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.     

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.     

Individual differences in authenticity and mindfulness as predictors of verbal defensiveness

We examined the extent to which individual differences in authenticity and mindfulness predicted verbal defensiveness. Participants first completed measures of authenticity [Kernis, M. H., & Goldman, B. M. (2006). A multicomponent conceptualization of authenticity: Theory and research. In M. P. Zanna (Ed.), Advances in Experimental Social Psychology, Vol. 38 (pp. 283–357).] and mindfulness [Brown, K. W., & Ryan, R. M. (2003). The benefits of being present: Mindfulness and its role in psychological well-being. Journal of Personality and Social Psychology, 84, 822–848]. Within the next few weeks, participants completed the Defensive Verbal Behavior Assessment [Feldman Barrett, L., Williams, N. L., & Fong, G. T. (2002). Defensive verbal behavior assessment. Personality and Social Psychology Bulletin, 28, 776–788]. Their responses to potentially self-threatening experiences subsequently were rated for the extent to which they reflected openness and honesty as opposed to defensiveness. Our findings indicated that authenticity and mindfulness correlated positively and that higher scores on each related to lower levels of verbal defensiveness. Additional analyses revealed that the relation between authenticity and verbal defensiveness was indirect, mediated by mindfulness. These findings support the view that higher authenticity and mindfulness relate to greater tendencies to engage self-relevant information in a relatively non-defensive manner.     

Relations between multidimensional perfectionism and burnout in junior-elite male athletes

ObjectivesThe purpose of the present investigation was to examine the moderating influence of perceptions of goal progress and achievement goal orientations on the relationship between multidimensional perfectionism and athlete burnout.Methods201 junior-elite male athletes, ranging from 11 to 21 years of age (M = 15.64, SD = 1.92), were recruited from professional sport clubs in the UK and completed a multi-section inventory assessing self-oriented and socially prescribed perfectionism [Hewitt, P. L., & Flett, G. L. (1991). Perfectionism in the self and social contexts: conceptualization, assessment, and association with psychopathology. Journal of Personality and Social Psychology, 60, 456–470], achievement goal orientations [Roberts, G. C., Treasure, D. C., & Balague, G. (1998). Achievement goals in sport: the development and validation of the perception of success questionnaire. Journal of Sport Sciences, 16, 337–347], perceived goal progress [Hill, A. P., Hall, H. K., Appleton, P. R., & Kozub, S. A. (2008). Perfectionism and burnout in junior-elite soccer players: the mediating influence of unconditional self-acceptance. Psychology of Sport and Exercise, 9, 630–644] and multidimensional athlete burnout [Raedeke, T. D., & Smith, A. L. (2001). Development and preliminary validation of an athlete burnout measure. Journal of Sport and Exercise Psychology, 23, 281–306].ResultsRegression analyses revealed that socially prescribed perfectionism demonstrated a significant positive association, and self-oriented perfectionism a significant negative association with burnout dimensions. However, the hypotheses for moderation of the perfectionism–burnout relationship were not supported.ConclusionsOverall, while there was no evidence to support the hypothesised moderation of the perfectionism–burnout relationship, the results provide support for a growing body of literature which indicates that maladaptive forms of perfectionism may contribute to burnout in elite junior athletes [Chen, L. H., Kee, Y. H., Chen, M., & Tsaim, Y. (2008). Relation of perfectionism with athletes' burnout: further examination. Perceptual and Motor Skills, 106, 811–820; Gould, D., Tuffey, S., Udrey, E., & Loehr, J. (1996). Burnout in competitive junior tennis players: II. Qualitative analysis. The Sport Psychologist, 10, 341–366; Gould, D., Udry, E., Tuffey, S., & Loehr, J. (1996). Burnout in competitive junior tennis players: I. A quantitative psychological assessment. The Sport Psychologist, 10, 332–340; Hall, H. K. (2006). Perfectionism: a hallmark quality of world class performers, or a psychological impediment to athletic development? In D. Hackfort, & G. Tenenbaum (Eds.), Perspectives in sport and exercise psychology: Essential processes for attaining peak performance (Vol. 1, pp. 178–211). Oxford, UK: Meyer & Meyer Publishers; Hill et al., 2008; Lemyre, P. N., Hall, H. K., & Roberts, G. C. (2008). A social cognitive approach to burnout in elite athletes. Scandinavian Journal of Medicine & Science in Sports, 18, 221–224].     

Is the relationship between self-esteem and physical aggression necessarily U-shaped?

Recent research by Perez, Vohs, and Joiner [Perez, M., Vohs, K. D., & Joiner, T. E., Jr. (2005). Discrepancies between self- and other-esteem as correlates of aggression. Journal of Social and Clinical Psychology, 24, 607–620] has supported a U-shaped curvilinear relationship between self-esteem and physical aggression in a sample of 140 undergraduates. The present study attempted to replicate this effect with a sample size more than 12 times larger. Thus, 1781 undergraduates completed items from Rosenberg’s [Rosenberg, M. (1965). Society and adolescent self-image. Princeton, NJ: Princeton University] Self-Esteem Scale and from the Physical Aggression subscale of Buss and Perry’s [Buss, A. H., & Perry, M. (1992). The Aggression Questionnaire. Journal of Personality and Social Psychology, 63, 452–459] Aggression Questionnaire. The results did not support a U-shaped relationship between self-esteem and physical aggression; if anything, they supported an inverted U-shaped one, such that the simple relationship between self-esteem and physical aggression became more negative and as self-esteem increased. Controlling for gender strengthened these effects, consistent with a pattern of mutual suppression between gender and self-esteem.     

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.     

The role of memory for compounds in cue competition

Revisions of common associative learning models incorporate a within-compound association mechanism in order to explain retrospective cue competition effects (e.g., [Dickinson, A., & Burke, J. (1996). Within-compound associations mediate the retrospective revaluation of causality judgements. Quarterly Journal of Experimental Psychology, 49B, pp. 60-80.]). These models predict a correlation between memory for compounds (as a measure for the strength of within-compound associations) and retrospective cue competition but not forward cue competition. This was indeed found in a study of [Melchers, K. G., Lachnit, H., & Shanks, D. (2004). Within-compound associations in retrospective revaluation and in direct learning: A challenge for comparator theory. Quarterly Journal of Experimental Psychology, 57B, pp. 25-54.]. We argue that a higher-order reasoning account of causal learning can also explain the evidence for the role of within-compound associations in cue competition. Moreover, this account predicts that if making inferences during the learning stage is impeded, a correlation between memory for compounds and forward cue competition should be found as well. The results of a new study confirmed this prediction.     

Lessons from a Faraway land: The effect of spatial distance on creative cognition

Recent research [Förster, J., Friedman, R. S., & Liberman, N. (2004). Temporal construal effects on abstract and concrete thinking: Consequences for insight and creative cognition. Journal of Personality and Social Psychology, 87, 177-189] has identified temporal distance as a situational moderator of creativity. According to Construal Level Theory [Liberman, N., Trope, Y., & Stephan, E. (2007). Psychological Distance. In E. T. Higgins & A. W. Kruglanski (Eds.), Social psychology: A handbook of basic principles (pp. 353-381). New York: Guilford Press], temporal distance is just one case of the broader construct of psychological distance. In the present research, we investigated the effect of another dimension of psychological distance, namely, spatial distance, on creative cognition and insight problem solving. In two studies, we demonstrate that when the creative task is portrayed as originating from a far rather than close location, participants provide more creative responses (Study 1) and perform better on a problem solving task that requires creative insight (Study 2). Both theoretical and practical implications of this finding are discussed.     

Assessing the sustainability of goal-based changes in adjustment over a four-year period

No published data addresses the longer-term sustainability of increases in well-being and adjustment derived from successful goal-striving. Although some theories and data suggest that such gains cannot be maintained, the sustainable happiness model [Lyubomirsky, S., Sheldon, K. M., & Schkade, D. (2005). Pursuing happiness: The architecture of sustainable change. Review of General Psychology, 9, 111–131.] suggests that successful intentional activity can indeed bring about lasting gains. In the current study, the “freshman goals” sample studied by Sheldon and Houser-Marko (Study 1) [Sheldon, K. M., & Houser-Marko, L. (2001). Self-concordance, goal-attainment, and the pursuit of happiness: Can there be an upward spiral? Journal of Personality and Social Psychology, 80, 152–165.] was revisited at the end of their senior year. Students who strove successfully during their freshman year, thereby enhancing their adjustment during that year, had maintained those gains three years later. Path modeling suggested that initial adjustment predicted subsequent goal-attainment which in turn predicted enhanced adjustment, combining recent theories concerning both the functional benefits of positive affect and the causes of sustainable gains in well-being.     

Are there multiple channels through which we connect with beauty and excellence?

This research answers the question whether there are multiple channels through which we connect with beauty and excellence, and thus contributes to the understanding of the structure of appreciation. Two models were examined: the appreciation of beauty and excellence (ABE) model [Haidt, J., & Keltner, D. (2004). Appreciation of beauty and excellence [awe, wonder, elevation]. In C. Peterson & M.E.P. Seligman (Eds.). Character strengths and virtues (pp. 537–551). New York, NY: Oxford University Press], and the engagement with beauty model [Diessner, R., Solom, R., Frost, N.K., Parsons, L., & Davidson, J. (2008). Engagement with beauty: Appreciating natural, artistic, and moral beauty. The Journal of Psychology: Interdisciplinary and Applied, 142, 303–329]. Study 1 describes the development and initial validation of the ABE Test (ABET), which assesses the types of appreciation included in Haidt and Keltner's (2004) model. In study 2, the ABE subscale of the Values In Action Inventory of Strengths [VIA-IS; Peterson, C., Park, N., & Seligman, M.E.P. (2005). Assessment of character strengths. In G.P. Koocher, J.C. Norcross, & S.S. Hill III (Eds.), Psychologists’ desk reference (Vol. 3, pp. 93–98). New York, NY: Oxford University Press], the Engagement with Beauty Scale (Diessner et al., 2008), and the ABET were included in a structural equation modeling analysis. Results suggested a new model encompassing the two previous ones, and distinguishing between natural beauty, artistic beauty, and non-aesthetic goodness.     

Motivational influences on compliance with and consequences of instructions to suppress stereotypes

Two experiments investigated differences in compliance with instructions to suppress stereotypes as a function of prejudice-related motivations. In Experiment 1, only participants identified as high in motivation to control prejudice [Dunton, B. C., & Fazio, R. H. (1997). An individual difference measure of motivation to control prejudiced reactions. Personality and Social Psychology Bulletin, 23, 316-326] complied with suppression instructions. These participants experienced post-suppression rebound effects, but only if they were also high in prejudice. In Experiment 2, only participants identified as high in external motivation to respond without prejudice [Plant, E. A., & Devine, P. G. (1998). Internal and external motivation to respond without prejudice. Journal of Personality and Social Psychology, 75, 811-832] complied with instructions to suppress. These participants later experienced stereotype rebound effects, but only if they were also low in internal motivation to respond without prejudice. These findings suggest that motivational factors play an important role in determining not only the outcome of suppression, but also the choice to attempt suppression in the first place.     

Ethnic differences in the stigma of obesity: Identification and engagement with a thin ideal

In the current research, components of disidentification theory [Steele, C. M., & Aronson, J. (1995). Stereotype vulnerability and the intellectual test performance of African–Americans. Journal of Personality and Social Psychology, 69, 797–811] are extended to the domain of body weight and provide an explanation as to why Black women typically do not – but under certain circumstances do – stigmatize obesity. Across three studies, results show that Black women are generally less likely to stigmatize obesity than are White women [see also Hebl, M., & Heatherton, T. F. (1997). The stigma of obesity: The differences are black and white. Personality and Social Psychology Bulletin, 24, 417–426]. Taken as a whole, the current research also provides preliminary evidence consistent with disidentification theory to demonstrate that there are situations in which Black women will re-engage with valuing thinness, particularly when re-engagement is tied to conceptions about the self.     

Mathematical psychology: Prospects for the 21st century: A guest editorial

The twenty-first century is certainly in progress by now, but hardly well underway. Therefore, I will take that modest elasticity in concept as a frame for this essay. This frame will serve as background for some of my hopes and gripes about contemporary psychology and mathematical psychology’s place therein. It will also act as platform for earnest, if wistful thoughts about what might have (and perhaps can still) aid us in forwarding our agenda and what I see as some of the promising avenues for the future. I loosely structure the essay into a section about mathematical psychology in the context of psychology at large and then a section devoted to prospects within mathematical psychology proper. The essay can perhaps be considered as in a similar spirit, although differing in content, to previous editorial-like reviews of general or specific aspects of mathematical psychology such as [Estes, W. K. (1975). Some targets for mathematical psychology. Journal of Mathematical Psychology, 12, 263-282; Falmagne, J. C. (2005). Mathematical psychology: A perspective. Journal of Mathematical Psychology, 49, 436-439; Luce, R. D. (1997). Several unresolved conceptual problems of mathematical psychology. Journal of Mathematical Psychology, 41, 79-87] that have appeared in this journal.     

The role of experience in location estimation: Target distributions shift location memory biases

Research based on the Category Adjustment model concluded that the spatial distribution of target locations does not influence location estimation responses [Huttenlocher, J., Hedges, L., Corrigan, B., & Crawford, L. E. (2004). Spatial categories and the estimation of location. Cognition, 93, 75–97]. This conflicts with earlier results showing that location estimation is biased relative to the spatial distribution of targets [Spencer, J. P., & Hund, A. M. (2002). Prototypes and particulars: Geometric and experience-dependent spatial categories. Journal of Experimental Psychology: General, 131, 16–37]. Here, we resolve this controversy by using a task based on Huttenlocher et al. (Experiment 4) with minor modifications to enhance our ability to detect experience-dependent effects. Results after the first block of trials replicate the pattern reported in Huttenlocher et al. After additional experience, however, participants showed biases that significantly shifted according to the target distributions. These results are consistent with the Dynamic Field Theory, an alternative theory of spatial cognition that integrates long-term memory traces across trials relative to the perceived structure of the task space.     

Investigating the causes of wrap-up effects: Evidence from eye movements and E-Z Reader

Wrap-up effects in reading have traditionally been thought to reflect increased processing associated with intra- and inter-clause integration (Just, M. A. & Carpenter, P. A. (1980). A theory of reading: From eye fixations to comprehension. Psychological Review,87(4), 329-354; Rayner, K., Kambe, G., & Duffy, S. A. (2000). The effect of clause wrap-up on eye movements during reading. The Quarterly Journal of Experimental Psychology,53A(4), 1061-1080; cf. Hirotani, M., Frazier, L., & Rayner, K. (2006). Punctuation and intonation effects on clause and sentence wrap-up: Evidence from eye movements. Journal of Memory and Language,54, 425-443). We report an eye-tracking experiment with a strong manipulation of integrative complexity at a critical word that was either sentence-final, ended a comma-marked clause, or was not comma-marked. Although both complexity and punctuation had reliable effects, they did not interact in any eye-movement measure. These results as well as simulations using the E-Z Reader model of eye-movement control (Reichle, E. D., Warren, T., & McConnell, K. (2009). Using E-Z Reader to model the effects of higher-level language processing on eye movements during reading. Psychonomic Bulletin & Review,16(1), 1-20) suggest that traditional accounts of clause wrap-up are incomplete.     

Negative congruency effects: a test of the inhibition account

Masked priming experiments occasionally revealed surprising effects: Participants responded slower for congruent compared to incongruent primes. This negative congruency effect (NCE) was ascribed to inhibition of prime-induced activation [Eimer, M., & Schlaghecken, F. (2003). Response faciliation and inhibition in subliminal priming. Biological Psychology, 64, 7-26.] that sets in if the prime activation is sufficiently strong. The current study tests this assumption by implementing manipulations designed to vary the amount of prime-induced activation in three experiments. In Experiments 1 and 3, NCEs were observed despite reduced prime-induced activation. Experiment 2 revealed no NCE with at least similar prime strength. Thus, the amount of prime activation did not predict whether or not NCEs occurred. The findings are discussed with regard to the inhibition account and the recently proposed account of mask-induced activation [cf. Lleras, A., & Enns, J. T. (2004). Negative compatibility or object updating? A cautionary tale of mask-dependent priming. Journal of Experimental Psychology: General, 133, 475-493; Verleger, R., Jaskowski, P., Aydemir, A., van der Lubbe, R. H. J., & Groen, M. (2004). Qualitative differences between conscious and nonconscious processing? On inverse priming induced by masked arrows. Journal of Experimental Psychology: General, 133, 494-515].     

Conceptualization of task boundaries preserves implicit sequence learning under dual-task conditions

Implicit learning in the serial reaction time (SRT) task is sometimes disrupted by the presence of a secondary distractor task (e.g., Schumacher & Schwarb Journal of Experimental Psychology: General 138:270–290, 2009) and at other times is not (e.g., Cohen, Ivry, & Keele Journal of Experimental Psychology: Learning, Memory, and Cognition 16:17–30, 1990). In the present study, we used an instructional manipulation to investigate how participants’ conceptualizations of the task affect sequence learning under dual-task conditions. Two experimental groups differed only in terms of the instructions and presequence training. One group was instructed that they were completing two separate tasks, whereas the other group was instructed that they were performing a single, integrated task. The separate group showed sequence learning, while the integrated group did not. These findings suggest that the conceptualization of task boundaries affects the availability of the sequential information necessary for implicit learning.     

Narcissism and self-presentation: The moderating effects of accountability and contingencies of self-worth

Using email, we investigated intra- and inter-personal variables that may influence self-presentational behavior of those high in narcissism. Participants rated themselves in domains requiring either external validation (e.g., attractiveness) or internal validation (e.g., morality), when either accountable or non-accountable to an evaluative audience. When degree of external self-worth contingency [Crocker, J., Luhtanen, R. K., Cooper, M. L., & Bouvrette, S. (2003). Contingencies of self-worth in college students: Theory and measurement. Journal of Personality and Social Psychology, 85, 894–908.] was high, those high in narcissism presented themselves in a self-enhancing manner in external domains, even when accountable, whereas those low in narcissism were more modest when accountable. Those high in narcissism may be chronically vigilant for self-enhancement opportunities, especially in external domains where their self-worth is contingent, but may be insensitive to social constraints and norms in their efforts to construct grandiose identities.