A new visualization and conceptualization of categorical longitudinal development: measurement invariance and change

The Overlapping Waves Model (OWM) is a metaphor introduced by Siegler (1996) to illustrate a typical sequence of increasing and decreasing use of strategies during development. Going beyond metaphor, a new model synthesized from Latent Growth Modeling (LGM) and Item Response Theory (IRT) will be presented to analyze such categorical longitudinal data. Use of strategies can be scored as a variable with only a few ordinal categories. IRT provides the means to relate the usage of strategies to position on an underlying developmental dimension. LGM allows to model movement of individuals along this dimension, acknowledging individual differences both in starting point and in speed of progress. Measuring and modeling such strategy development requires that at each time point the same categories are used, in the sense that item difficulties must remain invariant over time. Whether, discrimination can be relaxed is still an issue. The problem that had to be solved was disentangling the between-person-individual differences from real intra-individual developmental differences. Figures with polytomous or multi-category Item Characteristic Curves (ICC's) resemble the OWM in many respects. However, such figures are usually taken to represent inter-individual differences, whereas the OWM usually represents development (so intra-individual differences), and we cannot have both at the same time. The solution came from creating a framework with ability differences on one axis and the effect of time on another axis, resulting in a 3-D model. These (orthogonal) dimensions make it possible to adequately conceptualize measurement invariance in this complex context. As the result is difficult to conceptualize without extensive visualization, special 3-D figures will be used to illustrate and a dynamic (rotatable and scalable) version will be made available as Computable Document Format object (Mathematica). The model was successfully applied in several microgenetic studies.