Posterior predictive model checking for the diagnostic input noisy and gate model
Alagoz Ekici, Cigdem
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This study presents a posterior predictive model checking (PPMC) method for the deterministic inputs, noisy and gate (DINA) model. The potential of the PPMC method is examined for detecting problems with the DINA model. Χ2 statistics are calculated based on latent class and raw score groups to evaluate model fit and item fit. Then PPP-values are calculated using these Χ2 values as discrepancy measures for both item fit and model fit evaluation. Two problem conditions were simulated to study these fit indices. The first problem situation occurs, when the higher order structure among the attributes are ignored, when analyzing the data. The second problem situation occurs, when the Q-matrix is misspecified. The performance of the fit indices was evaluated under the presence of these two problem situations. Type I error rates and power were calculated. Χ2 is calculated based on latent classes. PPP-values based on this Χ2 produced small Type I error rates and very good power. On the other hand, Type I error rates and power from Χ2 calculated based on raw score groups and PPP-values based on this Χ2 were not in the acceptable range. Item fit indices successfully detected problems with the Q-matrix misspecification. This helped identify which items were misspecified. However, neither item fit nor model fit indices detected problems with the modeling of the attribute relationship structure. When the Q-matrix misspecification was small, model fit indices did not reject the model. When 5% or more of the Q-matrix were misspecified, the overall Χ2 calculated based on latent classes successfully rejected the model. A real data analysis was presented to demonstrate the application of these model and item fit indices for the DINA model.
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