Representation facilitates reasoning: what natural frequencies are and what they are not

When Cognitive Reflection Leads to Less Overall but More Systematic Judgment Bias: The Case of the Base Rates Fallacy

Although widely used in the judgment under uncertainty literature, the so-called Lawyer-Engineer problem does not have a Bayesian solution because the base rates typically oppose qualitative stereotypical information, which has an undefined diagnostic value. We propose an experimental paradigm that elicits participants' subjective estimates of the diagnosticity of stereotypical information and allows us to investigate the degree to which participants are able to integrate both sources of information (base rates and stereotypical descriptions) according to the Bayesian rule. This paradigm was used to test the hypothesis that the responses (probability estimates) to the Lawyer-Engineer problem from more rational individuals deviate from normative Bayesian solutions in a way that shows smaller but more systematic bias. The results further suggest that the estimates of less rational participants are noisier (less reliable) but may be more accurate when aggregated across several problems.

Comparing the impact of an icon array versus a bar graph on preference and understanding of risk information: Results from an online, randomized study

BACKGROUND

Few studies have examined the best way to convey the probability of serious events occurring in the future (i.e., risk of stroke or death) to persons with low numeracy or graph literacy proficiency. To address this gap, we developed and user-tested a bar graph and compared it to icon arrays to assess its impact on understanding and preference for viewing risk information.

OBJECTIVES

To determine the: (i) formats' impact on participants' understanding of risk information; (ii) formats' impact on understanding and format preference across numeracy and graph literacy subgroups; (iii) rationale supporting participants' preference for each graphical display format.

METHODS

An online sample (evenly made up of participants with high and low objective numeracy and graph literacy) was randomized to view either the icon array or the bar graph. Each format conveyed the risk of major stroke and death five years after choosing surgery, a stent, or medication to treat carotid artery stenosis. Participants answered questions to assess their understanding of the risk information. Lastly, both formats were presented in parallel, and participants were asked to identify their preferred format to view risk information and explain their preference.

RESULTS

Of the 407 participants, 197 were assigned the icon array and 210 the bar graph. Understanding of risk information and format preference did not differ significantly between the two trial arms, irrespective of numeracy and graph literacy proficiency. High numeracy and graph literacy proficiency was associated with high understanding (p<0.01) and a preference for the bar graph (p = 0.01).

CONCLUSION

We found no evidence to demonstrate the superiority of one format over another on understanding. The majority of participants preferred viewing the risk information using the bar graph format.

Tversky and Kahneman's Cognitive Illusions: Who Can Solve Them, and Why?

In the present paper we empirically investigate the psychometric properties of some of the most famous statistical and logical cognitive illusions from the "heuristics and biases" research program by Daniel Kahneman and Amos Tversky, who nearly 50 years ago introduced fascinating brain teasers such as the famous Linda problem, the Wason card selection task, and so-called Bayesian reasoning problems (e.g., the mammography task). In the meantime, a great number of articles has been published that empirically examine single cognitive illusions, theoretically explaining people's faulty thinking, or proposing and experimentally implementing measures to foster insight and to make these problems accessible to the human mind. Yet these problems have thus far usually been empirically analyzed on an individual-item level only (e.g., by experimentally comparing participants' performance on various versions of one of these problems). In this paper, by contrast, we examine these illusions as a group and look at the ability to solve them as a psychological construct. Based on an sample of N = 2,643 Luxembourgian school students of age 16-18 we investigate the internal psychometric structure of these illusions (i.e., Are they substantially correlated? Do they form a reflexive or a formative construct?), their connection to related constructs (e.g., Are they distinguishable from intelligence or mathematical competence in a confirmatory factor analysis?), and the question of which of a person's abilities can predict the correct solution of these brain teasers (by means of a regression analysis).

Perspectives on the 2 × 2 Matrix: Solving Semantically Distinct Problems Based on a Shared Structure of Binary Contingencies

Cognition 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 tasks, the model links several problems and semantic domains and provides a new perspective on representational accounts of cognition that recognizes representational isomorphs as opportunities, rather than as problems. The shared structural construct of a 2 × 2 matrix supports a set of generic tasks and semantic mappings that provide a unifying framework for understanding problems and defining scientific measures. Our model's key explanatory mechanism is the adoption of particular perspectives on a 2 × 2 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.

The Iowa Resistance to Sleeplessness Test (iREST)

INTRODUCTION

Despite considerable individual differences in the vulnerability vs resistance to effects of sleep loss, there is no practical self-report tool to predict these differences across domains and only limited evidence whether they are general or domain-specific. To address this need, we developed the Iowa Resistance to Sleeplessness Test (iREST).

METHODS

A construct-validation approach was employed. During the substantive phase, self-report items were generated to capture vulnerability vs resistance to sleep loss across various psycho-behavioral domains. During the structural phase, analyses identified the underlying factor structure and examined reliability of individual scale scores. Finally, the external phase used convergent and discriminant analyses to evaluate the factors in light of related sleep and personality measures, and tested criterion validity of the scale scores in predicting neurocognitive and affective responses to experimental sleep restriction (Total N = 1018).

RESULTS

Analyses yielded discriminant and reliable scale scores that reflected resistance across cognitive, affective, and somatic responses, while also marking a general resistance factor. Convergent and discriminant probes revealed moderate associations of scale scores with daytime sleepiness and sleep-related distress, but small to negligible associations with other measures of sleep behavior, perceptions, and personality. Critically, criterion analyses yielded validity evidence for predicting cognitive and affective impairments in response to experimental sleep loss.

CONCLUSION

Scores on the iREST show validity in capturing cognitive and affective resistance to moderate sleep loss among young adults, supporting its further exploration as a practical tool for predicting behavior due to lost sleep.

What facilitates Bayesian reasoning? A crucial test of ecological rationality versus nested sets hypotheses

Different theoretical views about Bayesian reasoning (ecological rationality and nested sets views) both claim support from results showing that natural sampling, whole numbers, and pictorial representations help with reasoning performance, although they differ in explaining how those results occur. Three studies (total N = 653) use minimally different numerical presentation formats-varying the singular or plural tense of the context story topic-and presence or absence of an additional icon array picture, to better understand the mechanisms driving these reasoning performance results. Plural wording, indicating a conceptual aggregation (i.e., frequencies) rather than just numerical whole numbers, consistently boosted performance. Icon arrays, in contrast, were helpful only when alongside single-tense information. These results fit more consistently with an ecological rationality view which has long argued that the mind is adapted to work best with frequentist information.

Clinical accuracy data presented as natural frequencies improve dentists' caries diagnostic inference: Evidence from a randomized controlled trial

BACKGROUND

The authors assessed whether dentists' diagnostic inferences differ when test accuracy information is communicated using natural frequencies versus conditional probabilities.

METHODS

A parallel, randomized controlled trial with dentists was carried out in Rio de Janeiro, Brazil. The dentists received a question on the probability of a patient having interproximal caries, given a positive bite-wing radiograph. This question was asked using information that was formulated into either natural frequencies or conditional probabilities.

RESULTS

Only 14 (13.9%) of the dentists gave the correct answer; 13 in the natural frequencies group, and 1 in the conditional probabilities group (P < .001). There were 7 nearly correct answers in the natural frequencies group and none in the conditional probabilities group (P = .005).

CONCLUSIONS

Representing diagnostic test accuracy in natural frequencies substantially helped dentists make diagnostic inferences. Nearly twice as many dentists overestimated the presence of interproximal caries when given information in conditional probabilities.

PRACTICAL IMPLICATIONS

Our study findings show information shared using natural frequencies may be more accurately interpreted by dentists than that based on conditional probabilities. Patients will probably receive different standards of care depending on the format in which dentists receive diagnostic test accuracy information.

Development and Usability of ADappt: Web-Based Tool to Support Clinicians, Patients, and Caregivers in the Diagnosis of Mild Cognitive Impairment and Alzheimer Disease

Background

As a result of advances in diagnostic testing in the field of Alzheimer disease (AD), patients are diagnosed in earlier stages of the disease, for example, in the stage of mild cognitive impairment (MCI). This poses novel challenges for a clinician during the diagnostic workup with regard to diagnostic testing itself, namely, which tests are to be performed, but also on how to engage patients in this decision and how to communicate test results. As a result, tools to support decision making and improve risk communication could be valuable for clinicians and patients.

Objective

The aim of this study was to present the design, development, and testing of a Web-based tool for clinicians in a memory clinic setting and to ascertain whether this tool can (1) facilitate the interpretation of biomarker results in individual patients with MCI regarding their risk of progression to dementia, (2) support clinicians in communicating biomarker test results and risks to MCI patients and their caregivers, and (3) support clinicians in a process of shared decision making regarding the diagnostic workup of AD.

Methods

A multiphase mixed-methods approach was used. Phase 1 consisted of a qualitative needs assessment among professionals, patients, and caregivers; phase 2, consisted of an iterative process of development and the design of the tool (ADappt); and phase 3 consisted of a quantitative and qualitative assessment of usability and acceptability of ADappt. Across these phases, co-creation was realized via a user-centered qualitative approach with clinicians, patients, and caregivers.

Results

In phase 1, clinicians indicated the need for risk calculation tools and visual aids to communicate test results to patients. Patients and caregivers expressed their needs for more specific information on their risk for developing AD and related consequences. In phase 2, we developed the content and graphical design of ADappt encompassing 3 modules: a risk calculation tool, a risk communication tool including a summary sheet for patients and caregivers, and a conversation starter to support shared decision making regarding the diagnostic workup. In phase 3, ADappt was considered to be clear and user-friendly.

Conclusions

Clinicians in a memory clinic setting can use ADappt, a Web-based tool, developed using multiphase design and co-creation, for support that includes an individually tailored interpretation of biomarker test results, communication of test results and risks to patients and their caregivers, and shared decision making on diagnostic testing.

Evidencing How Experience and Problem Format Affect Probabilistic Reasoning Through Interaction Analysis

This paper examines the role that lived experience plays in the human capacity to reason about uncertainty. Previous research shows that people are more likely to provide accurate responses in Bayesian tasks when the data are presented in natural frequencies, the problem in question describes a familiar event, and the values of the data are in line with beliefs. Precisely why these factors are important remains open to debate. We elucidate the issue in two ways. Firstly, we hypothesize that in a task that requires people to reason about conditional probabilities, they are more likely to respond accurately when the values of the problem reflect their own lived experience, than when they reflect the experience of the average participant. Secondly, to gain further understanding of the underlying reasoning process, we employ a novel interaction analysis method that tracks mouse movements in an interactive web application and applies transition analysis to model how the approach to reasoning differs depending on whether data are presented using percentages or natural frequencies. We find (1) that the closer the values of the data in the problem are to people's self-reported lived experience, the more likely they are to provide a correct answer, and (2) that the reasoning process employed when data are presented using natural frequencies is qualitatively different to that employed when data are presented using percentages. The results indicate that the benefits of natural frequency presentation are due to a clearer representation of the relationship between sets and that the prior humans acquire through experience has an overwhelming influence on their ability to reason about uncertainty.

Associations between Case Fatality Rates and Self-Reported Fear of Neoplasms and Circulatory Diseases

Background. According to decision by sampling theory, people store relative frequencies of events in memory, and these values constitute subjective representations of events. Because fear is a natural response to the threat of death, we hypothesized that case fatality rate (CFR) statistics, which represent how deadly a disease is, would be positively correlated with self-reported fear ratings of neoplasms and circulatory diseases. Methods. Participants (N = 239) were asked to rate various neoplasms and circulatory diseases (110 diseases in total) on fear, typicality, and disgust scales (e.g., 1 = no fear, 10 = intense fear). They also estimated mortality and morbidity rates for the same set of diseases. Finally, they completed the Berlin Numeracy Test. CFRs were obtained from the World Health Organization (WHO) database. The association between relative CFR and fear ratings was tested using correlation analyses and a multilevel linear model with Bayesian inference techniques. Results. We found that fear ratings were related to relative CFRs (r = 0.42, [0.25, 0.56], BF = 3511). This effect was present on aggregate and, to some extent, on individual levels, even after controlling for other ratings, morbidity rate, participants' estimates of mortality and morbidity statistics, numeracy, sex, age, and knowledge of WHO statistics. Also, women rated neoplasms as more frightening than circulatory diseases, and typicality ratings were related to morbidity rates. Limitations. Limited number of diagnostic entities and categories, lack of control over the technicality of disease names and participants' experience of diseases, and study sample (83% young women). Conclusions. We present initial evidence that implicit acquisition of CFRs of diseases through everyday experience may be related to the intensity of fear reactions to them.

Aiding Risk Information learning through Simulated Experience (ARISE): A Comparison of the Communication of Screening Test Information in Explicit and Simulated Experience Formats

OBJECTIVE

To determine whether the use of Aiding Risk Information learning through Simulated Experience (ARISE) to communicate conditional probabilities about maternal serum screening results for Down syndrome promotes more accurate positive predictive value (PPV) estimates and conceptual understanding of screening, compared with explicitly providing individuals with this information via numerical summary or icon array.

METHOD

In experiment 1, 582 participants completed an online study in which they were asked to estimate the PPV and rate their attitudes toward a screening test when information was presented in either a description (required calculation of the PPV), explicit (PPV was provided and had to be identified), or an ARISE format (PPV was inferred through experience-based learning). In experiment 2, 316 participants estimated the PPV and rated their attitudes toward screening based on information presented in either an icon array (identify the icons that represent the PPV) or ARISE format.

RESULTS

In experiment 1, ARISE elicited the most accurate PPV estimates compared with the description and explicit formats, and both the explicit and ARISE formats led to more unfavorable attitudes toward screening. In experiment 2, both the icon array and ARISE resulted in similar PPV estimates; however, ARISE led to more negative attitudes toward screening.

CONCLUSIONS

These findings suggest that ARISE may be superior to other formats in the communication of PPV information for screening tests. However, differences in the complexity of the formats vary and require further investigation.

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

For more than 20 years, research has proven the beneficial effect of natural frequencies when it comes to solving Bayesian reasoning tasks (Gigerenzer and Hoffrage, 1995). In a recent meta-analysis, McDowell and Jacobs (2017) showed that presenting a task in natural frequency format increases performance rates to 24% compared to only 4% when the same task is presented in probability format. Nevertheless, on average three quarters of participants in their meta-analysis failed to obtain the correct solution for such a task in frequency format. In this paper, we present an empirical study on what participants typically do wrong when confronted with natural frequencies. We found that many of them did not actually use natural frequencies for their calculations, but translated them back into complicated probabilities instead. This switch from the intuitive presentation format to a less intuitive calculation format will be discussed within the framework of psychological theories (e.g., the Einstellung effect).

Communicating risk of medication side-effects: role of communication format on risk perception

Background:

Medication side-effects often arouse fear in the minds of consumers and therefore need to be communicated in a manner such that the intended message is clearly understood, without causing undue fear.

Objectives:

Considering the message format and contextual factors that influence perceptions of risk, this study aimed at assessing the interaction effects of message format and contextual factors (rate of occurrence and severity) on risk perception of medication side-effects.

Methods:

Using Rhormann’s risk communication process model, a 2 (message format: words-only vs. words + numeric) X 2 (rate of occurrence: high vs low) X 2 (severity: mild vs severe) experimental factorial study was designed. Participants were presented with four of eight possible combinations of the three factors and were asked to indicate the risk perception with the associated side-effects. Repeated measures analysis was conducted while adjusting for control variables.

Results:

A total of 196 completed surveys were collected. Communication format did not have significant main effect on risk perception (P=0.4237) but demonstrated a significant interaction with rate of occurrence (P=0.0001). As compared to words-only format, least square means for words + numeric format were lower among low-rate side-effects but were higher among high-rate side-effects. Rate of occurrence (P<0.0001) and severity (P<0.0001) had significant main effects on risk perception as well as interaction effect with each other (P<0.0001).

Conclusions:

The results indicated that effect of communication format on risk perception of side-effect is dependent on the underlying rate of occurrence of side-effect. Healthcare providers should therefore carefully construct risk communication messages for effective communication with patients.

Assessing the effect of public health information by incentivised risk estimation: An example on Swedish snus

BACKGROUND

The provision of accurate information on health damaging behaviours and products is a widely accepted and widespread governmental task. It is easily mismanaged. This study demonstrates a simple method which can help to evaluate whether such information corrects recipient risk beliefs.

METHODS

Participants assess risks numerically, before and after being exposed to a relevant risk communication. Accuracy is incentivised by awarding financial prizes to answers closest to a pursued risk belief. To illustrate this method, 228 students from the University of Oslo, Norway, were asked to estimate the mortality risk of Swedish snus and cigarettes twice, before and after being exposed to one of three risk communications with information on the health dangers of snus.

RESULTS

The data allow us to measure how participants updated their risk beliefs after being exposed to different risk communications. Risk information from the government strongly distorted risk perceptions for snus. A newspaper article discussing the relative risks of cigarettes and snus reduced belief errors regarding snus risks, but increased belief errors regarding smoking. The perceived quality of the risk communication was not associated with decreased belief errors.

CONCLUSION

Public health information can potentially make the public less informed on risks about harmful products or behaviours. This risk can be reduced by targeting identified, measurable belief errors and empirically assessing how alternative communications affect these. The proposed method of incentivised risk estimation might be helpful in future assessments of risk communications.

Designing Visual Aids That Promote Risk Literacy: A Systematic Review of Health Research and Evidence-Based Design Heuristics

Background Effective risk communication is essential for informed decision making. Unfortunately, many people struggle to understand typical risk communications because they lack essential decision-making skills. Objective The aim of this study was to review the literature on the effect of numeracy on risk literacy, decision making, and health outcomes, and to evaluate the benefits of visual aids in risk communication. Method We present a conceptual framework describing the influence of numeracy on risk literacy, decision making, and health outcomes, followed by a systematic review of the benefits of visual aids in risk communication for people with different levels of numeracy and graph literacy. The systematic review covers scientific research published between January 1995 and April 2016, drawn from the following databases: Web of Science, PubMed, PsycINFO, ERIC, Medline, and Google Scholar. Inclusion criteria were investigation of the effect of numeracy and/or graph literacy, and investigation of the effect of visual aids or comparison of their effect with that of numerical information. Thirty-six publications met the criteria, providing data on 27,885 diverse participants from 60 countries. Results Transparent visual aids robustly improved risk understanding in diverse individuals by encouraging thorough deliberation, enhancing cognitive self-assessment, and reducing conceptual biases in memory. Improvements in risk understanding consistently produced beneficial changes in attitudes, behavioral intentions, trust, and healthy behaviors. Visual aids were found to be particularly beneficial for vulnerable and less skilled individuals. Conclusion Well-designed visual aids tend to be highly effective tools for improving informed decision making among diverse decision makers. We identify five categories of practical, evidence-based guidelines for heuristic evaluation and design of effective visual aids.

Another chance for good reasoning

Disagreement on the "probability status" of chances casts doubt on Girotto and Gonzalez's (2001) conclusion that the human mind can make sound Bayesian inferences involving single-event probabilities. The main objection raised has been that chances are de facto natural frequencies disguised as probabilities. In the present study, we empirically demonstrated that numbers of chances are perceived as being distinct from natural frequencies and that they have a facilitatory effect on Bayesian inference tasks that is completely independent from their (minor) frequentist readings. Overall, therefore, our results strongly disconfirm the hypothesis that natural frequencies are a privileged cognitive representational format for Bayesian inferences and suggest that a significant portion of laypeople adequately handle genuine single-event probability problems once these are rendered computationally more accessible by using numbers of chances.

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

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.

Replication Rate, Framing, and Format Affect Attitudes and Decisions about Science Claims

A series of five experiments examined how the evaluation of a scientific finding was influenced by information about the number of studies that had successfully replicated the initial finding. The experiments also tested the impact of frame (negative, positive) and numeric format (percentage, natural frequency) on the evaluation of scientific findings. In Experiments 1 through 4, an attitude difference score served as the dependent measure, while a measure of choice served as the dependent measure in Experiment 5. Results from a diverse sample of 188 non-institutionalized U.S. adults (Experiment 2) and 730 undergraduate college students (Experiments 1, 3, and 4) indicated that attitudes became more positive as the replication rate increased and attitudes were more positive when the replication information was framed positively. The results also indicate that the manner in which replication rate was framed had a greater impact on attitude than the replication rate itself. The large effect for frame was attenuated somewhat when information about replication was presented in the form of natural frequencies rather than percentages. A fifth study employing 662 undergraduate college students in a task in which choice served as the dependent measure confirmed the framing effect and replicated the replication rate effect in the positive frame condition, but provided no evidence that the use of natural frequencies diminished the effect.

Scientific uncertainty in media content: Introduction to this special issue

This introduction sets the stage for the special issue on the public communication of scientific uncertainty that follows by sketching the wider landscape of issues related to the communication of uncertainty and showing how the individual contributions fit into that landscape. The first part of the introduction discusses the creation of media content as a process involving journalists, scientific sources, stakeholders, and the responsive audience. The second part then provides an overview of the perception of scientific uncertainty presented by the media and the consequences for the recipients' own assessments of uncertainty. Finally, we briefly describe the six research articles included in this special issue.

Measures of Bayesian Reasoning Performance on 'Normal' and 'Natural' Frequency Tasks

While the majority of similar studies examining Bayesian reasoning investigate how participants avoid common errors such as base-rate neglect, the current research also examines whether different formats (frequency and probability) lead to a difference in levels of absolute accuracy. In Study One, older (≥60 years) and younger (18 to 29 years) participants completed tasks in probability and normalized frequency formats. In Study 2, participants completed tasks in probability and natural frequency formats. Findings are that frequencies lead to less over-estimation, particularly in natural frequency tasks, which also reveal an interaction between age and task format whereby older adults seem unaffected by format. There was no association found between format and the avoidance of errors such as base-rate neglect. Findings are discussed in the light of dual and multi-process theories of reasoning, having failed to support the theory that frequency formats elicit System 2 reasoning processes.

Helping Doctors and Patients Make Sense of Health Statistics

Many doctors, patients, journalists, and politicians alike do not understand what health statistics mean or draw wrong conclusions without noticing. Collective statistical illiteracy refers to the widespread inability to understand the meaning of numbers. For instance, many citizens are unaware that higher survival rates with cancer screening do not imply longer life, or that the statement that mammography screening reduces the risk of dying from breast cancer by 25% in fact means that 1 less woman out of 1,000 will die of the disease. We provide evidence that statistical illiteracy (a) is common to patients, journalists, and physicians; (b) is created by nontransparent framing of information that is sometimes an unintentional result of lack of understanding but can also be a result of intentional efforts to manipulate or persuade people; and (c) can have serious consequences for health.

The causes of statistical illiteracy should not be attributed to cognitive biases alone, but to the emotional nature of the doctor–patient relationship and conflicts of interest in the healthcare system. The classic doctor–patient relation is based on (the physician's) paternalism and (the patient's) trust in authority, which make statistical literacy seem unnecessary; so does the traditional combination of determinism (physicians who seek causes, not chances) and the illusion of certainty (patients who seek certainty when there is none). We show that information pamphlets, Web sites, leaflets distributed to doctors by the pharmaceutical industry, and even medical journals often report evidence in nontransparent forms that suggest big benefits of featured interventions and small harms. Without understanding the numbers involved, the public is susceptible to political and commercial manipulation of their anxieties and hopes, which undermines the goals of informed consent and shared decision making.

What can be done? We discuss the importance of teaching statistical thinking and transparent representations in primary and secondary education as well as in medical school. Yet this requires familiarizing children early on with the concept of probability and teaching statistical literacy as the art of solving real-world problems rather than applying formulas to toy problems about coins and dice. A major precondition for statistical literacy is transparent risk communication. We recommend using frequency statements instead of single-event probabilities, absolute risks instead of relative risks, mortality rates instead of survival rates, and natural frequencies instead of conditional probabilities. Psychological research on transparent visual and numerical forms of risk communication, as well as training of physicians in their use, is called for.

Statistical literacy is a necessary precondition for an educated citizenship in a technological democracy. Understanding risks and asking critical questions can also shape the emotional climate in a society so that hopes and anxieties are no longer as easily manipulated from outside and citizens can develop a better-informed and more relaxed attitude toward their health.

A levels of processing interpretation of dual-system theories of judgment and decision making

Recently there has been spirited disagreement about the merits of dual-system theories of higher cognition. I suggest that this dispute is very similar to the 1970s dispute between two-store theories of memory and levels of processing theory. The two-store or “box” theorists stipulated that short-term memory and long-term memory stores were quite dissimilar and therefore represented separate memory stores. Levels of processing theorists disputed the evidence for separate memory stores and asserted that memory was an epiphenomenon of the depth to which a stimulus was processed. I adopt the levels of processing approach to show how it can help clarify the phenomena previously described by dual-system theories. Furthermore, this proposed resolution to the controversy renders moot the serious disagreements about what features might characterize each of the two processes in dual-system theories.

Natural frequencies improve Bayesian reasoning in simple and complex inference tasks

Representing statistical information in terms of natural frequencies rather than probabilities improves performance in Bayesian inference tasks. This beneficial effect of natural frequencies has been demonstrated in a variety of applied domains such as medicine, law, and education. Yet all the research and applications so far have been limited to situations where one dichotomous cue is used to infer which of two hypotheses is true. Real-life applications, however, often involve situations where cues (e.g., medical tests) have more than one value, where more than two hypotheses (e.g., diseases) are considered, or where more than one cue is available. In Study 1, we show that natural frequencies, compared to information stated in terms of probabilities, consistently increase the proportion of Bayesian inferences made by medical students in four conditions-three cue values, three hypotheses, two cues, or three cues-by an average of 37 percentage points. In Study 2, we show that teaching natural frequencies for simple tasks with one dichotomous cue and two hypotheses leads to a transfer of learning to complex tasks with three cue values and two cues, with a proportion of 40 and 81% correct inferences, respectively. Thus, natural frequencies facilitate Bayesian reasoning in a much broader class of situations than previously thought.

Effects of visualizing statistical information - an empirical study on tree diagrams and 2 × 2 tables

In their research articles, scholars often use 2 × 2 tables or tree diagrams including natural frequencies in order to illustrate Bayesian reasoning situations to their peers. Interestingly, the effect of these visualizations on participants’ performance has not been tested empirically so far (apart from explicit training studies). In the present article, we report on an empirical study (3 × 2 × 2 design) in which we systematically vary visualization (no visualization vs. 2 × 2 table vs. tree diagram) and information format (probabilities vs. natural frequencies) for two contexts (medical vs. economical context; not a factor of interest). Each of N = 259 participants (students of age 16–18) had to solve two typical Bayesian reasoning tasks (“mammography problem” and “economics problem”). The hypothesis is that 2 × 2 tables and tree diagrams – especially when natural frequencies are included – can foster insight into the notoriously difficult structure of Bayesian reasoning situations. In contrast to many other visualizations (e.g., icon arrays, Euler diagrams), 2 × 2 tables and tree diagrams have the advantage that they can be constructed easily. The implications of our findings for teaching Bayesian reasoning will be discussed.