Modeling nonlinear conditional dependence between response time and accuracy

Abstract

The most common process variable available for analysis due to tests presented in a computerized form is response time. Psychometric models have been developed for joint modeling of response accuracy and response time in which response time is an additional source of information about ability and about the underlying response processes. While traditional models assume conditional independence between response time and accuracy given ability and speed latent variables (van der Linden, 2007), recently multiple studies (De Boeck and Partchev, 2012; Meng et al., 2015; Bolsinova et al., 2017a,b) have shown that violations of conditional independence are not rare and that there is more to learn from the conditional dependence between response time and accuracy. When it comes to conditional dependence between time and accuracy, authors typically focus on positive conditional dependence (i.e., relatively slow responses are more often correct) and negative conditional dependence (i.e., relatively fast responses are more often correct), which implies monotone conditional dependence. Moreover, most existing models specify the relationship to be linear. However, this assumption of monotone and linear conditional dependence does not necessarily hold in practice, and assuming linearity might distort the conclusions about the relationship between time and accuracy. In this paper we develop methods for exploring nonlinear conditional dependence between response time and accuracy. Three different approaches are proposed: (1) A joint model for quadratic conditional dependence is developed as an extension of the response moderation models for time and accuracy (Bolsinova et al., 2017b); (2) A joint model for multiple-category conditional dependence is developed as an extension of the fast-slow model of Partchev and De Boeck (2012); (3) An indicator-level nonparametric moderation method (Bolsinova and Molenaar, in press) is used with residual log-response time as a predictor for the item intercept and item slope. Furthermore, we propose using nonparametric moderation to evaluate the viability of the assumption of linearity of conditional dependence by performing posterior predictive checks for the linear conditional dependence model. The developed methods are illustrated using data from an educational test in which, for the majority of the items, conditional dependence is shown to be nonlinear.