2 Graphical Diagnostic Classification Model The present section elaborates on the GDCM framework after briefly introducing the DCM and the Markov network. Suppose a test consists of J items measuring K attributes, α$_{1}$ , . . . , α$_{K}$ . The cognitive status for each attribute is indicated by a binary value: 1 implying mastery of an attribute and 0 implying nonmastery of an attribute. Let X$_{j}$ be a response variable for item j , and x$_{j}$ be a realization of X$_{j}$ . An examinee’s latent cognitive profile is denoted by a 2 $^{K}$-dimensional binary vector, α = (1 , α$_{1}$, α$_{2}$, . . . , α$_{K}$, α$_{1}$α$_{2}$, . . . , α$_{1}$α$_{2}$ · · · α$_{K}$ ) $^{⊤}$. Likewise, statistical properties of item j are indicated by a vector of regression coefficients, β$_{j}$ = ( β$_{j}$$_{0}$, β$_{j}$$_{1}$, . . . , β$_{jK}$, β$_{j}$$_{12}$, . . . , β$_{j}$$_{12}$$_{···}$$_{K}$ ) $^{⊤}$, where β$_{j}$$_{0}$ is an intercept parameter; β$_{j}$$_{1}$ , . . . , β$_{jk}$ denote the main effects of the K attributes; β$_{j}$$_{12}$ shows an interaction effect of α$_{1}$ and α$_{2}$ ; and β$_{j}$$_{12}$$_{···}$$_{K}$ shows an interaction effect of α$_{1}$ , . . . , α$_{K}$ . The item response function of the DCM then admits the generalized linear form The current study assumes that the loading structure of test items on the latent attributes is properly implied by β$_{j}$ , for instance, via a J -by- K matrix Q = ( q$_{j}$ : j = 1 , . . . , J ) (Tatsuoka, 1985). Statistical methods for estimating Q have been discussed in many papers; see, e.g., Y. Chen, Liu, Xu, and Ying (2015), Y. Chen, Liu, and Ying (2015), Chiu (2013), de la Torre (2008), and J. Liu, Xu, and Ying (2012, 2013). To illustrate (1) via a specific model, let us consider the DINA model (Haertel, 1989; Junker & Sijtsma, 2001) as an example, one of the most popular and widely-studied models in the diagnostic modeling. The cognitive process underlying the DINA model is conjunctive in that, for responding an item correctly, an examinee needs to master all required attributes by the item. This results in two latent classes for each item: one with those who have mastered all desired attributes and the other with those who lack at least one of the measured attributes. Suppose item j requires 1 , 2 , . . . , K$_{j}$ attributes that, without loss of generality, can be recognized to the first K$_{j}$ positions of a row vector q$_{j}$ . The item parameter β$_{j}$ is then comprised of two nonzero values, β$_{j}$$_{0}$ and β$_{j}$$_{12}$$_{···}$$_{K}$$_{j}$ , and 0 otherwise. The intercept parameter β$_{j}$$_{0}$ characterizes the positive response probability for the baseline (nonmastery) group, and the interaction parameter β$_{j}$$_{12}$$_{···}$$_{K}$$_{j}$ differentiates the probability 5 logit Pr ( X$_{j}$ = 1 | α ; β$_{j}$ ) = β ⊤ j α . (1)