Several methods exist for developing strong measurement scales separately from the longitudinal model of stability or change (see Rost & Walter, chap. 18, this volume). These methods can be applied to dichotomous or ordinal data. The scales can be developed using the same or a different data set from that used to test the longitudinal model. The Rasch model (1-parameter; Rasch, 1960; Wright & Masters, 1982; Wright & Stone, 1979) provides interval-level measurement, and the 2-parameter logistic Item Response Theory model (IRT; see Embretson & Reise, 2000) provides a good approximation to interval-level measurement when the data are consistent with the model. These are probabilistic measurement models. For dichotomous items, equal changes in the underlying latent construct correspond to equal changes in the log of the odds of endorsing an item, for any level of the latent trait.
For items with multiple ordered response categories (1 = "not at all," 2, 3, 4, 5 = "very much) that typify Likert-type scales, there are extensions of both the Rasch and the 2-parameter IRT models.
A variety of polytomous models for multiple-ordered response categories have been developed. The Rasch extensions include the partial credit model (Masters, 1982) and the rating scale model (Andrich, 1978). The 2-parameter IRT extensions include the graded response model (Samejima, 1969) and the modified graded response model (Muraki, 1990). The basics of the Rasch model and its extensions are described and illustrated by Rost and Walter (chap. 18, this volume). Drasgow and Chuah (chap. 7, this volume) explain and illustrate the 2- and 3-parameter models in detail. In each of these models, there are multiple probability curves for each item, one for each response category. These probabilities provide information on how each category functions relative to other categories within an item. These models produce good approximations of interval level score estimates of the underlying construct while treating the response categories as ordinal. The interval level score estimates produced can be used to model longitudinal change.
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