Structural equation modeling permits simultaneous modeling of the measurement structure and the longitudinal model of stability or change. In the measurement portion of the model, each latent construct is hypothesized to be error free and normally distributed on an interval scale. The structural part represents the relationships between the latent constructs. This modeling approach can also be extended to two or more ordered categories (Muthen, 1984). This approach assumes that each dichotomous or ordered categorical measured variable is characterized by an underlying normally distributed continuous variable. For each measured variable, c-1 thresholds are estimated that separate each of the c categories (e.g., one threshold for a dichotomous variable). If the assumptions are met, then Muthen's approach will provide estimates of the underlying factors that approximate an interval-level scale of measurement. Indeed, Takane and de Leeuw (1987) have shown that 2-parameter IRT models and confirmatory factor models are identical for dichotomous items under certain conditions. Unfortunately, large sample sizes (e.g., 500-1,000
or more cases) are often required for the appropriate use of structural equation modeling approach to categorical data. Newer estimation methods may offer promise of adequate estimation with smaller sample sizes (Muthen & Muthen, 2004). However, separate scale development using external methods such as Rasch or IRT modeling will often be more efficient.
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