The four-parameter normal ogive model with response times

In recent years, interest in the four-parameter logistic (4PL) model (Barton and Lord, ETS Res Rep Ser 198(1):i-8, 1981), and its normal ogive equivalent, has been renewed (Culpepper, Psychometrika, 81(4):1142–1163, 2016; Feuerstahler and Waller (Multivar Behav Res 49(3):285–285, 2014)). The defining feature of this model is the inclusion of an upper asymptote parameter, in addition to those included in the more common three-parameter logistic (3PL) model. The use of the slipping parameter has come into contact with many assessment applications, such as high-stakes testing (Loken and Rulison, Br J Math Stat Psychol 63(3):509–525, 2010), low-stakes testing (Culpepper, Psychometrika, 81(4):1142–1163, 2016), and measuring psychopathology (Waller and Reise, Measuring psychopathology with nonstandard item response theory models: Fitting the four-parameter model to the Minnesota Multiphasic Personality Inventory, 2010). Yet as mentioned in Culpepper (Psychometrika, 81(4):1142–1163, 2016), the recovery of the slipping parameter also requires larger sample sizes and longer iterations for the sampling algorithm to converge. Response time (RT), which has already been widely utilized to study student behaviors, such as rapid-guessing, was included in our model to help recover the slipping parameter and the overall measurement accuracy. Based on the hierarchical framework of response and RT (van der Linden, Psychometrika 72(3):287–308, 2007), we extended the four-parameter normal ogive model by incorporating RT into the model formulation. A Gibbs sampling approach to estimation was developed and investigated.