Understanding conditional probabilities

In two experiments, subjects were asked to judge whether the probability of A given B was greater than, equal to, or less than the probability of B given A for various events A and B. In addition, in Experiment 2, subjects were asked to estimate the conditional probabilities and also to calculate conditional probabilities from contingency data. For problems in which one conditional probability was objectively larger than the other, performance ranged from about 25–80% correct, depending on the nature of A and B. Changes in the wording of problems also affected performance, although less dramatically. Patterns of responses consistent with the existence of a causal bias in judging probabilities were observed with one of the wordings used but not with the other. Several features of the data suggest that a major source of error was the confusion between conditional and joint probabilities.     

Temporal Estimation of Major News Events: Re-examining the Accessibility Principle

The way in which people estimate when an event occurred and the accuracy of their estimates are of concern both to psychologists interested in the structure of event memory and to other researchers who, for a variety of reasons, rely on the accuracy of people's temporal estimates. Over 2,000 subjects were asked to say when two major news events, Thatcher's resignation and the Hillsborough football disaster, happened. By embedding the questions in a face-to-face survey we hoped to maximize the probability of subjects using a relatively simple estimation heuristic, previously described as the accessibility principle. This predicts that underestimation of elapsed time should be associated with clearer memories. This hypothesis was not corroborated. We discuss which aspects of the accessibility principle can be retained. © 1997 by John Wiley & Sons, Ltd.     

A possible basis for conservatism in signal detection and probabilistic categorization tasks

It is commonly claimed that conservative placement of the criterion in signal detection is due to the form of the utility function of money, to conservatism in the estimation of prior probabilities, or to probability matching tendencies. This article shows how conservatism could be caused by a systematic misconception of the shape of the underlying distributions. An experiment is described in which subjects were asked to make posterior probability judgments after performing numerical analogues of signal detection. The posterior probability judgments were radical, i.e., high posterior probabilities were overestimated and low posterior probabilities were underestimated; if this pattern of radical probability estimation reflects the subjects’ understanding of the underlying distributions, it would account for conservative criterion placement.     

Probability judgment accuracy for general knowledge. Cross-national differences and assessment methods

In this study we compare the probability judgment accuracy of subjects from the United States and Turkey. Three different response modes were employed — numerical probabilities, pie diagrams, and odds. The questions employed in the study were restricted to two-alternative, general-knowledge items. The observed pattern of differences in the components of probability judgment accuracy paralleled those of studies that have compared Western and Asian subjects. In particular, Turkish subjects exhibited better discrimination but worse calibration than their US counterparts. This result persisted across all three response modes. These findings lend support to previous assertions that observed cross-national differences arise from socioeconomic rather than Asian versus Western cultural differences. However, the consistency of the observed differences across response modes refutes a previous assertion that observed cultural differences are merely the result of response bias.     

The effect of order of presentation of verbal probability expressions on numerical estimates in a medical context

Many medical decisions are made on a probability estimate. Models of risk benefit calculations in health behaviour rarely articulate how people perceive and appraise such probability. Cognitive processes may mediate the processing of probability expressions and may be important to understand the meaning or the range of meanings that probability statements portray to patients. Past studies have indicated that verbal expressions of probability are vague and subject to individual interpretation. On the theoretical level when subjects are asked to translate a set of verbal probability expressions, ranging from high to low, into their equivalent numerical expressions subjects usually produce a continuum of numerical equivalents also ranging from high to low. In practice clinicians frequently communicate information about uncertainty to the patients by verbal probability estimates. This study explored the effect of the order of presentation of the verbal expressions on the numerical probabilities produced by a group of medical students (n?=?87) in relation to medical probabilities. The results showed that the order of presentation (descending vs. random) of the verbal probability expressions was found to have a significant effect on three of the seven numerical probabilities produced by the subjects. The order effect is discussed to together with implications for clinical practice.     

Default Probability

A probability may be called “default” if it is neither derived from preestablished probabilities nor based on considerations of frequency or symmetry. Default probabilities presumably arise through reasoning based on causality and similarity. This article advances a model of default probability based on a featural approach to similarity. The accuracy of the model is assessed by comparing its predictions to the probabilities provided by undergraduates asked to reason about mammals.     

不确定性容忍度、损益概率及延迟风险对延迟选择的影响

研究通过两个实验考查了不确定性容忍度及相关变量对延迟选择的影响,其中实验1采用2（不确定性容忍度：高/低）×2（概率水平：高/中）被试间实验设计;研究2将实验任务设定在有延迟风险情景下中等概率水平,采用单因素（不确定性容忍度：高/低）被试间实验设计.实验1结果表明：不确定性容忍度与概率水平存在交互作用：中等概率时,低容忍度个体比高容忍度个体更偏好延迟选择,高概率时,两者的延迟选择无显著差异,都偏好于选择决策.实验2结果表明：在有延迟风险中等概率时,高、低容忍度个体的决策偏好无显著差异,都偏好选择决策.结论：不确定性容忍度对延迟选择存在影响且受概率水平和延迟风险的调节.     

EQUATING INVERSE PROBABILITIES IN IMPLICIT PERSONALITY JUDGMENTS

Abstract— Meehl and Rosen (1955) observed that their colleagues tended to equate the sensitivity of a sign (its probability given a diagnostic category) with its predictive accuracy (the probability of the diagnostic category given the sign). Such equating of inverse probabilities implies ignoring base rates. Later, many researchers demonstrated such neglect in a number of experimental contexts involving hypothetical judgments; others have found some use of base rates—but only when base rates were supplied by the experimenter, in which case even irrelevant base rates influenced judgment. The present study tested for equating conditional probabilities in a context similar to the contexts involved in the initial observations—that is, subjects strove for accuracy rather than attempted to make good judgments about hypothetical problems, and the base rates were generated by the subjects themselves. Neglect of base rates was revealed by within-individual comparisons showing subjects equated inverse probabilities without equating the corresponding unconditional ones.     

The same but different: an empirical investigation of the reducibility principle

According to classical utility theory the valuation of any lottery should depend only on its outcomes and their respective probabilities, and should not be affected by the nature, complexity and structure of the chance mechanism. Previous research has documented systematic violations of this principle. For example, most subjects prefer lotteries in which the payoffs are contingent on the joint occurrence of multiple (high probability) events to simple lotteries, and lotteries in which the earlier stages offer higher probabilities than the later stages. We review the various violations of this principle and suggest a classification into two major types associated with misunderstanding of chance mechanisms and attitude towards the chance mechanism and process. In the present study 40 subjects were presented with 30 pairs of binary gambles. In any given pair the lotteries had identical outcomes and equal ‘reduced’ probabilities (and thus equal expected values). However, the chance mechanisms varied along a variety of factors such as the size of the sample space, the number of stages, temporal ordering, order of probabilities, their transparency and time constraints. Half the subjects saw lotteries involving gains and the other half considered only losses. After choosing one lottery in each pair, the subjects were asked to explain and justify their choices. The findings revealed systematic violations of the reducibility principle: subjects displayed a preference for lotteries with larger sample spaces, and for lotteries that allow quicker resolutions in the earlier stages. A clear distinction between some patterns of preferences in the gains and loss domains was revealed. In gambles involving gains subjects preferred to have the highest probability on the first stage (and the lowest probability on the last stage), but displayed the opposite preferences for losses. A content analysis of the subjects' stated reasons for their choices identified eight major categories. The most frequently invoked were hope, fun, simplicity, stress and time. Copyright © 2001 John Wiley & Sons, Ltd.     

Measuring nominal scale agreement between a judge and a known standard

Two designs for comparing a judge's ratings with a known standard are presented and compared. Design A pertains to the situation where the judge is asked to categorize each ofN subjects into one ofr (known) classes with no knowledge of the actual number in each class. Design B is employed when the judge is given the actual number in each class and is asked to categorize the individuals subject to these constraints. The probability distribution of the total number of correct choices is developed in each case. A power comparison of the two procedures is undertaken.     

Risk, moral value of actions, and mood

In this article the acceptability of risk is related to the moral value of risky actions. Four widely different groups of subjects (clergy, MBA students, prison inmates and pregnant women) judged 20 individual and 20 collective acts with reference to acceptability of risk, moral value, and value and probability of negative and positive consequences. They also rated their current mood. Acceptability of risks was most strongly related to moral value, which thus was found to be an important determiner of acceptability of risk. A depressed mood was associated with a lenient attitude to risks. Societal acts were more negatively evaluated than individual acts. There was a strong positive correlation between values and probabilities of outcomes, both positive and negative.     

"Ratio" and "difference" judgments for length, area, and volume: are there two classes of sensory continua?

Subjects were required to judge ratios and differences of (a) line length for pairs of lines, (b) area for pairs of squares, and (c) volume for pairs of cubes. Nonmetric analyses of these judgments indicated that all subjects were able to make consistent ratio judgments for all three continua. Many of the subjects, when asked to judge subjective differences, however, performed as if they were judging subjective ratios rather than differences. The data for the few subjects who appeared to be judging subjective differences were not consistent across subjects and conditions. Previous studies of ratio and difference judgments of loudness and heaviness, on the other hand, showed the opposite pattern, in that subjects most often behaved as if they were judging sensory differences when asked to judge sensory ratios. We propose that ratio judgments are more natural to perceptual continua along which stimuli are easily "decomposed" into a number of smaller perceptual units.     

Studies in subjective probability III: The unimportance of alternatives

In four experiments, student subjects were asked to estimate probabilities for a list of two to ten exhaustive, non-chance events, covering a variety of situations, both of prediction and diagnosis. Only in the two-alternative case a majority gave estimates which add up to unity (or 100%). As the number of alternatives increased, the total probability increased far beyond 100%, indicating a non-distributional probability concept. For instance, the probability that a person has committed murder was considered to be quite independent of his being one of three or one of five suspects. Even when subjects were told that the list should be extended with two additional alternatives and were allowed to correct their earlier estimates, few thought it necessary to do so, and corrections went equally in both directions.     

Patterns of Preference for Numerical and Verbal Probabilities

We report results of an experiment designed to test a principle formulated by Budescu and Wallsten (1995), that, when communicating uncertainty information, mode choices are sensitive to sources and degrees of vagueness. In addition, we examined subjects’ efficacy in using such uncertainty information as a function of communication mode, source, and vagueness. In phase one of the experiment, subjects in a dyad used precise (numerical) or imprecise (verbal) expressions to communicate to a remote partner precise or vague uncertainty about the likelihoods of events. Spinner outcomes were used to generate precise uncertainty while answers to almanac questions were used to elicit vague uncertainty. In phase two, subjects saw the events paired with their partners’ estimates of similar events, and were asked to gamble on one event from each pair. Communication mode preferences were measured as the relative frequency that subjects chose the numerical mode to either express or receive uncertainty information regarding the events. Efficacy was measured as the relative frequency that subjects choose from the pair the event associated with the objectively more probable uncertainty expression. Underlying uncertainty interacted with direction of communication to affect preferences for modes of expression of the probabilities. Subjects preferred precise (numerical) information, especially for precise events (spinners). For vague events (questions), their preference for precise (numerical) information was stronger when receiving than when communicating information. Similar preferences were reflected in the efficiency of subsequent gamble decisions based on the probability estimates. Specifically, decisions were more efficacious (i.e. consistent with Expected Utility) when degrees of precision in events and estimates matched. © 1997 by John Wiley & Sons, Ltd.     

Heuristics and biases in diagnostic reasoning: II. Congruence,information, and certainty

Six experiments were carried out to examine possible heuristics and biases in the evaluation of yes-or-no questions for the purpose of hypothesis testing. In some experiments, the prior probability of the hypotheses and the conditional probabilities of the answers given each hypothesis were elicited from the subjects; in other experiments, they were provided. We found the following biases (systematic departures from a normative model), and interviews and justifications suggested that each was the result of a corresponding heuristic: Congruence bias. Subjects overvalued questions that have a high probability of a positive result given the most likely hypothesis. This bias was apparently reduced when alternative hypotheses or probabilities of negative results are explicitly stated. Information bias. Subjects evaluated questions as worth asking even when there is no answer that can change the hypothesis that will be accepted as a basis for action. Certainty bias. Subjects overvalued questions that have the potential to establish, or rule out, one or more hypotheses with 100% probability. These heuristics are explained in terms of the idea that people fail to consider certain arguments against the use of questions that seem initially worth asking, specifically, that a question may not distinguish likely hypotheses or that no answer can change the hypothesis accepted as a basis for action.     

Stability and coherence of health experts' upper and lower subjective probabilities about dose-response functions

As part of a method for assessing health risks associated with primary National Ambient Air Quality Standards. T. B. Feagans and W. F. Biller (Research Triangle Park, North Carolina. EPA Office of Air Quality Planning and Standards, May 1981) developed a technique for encoding experts' subjective probabilities regarding dose--response functions. The encoding technique is based on B. O. Koopman's (Bulletin of the American Mathematical Society, 1940, 46, 763-764; Annals of Mathematics, 1940, 41, 269-292) probability theory, which does not require probabilities to be sharp, but rather allows lower and upper probabilities to be associated with an event. Uncertainty about a dose--response function can be expressed either in terms of the response rate expected at a given concentration or, conversely, in terms of the concentration expected to support a given response rate. Feagans and Biller (1981, cited above) derive the relation between the two conditional probabilities, which is easily extended to upper and lower conditional probabilities. These relations were treated as coherence requirements in an experiment utilizing four ozone and four lead experts as subjects, each providing judgments on two separate occasions. Four subjects strongly satisfied the coherence requirements in both conditions. and three more did no in the second session only. The eighth subject also improved in Session 2. Encoded probabilities were highly correlated between the two sessions, but changed from the first to the second in a manner that improved coherence and reflected greater attention to certain parameters of the dose--response function.     

Seeing is believing: Priors,trust, and base rate neglect

Kahneman and Tversky (1973) described an effect they called ‘insensitivity to prior probability of outcomes’, later dubbed base rate neglect, which describes people’s tendency to underweight prior information in favor of new data. As probability theory requires that prior probabilities be taken into account, via Bayes’ theorem, the fact that most people fail to do so has been taken as evidence of human irrationality and, by others, of a mismatch between our cognitive processes and the questions being asked (Cosmides & Tooby, 1996). In contrast to both views, we suggest that simplistic Bayesian updating using base rates is not necessarily rational. To that end, we present experiments in which base rate neglect is often the right strategy, and show that people’s base rate usage varies systematically as a function of the extent to which the data that make up a base rate are perceived as trustworthy.     

Conditioned Assessment of Subjective Probabilities: Identifying the Benefits of Decomposition

Conditioned assessment of subjective probability decomposes an assessment task by using a related event to define a series of conditional probabilities. These component assessments are subsequently aggregated mathematically. Two experiments compared conditioned and unconditioned (direct) assessments of 32 general knowledge questions by graduate business students. Reliability was estimated through repeat administration of the questionnaire and accuracy was measured with a quadratic scoring rule. Conditioned assessments were more reliable and more accurate than direct assessments, confirming previous research on other decomposition approaches. Conditioned assessment appears to be effective because the mathematical combination of conditional probabilities reduces random response errors and reduces the tendency to produce overly extreme probability judgments. Implications for further research and application are considered.     

Children and chess expertise: The role of calibration

Summary Three studies of calibration are reported. Calibration refers to the accuracy with which one can predict one's own performance. In the first study child chess players, non-chess playing parents, and statistics students were asked to predict chances of winning chess games against hypothetical opponents. These subjective probabilities were compared to the actual probabilities, based on the Elo rating system. Better players' predictions were better calibrated. Confidence and ratings are negatively correlated, indicating that with lower ratings, players are overconfident. Skilled child players' predictions were better calibrated than any of the adults'. In the second study subjects were asked to estimate chances of winning in conjunctive situations, e. g., winning all the rounds in a tournament. Again, better child players were more accurate in their predictions and more accurate than adults. In the third study, child players were asked to predict their chances of winning in a non-chess domain after hearing a hypothetical win/loss history. Higher-rated players' predictions were again better calibrated, even though the domain was outside their expertise. The motivational and cognitive implications of calibration are discussed.     

Methods of data collection,perceptions of risks and losses,and motivation to give truthful answers to sensitive survey questions

Two experiments examined survey respondents' concerns about risks and losses when answering sensitive questions in a survey. In Experiment 1 subjects watched one of eight different versions of a videotaped staged interview of a woman being asked about abortion and drunk driving. Experiment 2 was an actual survey interview about women's health issues, including abortion. In Experiment 2, a large proportion of the sample had a recent abortion for which validation information from the clinic was available. Results of both experiments indicate that, when responding to questions about sensitive topics, survey respondents are concerned about risks and losses other than those covered by standard confidentiality assurances and that these other concerns affect truthful reporting. Experiment 1 showed the importance of privacy in surveys of sensitive behaviours. The experiments suggest that under some circumstances interviewer‐administration of sensitive questions may produce data as good as or better than self‐administration. Copyright © 1999 John Wiley & Sons, Ltd.