Simultaneous Ability and Difficulty Estimation Via the Linear Discriminant Function

In this paper, parameter estimation of the dichotomous Rasch model (Rasch, 1960) using the linear discriminant function (Fisher, 1936) is presented. This is accomplished by considering the scored item responses to be distinct groups and using a design matrix that is identical to one used in logistic regression for joint maximum likelihood estimation. The real dataset that was examined was the fraction subtraction dataset from Tatsuoka (1984). Through simulation parameter estimation accuracy using the linear discriminant function was compared to joint maximum likelihood estimation using logistic regression. Using the linear discriminant function person ability estimates from perfect total scores and total response scores of zero were estimable without using an ad hoc procedure, which is a well-known shortcoming of logistic regression based joint maximum likelihood estimation. Finally, computation of a closed-form solution for parameter estimation using the linear discriminant function is discussed.