Probability judgment and subadditivity: The role of working memory capacity and constraining retrieval

In this research, we examined the role that individual differences in working memory (WM) capacity, the strength of alternatives, and time constraints play in probability judgment and subadditivity. With a laboratory-based learning task, Experiment 1 revealed that the degree to which participants’ probability judgments were subadditive was negatively correlated with a measure of WM capacity, even when variance due to short-term memory capacity was removed. In addition, participants were more subadditive when the viable alternatives were all rather weak. Experiment 2 extended the WM-capacity-subadditivity correlation to a population judgment task and revealed that subadditivity increases when the judgment task is performed under time constraints. Results support a model that assumes that people make probability judgments by comparing the focal hypothesis with relevant alternatives retrieved from long-term memory and that people high in WM span include more alternatives in the comparison process. Time constraints are assumed to truncate the alternative generation process, leading to fewer alternatives being recalled from long-term memory.     

Differences between probability and frequency judgments: The role of individual differences in working memory capacity

Most theories of probability judgment assume that judgments are made by comparing the strength of a focal hypothesis relative to the strength of alternative hypotheses. In contrast, research suggests that frequency judgments are assessed using a non-comparative process; the strength of the focal hypothesis is assessed without comparing it to the strength of alternative hypotheses. We tested this distinction between probability and frequency judgments using the alternative outcomes paradigm (Windschitl, Young, & Jenson, 2002). Assuming that judgments of probability (but not judgments of frequency) entail comparing the focal hypothesis with alternative hypotheses, we hypothesized that probability judgments would be sensitive to the distribution of the alternative hypotheses and would be negatively correlated with individual differences in working memory (WM) capacity. In contrast, frequency judgments should be unrelated to the distribution of the alternatives and uncorrelated with WM-capacity. Results supported the hypotheses.     

Aging and conditional probability judgments: a global matching approach

Age differences in bias in conditional probability judgments were investigated based on predictions derived from the Minerva-Decision Making model (M. R. P. Dougherty, C. F. Gettys, & E. E. Ogden, 1999), a global matching model of likelihood judgment. In this study, 248 younger and older adults completed frequency judgment and conditional probability judgment tasks. Age differences in the frequency judgment task are interpreted as an age-related deficit in memory encoding. Older adults' stronger biases in the probability judgment task point to age differences in criterion setting. Age-related biases were eliminated when age groups were equated on memory encoding by means of study time manipulation. The authors conclude that older adults' stronger judgment biases are a function of memory impairment.     

MINERVA‐DM and subadditive frequency judgments

There is considerable evidence that frequency (and probability) judgments are often subadditive. That is, the frequency judgment assigned to an event is often less than the sum of the frequency judgments assigned to the mutually exclusive component events that together form it. Explanations for subadditive judgments have typically relied on relatively high‐level cognitive constructs such as the availability and representativeness heuristics. A lower‐level explanation of subadditivity is presented in this paper through a model of memory and judgment processes, MINERVA‐DM. Under MINERVA‐DM, subadditivity is influenced by the similarity of the representations of the judged component events in memory to one another and by the placement of decision criteria. Results from two experiments support the model predictions. The first examines the effects of component event similarity on subadditivity. The second replicates the first and also provides support for the model's prediction of the effects of payoffs on similarity criteria. Copyright © 2004 John Wiley & Sons, Ltd.     

Probability judgment in three-category classification learning

People give subadditive probability judgments--in violation of probability theory--when asked to assess each in a set of 3 or more mutually exclusive hypotheses, as indicated by their sum exceeding 1. Three potential evidential influences on subadditivity--cue conflict, cue frequency, and cue redundancy--are distinguished and tested in 5 experiments using a classification-learning task. Results indicate that (a) judgments of probability and of frequency are systematically subadditive even when the judgments are based on cues learned within the experimental context, (b) cue conflict has a reliable influence on the degree of subadditivity, and (c) judgments in this context are well described by a linear-discounting model within the framework of support theory.     

A pattern recognition account of decision making

In the domain of pattern recognition, experiments have shown that perceivers integrate multiple sources of information in an optimal manner. In contrast, other research has been interpreted to mean that decision making is nonoptimal. As an example, Tversky and Kahneman (1983) have shown that subjects commit a conjunction fallacy because they judge it more likely that a fictitious person named Linda is a bank teller and a feminist than just a bank teller. This judgment supposedly violates probability theory, because the probability of two events can never be greater than the probability of either event alone. The present research tests the hypothesis that subjects interpret this judgment task as a pattern recognition task. If this hypothesis is correct, subjects’ judgments should be described accurately by the fuzzy logical model of perception (FLMP)—a successful model of pattern recognition. In the first experiment, the Linda task was extended to an expanded factorial design with five vocations and five avocations. The probability ratings were described well by the FLMP and described poorly by a simple probability model. The second experiment included (1) two fictitious people, Linda and Joan, as response alternatives and (2) both ratings and categorization judgments. Although the ratings were accurately described by both the FLMP and an averaging of the sources of information, the categorization judgments were described better by the FLMP. These results reveal important similarities in recognizing patterns and in decision making. Given that the FLMP is an optimal method for combining multiple sources of information, the probability judgments appear to be optimal in the same manner as pattern-recognition judgments.     

Contrasting stochastic and support theory accounts of subadditivity

Numerous studies have found that likelihood judgment typically exhibits subadditivity in which judged probabilities of events are less than the sum of judged probabilities of constituent events. Whereas traditional accounts of subadditivity attribute this phenomenon to deterministic sources, this paper demonstrates both formally and empirically that subadditivity is systematically influenced by the stochastic variability of judged probabilities. First, making rather weak assumptions, we prove that regressive error (or variability) in mapping covert probability judgments to overt responses is sufficient to produce subadditive judgments. Experiments follow in which participants provided repeated probability estimates. The results support our model assumption that stochastic variability is regressive in probability estimation tasks and show the contribution of such variability to subadditivity. The theorems and the experiments focus on within-respondent variability, but most studies use between-respondent designs. Numerical simulations extend the work to contrast within- and between-respondent measures of subadditivity. Methodological implications of all the results are discussed, emphasizing the importance of taking stochastic variability into account when estimating the role of other factors (such as the availability bias) in producing subadditive judgments.     

Using the memory activation capture (MAC) procedure to investigate the temporal dynamics of hypothesis generation

Research investigating top-down capture has demonstrated a coupling of working memory content with attention and eye movements. By capitalizing on this relationship, we have developed a novel methodology, called the memory activation capture (MAC) procedure, for measuring the dynamics of working memory content supporting complex cognitive tasks (e.g., decision making, problem solving). The MAC procedure employs briefly presented visual arrays containing task-relevant information at critical points in a task. By observing which items are preferentially fixated, we gain a measure of working memory content as the task evolves through time. The efficacy of the MAC procedure was demonstrated in a dynamic hypothesis generation task in which some of its advantages over existing methods for measuring changes in the contents of working memory over time are highlighted. In two experiments, the MAC procedure was able to detect the hypothesis that was retrieved and placed into working memory. Moreover, the results from Experiment 2 suggest a two-stage process following hypothesis retrieval, whereby the hypothesis undergoes a brief period of heightened activation before entering a lower activation state in which it is maintained for output. The results of both experiments are of additional general interest, as they represent the first demonstrations of top-down capture driven by participant-established WM content retrieved from long-term memory.     

The relationship between repetition and depth of processing

The relationship between depth of information processing and stimulus repetitions was investigated using a simultaneous category judgment task. Several levels of processing were defined involving (1) physically identical items, (2) physically different but same-name items, (3) different items from the same semantic category, and (4) items from different categories. Stimulus pairs were represented by words and pictures, and each pair was presented one, three, or five times. Response times for categorization judgments increased with the level of processing and decreased with repetitions. Repetitions produced greater facilitation for decisions at deeper levels of processing. In a final incidental recall task, more items were remembered from category-same trials than from same-item trials, but level of processing did not interact with number of presentations. Repetitions produced an equivalent increase in final recall probability for items involved in all decision types, indicating that distributed repetitions can lead to the formation of stronger memory traces at several levels of processing.     

Cognitive processing of multidimensional stimuli in schizophrenia: formal modeling of judgment speed and content

This article presents a formal, mathematical account of relations between response times on simple cognitive tasks and content of complex judgments involving multiple stimulus dimensions for people with schizophrenia. Changes in multidimensional judgments were viewed as the result of interference from increased stages of encoding with respect to the individual dimensions. Information on dimensional properties encoded earlier in a judgment trial was considered to be more susceptible to loss over the rest of the trial, because of a larger number of encoding stages applied to the remaining dimensional properties. Model predictions were tested with samples of paranoid and nonparanoid schizophrenic participants and controls. Unidimensional encoding speed was assessed by reaction times in an explicit similarity ratings task, and multidimensional judgment content was assessed by the relative importance of different stimulus dimensions to participants' ratings in an implicit similarity ratings task. Results support validity of the model.     

Understanding conjunction effects in probability judgments: the role of implicit mental models

Subjective probability judgments often violate a normative principle in that the conjunction of two events is judged to be more likely than the probability of either of the two events occurring separately. Most previous explanations of these conjunction effects have assumed that probability judgments depend on some psychological relation (e.g. representativeness) between the constituents mentioned explicitly in the stimulus information. In contrast, the present approach highlights the fundamental role of implicitly inferred information. Participants are assumed to transform the explicit stimulus information into implicit mental models in their attempt to make sense of the experimental task. Probability judgments should then reflect the degree of activation of such a mental model in memory given a set of propositions, rather than the quantitative fit or likelihood of the propositions themselves. Two studies are reported which provide converging evidence for the proposed mental model approach. In the first study, using graded conjunctions of one to five propositions, probability judgments are shown to vary as a function of the activation of a mental model rather than the likelihood of the component events. In a second study, a priming procedure is employed to activate mental models that either fit an event conjunction or do not, leading to an increase or decrease of conjunction effects in probability judgment. Copyright © 1999 John Wiley & Sons, Ltd.     

The effect of attribute order on judgment in Chinese and English

The authors found that the order of attribute presentation had a stronger effect on judgment in English than in Chinese. In Experiment 1, with a sample of 102 female and 63 male bilingual Singaporeans, the authors found that participants' memory-based judgments showed a stronger primacy effect in English than in Chinese that was mediated by recall from long-term memory. In contrast, participants' online (immediate) judgments showed a primacy effect in both languages that was unmediated by recall from short-term memory. In Experiment 2, with a sample of 67 female and 53 male bilingual Singaporeans, the authors found that participants' online judgments were more influenced by the attribute order of a previously seen competitive advertisement in English than in Chinese. A cross-cultural field study in Mainland China and the United Kingdom provided external validity for the experimental results.     

The role of short-term memory capacity and task experience for overconfidence in judgment under uncertainty

Research with general knowledge items demonstrates extreme overconfidence when people estimate confidence intervals for unknown quantities, but close to zero overconfidence when the same intervals are assessed by probability judgment. In 3 experiments, the authors investigated if the overconfidence specific to confidence intervals derives from limited task experience or from short-term memory limitations. As predicted by the naive sampling model (P. Juslin, A. Winman, & P. Hansson, 2007), overconfidence with probability judgment is rapidly reduced by additional task experience, whereas overconfidence with intuitive confidence intervals is minimally affected even by extensive task experience. In contrast to the minor bias with probability judgment, the extreme overconfidence bias with intuitive confidence intervals is correlated with short-term memory capacity. The proposed interpretation is that increased task experience is not sufficient to cure the overconfidence with confidence intervals because it stems from short-term memory limitations.     

决策与判断中的分解效应

分解效应是人类主观判断中的一种较稳固的行为偏差,并且判断结果会对随后的决策产生重要影响,因此,对该领域研究成果的全面梳理具有重要的理论意义和实际价值。本文主要介绍了支持理论中的分解效应,总结梳理了概率判断与时间判断中分解效应的研究,综述了其他社会判断中的分解效应研究成果,展望了决策与判断中的分解效应的未来研究方向。通过上述内容的阐述,希望能为该领域研究者提供新的思路,推动国内相关领域研究的发展。     

Similarity, global matching, and judgments of frequency

In the test-pair similarity effect, forced-choice recognition is more accurate for similar test pairs, such as leopard-cheetah, than it is for unrelated test pairs, such as leopard-turnip. According to global matching models, this occurs because the retrieved familiarities of similar items are correlated. In the Minerva 2 model, global matching underlies frequency judgments as well as recognition memory. One implication of this model is that judged frequencies of similar items should be correlated. Another implication is that judgments of summed frequency for pairs of words (how many presentations were there of word1 and word2 combined?) should have higher variance when word1 and word2 are similar than when they are unrelated. These predictions were tested and confirmed in two experiments. A review of these and other results suggests that theories of recognition memory should also be applicable to frequency-judgment tasks.     

Effects of a Relative-Frequency Elicitation Question on Likelihood Judgment Accuracy: The Case of External Correspondence

In three experiments, college students judged the likelihood that they chose the correct alternative for each of 40 two-alternative, general-knowledge items. They responded either to a relative-frequency elicitation question (“Out of 100 questions for which you felt this certain of the answer, how many would you answer correctly?”) or to a probability elicitation question (“What is the probability that you chose the correct answer?”). Judgments in response to the relative-frequency elicitation question tended to be lower, exhibit less scatter, and express complete certainty less often than judgments in response to the probability elicitation question. Two types of explanation for these effects are considered. First, the effect of the relative-frequency elicitation question may be to reduce random response error in participants' likelihood judgments. Second, the relative-frequency elicitation question may encourage the use of frequency information and simpler algorithms for making likelihood judgments.     

The influence of improper sets of information on judgment: how irrelevant information can bias judged probability

This article introduces 2 new sources of bias in probability judgment, discrimination failure and inhibition failure, which are conceptualized as arising from an interaction between error prone memory processes and a support theory like comparison process. Both sources of bias stem from the influence of irrelevant information on participants' probability judgments, but they postulate different mechanisms for how irrelevant information affects judgment. The authors used an adaptation of the proactive interference (PI) and release from PI paradigm to test the effect of irrelevant information on judgment. The results of 2 experiments support the discrimination failure account of the effect of PI on probability judgment. In addition, the authors show that 2 commonly used measures of judgment accuracy, absolute and relative accuracy, can be dissociated. The results have broad implications for theories of judgment.     

The demands of an ongoing activity influence the success of event-based prospective memory

Four experiments were conducted to evaluate whether event-based prospective memory would be sensitive to the concurrent demands of the ongoing activity in which intention-related cues were embedded. In Experiments 1 and 2, random alternation between two judgments in the ongoing task reduced prospective memory as compared with having a single task throughout. In Experiment 3, participants' making two binary judgments on every trial resulted in worse prospective memory than did their making single four-alternative judgments. In Experiment 4, participants' making two related judgments resulted in better prospective memory than did their making two unrelated judgments. The results are consistent in spirit with a production rule account of the processing resources that are available when intention-related cues are encountered. Therefore, event-based prospective memory can inversely covary with the cognitive demands of the ongoing activity.     

Children's understanding of counterfactual emotions: Age differences,individual differences,and the effects of counterfactual‐information salience

The present study investigated developmental trends in the effects of the salience of counterfactual alternatives on judgments of others' counterfactual‐thinking‐based emotions. We also examined possible correlates of individual differences in the understanding of these emotions. Thirty‐four adults and 102 children, 5–8 years of age, were presented scenarios in which characters would be expected to experience regret. In one version of each scenario, the regret‐relevant counterfactual alternative was made more salient than was the case with the other version. Adults consistently judged that a character for whom a counterfactual course of events would have resulted in a better outcome would feel worse than a character for whom an alternative course of events would not have resulted in a more positive outcome. The majority of the children's judgments were not affected by the counterfactual alternatives. However, the judgments of the oldest children (the 8‐year‐olds) were significantly more adult‐like in the high‐salience than in the low‐salience condition. Although the three predictors examined in the present study (verbal ability, working memory capacity, second‐order false belief task performance) together accounted for significant variance in performance on the emotions judgment task, no single predictor alone accounted for significant unique variance in performance. The importance of different social cognitive abilities for understanding people's affective responses is discussed.