A context-free language for binary multinomial processing tree models

This paper provides a new formalization for the class of binary multinomial processing tree (BMPT) models, and theorems for the class are developed using the formalism. MPT models are a popular class of information processing models for categorical data in specific cognitive paradigms. They have a recursive structure that is productively described with the tools of formal language and computation theory. We provide an axiomatization that characterizes BMPT models as strings in a context-free language, and then we add model-theoretic axioms and definitions to interpret the strings as parameterized probabilistic models for categorical data. The language for BMPT models is related to the Full Binary Tree language, a well-studied context-free language. Once BMPT models are viewed from the perspective of the Full Binary Tree language, a number of theoretical and computational results can be developed. In particular, we have a number of results concerning the enumerations of BMPT models as well as the identifiability of subclasses of these models.     

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.     

Pooling data versus averaging model fits for some prototypical multinomial processing tree models

For a study with multinomial data where there are n_g individuals and with each person having n_r test trials, the question arises as to how to fit the parameters of a multinomial processing tree (MPT) model. Should each parameter be estimated for each individual and then averaged to obtain a group estimate, or should the frequencies in the multinomial categories be pooled so that the model is fit once for the entire group? This basic question is explored with a series of Monte Carlo simulations for some prototypical MPT models. There is a general finding of a pooling advantage for the case where there is a single experimental condition. Also when there are different experimental conditions, there is reduced bias for detecting condition differences for a method based on the pooled data. Although the focus of the paper is on multinomial models, a general theorem is advanced that establishes a basic condition that determines whether there is or is not a difference between the averaging of individual estimates and the estimate based on the pooled data.     

Parameterizing inequality constraints on unique variances in linear structural models

Current computer programs for analyzing linear structural models will apparently handle only two types of constraints: fixed parameters, and equality of parameters. An important constraint not handled is inequality; this is particularly crucial for preventing negative variance estimates. In this paper, a method is described for imposing several kinds of inequality constraints in models, without the necessity for having computer programs which explicitly allow such constraints. The examples discussed include the prevention of Heywood cases, extension to inequalities of parameters to be greater than a specified value, and imposing ordered inequalities. Work on this project was aided by the City University of New York—Professional Staff Congress Research Award Program Grant Number 13631.     

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.     

multiTree: A computer program for the analysis of multinomial processing tree models

Multinomial processing tree (MPT) models are a family of stochastic models for psychology and related sciences that can be used to model observed categorical frequencies as a function of a sequence of latent states. For the analysis of such models, the present article presents a platform-independent computer program called multiTree, which simplifies the creation and the analysis of MPT models. This makes them more convenient to implement and analyze. Also, multiTree offers advanced modeling features. It provides estimates of the parameters and their variability, goodness-of-fit statistics, hypothesis testing, checks for identifiability, parametric and nonparametric bootstrapping, and power analyses. In this article, the algorithms underlying multiTree are given, and a user guide is provided. The multiTree program can be downloaded from http://psycho3.uni-mannheim.de/multitree.     

The statistical analysis of general processing tree models with the EM algorithm

Multinomial processing tree models assume that an observed behavior category can arise from one or more processing sequences represented as branches in a tree. These models form a subclass of parametric, multinomial models, and they provide a substantively motivated alternative to loglinear models. We consider the usual case where branch probabilities are products of nonnegative integer powers in the parameters, 0_s1, and their complements, 1 - _s. A version of the EM algorithm is constructed that has very strong properties. First, the E-step and the M-step are both analytic and computationally easy; therefore, a fast PC program can be constructed for obtaining MLEs for large numbers of parameters. Second, a closed form expression for the observed Fisher information matrix is obtained for the entire class. Third, it is proved that the algorithm necessarily converges to a local maximum, and this is a stronger result than for the exponential family as a whole. Fourth, we show how the algorithm can handle quite general hypothesis tests concerning restrictions on the model parameters. Fifth, we extend the algorithm to handle the Read and Cressie power divergence family of goodness-of-fit statistics. The paper includes an example to illustrate some of these results.This research was supported by National Science Foundation Grant BNS-8910552 to William H. Batchelder and David M. Riefer. We are grateful to David Riefer for his useful comments, and to the Institute for Mathematical Behavior Sciences for its support.     

Likelihood inference on the underlying structure of IRT models

The assumptions underlying item response theory (IRT) models may be expressed as a set of equality and inequality constraints on the parameters of a latent class model. It is well known that the same assumptions imply that the parameters of the manifest distribution have to satisfy a more complicated set of inequality constraints which, however, are necessary but not sufficient. In this paper, we describe how the theory for likelihood-based inference under equality and inequality constraints may be used to test the underlying assumptions of IRT models. It turns out that the analysis based directly on the latent structure is simpler and more flexible. In particular, we indicate how several interesting extensions of the Rasch model may be obtained by partial relaxation of the basic constraints. An application to a data set provided by Educational Testing Service is used to illustrate the approach.We thank Dr. Gorman and Dr. Rogers of the Educational Testing Service for providing the data analyzed in Section 4. We also thank three reviewers for comments and suggestions.This revised article was published online in August 2005 with the PDF paginated correctly.     

Effects of resource demanding processing on context memory for context-related versus context-unrelated items

In two experiments, context memory was compared between conditions of higher versus lower cognitive load. The ease of the item–context binding was manipulated using words whose meaning was preexperimentally related versus unrelated to context (colour). Experiment 1 showed that an item generation task may lead to a decrease in memory for font colour without enhancing item recognition. Experiment 2 used a dual-task paradigm, that is, participants performed a concurrent task during the study phase of the memory task. The divided-attention condition decreased both item and context memory. However, in both experiments, the influence of resource demanding conditions on context memory was apparent only when the item–context binding process was not facilitated by preexperimental associations.     

A novel method for assessing rival models of recognition memory

A general comparison is made between the multinomial processing tree (MPT) approach and a strength-based approach for modeling recognition memory measurement. Strength models include the signal-detection model and the dual-process model. Existing MPT models for recognition memory and a new generic MPT model, called the Multistate (MS) model, are contrasted with the strength models. Although the ROC curves for the MS model and strength model are similar, there is a critical difference between existing strength models and MPT models that goes beyond the assessment of the ROC. This difference concerns the question of stochastic mixtures for foil test trials. The hazard function and the reverse hazard function are powerful methods for detecting the presence of a probabilistic mixture. Several new theorems establish a novel method for obtaining information about the hazard function and reverse hazard function for the latent continuous distributions that are assumed in the strength approach to recognition memory. Evidence is provided that foil test trials involve a stochastic mixture. This finding occurred for both short-term memory procedures, such as the Brown–Peterson task, and long-term list-learning procedures, such as the paired-associate task. The effect of mixtures on foil trials is problematic for existing strength models but can be readily handled by MPT models such as the MS model. Other phenomena, such as the mirror effect and the effect of target-foil similarity, are also predicted accurately by the MPT modeling framework.     

The online processing of only if and even if conditional statements: Implications for mental models

A sentential connective like only if or even if merges two simple propositions into a complex statement. This study used a visual world paradigm experiment to explore how this merging process proceeds online. We first presented participants with a short animation, illustrating different simple propositions that are possible to be merged by the sentential connectives. We then auditorily played an only if or an even if statement and recorded participants' eye movements on the concurrent test image. We observed that hearing the sentential connective results in more fixations on the tokens of the appropriate propositions that are eligible to be merged by the sentential connective. Each sentential connective elicited anticipatory effect suggests that once they heard the sentential connective, participants knew which propositions could be merged. We then discussed the implications of our results to the mental model theory of conditionals and the experimental studies reported in literature.     

The influence of contextual contrast on syntactic processing: evidence for strong-interaction in sentence comprehension

The present study compares the processing of unambiguous restrictive and non-restrictive relative clauses (RCs) within both a null context and a supportive discourse using a self-paced reading methodology. Individuals read restrictive RCs more slowly than non-restrictive RCs in a null context, but processed restrictive RCs faster than non-restrictive RCs in supportive context, resulting in an interaction between context and RC type. These results provide evidence for two theoretical points. First, principles analogous to those in referential theory [Altmann G. T. M., & Steedman, M. (1988). Interaction with context during human sentence processing. Cognition, 30, 191-238; Crain, S., & Steedman, M. (1985). On not being led up the garden path: The use of context by the psychological parser. In D. Dowty, L. Karttunnen, A. Zwicky (Eds.), Natural language parsing. Cambridge, UK: Cambridge University Press] apply not only in resolving ambiguity but also in processing unambiguous sentences. Second, the discourse context can guide and facilitate interpretive processing. This result suggests that intrasentential factors such as syntax are not autonomous from contextual processing, contrary to the modularity hypothesis [Fodor, J. A. (1983). Modularity of mind. Cambridge, MA: MIT Press].     

Font effects of Chinese characters and pseudo-characters on the N400: evidence for an orthographic processing view

Event-related brain potentials (ERPs) were recorded during a Chinese character decision task to examine whether N400 amplitude is modulated by stimulus font. Results revealed large negative-going ERPs in an N400 time window of 300-500ms to stimuli presented in degraded Xing Kai Ti (XKT) font compared with more intact Song Ti (ST) font regardless of whether the stimuli were real or pseudo-characters. ERPs for the pseudo-characters were more negative than for the real characters with similar timing and scalp distribution. The N400-like font effect on amplitude is interpreted as analogous to an N400 stimulus degradation effect, an extension to Holcomb (1993); the degraded perceptual cues provided by XKT supposedly account for this degradation effect. This effect is further interpreted to reflect relative difficulty, which results from orthographic processing difficulty, in retrieving the meaning of XKT stimuli compared with ST stimuli.     

A preliminary examination of the role of psychotherapist fidelity on outcomes of cognitive processing therapy during an RCT for military sexual trauma-related PTSD

While cognitive processing therapy (CPT) is an effective evidence-based treatment for many veterans with military-related post-traumatic stress disorder (PTSD), not all veterans experience therapeutic benefit. To account for the discrepancy in outcomes, researchers have investigated patient- and research design-related factors; however, therapist factors (e.g. fidelity) have received less attention. The present study is a preliminary examination of the effect of psychotherapists’ fidelity during CPT on clinical outcomes during a randomized clinical trial (RCT) for military sexual trauma-related PTSD. PTSD symptoms, trauma-related negative cognitions (NCs), and depression symptoms were assessed for 72 participants at baseline, and 1-week, 2-month, 4-month, and 6-month posttreatment. Of the four CPT therapists, two were found to have significantly poorer (i.e. “below average”) treatment fidelity scores compared to the other two therapists who had “good” treatment fidelity scores. To examine possible therapist effects on outcomes, hierarchical linear modeling was utilized with therapist fidelity entered as a Level 2 predictor. Participants treated by a therapist with “good” treatment fidelity experienced significantly greater reductions in PTSD symptoms, NCs, and depression symptoms than patients treated by a therapist with “below average” treatment fidelity. Our preliminary findings highlight the importance of monitoring, maintaining, and reporting fidelity in psychotherapy treatment RCTs.