Markov processes in learning theory

Consideration is given mathematical problems arising in two learning theories—one developed by Bush and Mosteller, the other developed by Estes. The theory of Bush and Mosteller leads to a class of Markov processes which have been studied in considerable detail (see [1] and [7]). The Estes model can be treated as a Markov chain, i.e., a Markov process with a finite number of states. For an important class of special cases, it is shown that the Bush-Mosteller model is, in a sense, a limiting form of the Estes model. The limiting probability distributions are derived for the cases treated in both models.This research was supported by the National Science Foundation through a grant given to the Dartmouth Mathematics Project.     

A response time model for choice behavior

A subject in a two-choice situation characteristically makes several observing responses before performing the final choice. This behavior can be described by means of a random walk model. The present paper explores some possibilities as to how this model can be extended to include choice time. The assumption is made that the duration of each step in the random walk is a random variable which is exponentially distributed. With this assumption, one can predict the probability distributions of the choice times as well as the moments of these distributions.The author gratefully acknowledges his debt to W. K. Estes and C. J. Burke. This study was initiated while the author held a USPHS postdoctoral fellowship at Indiana University.     

Context-sensitive elemental theory

My theories of associative learning, like those of N. J. Mackintosh and almost all learning theorists, have employed elemental representations of the stimuli involved. We must take notice when two important contributors to elemental theory, J. M. Pearce and W. K. Estes, find sufficient problems with the theory type to cause them to defect from it. I will describe some of the essential problems, concerning the substantial influence of context on learning and retrieval, characterize the different responses of Pearce and Estes, and, then, propose a variation on a recently developed elemental model that was similarly inspired. The resulting elemental theory has a close quantitative relationship to the “product-rule” of Estes and D. L. Medin, and may help us to rationalize how the same formal experimental design can sometimes produce results that favour the configural interpretation of Pearce and at other times the elemental interpretation of R. A. Rescorla and A. R. Wagner, as these have often been pitted against each other.     

Elemental representations of stimuli in associative learning

This article reviews evidence and theories concerning the nature of stimulus representations in Pavlovian conditioning. It focuses on the elemental approach developed in stimulus sampling theory (R. C. Atkinson & W. K. Estes, 1963; R. R. Bush & F. Mosteller, 1951b) and extended by I. P. L. McLaren and N. J. Mackintosh (2000, 2002) and contrasts this with models that invoke notions of configural representations that uniquely code for different patterns of stimulus inputs (e.g., J. M. Pearce, 1987, 1994; R. A. Rescorla & A. R. Wagner, 1972; A. R. Wagner & S. E. Brandon, 2001). The article then presents a new elemental model that emphasizes interactions between stimulus elements. This model is shown to explain a range of behavioral findings, including those (e.g., negative patterning and biconditional discriminations) traditionally thought to be beyond the explanatory capabilities of elemental models. Moreover, the model offers a ready explanation for recent findings reported by R. A. Rescorla (2000, 2001, 2002b) concerning the way that stimuli with different conditioning histories acquire associative strength when conditioned in compound.     

Comparing dependent correlations

In a recent article in The Journal of General Psychology, J. B. Hittner, K. May, and N. C. Silver (2003) described their investigation of several methods for comparing dependent correlations and found that all can be unsatisfactory, in terms of Type I errors, even with a sample size of 300. More precisely, when researchers test at the .05 level, the actual Type I error probability can exceed .10. The authors of this article extended J. B. Hittner et al.'s research by considering a variety of alternative methods. They found 3 that avoid inflating the Type I error rate above the nominal level. However, a Monte Carlo simulation demonstrated that when the underlying distribution of scores violated the assumption of normality, 2 of these methods had relatively low power and had actual Type I error rates well below the nominal level. The authors report comparisons with E. J. Williams' (1959) method.     

Deconstructing multicultural counseling competencies research: comment on Owen, Leach, Wampold, and Rodolfa (2011)

The present article offers a commentary and critique of the research presented by J. Owen, M. M. Leach, B. Wampold and E. Rodolfa (2011). Given the complexity of the research methodology, findings, and conclusions, the authors provide a concise summary of findings, study limitations, and conclusions followed by a detailed critique of the study. The authors respect and appreciate the ambitious efforts made by Owen et al. to address gaps in the literature regarding outcome research using client ratings of counselors' multicultural counseling competencies (MCCs). The authors provide a critical analysis of some of J. Owen et al.'s specific conclusions and offer alternative conclusions based on conceptual and methodological bases. The authors use the opportunity to comment on this study as a means of advancing recommendations regarding future research on MCCs that might contribute to substantive revisions to the long-standing theoretical foundation in this area.     

Logit models and logistic regressions for social networks: I. An introduction to Markov graphs andp

Spanning nearly sixty years of research, statistical network analysis has passed through (at least) two generations of researchers and models. Beginning in the late 1930's, the first generation of research dealt with the distribution of various network statistics, under a variety of null models. The second generation, beginning in the 1970's and continuing into the 1980's, concerned models, usually for probabilities of relational ties among very small subsets of actors, in which various simple substantive tendencies were parameterized. Much of this research, most of which utilized log linear models, first appeared in applied statistics publications.But recent developments in social network analysis promise to bring us into a third generation. The Markov random graphs of Frank and Strauss (1986) and especially the estimation strategy for these models developed by Strauss and Ikeda (1990; described in brief in Strauss, 1992), are very recent and promising contributions to this field. Here we describe a large class of models that can be used to investigate structure in social networks. These models include several generalizations of stochastic blockmodels, as well as models parameterizing global tendencies towards clustering and centralization, and individual differences in such tendencies. Approximate model fits are obtained using Strauss and Ikeda's (1990) estimation strategy.In this paper we describe and extend these models and demonstrate how they can be used to address a variety of substantive questions about structure in social networks.This research was supported by grants from the Australian Research Council and the National Science Foundation (#SBR93-10184). This paper was presented at the 1994 Annual Meeting of the Psychometric Society, Champaign, Illinois, June 1994. Special thanks go to Sarah Ardu for programming assistance, Laura Koehly and Garry Robins for help with this research, and to Shizuhiko Nishisato and three reviewers for their comments. INTERNETemail addresses:pattison@psych.unimelb.edu.au (PP);stanwass@uiuc.edu (SW). Affiliations: Department of Psychology, University of Melbourne (PP); Department of Psychology, Department of Statistics, and The Beckman Institute for Advanced Science and Technology, University of Illinois (SW).     

A stochastic theory of matching processes

A theory of matching processes is developed within which serial models and parallel models based on within-stage independent intercompletion times are defined. These models are then specialized to a class of processes possessing exponential intercompletion time densities and equivalence properties of parallel and serial models within this class are investigated. Nonequivalence theorems are proved that designate possible differences in “same” (+) and different (?) matching rates as important in testing parallel and serial models. An experimental paradigm is then derived with the property that if + rates are different from — rates, and if processing is self-terminating, then the parallel and serial models are distinguishable at the level of mean reaction times. The serial class of models that is tested by this paradigm includes a large number of stochastic distributions whose central assumption is additivity of element processing times. The corresponding parallel class of models is currently limited to those assuming exponential intercompletion times. A numerical example and an example of non-parametric relations predicted by the serial or parallel models are given. Some advantages and limitations of the present treatment are discussed.     

Exemplar and prototype models revisited: response strategies,selective attention,and stimulus generalization

J. D. Smith and colleagues (J. P. Minda & J. D. Smith, 2001; J. D. Smith & J. P. Minda, 1998,2000; J. D. Smith, M. J. Murray, & J. P. Minda, 1997) presented evidence that they claimed challenged the predictions of exemplar models and that supported prototype models. In the authors' view, this evidence confounded the issue of the nature of the category representation with the type of response rule (probabilistic vs. deterministic) that was used. Also, their designs did not test whether the prototype models correctly predicted generalization performance. The present work demonstrates that an exemplar model that includes a response-scaling mechanism provides a natural account of all of Smith et al.'s experimental results. Furthermore, the exemplar model predicts classification performance better than the prototype models when novel transfer stimuli are included in the experimental designs.     

Choice theories: What are they good for?

Simonson et al. present an ambitious sketch of an integrative theory of context. Provoked by this thoughtful proposal, I discuss what is the function of theories of choice in the coming decades. Traditionally, choice models and theory have attempted to predict choices as a function of the attributes of options. I argue that to be truly useful, they need to generate specific and quantitative predictions of the effect of the choice environment upon choice probability. To do this, we need to focus on rigorously modeling and measuring the underlying processes causing these effects, and use the Simonson et al. proposal to provide some examples. I also present some examples from research in decision-making and decision neuroscience, and argue that models that fail, and fail spectacularly are particularly useful. I close with a challenge: How would consumer researcher aid the design of real world choice environments such as the health exchanges under the Patient Protection and Affordable Care Act?     

On the form of ROCs constructed from confidence ratings

A classical question for memory researchers is whether memories vary in an all-or-nothing, discrete manner (e.g., stored vs. not stored, recalled vs. not recalled), or whether they vary along a continuous dimension (e.g., strength, similarity, or familiarity). For yes-no classification tasks, continuous- and discrete-state models predict nonlinear and linear receiver operating characteristics (ROCs), respectively (D. M. Green & J. A. Swets, 1966; N. A. Macmillan & C. D. Creelman, 1991). Recently, several authors have assumed that these predictions are generalizable to confidence ratings tasks (J. Qin, C. L. Raye, M. K. Johnson, & K. J. Mitchell, 2001; S. D. Slotnick, S. A. Klein, C. S. Dodson, & A. P. Shimamura, 2000, and A. P. Yonelinas, 1999). This assumption is shown to be unwarranted by showing that discrete-state ratings models predict both linear and nonlinear ROCs. The critical factor determining the form of the discrete-state ROC is the response strategy adopted by the classifier.     

How to hit Scylla without avoiding Charybdis: comment on Perruchet, Tyler, Galland, and Peereman (2004)

M. Pe?a, L. L. Bonatti, M. Nespor, and J. Mehler argued that humans compute nonadjacent statistical relations among syllables in a continuous artificial speech stream to extract words, but they use other computations to determine the structural properties of words. Instead, when participants are familiarized with a segmented stream, structural generalizations about words are quickly established. P. Perruchet, M. D. Tyler, N. Galland, and R. Peereman criticized M. Pe?a et al.'s work and dismissed their results. In this article, the authors show that P. Perruchet et al.'s criticisms are groundless.     

Evidence for independent parallel channels in tachistoscopic perception

This paper reviews the implications of a number of experimental results for a test between two different types of models of perceptual processing: models that assume a limitation of perceptual capacity vs models that postulate independent, parallel perceptual channels. The results of recent “detection” experiments (e.g., Estes & Taylor, 1966) have been consistent with the limited-capacity model developed by Rumelhart (1970). The same results, however, are predicted by the “ICC” independent-parallel-channels model, an extension of work by Eriksen and Spencer (1969) developed in the present paper. This model emphasizes the interaction of two factors: decisional properties of the detection task, plus perceptual confusion phenomena known to occur under tachistoscopic conditions.Two experiments relevant to a test between the ICC and Rumelhart conceptions are reported. Experiment I involved a “time-sharing” paradigm that varied the demands of two detection tasks performed within a single trial. Experiment II involved the elimination of certain perceptual confusion phenomena. The results of Expts I and II appeared consistent with the ICC model but inconsistent with the Rumelhart model. After consideration of alternative limited-capacity models, it was concluded that: (1) a decision-confusion interaction is a necessary component of information-processing models, and that (2) the ICC independent-parallel-channels model is sufficient to account for the broad spectrum of experimental results reviewed.     

Item-specific adaptation and the conflict-monitoring hypothesis: a computational model

M. M. Botvinick, T. S. Braver, D. M. Barch, C. S. Carter, and J. D. Cohen (2001) implemented their conflict-monitoring hypothesis of cognitive control in a series of computational models. The authors of the current article first demonstrate that M. M. Botvinick et al.'s (2001) conflict-monitoring Stroop model fails to simulate L. L. Jacoby, D. S. Lindsay, and S. Hessels's (2003) report of an item-specific proportion-congruent (ISPC) effect in the Stroop task. The authors then implement a variant of M. M. Botvinick et al.'s model based on the assumption that control must be able to operate at the item level. This model successfully simulates the ISPC effect. In addition, the model provides an alternative to M. M. Botvinick et al.'s explanation of the list-level proportion-congruent effect in terms of an ISPC effect. Implications of the present modeling effort are discussed.     

Experimental research on just-world theory: problems, developments, and future challenges

M. J. Lerner (1980) proposed that people need to believe in a just world; thus, evidence that the world is not just is threatening, and people have a number of strategies for reducing such threats. Early research on this idea, and on just-world theory more broadly, was reviewed in early publications (e.g., M. J. Lerner, 1980; M. J. Lerner & D. T. Miller, 1978). In the present article, focus is directed on the post-1980 experimental research on this theory. First, 2 conceptualizations of the term belief in a just world are described, the typical experimental paradigms are explained, and a general overview of the post-1980 experiments is provided. Second, problems with this literature are discussed, including the unsystematic nature of the research. Third, important developments that have occurred, despite the problems reviewed, are described. Finally, theoretical challenges that researchers should address if this area of inquiry is to advance in the future are discussed.     

Deciding about decision models of remember and know judgments: a reply to Murdock (2006)

B. B. Murdock (2006) has interpreted remember-know data within a decision space defined by item and associative information, the fundamental variables in his general recognition memory model TODAM (B. B. Murdock, 1982). He has related parameters of this extended model to stimulus characteristics for several classic remember-know data sets. The authors show that this accomplishment is shared by both one- and two-dimensional signal-detection-based remember-know models (J. C. Dunn, 2004; C. M. Rotello, N. A. Macmillan, & J. A. Reeder, 2004; J. T. Wixted & V. Stretch, 2004), which can be represented in this same decision space and can be related to stimulus characteristics with similar success. Murdock claims that his model, unlike its competitors, is a process model; however, the process aspects of TODAM are not used in his application, and the decision aspects are identical to a previously proposed model. Murdock's claim that one competing model (STREAK; C. M. Rotello et al., 2004) is not fully specified is shown to be false. The new model is not superior to existing ones, but comparisons among the models to date are not definitive. The authors describe several strategies that might be applied to distinguish among them.     

The generalization function is determined by one subject's probability learning data

In his recent review of probability learning, Estes (1972, p. 96) notes that “the chief limitation on effective application” of the models for learning with a continuum of responses to have emerged from the linear and pattern learning models is the failure to specify the smearing or generalization function. He notes that “a useful technique has been developed by Levine for estimating this function from individual data.” This paper describes, gives the rationale, and proves the validity of the previously unpublished technique.     

The repeated insertion model for rankings: Missing link between two subset choice models

Several probabilistic models for subset choice have been proposed in the literature, for example, to explain approval voting data. We show that Marley et al.'s latent scale model is subsumed by Falmagne and Regenwetter's size-independent model, in the sense that every choice probability distribution generated by the former can also be explained by the latter. Our proof relies on the construction of a probabilistic ranking model which we label the repeated insertion model. This model is a special case of Marden's orthogonal contrast model class and, in turn, includes the classical Mallows -model as a special case. We explore its basic properties as well as its relationship to Fligner and Verducci's multistage ranking model.The authors are grateful to the National Science Foundation for grants SES98-18756 to Regenwetter and Peke, and SBR97-30076 to Regenwetter. This collaborative research was carried out in the context of the conference Random Utility 2000 held at Duke University and sponsored by NSF, the Fuqua School of Business and the Center for International Business Education and Research. We thank the editor and four referees for helpful suggestions and we are grateful to Prof. J. I. Marden for providing useful information on contrast models. We thank Moon-Ho Ho for programming and running the data analyses.     

Tactical contingencies in the experimental analysis of reinforcement and operant classes

In his “Tactics of Scientific Research” (1960), his work on avoidance, his discovery of equivalence classes and his cautions on applications of coercion, Murray Sidman created high standards for behavior analytic research. I illustrate his influence in the context of three examples he might have characterized as pilot studies. Each examined trial N+1 response probabilities depending on whether trial N responding had produced a reinforcer. Differentially reinforced interresponse times, keys pecked in arbitrary matching, and two-key response sequences provided no robust evidence that reinforcing some response property on trial N raises the probability of responding with that property on trial N+1. These negative findings shed light on the nature of operant classes and on the relation of reinforcers to the responses that produced them. Through selection, reinforcers create operant classes and engender variations of the responses within those classes; operant classes are held together by common contingencies. Sidman extended our understanding of operant classes by expanding them to include equivalence relations.