An Extension of the Beta Response Model for Continuous Responses and Parameter Estimation via an EM Algorithm

Abstract

Item response theory (IRT) models have been proposed for continuous observed variables, such as response time and confidence in responses. The present paper extends Noel & Dauvier's (2007) model, according to which continuous responses were modeled with a beta distribution, and proposes a new item response theory model in which the estimation method for item parameters uses an EM algorithm from which asymptotic standard errors (SEs) are derived. The possibility of a linear transformation of parameters is also discussed. The performance of the proposed model was evaluated both in a simulation study and when applied to data from an actual math achievement test. The simulation revealed that the root-mean-square errors (RMSEs) between the estimates and the true values were about 0.1, and that the proposed method resulted in a more stable result than Noel & Dauvier's (2007) method, even with data from a test with a small number of items. Moreover, the application of the proposed method to actual data with a large number of observed categories, although not originally assumed to be continuous variables, yielded relatively small SEs of the item parameters and showed that the ability estimates were highly correlated with the estimates from a graded response model (GRM).