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/.     

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.     

Different storage and retrieval deficits in normal aging and mild cognitive impairment: a multinomial modeling analysis

The authors compared patients with mild cognitive impairment with healthy older adults and young control participants in a free recall test in order to locate potential qualitative differences in normal and pathological memory decline. Analysis with an extended multitrial version of W. H. Batchelder and D. M. Riefer's (1980) pair-clustering model revealed globally decelerated learning and an additional retrieval deficit in patients with mild cognitive impairment but not in healthy older adults. Results thus suggest differences in memory decline between normal and pathological aging that may be useful for the detection of risk groups for dementia, and they illustrate the value of model-based disentangling of processes and of multitrial tests for early detection of dementia.     

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.     

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.     

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.     

Encoding, maintenance, and retrieval processes in the lag effect: a multinomial processing tree analysis

Short-term studies on repeated learning of verbatim material have typically revealed an overall benefit of long lags compared to short lags between repetitions. This has been referred to as the lag effect. On educationally relevant time scales, however, an inverted-U-shaped relation between lag and memory performance is often observed. Recently, Cepeda et al. (2009) showed that the optimal lag for relearning depends heavily on the time interval between the last learning session and the final memory test (i.e., the retention interval; RI). In order to explore the cognitive mechanisms underlying this result in more detail we independently manipulated both the lag and the RI in a 3×2 experimental design and analysed our data using a multinomial processing tree model for free-then-cued-recall data. Our results reveal that the lag effect trends are mainly driven by encoding and maintenance processes rather than by retrieval mechanisms. Our findings have important implications for theories of the lag effect.     

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.     

Benefits of response time-extended multinomial processing tree models: A reply to Starns (2018)

Psychonomic Bulletin & Review - In his comment on Heck and Erdfelder (2016, Psychonomic Bulletin & Review, 23, 1440–1465), Starns (2018, Psychonomic Bulletin & Review, 25,...     

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.     

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).     

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.     

Cognitive processing deficits in reading disabilities: A prefrontal cortical hypothesis

Much research investigating the neuropsychological underpinnings of reading disabilities has emphasized posterior brain regions. However, recent evidence indicates that prefrontal cortex may also play a role. This study investigated cognitive processes that are associated with prefrontal and posterior brain functions. Subjects were 12-year-old reading disabled and nondisabled boys. Discriminant analysis procedures indicated that measures of prefrontal functions distinguished between the two groups better than measures of posterior functions. The results suggest that reading disabled boys have difficulty with cognitive processes involving selective and sustained attention, inhibition of routinized responses, set maintenance, flexibility in generating and testing alternative hypotheses, and phonemically based language production.     

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.     

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.