A framework for modeling componential data using item response theory model
s for polytomous items is presented. This framework models response accurac
ies on complex cognitive tasks, which are decomposed in terms of more basic
elements (components), such as knowledge structures, cognitive processes,
and strategies. Response categories of polytomous items correspond to possi
ble response patterns on the components. Models with only single component
effects are distinct from models that allow for component combination effec
ts. For the latter, a further distinction is made between constant and dime
nsion-dependent component combination effects as a function of whether thes
e effects depend on an examinee's ability. The proposed models are multidim
ensional, because separate abilities are assumed to be involved in their va
rious components. An example illustrates the approach.