A Hierarchical Model For Item Responses And Response Times Based On The Box-Cox Transformation

Response times on test items can be a valuable source of information on test takersand test items, especially this information is easily collected in modern computerized test-ing. And there is a relationship between responses on test items and response times appar-ently. For example, when a person has a high(low) ability, then his response time maybeshort(long); when a test item is more difficult(easy), then a subject maybe spend more(less)time on this item. Therefore, applying the response data of item to the response-time modelseems reasonable.A hierarchical framework for modelling response and response time is presented incurrent paper. The two-parameter normal ogive model can be used as response-time model,and in this paper, we develop a Box-Cox normal model for describing response times. TheBox-Cox transformation is a power transformation family and has stronger normality thanlogarithmic transformation.Then, an MCMC algorithm composed of Gibbs sampler and M-H sampler is used toobtain the Bayes estimates of parameters and the specific sampling procedures and iterativeformula are also provided. At last, a transformation-invariant implementation of the devianceinformation criterion(DIC) is developed that allows for comparing model fit between modelswith different transformation parameters.