An accumulator model for responses and response times in tests based on the proportional hazards model

Abstract

Latent trait models for responses and response times in tests often lack a substantial interpretation in terms of a cognitive process model. This is a drawback because process models are helpful in clarifying the meaning of the latent traits. In the present paper, a new model for responses and response times in tests is presented. The model is based on the proportional hazards model for competing risks. Two processes are assumed, one reflecting the increase in knowledge and the second the tendency to discontinue. The processes can be characterized by two proportional hazards models whose baseline hazard functions correspond to the temporary increase in knowledge and discouragement. The model can be calibrated with marginal maximum likelihood estimation and an application of the ECM algorithm. Two tests of model fit are proposed. The amenability of the proposed approaches to model calibration and model evaluation is demonstrated in a simulation study. Finally, the model is used for the analysis of two empirical data sets.

An Overview of Models for Response Times and Processes in Cognitive Tests

“…For the practical purpose of measurement and because it often fits the data very well, the lognormal distribution has become popular for cognitive test response times (van der Linden, 2006, 2007) without process interpretation claims. In some other applications, practical considerations have led to an approach based on the proportional hazard principle (e.g., Ranger and Kuhn, 2012, 2014; Ranger and Ortner, 2012; Wang and Xu, 2015; Kang, 2017). Burbeck and Luce (1982) explain that the normal, Gumbel, and ex-Gaussian distributions have a monotone non-decreasing hazard function, while the exponential distribution (a special case of the Weibull) has a constant hazard function, and the Weibull distribution can accommodate a decreasing, constant, and increasing function.…”

Analysing model fit of psychometric process models: An overview, a new test and an application to the diffusion model

“…Tests of item fit can be derived by using just the residuals from a single item. The test has not been used for the diffusion model so far; for applications to race models, see Ranger and Kuhn (2014a) and Ranger et al (2015).…”

“…The resulting model then serves as a measurement model by which the responses and response times in a test can be analysed. Several such models have lately been proposed, among them latent trait versions of the race model (Ranger & Kuhn, 2014a;Ranger, Kuhn, & Gaviria, 2015;Rouder, Province, Morey, Gomez, & Heathcote, 2015;Tuerlinckx & De Boeck, 2005) and of the diffusion model (Molenaar, Tuerlinckx, & van der Maas, 2015;Tuerlinckx & De Boeck, 2005;Tuerlinckx, Molenaar, & van der Maas, 2016;Vandekerckhove, Tuerlinckx, & Lee, 2011;van der Maas, Molenaar, Maris, Kievit, & Boorsboom, 2011).…”

Fitting Diffusion Item Response Theory Models for Responses and Response Times Using theRPackagediffIRT

“…These process IRT models mainly draw from process models that already exist in the field of mathematical psychology. For instance, Ranger and Kuhn (2014) proposed a model based on the proportional hazard model (see, e.g., Luce 1986), Tuerlinckx and De Boeck (2005) and Rouder, Province, Morey, Gomez, and Heathcote (2015) proposed extensions of the Race model (Audley and Pike 1965), and Tuerlinckx, Molenaar, and van der Maas (2016), Tuerlinckx and De Boeck (2005), and van der Maas et al (2011) proposed extensions of the so called diffusion model (Ratcliff 1978).…”