ABSTRACT
The importance of the statistical model for measurement was emphasized in the previous chapter. This chapter focusses on two key issues that must be addressed when one is measuring. First, the discussion about the statistical model in the previous chapter was confined to the dichotomous data case. This was a useful tactic for pedagogic purposes, but strongly limiting in terms of potential applications: it means that common response formats such as the Likert-style or Guttman-style formats for survey instruments and the partial credit format for achievement items are outside the scope of the statistical model. Here the scope is extended to the case where the data have more than two ordered categories—polytomous data—and the models are extended to polytomous models, specifically the partial credit model (PCM).
Second, given that a choice of model has been made, how does the measurer decide whether the specific data in hand are appropriate for the chosen statistical model? This will focus on the examination of “fit” for items and respondents. That is, are the estimated parameters of the item and the respondent, as embodied in the statistical model, working consistently with the observed data?
