of different, not directly neighbored paths include the information of a positive test. Figure 3 Relation of sets for applying Bayes rule in the diagnosis task visualized by the tree diagram, the 2x2-table and the unit square The research of Böcherer-Linder and Eichler (accepted) gave empirical evidence that the unit square is more efficient than the tree diagram in Bayesian situations. They found that the difference of the tree diagram and the unit square of making the relevant set-subset-relation transparent impacts significantly on the ability of students to solve tasks in Bayesian situations like the diagnosis task. Problems of changing the base rate in Bayesian situations Educational and psychological research proved that using natural frequencies in Bayesian situations increases the ability to solve tasks in Bayesian situations (e.g. Sedlmeier & Gigerenzer, 2001). However, acquiring conceptual knowledge about Bayesian situations in a globally aimed sense includes also dealing with relative frequencies or rather probabilities. Since it is possible to visualize relative frequencies and probabilities with each of the three visualizations, i.e. the tree diagram, the 2x2-table or the unit square, we will focus on visualization with regard to acquiring conceptual knowledge about Bayesian situations in this paragraph. One aspect of a conceptual knowledge about Bayesian situations is the structure of Bayes theorem itself. Using the tree diagram, the 2x2-table and the unit square, the visualization of the Bayes rule for the diagnosis task (representing situations with two variables and with each two categories) is shown in figure 4. Referring to the tree diagram, the multiplication term in the rule is represented by the multiplication rule for branches in a tree (e.g. Veaux et al., 2012). Referring to the unit square, the multiplication term is represented by calculating an area of a rectangle given the length of both sides. By contrast, the 2x2-table could only visualize the fraction but not the mentioned multiplication term (right hand side of the equation in fig. 4). For this reason, the tree diagram and the unit square seem to be more appropriate to visualize the development of Bayes rule.

5 Figure 4 Visual display of the Bayes theorem A further important aspect of a conceptual knowledge about Bayesian situations is the ability to estimate the impact of changes of the relevant values, and. Especially, the impact of the base rate was identified as being a main difficulty of understanding Bayesian situations (Kahneman & Tversky, 1973). For this reason we investigate a change of the base rate ( in the diagnosis task from 5% to 20%. Figure 5 A base rate change from 5% to 20% and the corresponding visualizations Of course, the numbers change in all three visual displays when the base rate is changed. However, it is evident that only the unit square shows additionally a changed shape for the different situations (figure 5). Here, one can get a graphically based impression that an increasing base rate is connected with an increasing value of by relating corresponding areas to each other. For this reason the unit square seems to be more appropriate to visualize the influence of the base rate that is a crucial aspect of a conceptual knowledge about Bayesian situations. This aspect is empirically proven by Böcherer-Linder, Eichler and Vogel (accepted). Visualizing independency A third relevant aspect of conceptual knowledge about Bayesian situations is the ability to distinguish dependent variables from independent variables. For this reason, we consider finally a visualization of independent variables within all three visual displays. For this we change the disease context arbitrarily: We assume that we use a test to detect brown eyes instead of using the diagnosis test. Of course, this eye color test is not appropriate for detecting the disease and thus, we can assume that this test will be positive in 90% of the cases (frequency of brown eyes in the world) and negative in 10% of the cases independently of the fact that the tested people have the disease or not. Hence, we have. The resulting test situation is visualized in figure 6. Similar to the discussion concerning the base rate, the tree diagram and the 2x2-table do not visualize independency directly. Thus, it is necessary to grasp the independency of two variables by evaluating the numbers shown in these two visual displays. Again, only the unit square directly visualizes independency: Independency is given if there is a horizontal line going through the whole unit square without being divided by the left and right part of the unit square.

6 Discussion Figure 6 Independency visualized by the tree diagram, the 2x2-table and the unit square Tree diagrams, 2x2-tables and unit squares differ regarding their characteristics. The unit square is a statistical graph (Tufte, 2013) that represents quantities by areas and, optionally, by numbers. By contrast, both 2x2-tables and tree diagrams are not statistical graphs since both visualizations represent quantities by numbers only. The tree diagram further implies a hierarchy or rather a sequence of events. For example, in the diagnosis task the first random experiment determines the health status, the second random experiment determines the test result. Finally, the 2x2-table represents a well-structured display of the relevant numbers in a Bayesian situation. The different characteristics of the visualizations impact on their helpfulness for solving tasks and for developing conceptual knowledge about Bayesian situations. First, the hierarchy of the tree diagram is a disadvantage when solving tasks that require applying Bayes rule since the relevant subset-setrelation and thus the nested sets structure for this task is not evident. Consequently, it is theoretically reasonable, and in fact, there is some empirical evidence (Böcherer-Linder & Eichler, accepted) that the 2x2-table and the unit square are more efficient when solving tasks in Bayesian situations. However, the characteristic of a statistical graph could also yield limitations for solving tasks in Bayesian situations. For example, the diagnosis test for HIV discussed in Sturm and Eichler (2014) has a sensitivity of and a specifity of. Whereas the sensitivity could be appropriately displayed with the unit square the rectangle representing the conjunction of disease and test positive ( ) could only be displayed almost as a line. This is the same if a base rate of a western country is considered that is mostly below 0.5% ( However, this limitation of the unit square is only a limitation of displaying extreme values but not a limitation of developing conceptual knowledge about Bayesian Situations neither when focusing on the structure of Bayes theorem nor when studying the impact of changes of the relevant values or when reflecting on questions of variables dependency. There is a difference between solving specific tasks in Bayesian situations on the one hand (we consider these activities as being more locally aimed) and developing conceptual knowledge about Bayesian reasoning on the other hand (which we consider as being more globally aimed). For communicating risk in Bayesian situations different forms of visualizations are proposed (e.g. Spiegelhalter & Gage, 2014). Some of these visualizations are obviously not appropriate for use in schools by constructing them in order to support the solution process of tasks in specific Bayesian situations. Even more, to facilitate conceptual learning, further characteristics of a visualization of

7 Bayesian situations are needed. Particularly for the combination of the latter two aspects, there are empirical reasons as well as theoretical reasons to consider the unit square as being appropriate to improve students learning. We discussed the advantages of the unit square for visualizing exemplarily fundamental aspects of Bayesian situations, namely the structure of Bayesian situations, the impact of the base rate and the construct of independence. Particularly concerning the latter two aspects, the unit square seems to be more efficient due to its characteristic as a statistical graph that potentially could be changed dynamically. Thus, there is some reason to consider teaching and learning of Bayesian situations using visualizations with the unit square. References Bea, W. (1995). Stochastisches Denken: Analysen aus Kognitionspsychologischer und Didaktischer Perspektive. Frankfurt am Main, Berlin: Lang. Biehler, R. & Burrill, G. (2011). Fundamental Statistical Ideas in the School Curriculum and in Training Teachers. In C. Batanero, G. Burrill, & C. Reading (Eds.), New ICMI Study Series: Vol. 14. Teaching Statistics in School Mathematics-Challenges for Teaching and Teacher Education. A Joint ICMI/IASE Study: The 18th ICMI Study (pp ). Dordrecht: Springer Science+Business Media B.V. Binder, K., Krauss, S., & Bruckmaier, G. (2015). Effects of Visualizing Statistical Information - an Empirical Study on Tree Diagrams and 2 2 Tables. Frontiers in Psychology, 6, Böcherer-Linder, K. & Eichler, A. (accepted). The impact of visualizing nested sets. An empirical study on tree diagrams and unit squares. Frontiers in Psychology. Böcherer-Linder, K., Eichler, A., & Vogel, M. (2015). Understanding conditional probability through visualization. In H. Oliveira, A. Henriques, A. P. Canavarro, C. Monteiro, C. Carvalho, J. P. Ponte, R. T. Ferreira, S. Colaço (Eds.), Proceedings of the International Conference Turning data into knowledge: New opportunities for statistics education (pp ). Lisbon, Portugal: Instituto de Educação da Universidade de Lisboa. Böcherer-Linder, K., Eichler, A., & Vogel, M. (accepted). The impact of visualization on flexible Bayesian reasoning. Accepted by AIEM Avances de Investigación en Educación Matemática. Brase, G. L. (2014). The Power of Representation and Interpretation: Doubling Statistical Reasoning Performance with Icons and Frequentist Interpretations of Ambiguous Numbers. Journal of Cognitive Psychology, 26(1), Duval, R. (2002). Representation, Vision and Visualization: Cognitive Functions in Mathematical Thinking. Basic Issues for Learning. In F. Hitt (Ed.), Representations and mathematics visualization. Papers presented in this Working Group of PME-NA, (pp ), Mexico: Cinestav - IPN. Eichler, A. & Vogel, M. (2013). Leitidee Daten und Zufall: Von konkreten Beispielen zur Didaktik der Stochastik. Wiesbaden: Springer.

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