How to confuse with statistics or: the use and misuse of conditional probabilities

The article shows by various examples how consumers of statistical information may be confused when this information is presented in terms of conditional probabilities. It also shows how this confusion helps others to lie with statistics, and it suggests how either confusion or lies can be avoided by using alternative modes of conveying statistical information. --

Peacemaking among inconsistent rationalities?

Kacelnik, Schuck-Paim and Pompilio (this volume, p. 377) show that rationality axioms from economics are neither necessary nor sufficient to guarantee that animal behavior is biologically adaptive. To illustrate that biological adaptiveness does not imply conformity with the consistency axioms of economics, Kacelnik et al describe animals that sensibly experiment with actions yielding sub-maximum levels of short-term energy intake to monitor their environments for change, leading to apparently intransitive patterns of choice that are nevertheless biologically adaptive. Invalidating the converse claim that economic rationality implies biological adaptiveness is Kacelnik et al’s example of female ruffs that are worse off when they conform to the constant-ratio rule, frequently interpreted as a normative consistency requirement of economic rationality. Together, the two examples demonstrate that axiomatic norms are both unnecessary and insufficient for determining whether a particular behavior is biologically adaptive. Additionally, Kacelnik et al call into question what has been reported in the animal behavior literature as preference reversals, such as risk attitudes among wild rufous hummingbirds or the food-hoarding propensities of grey jays. Kacelnik et al attribute apparent reversals to state-dependent fitness functions modulated by subtle differences in the training phase of animal experiments. For example, animals trained on menus that include a strictly dominated option will tend to have lower accumulated energy reserves and therefore exhibit systematically different patterns of choice––not because they fail to maximize, but because their training has induced systematically different nutritional states. Another possible explanation for preference reversals in animal studies with strictly dominated, or “decoy” options is that menus containing dominated items may convey valid information about future opportunities (Houston and McNamara, 1999). If menus are correlated through time, then menus with inferior options today predict scarcity in the future and imply a distinct optimal course of action, in violation of regularity assumptions that posit invariance with respect to the inclusion of strictly dominated alternatives. In environments with payoff structures that can be modeled as cooperative games, a family’s best response sometimes requires individual family members to behave suboptimally as part of a diversification strategy that reduces the risk of reproductive failure (Hutchinson, 1996). Futhermore, theoretical biologists have documented the fragility of expected fitness maximizing behaviour with respect to the assumption of stable environments. Once the model allows for shocks to the environment’s stochastic structure, simple behavior rules that are suboptimal (in terms of expected fitness) when viewed narrowly from the perspective of unchanging payoffs in a fixed environment may outperform rules based on maximazation within a static small world (Bookstaber and Langsam, 1985).Rationality, rationalities, irrationality, bounded rationality, biology, biological rationality

Does Consistency Predict Accuracy of Beliefs?: Economists Surveyed About PSA

Subjective beliefs and behavior regarding the Prostate Specific Antigen (PSA) test for prostate cancer were surveyed among attendees of the 2006 meeting of the American Economic Association. Logical inconsistency was measured in percentage deviations from a restriction imposed by Bayes’ Rule on pairs of conditional beliefs. Economists with inconsistent beliefs tended to be more accurate than average, and consistent Bayesians were substantially less accurate. Within a loss function framework, we look for and cannot find evidence that inconsistent beliefs cause economic losses. Subjective beliefs about cancer risks do not predict PSA testing decisions, but social influences do.logical consistency, predictive accuracy, elicitation, non-Bayesian, ecological rationality

Does consistency predict accuracy of beliefs?: Economists surveyed about PSA

Subjective beliefs and behavior regarding the Prostate Specific Antigen (PSA) test for prostate cancer were surveyed among attendees of the 2006 meeting of the American Economic Association. Logical inconsistency was measured in percentage deviations from a restriction imposed by Bayes’ Rule on pairs of conditional beliefs. Economists with inconsistent beliefs tended to be more accurate than average, and consistent Bayesians were substantially less accurate. Within a loss function framework, we look for and cannot find evidence that inconsistent beliefs cause economic losses. Subjective beliefs about cancer risks do not predict PSA testing decisions, but social influences do.logical consistency, predictive accuracy, elicitation, non-Bayesian, ecological rationality

Tools=Theories=Data? On Some Circular Dynamics in Cognitive Science

Algunos instrumentos han jugado un papel tan importante en la investigación psicológica que se han convertido en metáforas para el contenido mismo de la investigación, o han conducido a la generación de nuevos conceptos y teorías en psicología. Este capítulo pro-porciona dos casos prácticos relacionados que apoyan esta afirmación. Desde la llamada ―revolución cognitiva‖ de la década de 1960, la mente humana ha sido descrita teóricamen-te como un ―estadístico intuitivo‖ (una persona que piensa de forma intuitiva en términos estadísticos) o como un programa de ordenador. Estas teorías han sido fuertemente inspira-das por las herramientas introducidas en la investigación psicológica antes de la difusión de tales teorías en la comunidad psicológica: la estadística inferencial y el ordenador digital. Aquí discutimos los pros y los contras de las metáforas. Las metáforas pueden ser ventajo-sas, pues abren nuevas áreas de investigación, llevan a nuevas preguntas y datos; pero las metáforas pueden también incluir pérdidas, ya que siempre recalcan algunos aspectos y de-jan otros fuera. Además, en lugar de teorías que son evaluadas por datos recogidos median-te herramientas de investigación aparentemente neutrales, hay una tendencia hacia teorías que se ajustan al funcionamiento o a los usos de herramientas específicas de investigación. Tal tendencia se ve en ocasiones reforzada por los datos que sólo pueden ser producidos por las nuevas herramientas. Los científicos deberían ser conscientes de esta dinámica para evi-tar circularidades en su teorización.Some instruments have played such a strong role in psychological research that they have become metaphors for the subject matter of such research itself, or that they have lead to the generation of new psychological concepts and theories. This chapter provides two re-lated case studies for this claim. Since the so-called “cognitive revolution” of the 1960s, the human mind has been theoretically described as an “intuitive statistician” or as a com-puter program. Such theories have been strongly inspired by the tools introduced into psychological research somewhat before the spreading of such theories within the psychologi-cal community: inferential statistics and the digital computer. We discuss both the pros and cons of the metaphors. Metaphors can be advantageous, as they can open up new re-search areas, questions, and data; but metaphors can also include losses, because they al-ways emphasize some aspects and leave others out. Moreover, instead of the theories be-ing evaluated by data gathered by means of seemingly neutral research tools, there is a bi-as towards theories which match with the functioning or uses of specific research tools, a bias which is sometimes reinforced by data which can only be produced by the new tools. Scientists should be aware of such a dynamics in order to avoid circularities in their theo-rizing

The role of representation in Bayesian reasoning: Correcting common misconceptions

The terms nested sets, partitive frequencies, inside-outside view, and dual processes add little but confusion to our original analysis (Gigerenzer & Hoffrage 1995; 1999). The idea of nested set was introduced because of an oversight; it simply rephrases two of our equations. Representation in terms of chances, in contrast, is a novel contribution yet consistent with our computational analysis - it uses exactly the same numbers as natural frequencies. We show that non-Bayesian reasoning in children, laypeople, and physicians follows multiple rules rather than a general-purpose associative process in a vaguely specified "System 1.” It is unclear what the theory in "dual process theory” is: Unless the two processes are defined, this distinction can account post hoc for almost everything. In contrast, an ecological view of cognition helps to explain how insight is elicited from the outside (the external representation of information) and, more generally, how cognitive strategies match with environmental structure

¿Herramientas=teorías=datos? : Sobre cierta dinámica circular en la ciencia cognitiva

Algunos instrumentos han jugado un papel tan importante en la investigación psicológica que se han convertido en metáforas para el contenido mismo de la investigación, o han conducido a la generación de nuevos conceptos y teorías en psicología. Este capítulo pro-porciona dos casos prácticos relacionados que apoyan esta afirmación. Desde la llamada ―revolución cognitiva‖ de la década de 1960, la mente humana ha sido descrita teóricamen-te como un ―estadístico intuitivo‖ (una persona que piensa de forma intuitiva en términos estadísticos) o como un programa de ordenador. Estas teorías han sido fuertemente inspira-das por las herramientas introducidas en la investigación psicológica antes de la difusión de tales teorías en la comunidad psicológica: la estadística inferencial y el ordenador digital. Aquí discutimos los pros y los contras de las metáforas. Las metáforas pueden ser ventajo-sas, pues abren nuevas áreas de investigación, llevan a nuevas preguntas y datos; pero las metáforas pueden también incluir pérdidas, ya que siempre recalcan algunos aspectos y de-jan otros fuera. Además, en lugar de teorías que son evaluadas por datos recogidos median-te herramientas de investigación aparentemente neutrales, hay una tendencia hacia teorías que se ajustan al funcionamiento o a los usos de herramientas específicas de investigación. Tal tendencia se ve en ocasiones reforzada por los datos que sólo pueden ser producidos por las nuevas herramientas. Los científicos deberían ser conscientes de esta dinámica para evi-tar circularidades en su teorización.Some instruments have played such a strong role in psychological research that they have become metaphors for the subject matter of such research itself, or that they have lead to the generation of new psychological concepts and theories. This chapter provides two re-lated case studies for this claim. Since the so-called "cognitive revolution" of the 1960s, the human mind has been theoretically described as an "intuitive statistician" or as a com-puter program. Such theories have been strongly inspired by the tools introduced into psychological research somewhat before the spreading of such theories within the psychologi-cal community: inferential statistics and the digital computer. We discuss both the pros and cons of the metaphors. Metaphors can be advantageous, as they can open up new re-search areas, questions, and data; but metaphors can also include losses, because they al-ways emphasize some aspects and leave others out. Moreover, instead of the theories be-ing evaluated by data gathered by means of seemingly neutral research tools, there is a bi-as towards theories which match with the functioning or uses of specific research tools, a bias which is sometimes reinforced by data which can only be produced by the new tools. Scientists should be aware of such a dynamics in order to avoid circularities in their theo-rizing