Modeling Parental Involvement

Abstract

In this chapter, we first outline the models used and the estimation and testing procedures employed, and then summarize the results revealed by these models. Estimation and Testing ProceduresThe procedures we used for parameter estimation and evaluation of model fit are based on marginal maximum likelihood (MML). Most of the procedures we discuss are documented in more detail elsewhere (see Bock and Aitkin 1981;Bock et al. 1988;Gibbons and Hedeker 1992;Glas 1999;Adams and Wu 2006;De Jong et al. 2007;Jennrich and Bentler 2011;Glas and Jehangir 2014). We used the public domain software package MIRT (Glas 2010) in the calculations. Additional estimation and testing procedures were used for the bi-factor model, with unidimensional models as special cases, and random item parameters as a generalization. MML EstimationThe bi-factor model used in this study was in two parts: a measurement model (i.e., an IRT model) and a structural model. The measurement model pertains to a polytomously-scored response of a student n to an item i. The possible item scores range from 0 to m i and the score of student n on item i is denoted by the variables x nij (j = 1, …, m i ) where x nij = 1 if the response is in category 1 and zero otherwise. Note that m i has an index i, which indicates that the maximum score of items can differ.We describe the procedure for the bi-factor model, combined with the partial credit model (PCM;Masters 1982) and generalized partial credit model (GPCM; Muraki 1992) as IRT models, since these two models were the ones we selected for the present study. However, the theory also applies to other IRT models, such as the unidimensional PCM and GPCM, the graded response model (Samejima 1969), the