Limited-information goodness-of-fit testing of diagnostic classification item response models

• Published in: British journal of mathematical & statistical psychology Vol. 69; no. 3; pp. 225 - 252
• Main Authors: Hansen, Mark, Cai, Li, Monroe, Scott, Li, Zhen
• Format: Journal Article
• Language: English
• Published: England Blackwell Publishing Ltd 01.11.2016; British Psychological Society
• Subjects: Algorithms; Classification; Computer Simulation; Data Interpretation, Statistical; Diagnosis, Computer-Assisted - methods; diagnostic classification models; item response models; Likelihood Functions; limited-information goodness of fit; local item independence; Matrix; Measurement; Models, Statistical; Numerical Analysis, Computer-Assisted; Outcome Assessment (Health Care) - methods; Outcome Assessment (Health Care) - statistics & numerical data; Reproducibility of Results; Sample Size; Sensitivity and Specificity; Simulation; item response models; local item independence; limited-information goodness of fit; diagnostic classification models
• ISSN: 0007-1102; 2044-8317; 2044-8317
• DOI: 10.1111/bmsp.12074

Despite the growing popularity of diagnostic classification models (e.g., Rupp et al., 2010, Diagnostic measurement: theory, methods, and applications, Guilford Press, New York, NY) in educational and psychological measurement, methods for testing their absolute goodness of fit to real data remain relatively underdeveloped. For tests of reasonable length and for realistic sample size, full‐information test statistics such as Pearson's X2 and the likelihood ratio statistic G2 suffer from sparseness in the underlying contingency table from which they are computed. Recently, limited‐information fit statistics such as Maydeu‐Olivares and Joe's (2006, Psychometrika, 71, 713) M2 have been found to be quite useful in testing the overall goodness of fit of item response theory models. In this study, we applied Maydeu‐Olivares and Joe's (2006, Psychometrika, 71, 713) M2 statistic to diagnostic classification models. Through a series of simulation studies, we found that M2 is well calibrated across a wide range of diagnostic model structures and was sensitive to certain misspecifications of the item model (e.g., fitting disjunctive models to data generated according to a conjunctive model), errors in the Q‐matrix (adding or omitting paths, omitting a latent variable), and violations of local item independence due to unmodelled testlet effects. On the other hand, M2 was largely insensitive to misspecifications in the distribution of higher‐order latent dimensions and to the specification of an extraneous attribute. To complement the analyses of the overall model goodness of fit using M2, we investigated the utility of the Chen and Thissen (1997, J. Educ. Behav. Stat., 22, 265) local dependence statistic XLD2 for characterizing sources of misfit, an important aspect of model appraisal often overlooked in favour of overall statements. The XLD2 statistic was found to be slightly conservative (with Type I error rates consistently below the nominal level) but still useful in pinpointing the sources of misfit. Patterns of local dependence arising due to specific model misspecifications are illustrated. Finally, we used the M2 and XLD2 statistics to evaluate a diagnostic model fit to data from the Trends in Mathematics and Science Study, drawing upon analyses previously conducted by Lee et al., (2011, IJT, 11, 144).