Form Invariance - An Alternative Answer to the Measurement Problem of Item Response Theory

Abstract

The measurement problem of item response theory is the question of how to assign ability parameters <img src=image/13426621_01.gif> to persons and difficulty parameters <img src=image/13426621_02.gif> to items such that the comparison of abilities is independent of the specific set of difficulties <img src=image/13426621_02.gif>. Correspondingly, the comparison of difficulties <img src=image/13426621_02.gif> should be independent of the specific set of abilities <img src=image/13426621_01.gif>. These requirements are called specific objectivity. They are the basis of the Rasch model. It measures <img src=image/13426621_01.gif> and <img src=image/13426621_02.gif> on one and the same scale. The present paper asks the different question of how to assign ability parameters <img src=image/13426621_01.gif> to persons in a way that the comparison of abilities is independent of the position on the scale where the measurement takes place. Correspondingly, the comparison of difficulties <img src=image/13426621_02.gif> should also be independent of the position on the scale where the calibration of difficulties takes place. Again, <img src=image/13426621_01.gif> and <img src=image/13426621_02.gif> measured on one and the same scale. These requirements are called form invariance. They lead to an item response function (IRF) different from that of the Rasch model. It integrates information from <img src=image/13426621_01.gif> and <img src=image/13426621_02.gif> beyond the mere score dependence and also shows specific objectivity (in a generalized mathematical form). The properties of the form invariant item response function are compared to that of the Rasch model, and related to previous work by Warm, Jaynes and Samejima. Moreover, several numerical examples for the use of it are provided.