To proceed, we start with a description of the partial information of an item in the context of a general latent class model. Let α ∈ M = { 1 , ..., M } denote the latent class membership. The partial information of item j characterizes the latent classes it is capable of differentiating. Mathematically, we define an item-specific equivalence relation on M , denoted by “ j =”. For α $_{1}$, α 2 ∈ M , It is not hard to verify that “ j =” is an equivalence relation. We define the quotient set M / j = as the partial information of item j and use [ α ]$_{j}$ to denote the corresponding equivalence class that latent class α belongs to. The map [ · ]$_{j}$ is known as the canonical projection which leads to a partition of latent classes. Two latent classes are mapped to the same equivalence class, [ α $_{1}$]$_{j}$ = [ α $_{2}$]$_{j}$ if α 1 j = α 2 and, in this case, item j does not provide information to differentiate α 1 and α $_{2}$. From the modeling point of view, the Q -matrix along with a particular loading parameterization determines the partial information of each item. Consider a particular item j whose corresponding row vector in Q has the first l entries being one and others being zero. Then, the conditional response distribution reduces to Consider two attribute profiles α 1 and α $_{2}$. If their first l components are identical, then [ α $_{1}$]$_{j}$ = [ α $_{2}$]$_{j}$ . Under some simple loading structures (for instance, the DINO model), even if some of their first l elements are not identical, α 1 and α 2 may still belong to the same equivalence class. The following theorem presents identifiability of the partial information structure of each item. Theorem 5 Under each set of conditions of Theorems 1, 2, 3, or 4, the partial information of each item can be consistently estimated up to a permutation of the latent class label. That is, letting “ 〈 〉 $_{j}$” be the estimated canonical projection of item j , there exists a permutation of latent P ( Y j = y $^{j}$| α ) = P ( Y j = y $^{j}$| α $^{1}$, ..., α $^{l}$) . 15 α 1 j = α 2 if p y j $^{α}$1 = p y j $^{α}$2 for all y = 1 , ..., k$_{j}$ .