Estimating parameters of the diffusion model: approaches to dealing with contaminant reaction times and parameter variability

Three methods for fitting the diffusion model (Ratcliff, 1978) to experimental data are examined. Sets of simulated data were generated with known parameter values, and from fits of the model, we found that the maximum likelihood method was better than the chi-square and weighted least squares methods by criteria of bias in the parameters relative to the parameter values used to generate the data and standard deviations in the parameter estimates. The standard deviations in the parameter values can be used as measures of the variability in parameter estimates from fits to experimental data. We introduced contaminant reaction times and variability into the other components of processing besides the decision process and found that the maximum likelihood and chi-square methods failed, sometimes dramatically. But the weighted least squares method was robust to these two factors. We then present results from modifications of the maximum likelihood and chi-square methods, in which these factors are explicitly modeled, and show that the parameter values of the diffusion model are recovered well. We argue that explicit modeling is an important method for addressing contaminants and variability in nondecision processes and that it can be applied in any theoretical approach to modeling reaction time.     

Hierarchical Multinomial Processing Tree Models: A Latent-Class Approach

Multinomial processing tree models are widely used in many areas of psychology. Their application relies on the assumption of parameter homogeneity, that is, on the assumption that participants do not differ in their parameter values. Tests for parameter homogeneity are proposed that can be routinely used as part of multinomial model analyses to defend the assumption. If parameter homogeneity is found to be violated, a new family of models, termed latent-class multinomial processing tree models, can be applied that accommodates parameter heterogeneity and correlated parameters, yet preserves most of the advantages of the traditional multinomial method. Estimation, goodness-of-fit tests, and tests of other hypotheses of interest are considered for the new family of models. The author thanks Bill Batchelder, Edgar Erdfelder, Thorsten Meiser, and Christoph Stahl for helpful comments on a previous version of this paper. The author is also grateful to Edgar Erdfelder for making available the data set analyzed in this paper.     

To Pass or Not to Pass: A Mathematical Model for Competitive Interactions in Rugby Union

ABSTRACT. Predicting behavior has been a main challenge in human movement science. An important step within the theory of coordination dynamics is to find out the rules that govern human behavior by defining order parameters and control parameters that support mathematical models to predict the behavior of a system. Models to describe human coordination have been focused on interlimb coordination and on interpersonal coordination in affiliative tasks but not on competitive tasks. This article aims to present a formal model with two attractors to describe the interactive behavior on a 2v1 system in rugby union. Interpersonal distance and relative velocity critical values were empirically identified and were included as task constraints that define the attractor landscape. It is shown that using relative velocity as a control parameter the model offers reasonable prediction concerning the decision-making process. The model has the plasticity to adapt to other settings where interpersonal distances and relative velocities amongst system components act as significant task constraints.     

Global model analysis by parameter space partitioning

To model behavior, scientists need to know how models behave. This means learning what other behaviors a model can produce besides the one generated by participants in an experiment. This is a difficult problem because of the complexity of psychological models (e.g., their many parameters) and because the behavioral precision of models (e.g., interval-scale performance) often mismatches their testable precision in experiments, where qualitative, ordinal predictions are the norm. Parameter space partitioning is a solution that evaluates model performance at a qualitative level. There exists a partition on the model's parameter space that divides it into regions that correspond to each data pattern. Three application examples demonstrate its potential and versatility for studying the global behavior of psychological models.     

Latent Differential Equation Modeling of Self-Regulatory and Coregulatory Affective Processes

We examine emotion self-regulation and coregulation in romantic couples using daily self-reports of positive and negative affect. We fit these data using a damped linear oscillator model specified as a latent differential equation to investigate affect dynamics at the individual level and coupled influences for the 2 partners in each couple. Results indicate an absence of damping of relationship-specific affect within individuals in the sample. When both positive and negative affect are modeled at the individual level, the influence of positive affect is greater than that of negative affect. At the dyad level, the findings indicate coupled influences in both positive and negative affect between partners. With regard to positive affect, females are sensitive to their partners' overall displacement from average as well as their rate of change; males are sensitive only to their partners' displacement from average. For negative affect both partners are sensitive to each other's displacement from average, yet there are no coupled influences for rates of change in this dimension. We interpret the influence of the parameters on the system by examining the expected behavior of the system as a function of varying parameter values.     

Integrative processing of perception and reward in an auditory localization paradigm

Under natural conditions, human beings routinely have to choose among multiple alternatives which are associated with specific outcomes of varying desirability. Typically, decisions are based upon the processing of perceptual input, which introduces additional noise to the system. Bayesian decision theory (BDT) allows us to formalize decision under risk and to predict statistically optimal choice behavior. In the present study, human observers performed a classification task characterized by an extensive amount of perceptual uncertainty (auditory localization). In addition, a spatial reward function was imposed on the task. We set up a BDT model with no free parameters to serve as a benchmark for statistically optimal choices, and tested it against a purely perceptual model and a hybrid, heuristic model. In addition, we tested these three models with free rather than fixed parameters for the perceptual uncertainty and the peak of the reward function. The log likelihoods of the models given the empirical data were determined by means of Monte Carlo simulations. Bayesian model comparison (BMC) revealed that the BDT model with two free parameters was the most plausible among the tested models. The fitted parameter values for the peak of the reward function were consistently smaller than the actual peak reward communicated to the participants. The results are discussed in the context of an internal underweighting of the reward function.     

The many null distributions of person fit indices

This paper deals with the situation of an investigator who has collected the scores ofn persons to a set ofk dichotomous items, and wants to investigate whether the answers of all respondents are compatible with the one parameter logistic test model of Rasch. Contrary to the standard analysis of the Rasch model, where all persons are kept in the analysis and badly fittingitems may be removed, this paper studies the alternative model in which a small minority ofpersons has an answer strategy not described by the Rasch model. Such persons are called anomalous or aberrant. From the response vectors consisting ofk symbols each equal to 0 or 1, it is desired to classify each respondent as either anomalous or as conforming to the model. As this model is probabilistic, such a classification will possibly involve false positives and false negatives. Both for the Rasch model and for other item response models, the literature contains several proposals for a person fit index, which expresses for each individual the plausibility that his/her behavior follows the model. The present paper argues that such indices can only provide a satisfactory solution to the classification problem if their statistical distribution is known under the null hypothesis that all persons answer according to the model. This distribution, however, turns out to be rather different for different values of the person's latent trait value. This value will be called ability parameter, although our results are equally valid for Rasch scales measuring other attributes.As the true ability parameter is unknown, one can only use its estimate in order to obtain an estimated person fit value and an estimated null hypothesis distribution. The paper describes three specifications for the latter: assuming that the true ability equals its estimate, integrating across the ability distribution assumed for the population, and conditioning on the total score, which is in the Rasch model the sufficient statistic for the ability parameter.Classification rules for aberrance will be worked out for each of the three specifications. Depending on test length, item parameters and desired accuracy, they are based on the exact distribution, its Monte Carlo estimate and a new and promising approximation based on the moments of the person fit statistic. Results for the likelihood person fit statistic are given in detail, the methods could also be applied to other fit statistics. A comparison of the three specifications results in the recommendation to condition on the total score, as this avoids some problems of interpretation that affect the other two specifications.The authors express their gratitude to the reviewers and to many colleagues for comments on an earlier version.     

The structure of equivalence classes

THE STRUCTURE OF EQUIVALENCE CLASSES CAN BE COMPLETELY DESCRIBED BY FOUR PARAMETERS: class size, number of nodes, the distribution of "singles" among nodes, and directionality of training. Class size refers to the number of stimuli in a class. Nodes are stimuli linked by training to at least two other stimuli. Singles are stimuli linked by training to only one other stimulus. The distribution of singles refers to the number of singles linked by training to each node. Directionality of training refers to the use of stimuli as samples and as comparison stimuli in training. These four parameters define the different ways in which the stimuli in a class can be organized, and thus provide a basis for systematically characterizing the properties of stimuli in a given equivalence class. The four parameters can also be used to account for the development of individual differences that are commonly characterized in terms of "understanding" and connotative meaning.Methods are described for generating all possible combinations of parameter values, and a formula is introduced which specifies all of the parameter values for an equivalence class. Its utility for interrelating experimental procedures is demonstrated by analyzing a number of representative experiments that have addressed equivalence-class formation.     

The Neurodynamics of Cognition: A Tutorial on Computational Cognitive Neuroscience

Computational Cognitive Neuroscience (CCN) is a new field that lies at the intersection of computational neuroscience, machine learning, and neural network theory (i.e., connectionism). The ideal CCN model should not make any assumptions that are known to contradict the current neuroscience literature and at the same time provide good accounts of behavior and at least some neuroscience data (e.g., single-neuron activity, fMRI data). Furthermore, once set, the architecture of the CCN network and the models of each individual unit should remain fixed throughout all applications. Because of the greater weight they place on biological accuracy, CCN models differ substantially from traditional neural network models in how each individual unit is modeled, how learning is modeled, and how behavior is generated from the network. A variety of CCN solutions to these three problems are described. A real example of this approach is described, and some advantages and limitations of the CCN approach are discussed.     

Inferring the structure of latent class

Present optimization techniques in latent class analysis apply the expectation maximization algorithm or the Newton-Raphson algorithm for optimizing the parameter values of a prespecified model. These techniques can be used to find maximum likelihood estimates of the parameters, given the specified structure of the model, which is defined by the number of classes and, possibly, fixation and equality constraints. The model structure is usually chosen on theoretical grounds. A large variety of structurally different latent class models can be compared using goodness-of-fit indices of the chi-square family, Akaike’s information criterion, the Bayesian information criterion, and various other statistics. However, finding the optimal structure for a given goodness-of-fit index often requires a lengthy search in which all kinds of model structures are tested. Moreover, solutions may depend on the choice of initial values for the parameters. This article presents a new method by which one can simultaneously infer the model structure from the data and optimize the parameter values. The method consists of a genetic algorithm in which any goodness-of-fit index can be used as a fitness criterion. In a number of test cases in which data sets from the literature were used, it is shown that this method provides models that fit equally well as or better than the models suggested in the original articles.     

Cognitive models of risky choice: parameter stability and predictive accuracy of prospect theory

In the behavioral sciences, a popular approach to describe and predict behavior is cognitive modeling with adjustable parameters (i.e., which can be fitted to data). Modeling with adjustable parameters allows, among other things, measuring differences between people. At the same time, parameter estimation also bears the risk of overfitting. Are individual differences as measured by model parameters stable enough to improve the ability to predict behavior as compared to modeling without adjustable parameters? We examined this issue in cumulative prospect theory (CPT), arguably the most widely used framework to model decisions under risk. Specifically, we examined (a) the temporal stability of CPT’s parameters; and (b) how well different implementations of CPT, varying in the number of adjustable parameters, predict individual choice relative to models with no adjustable parameters (such as CPT with fixed parameters, expected value theory, and various heuristics). We presented participants with risky choice problems and fitted CPT to each individual’s choices in two separate sessions (which were 1 week apart). All parameters were correlated across time, in particular when using a simple implementation of CPT. CPT allowing for individual variability in parameter values predicted individual choice better than CPT with fixed parameters, expected value theory, and the heuristics. CPT’s parameters thus seem to pick up stable individual differences that need to be considered when predicting risky choice.     

Prior sensitivity in theory testing: An apologia for the Bayes factor

A commonly voiced concern with the Bayes factor is that, unlike many other Bayesian and non-Bayesian quantitative measures of model evaluation, it is highly sensitive to the parameter prior. This paper argues that, when dealing with psychological models that are quantitatively instantiated theories, being sensitive to the prior is an attractive feature of a model evaluation measure. This assertion follows from the observation that in psychological models parameters are not completely unknown, but correspond to psychological variables about which theory often exists. This theory can be formally captured in the prior range and prior distribution of the parameters, indicating which parameter values are allowed, likely, unlikely and forbidden. Because the prior is a vehicle for expressing psychological theory, it should, like the model equation, be considered as an integral part of the model. It is argued that the combined practice of building models using informative priors, and evaluating models using prior sensitive measures advances knowledge.     

Minimax d-optimal designs for item response theory models

Various different item response theory (IRT) models can be used in educational and psychological measurement to analyze test data. One of the major drawbacks of these models is that efficient parameter estimation can only be achieved with very large data sets. Therefore, it is often worthwhile to search for designs of the test data that in some way will optimize the parameter estimates. The results from the statistical theory on optimal design can be applied for efficient estimation of the parameters.A major problem in finding an optimal design for IRT models is that the designs are only optimal for a given set of parameters, that is, they are locally optimal. Locally optimal designs can be constructed with a sequential design procedure. In this paper minimax designs are proposed for IRT models to overcome the problem of local optimality. Minimax designs are compared to sequentially constructed designs for the two parameter logistic model and the results show that minimax design can be nearly as efficient as sequentially constructed designs.     

Beta-MPT: Multinomial processing tree models for addressing individual differences

Traditionally, multinomial processing tree (MPT) models are applied to groups of homogeneous participants, where all participants within a group are assumed to have identical MPT model parameter values. This assumption is unreasonable when MPT models are used for clinical assessment, and it often may be suspect for applications to ordinary psychological experiments. One method for dealing with parameter variability is to incorporate random effects assumptions into a model. This is achieved by assuming that participants’ parameters are drawn independently from some specified multivariate hyperdistribution. In this paper we explore the assumption that the hyperdistribution consists of independent beta distributions, one for each MPT model parameter. These beta-MPT models are ‘hierarchical models’, and their statistical inference is different from the usual approaches based on data aggregated over participants. The paper provides both classical (frequentist) and hierarchical Bayesian approaches to statistical inference for beta-MPT models. In simple cases the likelihood function can be obtained analytically; however, for more complex cases, Markov Chain Monte Carlo algorithms are constructed to assist both approaches to inference. Examples based on clinical assessment studies are provided to demonstrate the advantages of hierarchical MPT models over aggregate analysis in the presence of individual differences.     

Ambiguous comparative judgment: Fuzzy set model and data analysis

Abstract:  This paper proposed two types of fuzzy set models for ambiguous comparative judgments, which did not always hold transitivity and comparability properties. The first type of model was a fuzzy theoretical extension of the additive difference model for preference that was used to explain ambiguous preference strength. The second was a fuzzy logic model for explaining ambiguous preference in which preference strength was bounded, such as a probability measure. In both models, multi-attribute weighting parameters and all attribute values were assumed to be asymmetric fuzzy L-R numbers. For each model, a method of parameter estimation using fuzzy regression analysis was proposed. Numerical examples were also provided for comparison. Finally, the theoretical and practical implications of the proposed models were discussed.     

Bridging the gap between physiology and behavior: evidence from the sSoTS model of human visual attention

We present the case for a role of biologically plausible neural network modeling in bridging the gap between physiology and behavior. We argue that spiking-level networks can allow "vertical" translation between physiological properties of neural systems and emergent "whole-system" performance-enabling psychological results to be simulated from implemented networks and also inferences to be made from simulations concerning processing at a neural level. These models also emphasize particular factors (e.g., the dynamics of performance in relation to real-time neuronal processing) that are not highlighted in other approaches and that can be tested empirically. We illustrate our argument from neural-level models that select stimuli by biased competition. We show that a model with biased competition dynamics can simulate data ranging from physiological studies of single-cell activity (Study 1) to whole-system behavior in human visual search (Study 2), while also capturing effects at an intermediate level, including performance breakdown after neural lesion (Study 3) and data from brain imaging (Study 4). We also show that, at each level of analysis, novel predictions can be derived from the biologically plausible parameters adopted, which we proceed to test (Study 5). We argue that, at least for studying the dynamics of visual attention, the approach productively links single-cell to psychological data.     

Model inference or model selection: discussion of Klugkist, Laudy, and Hoijtink (2005)

I. Klugkist, O. Laudy, and H. Hoijtink (2005) presented a Bayesian approach to analysis of variance models with inequality constraints. Constraints may play 2 distinct roles in data analysis. They may represent prior information that allows more precise inferences regarding parameter values, or they may describe a theory to be judged against the data. In the latter case, the authors emphasized the use of Bayes factors and posterior model probabilities to select the best theory. One difficulty is that interpretation of the posterior model probabilities depends on which other theories are included in the comparison. The posterior distribution of the parameters under an unconstrained model allows one to quantify the support provided by the data for inequality constraints without requiring the model selection framework.     

An associative model of geometry learning: a modified choice rule

In a recent article, the authors (Miller & Shettleworth, 2007) showed how the apparently exceptional features of behavior in geometry learning ("reorientation") experiments can be modeled by assuming that geometric and other features at given locations in an arena are learned competitively as in the Rescorla-Wagner model and that the probability of visiting a location is proportional to the total associative strength of cues at that location relative to that of all relevant locations. Reinforced or unreinforced visits to locations drive changes in associative strengths. Dawson, Kelly, Spetch, and Dupuis (2008) have correctly pointed out that at parameter values outside the ranges the authors used to simulate a body of real experiments, our equation for choice probabilities can give impossible and/or wildly fluctuating results. Here, the authors show that a simple modification of the choice rule eliminates this problem while retaining the transparent way in which the model relates spatial choice to competitive associative learning of cue values.     

An Idiographic Approach to Estimating Models of Dyadic Interactions with Differential Equations

We present an idiographic approach to modeling dyadic interactions using differential equations. Using data representing daily affect ratings from romantic relationships, we examined several models conceptualizing different types of dyadic interactions. We fitted each model to each of the dyads and the resulting AICc values were used to classify the most likely configuration of interaction for each dyad. Additionally, the AICc from the different models were used in parameter averaging across models. Averaged parameters were used in models involving predictors of relationship dynamics, as indexed by these parameters, as well as models wherein the parameters predicted distal outcomes of the dyads such as relationship satisfaction and status. Results indicated that, within our sample, the most likely interaction style was that of independence, without evidence of emotional interrelations between the two individuals in the couple. Attachment-related avoidance and anxiety showed significant relations with model parameters, such that ideal levels of affect for males were negatively influenced by higher levels of avoidance from their partner while their own levels of anxiety had positive effects on their levels of dyadic coregulation. For females coregulation was negatively influenced by both time in the relationship and their partner’s level of avoidance. Analysis involving distal outcomes showed modest influences from the individual’s level of ideal affect.     

Extending General Processing Tree Models to Analyze Reaction Time Experiments

General processing tree (GPT) models are usually used to analyze categorical data collected in psychological experiments. Such models assume functional relations between probabilities of the observed behavior categories and the unobservable choice probabilities involved in a cognitive task. This paper extends GPT models for categorical data to the analysis of continuous data in a class of response time (RT) experiments in cognitive psychology. Suppose that a cognitive task involves several discrete processing stages and both accuracy (categorical) and latency (continuous) measures are obtained for each of the response categories. Furthermore, suppose that the task can be modeled by a GPT model that assumes serialization among the stages. The observed latencies of the response categories are functions of the choice probabilities and processing times (PT) at each of the processing stages. The functional relations are determined by the processing structure of the task. A general framework is presented and it is applied to a set of data obtained from a source monitoring experiment. Copyright 2001 Academic Press.