Formulating latent growth using an explanatory item response model approach

In this paper, we present a way to extend the Hierarchical Generalized Linear Model (HGLM; Kamata (2001), Raudenbush (1995)) to include the many forms of measurement models available under the formulation known as the Random Coefficients Multinomial Logit (MRCML) Model (Adams, Wilson and Wang, 1997), and apply that to growth modeling. First, we review two different traditions in modeling growth studies: the first is based in the hierarchical linear modeling (HLM) tradition, and the second, which is the topic of this paper, is rooted in the Rasch measurement tradition - this is the linear Latent Growth Item Response Model (LG-IRM). Going beyond the linear case, the LG-IRM approach allows us to considerably extend the range of models available in the HLM tradition to incorporate several of the extensions of IRT models that are used in creating explanatory item response models (EIRM; De Boeck and Wilson, 2004). We next present a number of extensions - including polynomial growth modeling, differential item functioning (DIF) effects, growth functions that can be approximated by polynomial expressions, provision for polytomous responses, person and item covariates (and time varying covariates), and multiple dimensions of growth. We provide two empirical examples to illustrate several of the models, using the ConQuest software (Wu, Adams, Wilson and Haldane, 2008) to carry out the analyses. We also provide several simulations to investigate the success of the estimation procedures.