Some probabilistic models of best, worst, and best-worst choices

Over the past decade or so, a choice design in which a person is asked to select both the best and the worst option in an available set of options has been gaining favor over more traditional designs, such as where the person is asked, for instance, to: select the best option; select the worst option; rank the options; or rate the options. In this paper, we develop theoretical results for three overlapping classes of probabilistic models for best, worst, and best-worst choices, with the models in each class proposing specific ways in which such choices might be related. The models in these three classes are called random ranking and random utility, joint and sequential, and ratio scale. We include some models that belong to more than one class, with the best known being the maximum-difference (maxdiff) model, summarize estimation issues related to the models, and formulate a number of open theoretical problems.     

Sources and influence of perceptual variance: Comment on Dzhafarov's Regular Minimality Principle

Dzhafarov [(2002). Multidimensional Fechnerian scaling: Pairwise comparisons, regular minimality, and nonconstant self-similarity. Journal of Mathematical Psychology, 46, 583-608] claims that Regular Minimality (RM) is a fundamental property of “same-different” discrimination probabilities and supports his claim with some empirical evidence. The key feature of RM is that the mapping, h, between two observation areas based on minimum discrimination probability is invertible. Dzhafarov [(2003a). Thurstonian-type representations for “same-different” discriminations: Deterministic decisions and independent images. Journal of Mathematical Psychology, 47, 184-204; (2003b). Thurstonian-type representations for “same-different” discriminations: Probabilistic decisions and interdependent images. Journal of Mathematical Psychology, 47, 229-243] also demonstrates that well-behaved Thurstonian models of “same-different” judgments are incompatible with RM and Nonconstant Self-Similarity (NCSS). There is extensive empirical support for the latter. Stimulus and neural sources of perceptual noise are discussed and two points are made:

Point 1: Models that require discrimination probabilities for noisy stimuli to possess the property that h is invertible would be too restrictive.

Point 2: In the absence of stimulus noise, violations of RM may be so subtle that their detection would be unlikely.

     

Extending probabilistic dynamic epistemic logic

This paper aims to extend in two directions the probabilistic dynamic epistemic logic provided in Kooi’s paper (J Logic Lang Inform 12(4):381–408, 2003) and to relate these extensions to ones made in van Benthem et al. (Proceedings of LOFT’06. Liverpool, 2006). Kooi’s probabilistic dynamic epistemic logic adds to probabilistic epistemic logic sentences that express consequences of public announcements. The paper (van Benthem et al., Proceedings of LOFT’06. Liverpool, 2006) extends (Kooi, J Logic Lang Inform 12(4):381–408, 2003) to using action models, but in both papers, the probabilities are discrete, and are defined on trivial σ-algebras over finite sample spaces. The first extension offered in this paper is to add a previous-time operator to a probabilistic dynamic epistemic logic similar to Kooi’s in (J Logic Lang Inform 12(4):381–408, 2003). The other is to involve non-trivial σ-algebras and continuous probabilities in probabilistic dynamic epistemic logic.     

A cross-cultural study of hindsight bias and conditional probabilistic reasoning

Hindsight bias is a mistaken belief that one could have predicted a given outcome once the outcome is known. Choi and Nisbett (2000 Choi, I. and Nisbett, R. E. 2000. Cultural psychology of surprise: Holistic theories and recognition of contradiction. Journal of Personality and Social Psychology, 79: 890–905. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) reported that Koreans showed stronger hindsight bias than Americans, and explained the results using the distinction between analytic cognition (Westerners) and holistic cognition (Easterners). The purpose of the present study was to see whether hindsight bias is stronger among Easterners than among Westerners using a probability judgement task, and to test an “explicit–implicit” hypothesis and a “rule-dialectics” hypothesis. We predicted that the implicit process is more active among Easterners to generate hindsight bias, and that Easterners are more dialectical thinkers, whereas Westerners are more rule-based thinkers. French, British, Japanese, and Korean participants were asked to make probabilistic judgements in a Good Samaritan scenario (Experiment 1) and in a scenario including conditional probabilistic judgement (Experiment 2). In both Experiments, we presume that the implicit revision of causal models is made just by being given unexpected outcome information, and that explicit revision is made by being asked to point out possible factors for an unexpected outcome. In the results Easterners showed greater hindsight bias generally and it was greater in the Good Samaritan scenario. We conclude that the reason why hindsight bias was lower among Westerners is primarily that they tried to follow a rule to suppress the bias.     

The effects of endowment on the demand for probabilistic information

Research shows that individuals are ambiguity averse: they choose unambiguous over equivalent ambiguous prospects and price them higher (either as buyers or sellers). Moreover, it is often assumed that ambiguity averse individuals are willing to pay an ambiguity premium for information that reduces ambiguity [Camerer, C. F., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of Risk and Uncertainty, 5(4), 325–370]. However, when people are asked to exchange an ambiguous alternative in their possession for an equivalent unambiguous one, they prefer to retain the former [Roca, M., Hogarth, R. M., & Maule, A. J. (2006). Ambiguity seeking as a result of the status quo bias. Journal of Risk and Uncertainty, 32(3), 175–194]. We present three experiments investigating the economic effects of endowment on attitudes towards ambiguity and the ambiguity premium. The experiments, based on a [Becker, G., DeGroot, M., & Marschak, J. (1964). Measuring utility by a single-response sequential method. Science, 9, 3] procedure, show that the value attributed to ambiguity reducing information is substantially affected by the status quo of the individual.     

The logical syntax of number words: Theory, acquisition and processing

Recent work on the acquisition of number words has emphasized the importance of integrating linguistic and developmental perspectives [Musolino, J. (2004). The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. Cognition93, 1-41; Papafragou, A., Musolino, J. (2003). Scalar implicatures: Scalar implicatures: Experiments at the semantics-pragmatics interface. Cognition, 86, 253-282; Hurewitz, F., Papafragou, A., Gleitman, L., Gelman, R. (2006). Asymmetries in the acquisition of numbers and quantifiers. Language Learning and Development, 2, 76-97; Huang, Y. T., Snedeker, J., Spelke, L. (submitted for publication). What exactly do numbers mean?]. Specifically, these studies have shown that data from experimental investigations of child language can be used to illuminate core theoretical issues in the semantic and pragmatic analysis of number terms. In this article, I extend this approach to the logico-syntactic properties of number words, focusing on the way numerals interact with each other (e.g. Three boys are holding two balloons) as well as with other quantified expressions (e.g. Three boys are holding each balloon). On the basis of their intuitions, linguists have claimed that such sentences give rise to at least four different interpretations, reflecting the complexity of the linguistic structure and syntactic operations involved. Using psycholinguistic experimentation with preschoolers (n = 32) and adult speakers of English (n = 32), I show that (a) for adults, the intuitions of linguists can be verified experimentally, (b) by the age of 5, children have knowledge of the core aspects of the logical syntax of number words, (c) in spite of this knowledge, children nevertheless differ from adults in systematic ways, (d) the differences observed between children and adults can be accounted for on the basis of an independently motivated, linguistically-based processing model [Geurts, B. (2003). Quantifying kids. Language Acquisition, 11(4), 197-218]. In doing so, this work ties together research on the acquisition of the number vocabulary with a growing body of work on the development of quantification and sentence processing abilities in young children [Geurts, 2003; Lidz, J., Musolino, J. (2002). Children’s command of quantification. Cognition, 84, 113-154; Musolino, J., Lidz, J. (2003). The scope of isomorphism: Turning adults into children. Language Acquisition, 11(4), 277-291; Trueswell, J., Sekerina, I., Hilland, N., Logrip, M. (1999). The kindergarten-path effect: Studying on-line sentence processing in young children. Cognition, 73, 89-134; Noveck, I. (2001). When children are more logical than adults: Experimental investigations of scalar implicature. Cognition, 78, 165-188; Noveck, I., Guelminger, R., Georgieff, N., & Labruyere, N. (2007). What autism can tell us about every . . . not sentences. Journal of Semantics,24(1), 73-90. On a more general level, this work confirms the importance of integrating formal and developmental perspectives [Musolino, 2004], this time by highlighting the explanatory power of linguistically-based models of language acquisition and by showing that the complex structure postulated by linguists has important implications for developmental accounts of the number vocabulary.     

On the existence of extremal cones and comparative probability orderings

We study the recently discovered phenomenon [Conder, M. D. E., & Slinko, A. M. (2004). A counterexample to Fishburn's conjecture. Journal of Mathematical Psychology, 48(6), 425-431] of existence of comparative probability orderings on finite sets that violate the Fishburn hypothesis [Fishburn, P. C. (1996). Finite linear qualitative probability. Journal of Mathematical Psychology, 40, 64-77; Fishburn, P. C. (1997). Failure of cancellation conditions for additive linear orders. Journal of Combinatorial Designs, 5, 353-365]—we call such orderings and the discrete cones associated with them extremal. Conder and Slinko constructed an extremal discrete cone on a set of n=7 elements and showed that no extremal cones exist on a set of n?6 elements. In this paper we construct an extremal cone on a finite set of prime cardinality p if p satisfies a certain number theoretical condition. This condition has been computationally checked to hold for 1725 of the 1842 primes between 132 and 16,000, hence for all these primes extremal cones exist.     

How checking breeds doubt: Reduced performance in a simple workingmemory task

A paradox of memory research is that repeated checking results in a decrease in memory certainty, memory vividness and confidence [van den Hout, M. A., & Kindt, M. (2003a). Phenomenological validity of an OCD-memory model and the remember/know distinction. Behaviour Research and Therapy, 41, 369-378; van den Hout, M. A., & Kindt, M. (2003b). Repeated checking causes memory distrust. Behaviour Research and Therapy, 41, 301-316]. Although these findings have been mainly attributed to changes in episodic long-term memory, it has been suggested [Shimamura, A. P. (2000). Toward a cognitive neuroscience of metacognition. Consciousness and Cognition, 9, 313-323] that representations in working memory could already suffer from detrimental checking. In two experiments we set out to test this hypothesis by employing a delayed-match-to-sample working memory task. Letters had to be remembered in their correct locations, a task that was designed to engage the episodic short-term buffer of working memory [Baddeley, A. D. (2000). The episodic buffer: a new component in working memory? Trends in Cognitive Sciences, 4, 417-423]. Of most importance, we introduced an intermediate distractor question that was prone to induce frustrating and unnecessary checking on trials where no correct answer was possible. Reaction times and confidence ratings on the actual memory test of these trials confirmed the success of this manipulation. Most importantly, high checkers [cf. VOCI; Thordarson, D. S., Radomsky, A. S., Rachman, S., Shafran, R, Sawchuk, C. N., & Hakstian, A. R. (2004). The Vancouver obsessional compulsive inventory (VOCI). Behaviour Research and Therapy, 42(11), 1289-1314] were less accurate than low checkers when frustrating checking was induced, especially if the experimental context actually emphasized the irrelevance of the misleading question. The clinical relevance of this result was substantiated by means of an extreme groups comparison across the two studies. The findings are discussed in the context of detrimental checking and lack of distractor inhibition as a way of weakening fragile bindings within the episodic short-term buffer of Baddeley's (2000) model. Clinical implications, limitations and future research are considered.     

Estimation of careless error and lucky guess probabilities for dichotomous test items: A psychometric application of a biometric latent class model with random effects

Medical research has extensively dealt with the estimation of the accuracy (sensitivity and specificity) of a diagnostic test for screening individuals. In this paper we apply the biometric latent class model with random effects by Qu, Tan, and Kutner [(1996). Random effects models in latent class analysis for evaluating accuracy of diagnostic tests. Biometrics, 52, 797-810] to estimate the response error (careless error and lucky guess) probabilities for dichotomous test items in the psychometric theory of knowledge spaces. The approach is illustrated with simulated data. In particular, we extend this approach to give a generalization of the basic local independence model in knowledge space theory. This allows for local dependence among the indicators given the knowledge state of an examinee and/or for the incorporation of covariates.     

Mixture Models of Categorization

Many currently popular models of categorization are either strictly parametric (e.g., prototype models, decision bound models) or strictly nonparametric (e.g., exemplar models) (F. G. Ashby & L. A. Alfonso-Reese, 1995, Journal of Mathematical Psychology, 39, 216-233). In this article, a family of semiparametric classifiers is investigated where categories are represented by a finite mixture distribution. The advantage of these mixture models of categorization is that they contain several parametric models and nonparametric models as a special case. Specifically, it is shown that both decision bound models (F. G. Ashby & W. T. Maddox, 1992, Journal of Experimental Psychology: Human Perception and Performance, 16, 598-612; 1993, Journal of Mathematical Psychology, 37, 372-400) and the generalized context model (R. M. Nosofsky, 1986, Journal of Experimental Psychology: General, 115, 39-57) can be interpreted as two extreme cases of a common mixture model. Furthermore, many other (semiparametric) models of categorization can be derived from the same generic mixture framework. In this article, several examples are discussed and a parameter estimation procedure for fitting these models is outlined. To illustrate the approach, several specific models are fitted to a data set collected by S. C. McKinley and R. M. Nosofsky (1995, Journal of Experimental Psychology: Human Perception and Performance, 21, 128-148). The results suggest that semi-parametric models are a promising alternative for future model development.     

Language, thought, and real nouns

We test the claim that acquiring a mass-count language, like English, causes speakers to think differently about entities in the world, relative to speakers of classifier languages like Japanese. We use three tasks to assess this claim: object-substance rating, quantity judgment, and word extension. Using the first two tasks, we present evidence that learning mass-count syntax has little effect on the interpretation of familiar nouns between Japanese and English, and that speakers of these languages do not divide up referents differently along an individuation continuum, as claimed in some previous reports [Gentner, D., & Boroditsky, L. (2001). Individuation, relativity, and early word learning. In M. Bowerman, & S. Levinson (Eds.), Language acquisition and conceptual development (pp. 215-256). Cambridge University Press]. Instead, we argue that previous cross-linguistic differences [Imai, M., & Gentner, D. (1997). A cross-linguistic study of early word meaning: Universal ontology and linguistic influence. Cognition, 62, 169-200] are attributable to “lexical statistics” [Gleitman, L., & Papafragou, A. (2005). Language and thought. In K. Holyoak, & R. Morrison (Eds.), Cambridge handbook of thinking and reasoning (pp. 633-661). Cambridge University Press]. Speakers of English are more likely to think that a novel ambiguous expression like “the blicket” refers to a kind of object (relative to speakers of Japanese) because speakers of English are likely to assume that “blicket” is a count noun rather than a mass noun, based on the relative frequency of each kind of word in English. This is confirmed by testing Mandarin-English bilinguals with a word extension task. We find that bilinguals tested in English with mass-count ambiguous syntax extend novel words like English monolinguals (and assume that a word like “blicket” refers to a kind of object). In contrast, bilinguals tested in Mandarin are significantly more likely to extend novel words by material. Thus, online lexical statistics, rather than non-linguistic thought, mediate cross-linguistic differences in word extension. We suggest that speakers of Mandarin, English, and Japanese draw on a universal set of lexical meanings, and that mass-count syntax allows speakers of English to select among these meanings.     

Sensitivity to probabilistic orthographic cues to lexical stress in adolescent speakers with autism spectrum disorder and typical peers

Lexical stress refers to the opposition of strong and weak syllables within polysyllabic words and is a core feature of the English prosodic system. There are probabilistic cues to lexical stress present in English orthography. For example, most disyllabic English words ending with the letters “-ure” have first-syllable stress (e.g., “pasture”, but note words such as “endure”), whereas most ending with “-ose” have second-syllable stress (e.g., “propose”, but note examples such as “glucose”). Adult native speakers of English are sensitive to these probabilities during silent reading. During testing, they tend to assign first-syllable stress when reading a nonword such as “lenture” but second-syllable stress when reading “fostpose” (Arciuli & Cupples, 2006 Arciuli, J. and Cupples, L. 2006. The processing of lexical stress during visual word recognition: Typicality effects and orthographic correlates. Quarterly Journal of Experimental Psychology, 59(5): 920–948. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Difficulties with prosody, including problems processing lexical stress, are a notable feature of autism spectrum disorder (ASD). The current study investigated the ability of adolescents with ASD (13–17 years of age) to show this sensitivity compared with a group of typically developing peers. Results indicated reduced sensitivity to probabilistic cues to lexical stress in the group with ASD. The implications of these findings are discussed.     

Assessing the local identifiability of probabilistic knowledge structures

Given a collection Q of problems, in knowledge space theory Doignon & Falmagne, (International Journal of Man–Machine Studies 23:175–196, 1985) the knowledge state of a student is the collection K ? Q of all problems that this student is capable of solving. A knowledge structure is a pair (Q, ), where is a collection of knowledge states that contains at least the empty set and Q. A probabilistic knowledge structure (PKS) is a knowledge structure (Q, , π), where π is a probability distribution on the knowledge states. The PKS that has received the most attention is the basic local independence model BLIM; Falmagne & Doignon, (British Journal of Mathematical and Statistical Psychology 41:1–23, 1988a, Journal of Mathematical Psychology 32:232–258, 1988b). To the best of our knowledge, systematic investigations in the literature concerning the identifiability of the BLIM are totally missing. Based on the theoretical work of Bamber and van Santen (Journal of Mathematical Psychology 29:443–473, 1985), the present article is aimed to present a method and a corresponding computerized procedure for assessing the local identifiability of the BLIM, which is applicable to any finite knowledge structure of moderate size.     

Working memory involved in predicting future outcomes based on past experiences

Deficits in working memory have been shown to contribute to poor performance on the Iowa Gambling Task [IGT: Bechara, A., &; Martin, E.M. (2004). Impaired decision making related to working memory deficits in individuals with substance addictions. Neuropsychology, 18, 152–162]. Similarly, a secondary memory load task has been shown to impair task performance [Hinson, J., Jameson, T. &; Whitney, P. (2002). Somatic markers, working memory, and decision making. Cognitive, Affective, &; Behavioural Neuroscience, 2, 341–353]. In the present study, we investigate whether the latter findings were due to increased random responding [Franco-Watkins, A. M., Pashler, H., &; Rickard, T. C. (2006). Does working memory load lead to greater impulsivity? Commentary on Hinson, Jameson, and Whitney’s (2003). Journal of Experimental Psychology: Learning, Memory &; Cognition, 32, 443–447]. Participants were tested under Low Working Memory (LWM; n = 18) or High Working Memory (HWM; n = 17) conditions while performing the Reversed IGT in which punishment was immediate and reward delayed [Bechara, A., Dolan, S., &; Hindes, A. (2002). Decision making and addiction (part II): Myopia for the future or hypersensitivity to reward? Neuropsychologia, 40, 1690–1705]. In support of a role for working memory in emotional decision making, compared to the LWM condition, participants in the HWM condition made significantly greater number of disadvantageous selections than that predicted by chance. Performance by the HWM group could not be fully explained by random responding.     

The tune in-and-out model: A random walk and its application to a presidential election survey

The stochastic model for the evolution of preferences proposed by Falmagne, Regenwetter, and Grofman [1997. Journal of Mathematical Psychology, 41, 129-143] and tested by Regenwetter, Falmagne, and Grofman [1999. Psychological Review, 106, 362-384], as well as the alternative Thurstonian model of Böckenholt [Falmagne, J.-C., Regenwetter, M., & Grofman, B. (1997). A stochastic model for the evolution of preferences. In A. A. J. Marley (Ed.), Choice, decision and measurement: Essays in honor of R. Duncan Luce (pp. 113-131). Mahwah, NJ: Lawrence Erlbaum.], gave a good statistical account of attitudinal panel data from the 1992 US presidential election. We show, however, that both models have the defect of underestimating the number of respondents who did not change their order of preference for the candidates across different polls. We present a generalization of Falmagne et al.'s model based on the idea that some individuals may become momentarily impervious to all matters related to the campaign and ‘tune out.’ This behavior could be triggered by some personal reason or by some external event related to the campaign. Like the original model, the resulting model is a random walk, but on an augmented set of states. A respondent in a ‘live’ state behaves as in the previous model, except when receiving a ‘tune-out’ token, which effectively freezes the respondent's preference state until it is reversed by a ‘tune-in’ token. We describe and successfully test the new model on the same 1992 National Election Study panel data as those used by Böckenholt (2002) and Regenwetter et al. (1999).     

Analysis of multinomial models under inequality constraints: Applications to measurement theory

Multinomial random variables are used across many disciplines to model categorical outcomes. Under this framework, investigators often use a likelihood ratio test to determine goodness-of-fit. If the permissible parameter space of such models is defined by inequality constraints, then the maximum likelihood estimator may lie on the boundary of the parameter space. Under this condition, the asymptotic distribution of the likelihood ratio test is no longer a simple χ² distribution. This article summarizes recent developments in the constrained inference literature as they pertain to the testing of multinomial random variables, and extends existing results by considering the case of jointly independent mutinomial random variables of varying categorical size. This article provides an application of this methodology to axiomatic measurement theory as a means of evaluating properly operationalized measurement axioms. This article generalizes Iverson and Falmagne’s [Iverson, G. J. & Falmagne, J. C. (1985). Statistical issues in measurement. Mathematical Social Sciences, 10, 131-153] seminal work on the empirical evaluation of measurement axioms and provides a classical counterpart to Myung, Karabatsos, and Iverson’s [Myung, J. I., Karabatsos, G. & Iverson, G. J. (2005). A Bayesian approach to testing decision making axioms. Journal of Mathematical Psychology, 49, 205-225] Bayesian methodology on the same topic.     

Probabilistic choice (models) as a result of balancing multiple goals

We conceptualize probabilistic choice as the result of the simultaneous pursuit of multiple goals in a vector optimization representation, which is reduced to a scalar optimization that implies goal balancing. The majority of prior theoretical and empirical work on such probabilistic choice is based on random utility models, the most basic of which assume that each choice option has a valuation that has a deterministic (systematic) component plus a random component determined by some specified distribution. An alternate approach to probabilistic choice has considered maximization of one quantity (e.g., utility), subject to constraints on one or more other quantities (e.g., cost). The multiple goal perspective integrates the results regarding the well-studied multinomial logit model of probabilistic choice that has been derived from each of the above approaches; extends the results to other models in the generalized extreme value (GEV) class; and relates them to recent axiomatic work on the utility of gambling.     

Progressing measurement in sport motivation with the SMS-6: A response to Pelletier,Vallerand, and Sarrazin

Measurement in sport psychology is a major issue and attempts to progress measurement should be valued and encouraged. Construct validation is an ongoing process [Marsh, H. W., & Jackson, S. A. (1999). Flow experience in sport: Construct validation of multidimensional hierarchical state and trait responses. Structural Equation Modeling, 6, 343–371] and the publication of the SMS-6 [Mallett, C. J., Kawabata, M., Newcombe, P., Otero-Ferero, A., & Jackson, S. A. (2007). Sports Motivation Scale-6 (SMS-6): A revised six-factor Sport Motivation Scale. Psychology of Sport and Exercise, 8, 600–614] was an attempt to progress measurement in contextual sport motivation using self-determination theory (SDT) [Deci, E. L., & Ryan, R. M. (1985). Intrinsic motivation and self-determination in human behaviour. New York: Plenum Press]. Mallett et al. argued for the need to re-develop the Sport Motivation Scale (SMS) [Pelletier, L. G., Fortier, M. S., Vallerand, R. J., Tuson, K. M., Brière, N. M., & Blais, M. R. (1995). Toward a new measure of intrinsic motivation, extrinsic motivation, and amotivation in sports: The Sport Motivation Scale (SMS). Journal of Sport & Exercise Psychology, 17, 35–53], primarily because the SMS did not measure integrated regulation, and secondly due to consistently reported problems with the lack of convergent and discriminant validity. Pelletier, Vallerand, and Sarrazin's [Pelletier, L. G., Vallerand, R. J., & Sarrazin, P. (2007b). Something old, something new, and something borrowed. Psychology of Sport and Exercise, 8, 615–621] commentary on the SMS-6, focused on two legitimate questions: “Does the SMS need to be revised?”, and “Is the revised 6-factor SMS a better scale?” In addressing the first question, this article provides strong evidence that supports the arguments that the SMS does need revision. In addressing the second question, we provided sound arguments for the development of the SMS-6. However, we concluded that the superiority of the SMS-6 in measuring contextual sport motivation across diverse age and cultural groups is a question for future and continuing research. Further examination of the SMS-6 is necessary before such claims can be endorsed.