EstCRM: An R Package for Samejima's Continuous IRT Model

Samejima (1973) proposed an item response theory (IRT) model for continuous item scores as a limiting form of the graded response model. Although the continuous response model (CRM) is as old as its counterparts for the binary and polytomous response formats, it is not commonly used in practice, nor is it well known by practitioners. One of the reasons for the unpopularity of the CRM is probably the lack of accessible software to estimate the model parameters. The first software to estimate the CRM item parameters was the EM2 program (Wang & Zeng, 1998) written in C language and available as an executable file. The EM2 program estimates item parameters using the marginal maximum likelihood method and the expectation-maximization (EM) algorithm. EM2 uses Gaussian quadrature points for approximating the integration over the posterior ability distribution to compute the expected log likelihood at the E step and applies the Newton–Raphson method to solve for the item parameters maximizing the expected log likelihood at the M step. The available version of EM2 works only with 32-bit computers. Another procedure to estimate the CRM item parameters is described by Shojima (2005) and implements a simplified EM algorithm. Shojima (2005) showed that the expected log-likelihood at the E step can be obtained explicitly without approximation, and the item parameter equations at the M step can be analytically solved by assuming flat priors on item parameters. As a result of a simulation study, Shojima (2005) reported that the simplified EM algorithm worked as well as the procedure implemented by the EM2 program. A new R (R Development Core Team, 2011) package, EstCRM, that implements the simplified EM algorithm as proposed by Shojima (2005) is now available to estimate the item parameters for the CRM. The standard errors of the estimated item parameter are approximated by using a nonparametric bootstrap procedure. In addition to estimating the item parameters, the EstCRM package includes functions to obtain the maximum likelihood estimates of the person parameters from a closed formula derived by Samejima (1973), to compute the standardized item-fit residual statistics based on a procedure proposed by Ferrando (2002), to draw empirical three-dimensional item category response curves from sample proportions, and to draw theoretical three-dimensional item category response curves based on the estimated model