Chapter 9 Latent item predictors with fixed effects
Dirk Smits, Stephen H. Moore · 2004
The Rasch model (Rasch, 1960) and the linear logistic test model (LLTM, Fischer, 1973, 1977) are two commonly used item response models. Both models are discussed in Chapter 2. The Rasch model assumes item indica tors as predictors, so that each item has a specific effect, the weight of the corresponding item indicator. The LLTM explains these effects in terms of item properties, or in other words item properties are used as item predic tors. Therefore, the LLTM may be considered an item explanatory model, in contrast with the Rasch model which is descriptive. The requirement that the values of the item properties be known in advance for every item is both a strength and a limitation of the LLTM. The strength is that the model supports a more parsimonious account of item effects, but the limitation is that the values specified for items on the properties imply additional model assumptions. In the current chapter, a model will be introduced with latent item properties: The values of these latent item properties do not have to be known apriori, but they may have unknown values that are estimated as model parameters. The model introduced in this chapter is called the model with internal restrietion on item difficulties (MIRID; Butter, De Boeck, & Verhelst, 1998). The MIRID was originally published by Butter et al. (1998), based on Butter (1994). In Butter et al. (1998), a conditional maximum likelihood formulation and estimation method was explained, and the results of a simulation study were presented. An application of the MIRID and an ex tension of the MIRID to the OPLM-MIRID (originally described by Butter, 1994) and the 2PL-MIRID can be found in Smits and De Boeck (2003). A comparison between two estimation methods for the MIRID and the OPLM-MIRID - a conditional maximum likelihood estimation (Smits, De Boeck, Verhelst, & Butter, 2001) and a marginal maximum likelihood es timation, implemented within PROC NLMIXED - can be found in Smits, De Boeck, and Verhelst (2003). The MIRID in its standard form will be explained in the first part of