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

Psychon Bull Rev (2016) 23:1440–1465 DOI 10.3758/s13423-016-1025-6 THEORETICAL REVIEW Extending multinomial processing tree models to measure the relative speed of cognitive processes Daniel W. Heck1 · Edgar Erdfelder1 Published online: 16 June 2016 © Psychonomic Society, Inc. 2016 Abstract Multinomial processing tree (MPT) models account for observed categorical responses by assuming a finite number of underlying cognitive processes. We propose a general method that allows for the inclusion of response times (RTs) into any kind of MPT model to measure the relative speed of the hypothesized processes. The approach relies on the fundamental assumption that observed RT distributions emerge as mixtures of latent RT distributions that correspond to different underlying processing paths. To avoid auxiliary assumptions about the shape of these latent RT distributions, we account for RTs in a distribution-free way by splitting each observed category into several bins from fast to slow responses, separately for each individual. Given these data, latent RT distributions are parameterized by probability parameters for these RT bins, and an extended MPT model is obtained. Hence, all of the statistical results and software available for MPT models can easily be used to fit, test, and compare RT-extended MPT models. We demonstrate the proposed method by applying it to the two-high-threshold model of recognition memory. Keywords Cognitive modeling · Response times · Mixture models · Processing speed Electronic supplementary material The online version of this article (doi:10.1007/s13423-016-1025-6) contains supplementary material, which is available to authorized users.  Daniel W. Heck 1 Department of Psychology, School of Social Sciences, University of Mannheim, Schloss EO 254, 68131 Mannheim, Germany Many substantive psychological theories assume that observed behavior results from one or more latent cognitive processes. Because these hypothesized processes can often not be observed directly, measurement models are important tools to test the assumed cognitive structure and to obtain parameters quantifying the probabilities that certain underlying processing stages take place or not. Multinomial processing tree models (MPT models; Batchelder & Riefer, 1990) provide such a means by modeling observed, categorical responses as originating from a finite number of discrete, latent processing paths. MPT models have been successfully used to explain behavior in many areas such as memory (Batchelder & Riefer, 1986, 1990), decision making (Erdfelder, Castela, Michalkiewicz, & Heck, 2015; Hilbig, Erdfelder, & Pohl, 2010), reasoning (Klauer, Voss, Schmitz, & Teige-Mocigemba, 2007), perception (Ashby, Prinzmetal, Ivry, & Maddox, 1996), implicit attitude measurement (Conrey, Sherman, Gawronski, Hugenberg, & Groom, 2005; Nadarevic & Erdfelder, 2011), and processing fluency (Fazio, Brashier, Payne, & Marsh, 2015; Unkelbach & Stahl, 2009). Batchelder & Riefer (1999) and Erdfelder et al. (2009) reviewed the literature and showed the usefulness and broad applicability of the MPT model class. In the present paper, we introduce a simple but general approach to include information about response times (RTs) into any kind of MPT model. As a running example, we will use one of the most simple MPT models, the two-high-threshold model of recognition memory (2HTM; Bröder & Schütz, 2009; Snodgrass & Corwin, 1988). The 2HTM accounts for responses in a binary recognition paradigm. In such an experiment, participants first learn a list of items and later are prompted to categorize old and new items as such. Hence, one obtains frequencies of hits (correct old), misses (incorrect new), false alarms (incorrect old), and correct rejections (correct Psychon Bull Rev (2016) 23:1440–1465 1441 do Old Targets 1 – do g Old 1 – dn 1–g New Lures dn New Fig. 1 The two-high threshold model of recognition memory new responses). The 2HTM, shown in Fig. 1, assumes that hits emerge from two distinct processes: Either a memory signal is sufficiently strong to exceed a high threshold and the item is recognized as old, or the signal is too weak, an uncertainty state is entered, and respondents only guess old. The two processing stages of target detection and guessing conditional on the absence of detection are parameterized by the probabilities of their occurrence do and g, respectively. Given that the two possible processing paths are disjoint, the overall probability of an old response to an old item is given by the sum do +(1−do )g. Similarly, correct rejections can emerge either from lure detection with probability dn or from guessing new conditional on nondetection with probability 1 − g. In contrast, incorrect old and new responses always result from incorrect guessing. The validity of the 2HTM has often been tested in experiments by manipulating the base rate of learned items, which should only affect response bias and thus the guessing parameter g (Bröder & Schütz, 2009; Dube, Starns, Rotello, & Ratcliff, 2012). If the memory strength remains constant, the model predicts a linear relation between the probabilities of hits and false alarms (i.e., a linear receiver-operating characteristic, or ROC, curve; Bröder & Schütz, 2009; Kellen, Klauer, & Bröder, 2013). The 2HTM is at the core of many other MPT models that account for more complex memory paradigms such as source memory (Bayen, Murnane, & Erdfelder, 1996; Klauer & Wegener, 1998; Meiser & Böder, 2002) or process dissociation (Buchner, Erdfelder, Steffens, & Martensen, 1997; Jacoby, 1991; Steffens, Buchner, Martensen, & Erdfelder, 2000). These more complex models have a structure similar to the 2HTM because they assume that correct responses either result from some memory processes of theoretical interest or from some kind of guessing. Whereas MPT models are valuable tools to disentangle cognitive processes based on categorical data, they lack the ability to account for response times (RTs). Hence, MPT models cannot be used to test hypotheses about the speed of the assumed cognitive processes, for example, whether one underlying process is faster than another one. However, modeling RTs has a long tradition in experimental psychology, for instance, in testing whether cognitive processes occur serially or in parallel (Luce, 1986; Townsend & Ashby, 1983). Given that many MPT models have been developed for cognitive experiments that are conducted with the help of computers under controlled conditions, recording RTs in addition to categorical responses comes at a small cost. Even more importantly, substantive theories implemented as MPT models might readily provide predictions about the relative speed of the hypothesized processes or about the effect of experimental manipulations on processing speeds. For instance, the 2HTM can be seen as a twostage serial process model in which guessing occurs only after unsuccessful detection attempts (see, (...truncated)