A person fit test for IRT models for polytomous items

A person fit test based on the Lagrange multiplier test is presented for three item response theory models for polytomous items: the generalized partial credit model, the sequential model, and the graded response model. The test can also be used in the framework of multidimensional ability parameters. It is shown that the Lagrange multiplier statistic can take both the effects of estimation of the item parameters and the estimation of the person parameters into account. The Lagrange multiplier statistic has an asymptotic χ2-distribution. The Type I error rate and power are investigated using simulation studies. Results show that test statistics that ignore the effects of estimation of the persons’ ability parameters have decreased Type I error rates and power. Incorporating a correction to account for the effects of the estimation of the persons’ ability parameters results in acceptable Type I error rates and power characteristics; incorporating a correction for the estimation of the item parameters has very little additional effect. It is investigated to what extent the three models give comparable results, both in the simulation studies and in an example using data from the NEO Personality Inventory-Revised.