A framework for modeling componential data using item response theory models for polytomous items is presented. This framework models response accuracies 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 possible response patterns on the components. Models with only single component effects are distinct from models that allow for component combination effects. For the latter, a further distinction is made between constant and dimension-dependent component combination effects as a function of whether these effects depend on an examinee's ability. The proposed models are multidimensional, because separate abilities are assumed to be involved in their various components. An example illustrates the approach.