Frequency versus probability formats in statistical word problems

Previous beliefs affect Bayesian reasoning in conditions fostering gist comprehension

It has been shown that Bayesian reasoning is affected by the believability of the data, but it is unknown which conditions could potentiate or reduce such belief effect. Here, we tested the hypothesis that the belief effect would mainly be observed in conditions fostering a gist comprehension of the data. Accordingly, we expected to observe a significant belief effect in iconic rather than in textual presentations and, in general, when nonnumerical estimates were requested. The results of three studies showed more accurate Bayesian estimates, either expressed numerically or nonnumerically, for icons than for text descriptions of natural frequencies. Moreover, in line with our expectations, nonnumerical estimates were, in general, more accurate for believable rather than for unbelievable scenarios. In contrast, the belief effect on the accuracy of the numerical estimates depended on the format and on the complexity of the calculation. The present findings also showed that single-event posterior probability estimates based on described frequencies were more accurate when expressed nonnumerically rather than numerically, opening new avenues for the development of interventions to improve Bayesian reasoning.

Source oxygen contributions of primary nitrate emitted from biomass burning

Atmospheric nitrate (NO3-) produced by photochemical oxidation in the atmosphere has high oxygen isotope ratios (δ18O values). Recently, the primary NO3- emitted from combustion sources was found to have much lower δ18O values. However, it is unclear how and to what extents the low δ18O signatures were controlled by major O sources during the primary NO3- formation of combustion processes. Here, we first measured concentrations and δ18O values of NO3- from burning five biomass materials (bb-NO3- and δ18Obb-NO3-, respectively) in China. Distinctly higher concentration levels of the bb-NO3- emissions (42.1 ± 8.1 μmol m-3) than ambient NO3- suggest it is a potential source of atmospheric NO3- pollution. Much lower δ18Obb-NO3- signatures (27.6 ± 2.7 ‰) than ambient NO3- support it as a primary emission source with different O sources and formation mechanism from secondary NO3-. Isotope mass-balance modeling revealed that atmospheric O2 and the biomass O dominated the O of bb-NO3- (53 ± 7 % and 40 ± 4 %, respectively) over the aqueous vapor (7 ± 3 %). Besides, we found increasing δ18Obb-NO3- values with the biomass N contents and relatively lower δ18Obb-NO3- values for biomasses with higher carbon (C) and lower O contents, indicating that biomass C, N, and O contents may influence the source O contributions of the bb-NO3-. This work provides a novel isotope analysis on the O source contribution of the bb-NO3-, which is useful for understanding the formation mechanism of combustion-related NO3- sources and evaluating the primary NO3- emissions.

A Generative View of Rationality and Growing Awareness†

In this paper we contrast bounded and ecological rationality with a proposed alternative, generative rationality. Ecological approaches to rationality build on the idea of humans as "intuitive statisticians" while we argue for a more generative conception of humans as "probing organisms." We first highlight how ecological rationality's focus on cues and statistics is problematic for two reasons: (a) the problem of cue salience, and (b) the problem of cue uncertainty. We highlight these problems by revisiting the statistical and cue-based logic that underlies ecological rationality, which originate from the misapplication of concepts in psychophysics (e.g., signal detection, just-noticeable-differences). We then work through the most popular experimental task in the ecological rationality literature-the city size task-to illustrate how psychophysical assumptions have informally been linked to ecological rationality. After highlighting these problems, we contrast ecological rationality with a proposed alternative, generative rationality. Generative rationality builds on biology-in contrast to ecological rationality's focus on statistics. We argue that in uncertain environments cues are rarely given or available for statistical processing. Therefore we focus on the psychogenesis of awareness rather than psychophysics of cues. For any agent or organism, environments "teem" with indefinite cues, meanings and potential objects, the salience or relevance of which is scarcely obvious based on their statistical or physical properties. We focus on organism-specificity and the organism-directed probing that shapes awareness and perception. Cues in teeming environments are noticed when they serve as cues-for-something, requiring what might be called a "cue-to-clue" transformation. In this sense, awareness toward a cue or cues is actively "grown." We thus argue that perception might more productively be seen as the presentation of cues and objects rather than their representation. This generative approach not only applies to relatively mundane organism (including human) interactions with their environments-as well as organism-object relationships and their embodied nature-but also has significant implications for understanding the emergence of novelty in economic settings. We conclude with a discussion of how our arguments link with-but modify-Herbert Simon's popular "scissors" metaphor, as it applies to bounded rationality and its implications for decision making in uncertain, teeming environments.

Reference Dependence in Bayesian Reasoning: Value Selection Bias, Congruence Effects, and Response Prompt Sensitivity

This work examines the influence of reference dependence, including value selection bias and congruence effects, on diagnostic reasoning. Across two studies, we explored how dependence on the initial problem structure influences the ability to solve simplified precursors to the more traditional Bayesian reasoning problems. Analyses evaluated accuracy and types of response errors as a function of congruence between the problem presentation and question of interest, amount of information, need for computation, and individual differences in numerical abilities. Across all problem variations, there was consistent and strong evidence of a value selection bias in that incorrect responses almost always conformed to values that were provided in the problem rather than other errors including those related to computation. The most consistent and unexpected error across all conditions in the first experiment was that people were often more likely to utilize the superordinate value (N) as part of their solution rather than the anticipated reference class values. This resulted in a weakened effect of congruence, with relatively low accuracy even in congruent conditions, and a dominant response error of the superordinate value. Experiment 2 confirmed that the introduction of a new sample drew attention away from the provided reference class, increasing reliance on the overall sample size. This superordinate preference error, along with the benefit of repeating the PPV reference class within the question, demonstrated the importance of reference dependence based on the salience of information within the response prompt. Throughout, higher numerical skills were generally associated with higher accuracy, whether calculations were required or not.

Which cognitive individual differences predict good Bayesian reasoning? Concurrent comparisons of underlying abilities

We know a lot about how to present Bayesian reasoning tasks in order to aid performance, but less about underlying individual differences that can account for interindividual variability on the same tasks. Such information would be useful for both theoretical and practical reasons. Two theoretical positions, ecological rationality and nested set views, generate multiple hypotheses about which individual difference traits should be most relevant as underlying Bayesian reasoning performance. However, because many of these traits are somewhat overlapping, testing variables in isolation can yield misleading results. The present research assesses Bayesian reasoning abilities in conjunction with multiple individual different measures. Across three experiments, Bayesian reasoning was best predicted by measures of numerical literacy and visuospatial ability, as opposed to several different measures of cognitive thinking dispositions/styles, ability to conceptually model set-theoretic relationships, or cognitive processing ability (working memory span). These results support an ecological rationality view of Bayesian reasoning, rather than nested sets views. There also was some predictive ability for the Cognitive Reflection Task, which was only partially due to the numeracy aspects of that instrument, and further work is needed to clarify if this is a distinct factor. We are now beginning to understand not only how to build Bayesian reasoning tasks, but also how to build good Bayesian reasoners.

Isotopic source analysis of nitrogen-containing aerosol: A study of PM2.5 in Guiyang (SW, China)

The source of fine particulate matter (PM_2.5) has been a longstanding subject of debate, the nitrogen-15 isotope (δ¹⁵N) has been used to identify the major sources of atmospheric nitrogen. In this study, PM_2.5 samples (n = 361) were collected from September 2017 to August 2018 in the urban area of Guiyang (SW, China), to investigate the chemical composition and potential sources of PM_2.5. The results showed an average PM_2.5 of 33.0 μg m^-3 ± 20.0 μg m^-3. The concentration of PM_2.5 was higher in Winter, lower in Summer. The major water resolved inorganic ions (WSIIs) were Ca²⁺, NH₄⁺, Na⁺, SO₄^2-, NO₃^-, Cl^-. Nitrogen-containing aerosols (i.e., NO₃^- and NH₄⁺) suddenly strengthened during the winter, when NO₃^- became the dominant contributor. Over the sampling period, the molar ratio of NH₄⁺/(NO₃^- + 2 × SO₄^2-) ranged from 0.1 to 0.9, thus indicating the full fixation of NH₄⁺ by existing NO₃^- and SO₄^2- in PM_2.5. The annual value of NOR was 0.1 while rised to 0.5 in Winter. The variations of NOR (Nitrogen oxidation ratio) (0.1-0.5) values suggest that the secondary formation of NO₃^- occurred every season and was most influential during the winter. The total particulate nitrogen (TN) δ¹⁵N value of PM_2.5 ranged from -5.9‰ to 25.3‰ over the year with annual mean of +11.8‰ ± 4.7‰, whereas it was between -5.9‰ and 14.3‰ during the winter with mean of 7.0‰ ± 3.8‰. A Bayesian isotope mixing model (Stable Isotope Analysis in R; SIAR) was applied to analyze the nitrogen sources. The modeling results showed that 29%, 21%, and 40% of TN in PM_2.5 during the winter in Guiyang was due to nitrogen-emissions from coal combustion, vehicle exhausts, and biomass burning, respectively. Our results demonstrate that biomass burning was the main contributor to PM during the winter, 80% of the air mass comes from rural areas of Guizhou border, this transport process can increase the risk of particulate pollution in Guiyang.

Visualizing Uncertainty for Non-Expert End Users: The Challenge of the Deterministic Construal Error

There is a growing body of evidence that numerical uncertainty expressions can be used by non-experts to improve decision quality. Moreover, there is some evidence that similar advantages extend to graphic expressions of uncertainty. However, visualizing uncertainty introduces challenges as well. Here, we discuss key misunderstandings that may arise from uncertainty visualizations, in particular the evidence that users sometimes fail to realize that the graphic depicts uncertainty. Instead they have a tendency to interpret the image as representing some deterministic quantity. We refer to this as the deterministic construal error. Although there is now growing evidence for the deterministic construal error, few studies are designed to detect it directly because they inform participants upfront that the visualization expresses uncertainty. In a natural setting such cues would be absent, perhaps making the deterministic assumption more likely. Here we discuss the psychological roots of this key but underappreciated misunderstanding as well as possible solutions. This is a critical question because it is now clear that members of the public understand that predictions involve uncertainty and have greater trust when uncertainty is included. Moreover, they can understand and use uncertainty predictions to tailor decisions to their own risk tolerance, as long as they are carefully expressed, taking into account the cognitive processes involved.

Important contributions of non-fossil fuel nitrogen oxides emissions

Since the industrial revolution, it has been assumed that fossil-fuel combustions dominate increasing nitrogen oxide (NO_x) emissions. However, it remains uncertain to the actual contribution of the non-fossil fuel NO_x to total NO_x emissions. Natural N isotopes of NO₃⁻ in precipitation (δ¹⁵N_w-NO3−) have been widely employed for tracing atmospheric NO_x sources. Here, we compiled global δ¹⁵N_w-NO3− observations to evaluate the relative importance of fossil and non-fossil fuel NO_x emissions. We found that regional differences in human activities directly influenced spatial-temporal patterns of δ¹⁵N_w-NO3− variations. Further, isotope mass-balance and bottom-up calculations suggest that the non-fossil fuel NO_x accounts for 55 ± 7% of total NO_x emissions, reaching up to 21.6 ± 16.6Mt yr⁻¹ in East Asia, 7.4 ± 5.5Mt yr⁻¹ in Europe, and 21.8 ± 18.5Mt yr⁻¹ in North America, respectively. These results reveal the importance of non-fossil fuel NO_x emissions and provide direct evidence for making strategies on mitigating atmospheric NO_x pollution.

This study investigates in the importance of non-fossil fuel NO_x emissions in the surface-earth-nitrogen cycle. The study shows how changes of regional human activities directly influence δ¹⁵N signatures of deposited NO_x to terrestrial environments and that emissions have largely been underestimated.

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.

Causal explanation improves judgment under uncertainty, but rarely in a Bayesian way

Three studies reexamined the claim that clarifying the causal origin of key statistics can increase normative performance on Bayesian problems involving judgment under uncertainty. Experiments 1 and 2 found that causal explanation did not increase the rate of normative solutions. However, certain types of causal explanation did lead to a reduction in the magnitude of errors in probability estimation. This effect was most pronounced when problem statistics were expressed in percentage formats. Experiment 3 used process-tracing methods to examine the impact of causal explanation of false positives on solution strategies. Changes in probability estimation following causal explanation were the result of a mixture of individual reasoning strategies, including non-Bayesian mechanisms, such as increased attention to explained statistics and approximations of subcomponents of Bayes' rule. The results show that although causal explanation of statistics can affect the way that a problem is mentally represented, this does not necessarily lead to an increased rate of normative responding.

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).

From reading numbers to seeing ratios: a benefit of icons for risk comprehension

Promoting a better understanding of statistical data is becoming increasingly important for improving risk comprehension and decision-making. In this regard, previous studies on Bayesian problem solving have shown that iconic representations help infer frequencies in sets and subsets. Nevertheless, the mechanisms by which icons enhance performance remain unclear. Here, we tested the hypothesis that the benefit offered by icon arrays lies in a better alignment between presented and requested relationships, which should facilitate the comprehension of the requested ratio beyond the represented quantities. To this end, we analyzed individual risk estimates based on data presented either in standard verbal presentations (percentages and natural frequency formats) or as icon arrays. Compared to the other formats, icons led to estimates that were more accurate, and importantly, promoted the use of equivalent expressions for the requested probability. Furthermore, whereas the accuracy of the estimates based on verbal formats depended on their alignment with the text, all the estimates based on icons were equally accurate. Therefore, these results support the proposal that icons enhance the comprehension of the ratio and its mapping onto the requested probability and point to relational misalignment as potential interference for text-based Bayesian reasoning. The present findings also argue against an intrinsic difficulty with understanding single-event probabilities.

Nitrate is an important nitrogen source for Arctic tundra plants

Plant nitrogen (N) use is a key component of the N cycle in terrestrial ecosystems. The supply of N to plants affects community species composition and ecosystem processes such as photosynthesis and carbon (C) accumulation. However, the availabilities and relative importance of different N forms to plants are not well understood. While nitrate (NO3-) is a major N form used by plants worldwide, it is discounted as a N source for Arctic tundra plants because of extremely low NO3- concentrations in Arctic tundra soils, undetectable soil nitrification, and plant-tissue NO3- that is typically below detection limits. Here we reexamine NO3- use by tundra plants using a sensitive denitrifier method to analyze plant-tissue NO3- Soil-derived NO3- was detected in tundra plant tissues, and tundra plants took up soil NO3- at comparable rates to plants from relatively NO3--rich ecosystems in other biomes. Nitrate assimilation determined by 15N enrichments of leaf NO3- relative to soil NO3- accounted for 4 to 52% (as estimated by a Bayesian isotope-mixing model) of species-specific total leaf N of Alaskan tundra plants. Our finding that in situ soil NO3- availability for tundra plants is high has important implications for Arctic ecosystems, not only in determining species compositions, but also in determining the loss of N from soils via leaching and denitrification. Plant N uptake and soil N losses can strongly influence C uptake and accumulation in tundra soils. Accordingly, this evidence of NO3- availability in tundra soils is crucial for predicting C storage in tundra.

Natural Frequencies Do Foster Public Understanding of Medical Tests: Comment on Pighin, Gonzalez, Savadori, and Girotto (2016)

Patients and doctors often need to make decisions based on the results of medical tests. When these results are presented in the form of conditional probabilities, even doctors find it difficult to interpret them correctly. There is over 20 y of research supporting the finding that people are better able to calculate the correct positive predictive value of a test when given information in natural frequencies, as opposed to conditional probabilities. Natural frequencies are one of a few psychological tools that have made it into evidence-based medicine. Recently, Pighin and others (Med Decis Making 2016;36:686-91) argued that natural frequencies could hinder informed decision making, a critique based on a single task and a crude scoring criterion we refer to as the 50%-Split. Our commentary addresses these criticisms based on three analyses. First, we show how the 50%-Split scoring used by Pighin and others misclassifies known errors, such as solely attending to the hit rate (true-positive rate) of the test, as strategies that support understanding. Second, we reanalyze data from 21 additional problems completed by various participant groups to show that their scoring criterion does not support their results in 19 out of 21 cases. Third, we apply the mean deviation scoring method and show that, when given information in natural frequency formats, participants provide estimates that are closer to the correct Bayesian solution than for conditional probability formats. In each analysis, natural frequencies lead to more correct judgements and therefore promote informed decision making relative to conditional probabilities. We welcome further discussions of performance metrics that can provide insight into how the public and therefore patients understand the implications of medical test results.

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.

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.

Comprehension and computation in Bayesian problem solving

Humans have long been characterized as poor probabilistic reasoners when presented with explicit numerical information. Bayesian word problems provide a well-known example of this, where even highly educated and cognitively skilled individuals fail to adhere to mathematical norms. It is widely agreed that natural frequencies can facilitate Bayesian inferences relative to normalized formats (e.g., probabilities, percentages), both by clarifying logical set-subset relations and by simplifying numerical calculations. Nevertheless, between-study performance on “transparent” Bayesian problems varies widely, and generally remains rather unimpressive. We suggest there has been an over-focus on this representational facilitator (i.e., transparent problem structures) at the expense of the specific logical and numerical processing requirements and the corresponding individual abilities and skills necessary for providing Bayesian-like output given specific verbal and numerical input. We further suggest that understanding this task-individual pair could benefit from considerations from the literature on mathematical cognition, which emphasizes text comprehension and problem solving, along with contributions of online executive working memory, metacognitive regulation, and relevant stored knowledge and skills. We conclude by offering avenues for future research aimed at identifying the stages in problem solving at which correct vs. incorrect reasoners depart, and how individual differences might influence this time point.

What makes us think? A three-stage dual-process model of analytic engagement

The distinction between intuitive and analytic thinking is common in psychology. However, while often being quite clear on the characteristics of the two processes ('Type 1' processes are fast, autonomous, intuitive, etc. and 'Type 2' processes are slow, deliberative, analytic, etc.), dual-process theorists have been heavily criticized for being unclear on the factors that determine when an individual will think analytically or rely on their intuition. We address this issue by introducing a three-stage model that elucidates the bottom-up factors that cause individuals to engage Type 2 processing. According to the model, multiple Type 1 processes may be cued by a stimulus (Stage 1), leading to the potential for conflict detection (Stage 2). If successful, conflict detection leads to Type 2 processing (Stage 3), which may take the form of rationalization (i.e., the Type 1 output is verified post hoc) or decoupling (i.e., the Type 1 output is falsified). We tested key aspects of the model using a novel base-rate task where stereotypes and base-rate probabilities cued the same (non-conflict problems) or different (conflict problems) responses about group membership. Our results support two key predictions derived from the model: (1) conflict detection and decoupling are dissociable sources of Type 2 processing and (2) conflict detection sometimes fails. We argue that considering the potential stages of reasoning allows us to distinguish early (conflict detection) and late (decoupling) sources of analytic thought. Errors may occur at both stages and, as a consequence, bias arises from both conflict monitoring and decoupling failures.

Good fences make for good neighbors but bad science: a review of what improves Bayesian reasoning and why

Bayesian reasoning, defined here as the updating of a posterior probability following new information, has historically been problematic for humans. Classic psychology experiments have tested human Bayesian reasoning through the use of word problems and have evaluated each participant’s performance against the normatively correct answer provided by Bayes’ theorem. The standard finding is of generally poor performance. Over the past two decades, though, progress has been made on how to improve Bayesian reasoning. Most notably, research has demonstrated that the use of frequencies in a natural sampling framework—as opposed to single-event probabilities—can improve participants’ Bayesian estimates. Furthermore, pictorial aids and certain individual difference factors also can play significant roles in Bayesian reasoning success. The mechanics of how to build tasks which show these improvements is not under much debate. The explanations for why naturally sampled frequencies and pictures help Bayesian reasoning remain hotly contested, however, with many researchers falling into ingrained “camps” organized around two dominant theoretical perspectives. The present paper evaluates the merits of these theoretical perspectives, including the weight of empirical evidence, theoretical coherence, and predictive power. By these criteria, the ecological rationality approach is clearly better than the heuristics and biases view. Progress in the study of Bayesian reasoning will depend on continued research that honestly, vigorously, and consistently engages across these different theoretical accounts rather than staying “siloed” within one particular perspective. The process of science requires an understanding of competing points of view, with the ultimate goal being integration.

The effect of different graphical and numerical likelihood formats on perception of likelihood and choice

BACKGROUND

Quantitative risk information plays an important role in decision making about health. This study focuses on commonly used numerical and graphical formats and examines their effect on perception of different likelihoods and choice preferences.

METHODS

An experimental study was conducted with 192 participants, who evaluated 2 sets of 4 lotteries. Numerical formats to describe likelihood varied systematically between participants (X%, X-in-100, or 1-in-X). The effect of graphic formats (bar charts, icon charts) was assessed as a within-subjects factor. Dependent measures included perceived likelihood, choice preferences about participating in the lottery, and processing times.

RESULTS

Numerical likelihoods presented as 1-in-X were processed fastest and were perceived as conveying larger likelihoods than the X-in-100 and percentages formats (mean response times in seconds: 5.65 v. 7.31 and 6.50; mean rating on a 1-9 scale: 4.38 v. 3.30 and 3.31, respectively). The 1-in-X format also evoked a stronger willingness to participate in a lottery than the 2 other numerical formats. The effect of adding graphs on perceived likelihood was moderated by numerical aptitude. Graphs reduced ratings of perceived likelihood of participants with lower numeracy, while there was no overall effect for participants with higher numeracy.

CONCLUSION

Perception of likelihood differs significantly depending on the numerical format used. The 1-in-X format yields higher perceived likelihoods and it appears to be the easiest format to interpret. Graphs primarily affect perception of likelihood of people with lower numerical aptitude. These effects should be taken into account when discussing medical risks with patients.

The effects of mental steps and compatibility on Bayesian reasoning

Four laboratory studies were conducted to test the hypothesis that correct Bayesian reasoning can be predicted by two factors of task complexity — the number of mental steps required to reach the normative solution, and the compatibility between the framing of data presented and the framing of the question posed. The findings show that participants performed better on frequency format questions only when one mental step was required to solve the task and when the data were in a compatible frequency format. By contrast, participants performed more poorly on more complicated tasks which required more mental steps (in a compatible frequency or probability format) or when the data and question formats were incompatible (Studies 1 and 2). Incompatibility between data and question formats was also associated with higher reaction times (Study 2b). Furthermore, on problems that incorporated incompatibility between the data sample size and the target (question) sample size, participants performed better on the probability question than the frequency question, regardless of data format (Study 3). The latter findings highlight the ecological advantage of translating data into probability terms, which are normalized in a range between 0 and 1, and thus can be transferred from one situation to another.

Quantifying the relative risk of sex offenders: risk ratios for static-99R

Given the widespread use of empirical actuarial risk tools in corrections and forensic mental health, it is important that evaluators and decision makers understand how scores relate to recidivism risk. In the current study, we found strong evidence for a relative risk interpretation of Static-99R scores using 8 samples from Canada, United Kingdom, and Western Europe (N = 4,037 sex offenders). Each increase in Static-99R score was associated with a stable and consistent increase in relative risk (as measured by an odds ratio or hazard ratio of approximately 1.4). Hazard ratios from Cox regression were used to calculate risk ratios that can be reported for Static-99R. We recommend that evaluators consider risk ratios as a useful, nonarbitrary metric for quantifying and communicating risk information. To avoid misinterpretation, however, risk ratios should be presented with recidivism base rates.

Sample representativeness affects whether judgments are influenced by base rate or sample size

We investigated how people use base rates and sample size information when combining data to make overall probability judgments. Participants considered two samples from an animal population in order to estimate the probability of that animal being aggressive. Participants' judgments were influenced by subpopulation base rates when they were provided and linked to specific samples. When samples were not identified as coming from different subpopulations, judgments typically reflected sample size information. We conclude that 1) People can use base rates when combining samples to make an inference; 2) People can correctly use sampling information to determine when to use base rates, and 3) People are able to consider base rate and sample size information at the same time. Additionally, we found that individuals' numeracy correlates with the extent to which base rate and sample size information is used.

When and for whom do frequencies facilitate performance? On the role of numerical literacy

The thesis that the mind is better prepared to process frequencies-as compared to other numerical formats-continues to be debated. A recent aspect of this issue is the role of numeracy (numerical literacy; one's ability to understand and work with numerical information) and specifically the argument that individual differences in numeracy interact with numerical formats. This interaction, either that frequencies improve performance only for those of low numeracy or that frequencies work only for those of high numeracy, would suggest that better performance using frequencies could be due to (nonevolutionary) numeracy effects. The three present studies revisited prior work with cumulative probability, Bayesian reasoning, and scenario risk assessments to study the effects of numeracy on frequency facilitation. Results from these experiments consistently failed to replicate previous findings of interactions; however, a more consistent finding emerged of a straightforward frequency effect. The lack of interactions and observations of frequency main effects lend support to the evolutionary explanation of the frequency effect. In addition, some possible statistical processes are proposed to explain the observation of interactions in past studies.

Towards a descriptivist psychology of reasoning and decision making

Our target article identifiednormativismas the view that rationality should be evaluated against unconditional normative standards. We believe this to be entrenched in the psychological study of reasoning and decision making and argued that it is damaging to this empirical area of study, calling instead for a descriptivist psychology of reasoning and decision making. The views of 29 commentators (from philosophy and cognitive science as well as psychology) were mixed, including some staunch defences of normativism, but also a number that were broadly supportive of our position, although critical of various details. In particular, many defended a position that we call “soft normativism,” which sees a role for normative evaluation within boundaries alongside more descriptive research goals. In this response, we clarify our use of the term “instrumental rationality” and add discussion of “epistemic rationality,” defining both as descriptive and non-normative concepts. We consider the debate with reference to dual-process theory, the “new paradigm” psychology of reasoning, and empirical research strategy in these fields. We also discuss cognitive variation by age, intelligence, and culture, and the issue of relative versus absolute definitions of norms. In conclusion, we hope at least to have raised consciousness about the important boundaries between norm and description in the psychology of thinking.

Norms and high-level cognition: Consequences, trends, and antidotes

We are neither as pessimistic nor as optimistic as Elqayam & Evans (E&E). The consequences of normativism have not been uniformly disastrous, even among the examples they consider. However, normativism won't be going away any time soon and in the literature on causal Bayes nets new debates about normativism are emerging. Finally, we suggest that to concentrate on expert reasoners as an antidote to normativism may limit the contribution of research on thinking to basic psychological science.

Subtracting “ought” from “is”: Descriptivism versus normativism in the study of human thinking

We propose a critique ofnormativism, defined as the idea that human thinking reflects a normative system against which it should be measured and judged. We analyze the methodological problems associated with normativism, proposing that it invites the controversial “is-ought” inference, much contested in the philosophical literature. This problem is triggered when there are competing normative accounts (the arbitration problem), as empirical evidence can help arbitrate between descriptive theories, but not between normative systems. Drawing on linguistics as a model, we propose that a clear distinction between normative systems and competence theories is essential, arguing that equating them invites an “is-ought” inference: to wit, supporting normative “ought” theories with empirical “is” evidence. We analyze in detail two research programmes with normativist features – Oaksford and Chater's rational analysis and Stanovich and West's individual differences approach – demonstrating how, in each case, equating norm and competence leads to an is-ought inference. Normativism triggers a host of research biases in the psychology of reasoning and decision making: focusing on untrained participants and novel problems, analyzing psychological processes in terms of their normative correlates, and neglecting philosophically significant paradigms when they do not supply clear standards for normative judgement. For example, in a dual-process framework, normativism can lead to a fallacious “ought-is” inference, in which normative responses are taken as diagnostic of analytic reasoning. We propose that little can be gained from normativism that cannot be achieved by descriptivist computational-level analysis, illustrating our position with Hypothetical Thinking Theory and the theory of the suppositional conditional. We conclude that descriptivism is a viable option, and that theories of higher mental processing would be better off freed from normative considerations.

Discussing side effects of over-the-counter medicines: impact of adding percentage data

OBJECTIVES

Discussing side effects with patients continues to be a difficult area of practice. Questions arise as to how many should be mentioned and which ones. The way such information is presented can affect drug-taking decisions. This study examined how over-the-counter (OTC) medicine users are influenced by numerical risk estimates of side effects.

METHODS

As part of a larger study on patient decision-making, 30 participants aged over 50 years were asked to consider three OTC headache medicines. They responded to one of two headache scenarios, one with symptoms described as mild but common and the other severe but rare. Participants made their selection based on drug efficacy and side effects, at first not linked to occurrence rates and then with this information provided.

KEY FINDINGS

Average age was 66.6 years and the majority were female. Most were currently using some form of drug therapy. Drug choices differed in relation to mild versus severe headache scenarios. A stronger preference for drug X (50% effective and two side effects) was evident when the headaches were mild, shifting to a more effective agent (but with more side effects) when more severe. Addition of occurrence rates to the side effects had the greatest effect within the severe headache scenario, where more participants opted for the most effective agent (drug Z at 100% effective but six side effects) upon seeing the numbers. Overall, however, most kept the same drug in spite of the numerical information.

CONCLUSIONS

Inclusion of numerical data for side effects did not negatively influence potential OTC medicine users. For most, effectiveness and side effects were the concern before receiving the percentages, while effectiveness became more important when the frequency data seemed to instil a sense of reassurance.

How numeracy influences risk comprehension and medical decision making

We review the growing literature on health numeracy, the ability to understand and use numerical information, and its relation to cognition, health behaviors, and medical outcomes. Despite the surfeit of health information from commercial and noncommercial sources, national and international surveys show that many people lack basic numerical skills that are essential to maintain their health and make informed medical decisions. Low numeracy distorts perceptions of risks and benefits of screening, reduces medication compliance, impedes access to treatments, impairs risk communication (limiting prevention efforts among the most vulnerable), and, based on the scant research conducted on outcomes, appears to adversely affect medical outcomes. Low numeracy is also associated with greater susceptibility to extraneous factors (i.e., factors that do not change the objective numerical information). That is, low numeracy increases susceptibility to effects of mood or how information is presented (e.g., as frequencies vs. percentages) and to biases in judgment and decision making (e.g., framing and ratio bias effects). Much of this research is not grounded in empirically supported theories of numeracy or mathematical cognition, which are crucial for designing evidence-based policies and interventions that are effective in reducing risk and improving medical decision making. To address this gap, we outline four theoretical approaches (psychophysical, computational, standard dual-process, and fuzzy trace theory), review their implications for numeracy, and point to avenues for future research.

How different types of participant payments alter task performance

Researchers typically use incentives (such as money or course credit) in order to obtain participants who engage in the specific behaviors of interest to the researcher. There is, however, little understanding or agreement on the effects of different types and levels of incentives used. Some results in the domain of statistical reasoning suggest that performance differences — previously deemed theoretically important — may actually be due to differences in incentive types across studies. 704 participants completed one of five variants of a statistical reasoning task, for which they received either course credit, flat fee payment, or performance-based payment incentives. Successful task completion was more frequent with performance-based incentives than with either of the other incentive types. Performance on moderately difficult tasks (compared to very easy and very hard tasks) was most sensitive to incentives. These results can help resolve existing debates about inconsistent findings, guide more accurate comparisons across studies, and be applied beyond research settings.

Dual-processing accounts of reasoning, judgment, and social cognition

This article reviews a diverse set of proposals for dual processing in higher cognition within largely disconnected literatures in cognitive and social psychology. All these theories have in common the distinction between cognitive processes that are fast, automatic, and unconscious and those that are slow, deliberative, and conscious. A number of authors have recently suggested that there may be two architecturally (and evolutionarily) distinct cognitive systems underlying these dual-process accounts. However, it emerges that (a) there are multiple kinds of implicit processes described by different theorists and (b) not all of the proposed attributes of the two kinds of processing can be sensibly mapped on to two systems as currently conceived. It is suggested that while some dual-process theories are concerned with parallel competing processes involving explicit and implicit knowledge systems, others are concerned with the influence of preconscious processes that contextualize and shape deliberative reasoning and decision-making.

"At least one" problem with "some" formal reasoning paradigms

In formal reasoning, the quantifier "some" means "at least one and possibly all." In contrast, reasoners often pragmatically interpret "some" to mean "some, but not all" on both immediate-inference and Euler circle tasks. It is still unclear whether pragmatic interpretations can explain the high rates of errors normally observed on syllogistic reasoning tasks. To address this issue, we presented participants (reasoners) in the present experiments either standard quantifiers or clarified quantifiers designed to precisely articulate the quantifiers' logical interpretations. In Experiment 1, reasoners made significantly more logical responses and significantly fewer pragmatic responses on an immediate-inference task when presented with logically clarified as opposed to standard quantifiers. In Experiment 2, this finding was extended to a variant of the immediate-inference task in which reasoners were asked to deduce what followed from premises they were to assume to be false. In Experiment 3, we used a syllogistic reasoning task and observed that logically clarified premises reduced pragmatic and increased logical responses relative to standard ones, providing strong evidence that pragmatic responses can explain some aspects of the errors made in the syllogistic reasoning task. These findings suggest that standard quantifiers should be replaced with logically clarified quantifiers in teaching and in future research.

Frequency formats, probability formats, or problem structure? A test of the nested-sets hypothesis in an extensional reasoning task

Five experiments addressed a controversy in the probability judgment literature that centers on the efficacy of framing probabilities as frequencies. The natural frequency view predicts that frequency formats attenuate errors, while the nested-sets view predicts that highlighting the set-subset structure of the problem reduces error, regardless of problem format. This study tested these predictions using a conjunction task. Previous studies reporting that frequency formats reduced conjunction errors confounded reference class with problem format. After controlling this confound, the present study’s findings show that conjunction errors can be reduced using either a probability or a frequency format, that frequency effects depend upon the presence of a reference class, and that frequency formats do not promote better statistical reasoning than probability formats.