## F. Implementation of Parameter Estimation Under the Graded Response Model

The equations for the estimation of item and ability parameters under the graded response model were presented in Chapter 8. In the present appendix, these two building blocks for the graded response model will not be implemented within the joint maximum likelihood estimation (JMLE) paradigm. Separate implementation of the two building blocks reduces the complexity and the length of the required computer software. Because of the formulation of the parameter estimation process in terms of boundary curve parameters, the mathematics of the graded response case appears complex. However, the implementation is quite straightforward as the primary terms needed are simply the P and W yielded by a boundary curves at a given ability level. Section F.2 describes the implementation of estimation of item parameters under the graded response model that is based on Equation 8.7. The solution of this Newton-Raphson (Fisher scoring) equation yields the estimates of the item intercept parameters for the m - 1 boundary curves and the common slope. Using the relationships presented in Chapter 8, the estimates of the m item difficulty parameters can be obtained. Section F.3 describes the implementation of the estimation of a single examinee's ability using the responses to a five-item test scored under the graded response model.