A3 π α > 0 for α = 1 , 2 , . . . , M . 

Then, the item parameters p$_{j}$ α and the latent class population π α are identifiable up to a permuta- tion of the class label. 

For the latent class models, the class labels are not identifiable based on the data. The identifiability up to a permutation of class label refers to the following fact. Suppose that there exist a set of item parameters and attribute prior distribution, denoted by ˜ p k j α and ˜ π α , yielding the same marginal distribution as that in (1) with parameters p k j α and π α . Then, there exists a permutation of the class labels λ such that 

˜ p k jλ ( α ) = p k j α and ˜ π$_{λ}$$_{(}$ α ) = π α . 

Theorem 1 relies on several assumptions. Assumption A 2 requires that each item response function take only two values. This is applicable to simple models, such as the DINA and the DINO model. In the context of educational testing, the population is often split to two groups: capable and incapable. In psychiatric assessments, the partition corresponds to the presence or absence of mental disorders. Regarding assumption A 1 for a test, it is necessary to have at least one set of items I such that the conditional distributions of ( Y$_{j}$ : j ∈ I ) are different for all α , that is, the items in I differentiate among all latent classes; otherwise, it is always possible to merge some latent classes and reduce the model to satisfy the condition. Following the idea of repeated measurements [34], Assumption A 1 requires three such sets of items. It is often satisfied for tests with reasonably high accuracy. 

The following corollary of Theorem 1 presents easy-to-check conditions for diagnostic classifi- cation models. The corollary does not assume a particular paramerization. 

Corollary 1 Consider a diagnostic classification model for J binary responses with K binary at- tributes. Suppose that we can rearrange the columns and rows of Q such that it contains three distinct identity submatrices. Then, the item response function and the attribute population are identifiable up to a relabeling of the latent classes. 

