An Application of <i>M</i> ₂ Statistic to Evaluate the Fit of Cognitive Diagnostic Models

Abstract

The fit of cognitive diagnostic models (CDMs) to response data needs to be evaluated, since CDMs might yield misleading results when they do not fit the data well. Limited-information statistic M 2 and the associated root mean square error of approximation (RMSEA 2 ) in item factor analysis were extended to evaluate the fit of CDMs. The findings suggested that the M 2 statistic has proper empirical Type I error rates and good statistical power, and it could be used as a general statistical tool. More importantly, we found that there was a strong linear relationship between mean marginal misclassification rates and RMSEA 2 when there was model–data misfit. The evidence demonstrated that .030 and .045 could be reasonable thresholds for excellent and good fit, respectively, under the saturated log-linear cognitive diagnosis model.