Two-Parameter IRT Models

Abstract

The Rasch model is sometimes also called the one-parameter IRT model in that the probability of success as a function of the ability $\theta$ has only one parameter (the item difficulty parameter) estimated for each item, as shown in Eq. (10.1).

Appendices

Discussion Points

• 1. Discuss the differences between the Rasch model and the 2PL model. What are the differences in terms of the mathematical formulations of the models? What are the differences in interpreting results of the two models? What are the differences in building measuring instruments?

• 2. How are the Rasch partial credit model and the 2PL model linked? Why does Chap. 9 say “In fact, when we use the partial credit model and have the task of assigning a weight (maximum score) to an item, we are already thinking in the framework of the two-parameter model?”

Exercises

• Q1. In EXCEL, plot ICCs of 2PL items. Vary the difficulty and slope parameters to see the differences. For example

  

• Q2. Indicate whether you agree or disagree with each of the following statements

  Item response data fitted with the Rasch model will always have better measurement properties than data fitted with the 2PL model	Agree/Disagree
  Residual-based fit statistics are useful for detecting mis-fits of items to the 2PL model	Agree/Disagree
  If the slope parameter is 0.6 for Item A and 1.2 for Item B, then, under 2PL, a student who obtained the correct answer for A but incorrect answer for B will have a lower ability estimate than a student who obtained the incorrect answer for A but correct answer for B	Agree/Disagree
  If the item difficulty parameter is 0.6 for Item A and 1.2 for Item B, then, under 2PL, a student who obtained the correct answer for A but incorrect answer for B will have a lower ability estimate than a student who obtained the incorrect answer for A but correct answer for B	Agree/Disagree
  If the item difficulty parameter is 0.6 for Item A and 1.2 for Item B, then, under the Rasch model, a student who obtained the correct answer for A but incorrect answer for B will have a lower ability estimate than a student who obtained the incorrect answer for A but correct answer for B	Agree/Disagree
  If more students obtained the correct answer to Item A than to Item B, then Item A is likely to be more discriminating than Item B	Agree/Disagree
  If a set of items fit the Rasch model so the observed ICCs are approximately parallel, then fitting a 2PL model to the same data will make the observed ICCs criss-cross each other	Agree/Disagree