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.     

Parameter identification in multinomial processing tree models

Multinomial processing tree models form a popular class of statistical models for categorical data that have applications in various areas of psychological research. As in all statistical models, establishing which parameters are identified is necessary for model inference and selection on the basis of the likelihood function, and for the interpretation of the results. The required calculations to establish global identification can become intractable in complex models. We show how to establish local identification in multinomial processing tree models, based on formal methods independently proposed by Catchpole and Morgan (1997) and by Bekker, Merckens, and Wansbeek (1994). This approach is illustrated with multinomial processing tree models for the source-monitoring paradigm in memory research.     

MPTinR: Analysis of multinomial processing tree models in R

We introduce MPTinR, a software package developed for the analysis of multinomial processing tree (MPT) models. MPT models represent a prominent class of cognitive measurement models for categorical data with applications in a wide variety of fields. MPTinR is the first software for the analysis of MPT models in the statistical programming language R, providing a modeling framework that is more flexible than standalone software packages. MPTinR also introduces important features such as (1) the ability to calculate the Fisher information approximation measure of model complexity for MPT models, (2) the ability to fit models for categorical data outside the MPT model class, such as signal detection models, (3) a function for model selection across a set of nested and nonnested candidate models (using several model selection indices), and (4) multicore fitting. MPTinR is available from the Comprehensive R Archive Network at http://cran.r-project.org/web/packages/MPTinR/.     

HMMTree: A computer program for latent-class hierarchical multinomial processing tree models

Latent-class hierarchical multinomial models are an important extension of the widely used family of multinomial processing tree models, in that they allow for testing the parameter homogeneity assumption and provide a framework for modeling parameter heterogeneity. In this article, the computer program HMMTree is introduced as a means of implementing latent-class hierarchical multinomial processing tree models. HMMTree computes parameter estimates, confidence intervals, and goodness-of-fit statistics for such models, as well as the Fisher information, expected category means and variances, and posterior probabilities for class membership. A brief guide to using the program is provided.     

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.     

Bounds on variances of estimators for multinomial processing tree models

When there are order constraints among the parameters of a binary, multinomial processing tree (MPT) model, methods have been developed for reparameterizing the constrained MPT into an equivalent unconstrained MPT. This note provides a theorem that is useful in computing bounds on the estimator variances for the parameters of the constrained model in terms of estimator variances of the parameters of the unconstrained model. In particular, we show that if X and Y are random variables taking values in [0,1], then Var[XY]?2(Var[X]+Var[Y]).     

Representing parametric order constraints in multi-trial applications of multinomial processing tree models

Binary multinomial processing tree (MPT) models parameterize the multinomial distribution over a set of J categories, such that each of its parameters, θ₁,θ₂,…,θ_S, is functionally independent and free to vary in the interval [0,1]. This paper analyzes binary MPT models subject to parametric order-constraints of the form 0?θ_s?θ_t?1. Such constraints arise naturally in multi-trial learning and memory paradigms, where some parameters representing cognitive processes would naturally be expected to be non-decreasing over learning trials or non-increasing over forgetting trials. The paper considers the case of one or more, non-overlapping linear orders of parametric constraints. Several ways to reparameterize the model to reflect the constraints are presented, and for each it is shown how to construct a new binary MPT that has the same number of parameters and is statistically equivalent to the original model with the order constraints. The results both extend the mathematical analysis of the MPT class as well as offering an approach to order restricted inference at the level of the entire class. An empirical example of this approach is provided.     

Appletree: A multinomial processing tree modeling program for macintosh computers

Multinomial processing tree (MPT) models are statistical models that allow for the prediction of categorical frequency data by sets of unobservable (cognitive) states. In MPT models, the probability that an event belongs to a certain category is a sum of products of state probabilities. AppleTree is a computer program for Macintosh for testing user-defined MPT models. It can fit model parameters to empirical frequency data, provide confidence intervals for the parameters, generate tree graphs for the models, and perform identifiability checks. In this article, the algorithms used by AppleTree and the handling of the program are described.     

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.     

Cognitive psychometrics: assessing storage and retrieval deficits in special populations with multinomial processing tree models

This article demonstrates how multinomial processing tree models can be used as assessment tools to measure cognitive deficits in clinical populations. This is illustrated with a model developed by W. H. Batchelder and D. M. Riefer (1980) that separately measures storage and retrieval processes in memory. The validity of the model is tested in 2 experiments, which show that presentation rate affects the storage of items (Experiment 1) and part-list cuing hurts item retrieval (Experiment 2). Experiments 3 and 4 examine 2 clinical populations: schizophrenics and alcoholics with organic brain damage. The model reveals that each group exhibits deficits in storage and retrieval, with the retrieval deficits being stronger and occurring more consistently over trials. Also, the alcoholics with organic brain damage show no improvement in retrieval over trials, although their storage improves at the same rate as a control group.     

A multinomial processing tree model for degradation and redintegration in immediate recall

When items are presented for immediate recall, a verbal trace is formed and degrades quickly, becoming useless after about 2 sec. The span for items such as digits equals the number of items that canbe pronounced in the available time. The length of the items affects span by affecting pronunciation rate. Other properties, such as phonological similarity and lexicality, can affect span without affecting pronunciation rate. These properties change the trace's useful lifetime by affecting redintegration. An analogy is drawn between trace reconstruction and repair of errors in speech. When a trace is degraded, one process attempts to form a phoneme string, and another process attempts to form a word. The two processes are autonomous and can be selectively influenced by lexicality and phonological similarity. The resulting processing tree models make simple predictions that depend on whether or not the influenced processes are sequential. The results are illustrated with data from experiments by Besner and Davelaar (1982).     

The impact of subjective recognition experiences on recognition heuristic use: A multinomial processing tree approach

The recognition heuristic (RH) theory states that, in comparative judgments (e.g., Which of two cities has more inhabitants?), individuals infer that recognized objects score higher on the criterion (e.g., population) than unrecognized objects. Indeed, it has often been shown that recognized options are judged to outscore unrecognized ones (e.g., recognized cities are judged as larger than unrecognized ones), although different accounts of this general finding have been proposed. According to the RH theory, this pattern occurs because the binary recognition judgment determines the inference and no other information will reverse this. An alternative account posits that recognized objects are chosen because knowledge beyond mere recognition typically points to the recognized object. A third account can be derived from the memory-state heuristic framework. According to this framework, underlying memory states of objects (rather than recognition judgments) determine the extent of RH use: When two objects are compared, the one associated with a “higher” memory state is preferred, and reliance on recognition increases with the “distance” between their memory states. The three accounts make different predictions about the impact of subjective recognition experiences—whether an object is merely recognized or recognized with further knowledge—on RH use. We estimated RH use for different recognition experiences across 16 published data sets, using a multinomial processing tree model. Results supported the memory-state heuristic in showing that RH use increases when recognition is accompanied by further knowledge.     

Multinomial processing tree models: An implementation

Multinomial processing tree (MPT) models have been widely used by researchers in cognitive psychology. This paper introduces MBT.EXE, a computer program that makes MPT easy to use for researchers. MBT.EXE implements the statistical theory developed by Hu and Batchelder (1994). This user-friendly software can be used to construct MPT models and conduct statistical inferences, including point and interval estimation, hypothesis testing, and goodness of fit. Furthermore, this program can be used to examine the robustness of MPT models. Algorithms for parameter estimation, hypothesis testing, and Monte Carlo simulation are presented.     

Serial processing in reading aloud: reply to Zorzi (2000)

K. Rastle and M. Coltheart (1999; see also M. Coltheart & K. Rastle, 1994) reported data demonstrating that the cost of irregularity in reading aloud low-frequency exception words is modulated by the position of the irregularity in the word. They argued that these data implicated a serial process and falsified all models of reading aloud that operate solely in parallel, a conclusion that M. Zorzi (2000) challenged by successfully simulating the position of irregularity effect with such a model. Zorzi (2000) further claimed that a reanalysis of K. Rastle and M. Coltheart's (1999) data demonstrates sensitivity to grapheme-phoneme consistency (which he claimed was confounded across the position of irregularity manipulation) rather than the use of a serial process. Here, the authors argue that M. Zorzi's (2000) reanalyses were inappropriate and reassert that K. Rastle and M. Coltheart's (1999) findings are evidence for serial processing.     

The response of counsellors to apartheid: A reply to Dryden (1990) and De Jager (1992)

This reply to Dryden (1990) and De Jager (1992) explores the role of organised counselling psychology in challenging, maintaining or promoting apartheid. Counsellors have an important role to play in newly racially integrated tertiary institutions. This role should be informed by an acknowledgement of the central involvement of race and politics within counselling psychology so that appropriate politically contextualised organisational interventions can be implemented.     

Redintegration and the benefits of long-term knowledge in verbal short-term memory: an evaluation of Schweickert's (1993) multinomial processing tree model

The impact of four long-term knowledge variables on serial recall accuracy was investigated. Serial recall was tested for high and low frequency words and high and low phonotactic frequency nonwords in 2 groups: monolingual English speakers and French-English bilinguals. For both groups the recall advantage for words over nonwords reflected more fully correct recalls with fewer recall attempts that consisted of fragments of the target memory items (one or two of the three target phonemes recalled correctly); completely incorrect recalls were equivalent for the 2 list types. However, word frequency (for both groups), nonword phonotactic frequency (for the monolingual group), and language familiarity all influenced the proportions of completely incorrect recalls that were made. These results are not consistent with the view that long-term knowledge influences on immediate recall accuracy can be exclusively attributed to a redintegration process of the type specified in multinomial processing tree model of immediate recall. The finding of a differential influence on completely incorrect recalls of these four long-term knowledge variables suggests instead that the beneficial effects of long-term knowledge on short-term recall accuracy are mediated by more than one mechanism.     

Compounds,confounds, and models in developmental information processing: A reply to Trabasso and Foellinger

A point by point examination of Trabasso and Foellinger's paper shows their criticisms of my 1970 work to be based on errors of fact (regarding the data I reported) and errors of method (regarding proper procedures for model evaluation). Factual errors are refuted by summarizing crucial data reported in 1970 but ignored by my critics. Errors of method are refuted by contrasting my $\text{CSVI}\text{BE}$ model and data with Trabasso and Foellinger's model and data in the light of the scientific epistemology of model evaluation. The issues of “general” versus “local” models, their “empirical scope,” “number of empirical parameters,” and “simplifying assumptions” are examined. Five ways of evaluating models are distinguished, and in all five my 1970 model is shown to be superior to that of Trabasso and Foellinger. The 1970 data (in light of the controls built into the CSVI task) and new unreported data exhibiting developmental step functions confirm the model of a developmental growth of M (mental energy, working memory), which occurs concurrently with and independently from the growth of executive/control structures. Executive growth alone cannot explain the obtained results.     

The role of orthographic and phonological codes in the word and the pseudoword superiority effect: an analysis by means of multinomial processing tree models

Central to the current accounts of the word and the pseudoword superiority effect (WSE and PWSE, respectively) is the concept of a unitized code that is less susceptible to masking than single-letter codes. Current explanations of the WSE and PWSE assume that this unitized code is orthographic, explaining these phenomena by the assumption of dual read-out from unitized and single-letter codes. In this article, orthographic dual read-out models are compared with a phonological dual read-out model (which is based on the assumption that the 1st unitized code is phonological). From this phonological code, an orthographic code is derived, through either lexical access or assembly. Comparison of the orthographic and phonological dual read-out models was performed by formulating both models as multinomial processing tree models. From an application of these models to the data of 2 letter identification experiments, it was clear that the orthographic dual read-out models are insufficient as an explanation of the PWSE, whereas the phonological dual read-out model is sufficient.