Assessing Approximate Fit in Categorical Data Analysis.

A family of Root Mean Square Error of Approximation (RMSEA) statistics is proposed for assessing the goodness of approximation in discrete multivariate analysis with applications to item response theory (IRT) models. The family includes RMSEAs to assess the approximation up to any level of association of the discrete variables. Two members of this family are RMSEA2, which uses up to bivariate moments, and the full information RMSEAn. The RMSEA2 is estimated using the M2 statistic of Maydeu-Olivares and Joe (2005, 2006), whereas for maximum likelihood estimation, RMSEAn is estimated using Pearson's X(2) statistic. Using IRT models, we provide cutoff criteria of adequate, good, and excellent fit using the RMSEA2. When the data are ordinal, we find a strong linear relationship between the RMSEA2 and the Standardized Root Mean Squared Residual goodness-of-fit index. We are unable to offer cutoff criteria for the RMSEAn as its population values decrease as the number of variables and categories increase.