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Limited‐information goodness‐of‐fit testing of diagnostic classification item response models
Author(s) -
Hansen Mark,
Cai Li,
Monroe Scott,
Li Zhen
Publication year - 2016
Publication title -
british journal of mathematical and statistical psychology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.157
H-Index - 51
eISSN - 2044-8317
pISSN - 0007-1102
DOI - 10.1111/bmsp.12074
Subject(s) - goodness of fit , statistics , local independence , item response theory , statistic , mathematics , contingency table , econometrics , complement (music) , range (aeronautics) , independence (probability theory) , latent variable , psychometrics , materials science , composite material , biochemistry , chemistry , complementation , gene , phenotype
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 X 2 and the likelihood ratio statistic G 2 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) M 2 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) M 2 statistic to diagnostic classification models. Through a series of simulation studies, we found that M 2 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, M 2 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 M 2 , we investigated the utility of the Chen and Thissen (1997, J. Educ. Behav. Stat ., 22, 265) local dependence statistic X LD 2 for characterizing sources of misfit, an important aspect of model appraisal often overlooked in favour of overall statements. The X LD 2 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 M 2 and X LD 2 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).