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classes. We defined the T -matrix as a κ × M matrix. Each column of the matrix corresponds to one attribute profile or one latent class and each row corresponds to one response pattern. The particular order of the columns and rows of the T -matrix does not affect the results. For technical convenience, we use α and y = ( y $^{1}$, ..., y $^{J}$) to label its columns and rows, that is, t y α is the element in row y and column α . 

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Very often, we construct a T -matrix for a subset of items I ⊂ { 1 , ..., J } . We use T$_{I}$ to denote the corresponding matrix of the items in I and T to denote the matrix to all items. We first present a theorem that is essentially applicable to all situations, but its conditions are sometimes difficult to check and then we provide a few propositions for specific situations. 

Theorem 3 Consider a diagnostic classification model for J items with K attributes. Suppose that we are able split the items into three nonoverlap subsets I$_{1}$ , I$_{2}$ , and I$_{3}$ , that is, I$_{1}$ ∪ I$_{2}$ ∪ I$_{3}$ = { 1 , ..., J } . If T$_{I}$$_{1}$ , T$_{I}$$_{2}$ , and T$_{I}$$_{3}$ , the T -matrices corresponding to each of the three subsets, are all of full column rank. The probability of each latent attribute profile class π α is positive. Then the item parameters p k j α and the latent class population π α are identifiable up to a permutation of the class labels. 

The above theorem provides sufficient conditions for the identifiability. These conditions are very weak but are sometimes difficult to check in practice. This is mostly due to the fact that the T -matrix is computationally costly to construct. For instance, consider a subset of 20 items having binary responses and its T -matrix has 2 20 = 1 , 048 , 576 rows. Thus, construction of T -matrix for reasonably large-scale studies is impossible. For this concern, we present the following theorem that provides stronger but much easier to check conditions. It requires the constructions of much smaller T -matrices. 

Theorem 4 Consider a diagnostic classification model for J items with K multi-category at- tributes. For each attribute k = 1 , ..., K , there exist three nonoverlap subsets of items I$_{1}$$_{,k}$ , I$_{2}$$_{,k}$ , and I$_{3}$$_{,k}$ satisfying the following conditions. 

