## Multilevel IRT Modeling

Item response theory modeling is a widely utilized class of traditional measurement models. For dichotomously scored test items, there are several well-recognized IRT models, such as the Rasch model, the twoparameter logistic model, and the threeparameter logistic model. For example, the two-parameter logistic model can be written as

+[ ] + +[ ] exp

exp ,

α θ δ α θ δ1

(3.1)

where θp is the ability of examinee p, αi is the discrimination power of item i, and δi is the threshold or location of item i. In IRT applications, the threshold is typically transformed into the difficulty parameter βi by βi = –δi/αi, such that the exponential function has a form of αi(θp – βi). However, in this chapter we will use the threshold parameter directly for simplicity from a modeling perspective. The metric of θp and –δi/αi are typically in a standardized scale, where 0 is the center of the distribution with a standard deviation of 1. When discrimination power is assumed to be equal for all items in the instrument and constrained to be 1, the model becomes

+[ ] + +[ ] exp

exp ,

θ δ θ δ1

(3.2)

and is known as the Rasch model. The difference between θp and –δi = βi is directly a logit quantity, where θ indicates a typical ability or difficulty, respectively. Furthermore, the two-parameter logistic

model can be extended to the three-parameter logistic model

i p i = + −

+[ ] + +[ ]γ γ

α θ δ α θ δ

( ) exp

exp ,1

1 (3.3)

where γi is the lower asymptote of the logistic curve and known as the pseudo guessing parameter. Under the three-parameter logistic model, we assume a nonzero lower asymptote, indicating a nonzero probability of endorsing an item for examinees with any ability level.