Lognormal Race Psychometric Model 1 Running head: LOGNORMAL RACE PSYCHOMETRIC MODELS The Lognormal Race: A Cognitive-Process Model of Choice and Latency with Desirable Psychometric Properties. Jeffrey N. Rouder and Jordan M. Province University of Missouri

We present a cognitive process model of response choice and response time performance data that has excellent psychometric properties and may be used in a wide variety of contexts. In the model there is an accumulator associated with each response option. These accumulators have bounds, and the first accumulator to reach its bound determines the response time and response choice. The times each accumulator reaches its bound is assumed to be lognormally distributed, hence the model is race or minima process among lognormal variables. A key property of the model is that it is relatively straightforward to place a wide variety of models on the logarithm of these finishing times including linear models, structural equation models, auto-regressive models, growth-curve models, etc. Consequently, the model has excellent statistical and psychometric properties and can be used in a wide range of contexts, from laboratory experiments to high-stakes testing, to assess performance. We provide a Bayesian hierarchical analysis of the model, and illustrate its flexibility with an application in testing and one in lexical decision making, a reading skill. Lognormal Race Psychometric Model 3 The Lognormal Race: A Cognitive-Process Model of Choice and Latency with Desirable Psychometric Properties. Cognitive modelers and psychometricians analyze human performance data to measure individual abilities, to measure the effects of covariates, and to assess latent structure in these data. Yet, there is relatively little overlap between psychometric models and cognitive-process models. This lack of overlap is in many ways understandable. Cognitive process modelers and psychometricians not only have different goals, they come from different traditions. For cognitive modelers, the emphasis is on using structure in data to draw inferences about the processes and mental representations underlying perception, memory, attention, decision-making and information-processing activities. These researchers specify complex, detailed nonlinear models that incorporate theoretical insights on the acquisition, storage, and processing of mental information. The goal of these researchers is to uncover the true structures underlying mental life, and the usefulness of models is measured by how well they approximate these truths. Consequently, models are benchmarked on their ability to fit fine-grained features in the data. A well-known cognitive-process model that exemplifies these points is McClelland and Rumelhart’ interactive activation mode (McClelland & Rumelhart, 1981; Rumelhart & McClelland, 1982) . The goal of the model is to describe the processes in word recognition. The model stipulates that features, letters, and words are represented by nodes, which are hierarchically organized. When a word is presented, features first become active, and then this activity propagates to supported letters, and finally to supported words. There are a several key specifications in the model: I. The activity of a node is a sigmoidal function of its input. II. There is feedback between words and letters where activated words will support activation in constituent letter. III. Activation of a word node inhibits visually Lognormal Race Psychometric Model 4 similar words, e.g., activation of the node for work inhibits the activation of the node for worm; there is no inhibition at other levels. V. A response is made the first time a word node reaches its bounds. The success of this model is measured by ability to predict a wealth of empirical results including the accuracy and speed of word identification for various words, the effects of word frequency1 on identification, and the effects of degrading the stimulus through masking. The strength of the model are the detailed specification of representations and processes, and the ability to account for a number of phenomena. Balanced against these strengths are a few disadvantages: It is difficult to analyze the model and parameter estimation proceeds by trial-and-error; it is difficult to assess how robust the model is to misspecification; it is difficult to asses misfit; it is difficult to assess the complexity of the model and characterize what types of data patterns could not be fit (Roberts & Pashler, 2000); and it is not clear how the model may be extended to account for covariates or used in a psychometric setting to understand variation across people. Psychometricians, on the other hand, place an emphasis on the usefulness of models. Good models may not fit the data in the finest detail, but they are statistically tractable, convenient to analyze, incorporate covariates of interest, and may be used to understand variation in latent traits and abilities across people. Psychometricians retain these desirable properties by using models that cognitive modelers would consider simple. Consider, for example, the Rasch IRT model in which the key structural specification is that performance reflects the difference between participant ability and item difficulty (Rasch, 1960). Because this specification is simple, the model retains desirable statistical properties including ease of analysis, the ability to understand misspecification. Moreover, because the base model is so simple, it is straightforward to integrate sophisticated models on item and participant effects including factor models and structural equation models (Skrondal & Rabe-Hesketh, 2004; Embretson, 1991). In our view cognitive modelers and psychometricians make different trade-offs: Lognormal Race Psychometric Model 5 psychometricians may gain the ability to measure latent abilities by stipulating simple psychological processes and representations; cognitive modelers may gain the ability to understand the details of these processes and representation by stipulating detailed and complex models, but at the expense of statistical tractability, robustness to misspecification, and the ability to efficiently measure individual variation. There is, however, a fertile middle ground of models, sometimes called cognitive psychometrics (Riefer, Knapp, Batchelder, Bamber, & Manifold, 2002), where the goal is to develop psychological process models with good psychometric and statistical properties. In this paper, we develop a middle-ground model for the joint distribution of response choice and response time (RT). The model captures a processing commitment that evidence for a response grows gradually in time in separate, independent accumulators. Each accumulator has a bound, and accumulators race with the winner being the one that first reaches its bound. The response made is that associated with the winning accumulator, and the response is related to the finishing time of the winning accumulator. We adopt perhaps the simplest version of this race-between-accumulators theme, and show how it retains the above desirable properties of useful psychometric models. Not only is our model statistically tractable, it is straightforward to place sophisticated model components, such as auto-regressive components, factor components, or structural-equation components, on model parameters. In the next section we provide a brief and selective overview of the literature on accounting for response time and response choice to provide context for our approach. We then present a base model that accounts for response choice and response time for a single participant observing a number of replicate trials. Following that, analysis of base model through Bayesian methods is discussed. The base model is easily extended for real-world applications, and we provide two examples. The first is a Rasch-model equivalent for psychometric testing. In addition to measuring item difficulty and participant ability, the Lognormal Race Psychometric Model 6 model provides estimates of participant and item speed. The second is an application in lexical decision, where participants decide if a presented letter string is a word or a nonword. The model provides measures how latent accumulation for word and nonword information differ across various experimental conditions and across people. When fit to experimental data, the model reveals the novel insight that people’s ability to identify nonwords is not correlated with their ability to identify words. Modeling Response Time and Response Choice Jointly Making choices always produces two dependent measures, the choice selected and response time (RT), the time it takes to make the selection. Given that experimenting and testing is often carried out via computer, RT is often readily available. The most popular approach in experimental psychology and psychometrics is to model one dependent measure, either RT or choice, and ignore the other. Although ignoring one of the measures may seem naive, it is often natural and appropriate in real-world settings. Analyzing one measure seems appropriate when there is variation in only one, or where the joint variation is so thoroughly correlated that all the information is effectively contained in one measure. Modeling both RT and accuracy jointly entails additional complications over modeling just one measure, and in some contexts the gain may be marginal. Modeling RT and choice jointly seems appropriate when there is substantial variation in both that is not redundant. One area where this is likely is the study of performance across individuals. Here, it is reasonable to expect that different people have different characteristic trade-offs so that the RT and accuracy are not redundant. Two people may have comparable skills yet one person may stress accuracy while the other may stress speed, and consideration of one measure alone would not reveal the comparability of their skills. For the remainder of this paper, we assume that the consideration of RT and choice jointly indeed provide insights that could not be provided Lognormal Race Psychometric Model 7 by consideration of either alone. One conventional psychometric approac