Parameter identification in multinomial processing tree models

Abstract

Multinomial processing tree (MPT) models form an increasingly popular class of stochastic models for categorical data that have applications in a variety of research areas in cognitive, differential, and social psychology (e.g., Batchelder & Riefer, 1999Erdfelder et al., 2009;Stahl, 2006). For example, Batchelder and Riefer (1999) discussed over 80 applications of MPT models in various areas of psychology, including memory, perception, and reasoning. The statistical theory of MPT models, including maximum likelihood (ML) parameter estimation, overall model testing, and tests of specific hypotheses within models, has been discussed by Hu and Batchelder (1994) and by Riefer and Batchelder (1988). Flexible software to fit MPT models by means of ML estimation has been developed by Hu and Phillips (1999), Moshagen (2010), Rothkegel (1999), and Stahl and Klauer (2007. MPT models entail a reparameterization of the cell probabilities of the multinomial or product-multinomial distribution (Andersen, 1980;Bishop, Fienberg, & Holland, 1975) in terms of parameters assumed to represent the probabilities of underlying cognitive processes. The underlying cognitive architecture is represented as a rooted tree, in which each branch represents a processing sequence leading to an observable categorical response. More than one branch may terminate in a given response.In MPT models, as in all statistical models, global model identification (the ability to infer unique parameter values from data) is an important issue. A common cause of nonidentifiability of a model is parameter redundancy, in which case the likelihood of the model can be expressed as a function of fewer than the original number of parameters (Catchpole & Morgan, 1997). If a model is not parameter redundant, it is locally, and possibly even globally, identified; we go into the details of local and global identification in terms of MPT models below.In an MPT model, there is a function f ( ) that maps the model's parameter space into the set of possible multinomial probability distributions. Global identification can be established by proving that inequality of two parameter vectors implies inequality of the modeled multi nomial cell probability vectors f ( ) f ( ) for all parameter vectors and in the parameter space Meiser, 2005;Riefer & Batchelder, 1988; see also Bishop et al., 1975, p. 510). The required calculations for establishing global identification can become tedious, and even intractable, in complex models. In this case, the next best thing is to establish local identification; that is, implies f ( ) f ( ) for all in an open neighborhood around . One can empirically investigate local identification at a specific point G in the parameter space by fitting the generating MPT model to artificial (simulated) data generated by G , or to exact summary statistics-that is, the expected cell counts, derived from the probabilities f ( G ) implied by the generating MPT model. Obtaining parameter estimates sufficiently close to the true values G (in the case of simulated...

A New Model for Acquiescence at the Interface of Psychometrics and Cognitive Psychology

“…Given these expected category probabilities, the observed response frequencies are assumed to follow a multinomial distribution. To estimate the model parameters ξ p , they need to be identifiable, which means that identical expected category probabilities Pr(ξ) = Pr(ξ ) must imply identical parameter values ξ = ξ (Batchelder & Riefer, 1999;Schmittmann, Dolan, Raijmakers, & Batchelder, 2010). A necessary (but not sufficient) condition for the identifiability of multinomial models is that the number of parameters does not exceed the number of free categories.…”

MPTinR: Analysis of multinomial processing tree models in R

“…Model identifiability concerns the property that a set of predicted response probabilities Θ lies (Schmittmann, Dolan, Raijmakers, & Batchelder, 2010). An important aspect is that the degrees of freedom provided by a data set provide the upper bound for the number of potentially identifiable free parameters in an model-that is,…”

Extending multinomial processing tree models to measure the relative speed of cognitive processes

“…Moreover, computer algebra software is available facilitating the search for a set of restrictions on the latent RT distributions that ensures identifiability (cf. Schmittmann et al, 2010).…”

“…Therefore, in experiments, it is important to ensure that all of the hypothesized cognitive processes actually occur. Otherwise, it is possible that some of the latent RT distributions are empirically not identified (Schmittmann et al, 2010). One solution to this issue is the analysis of the group frequencies aggregated across participants.…”

“…For larger, more complex MPT models, it can be difficult to proof this one-to-one mapping of parameters to category probabilities analytically (but see Batchelder & Riefer, 1990;Meiser, 2005 for examples of analytical solutions). Therefore, numerical methods based on simulated model identifiability (Moshagen, 2010) or the rank of the Jacobian matrix Schmittmann, Dolan, Raijmakers, & Batchelder, (2010) have been developed to check the local identifiability of MPT models, which ensures the one-to-one relation only in the proximity of a specific set of parameters. All of these methods can readily be applied to RT-extended MPT models.…”