Investigating the Behaviors of M₂ and RMSEA₂ in Fitting a Unidimensional Model to Multidimensional Data

It has been widely known that the Type I error rates of goodness-of-fit tests using full information test statistics, such as Pearson's test statistic χ² and the likelihood ratio test statistic G², are problematic when data are sparse. Under such conditions, the limited information goodness-of-fit test statistic M₂ is recommended in model fit assessment for models with binary response data. A simulation study was conducted to investigate the power and Type I error rate of M₂ in fitting unidimensional models to many different types of multidimensional data. As an additional interest, the behavior of RMSEA₂ was also examined, which is the root mean square error approximation (RMSEA) based on M₂. Findings from the current study showed that M₂ and RMSEA₂ are sensitive in detecting the misfits due to varying slope parameters, the bifactor structure, and the partially (or completely) simple structure for multidimensional data, but not the misfits due to the within-item multidimensional structures.