Simultaneous Ability and Difficulty Estimation Via the Linear Discriminant Function

Area of Science:

• Psychometrics
• Statistical Modeling
• Educational Measurement

Background:

• The dichotomous Rasch model is a key tool in psychometrics.
• Joint maximum likelihood estimation (JMLE) using logistic regression has limitations, particularly with extreme scores.
• Parameter estimation accuracy is crucial for reliable measurement.

Purpose of the Study:

• To present parameter estimation for the dichotomous Rasch model using Fisher's linear discriminant function (LDF).
• To compare the accuracy of LDF estimation with JMLE via logistic regression.
• To address the estimability issue of extreme scores in Rasch model parameter estimation.

Main Methods:

• Utilized Fisher's linear discriminant function for parameter estimation.
• Employed a design matrix similar to logistic regression for JMLE.
• Compared LDF and JMLE using simulations and the Tatsuoka (1984) fraction subtraction dataset.

Main Results:

• Parameter estimation accuracy using LDF was compared to JMLE.
• LDF allowed estimation of person ability for perfect total scores and zero total response scores.
• This resolved a known shortcoming of JMLE in logistic regression.

Conclusions:

• Fisher's linear discriminant function offers a viable alternative for Rasch model parameter estimation.
• LDF overcomes limitations of JMLE regarding extreme score estimation.
• The study discusses the computation of a closed-form solution for LDF parameter estimation.