Restricted Latent Class Models for Nominal Response Data: Identifiability and Estimation

Abstract

Restricted latent class models (RLCMs) provide an important framework for diagnosing and classifying respondents on a collection of multivariate binary responses. Recent research made significant advances in theory for establishing identifiability conditions for RLCMs with binary and polytomous response data. Multiclass data, which are unordered nominal response data, are also widely collected in the social sciences and psychometrics via forced-choice inventories and multiple choice tests. We establish new identifiability conditions for parameters of RLCMs for multiclass data and discuss the implications for substantive applications. The new identifiability conditions are applicable to a wealth of RLCMs for polytomous and nominal response data. We propose a Bayesian framework for inferring model parameters, assess parameter recovery in a Monte Carlo simulation study, and present an application of the model to a real dataset.

Appendices

Appendix A Proof of Theorems

1.1 Preliminary Results

We start the proof with introducing some notation.

Definition 4

For a matrix $\varvec{M}$, the Kruskal rank of $\varvec{M}$ is the largest number I such that every set of I columns in $\varvec{M}$ are linearly independent.

Remark 7

Compared with the rank of a matrix M, we have $rank_K(M)\le rank(M)$. If M has full column rank, then the equality holds.

Consider a tripartition of the set ${\mathbb {J}}=\{1,2,\ldots ,J\}$ into three disjoint, non-empty subsets ${\mathbb {J}}_1=\{1,2,\ldots ,K\}$, ${\mathbb {J}}_2=\{K+1,\ldots ,2K\}$ and ${\mathbb {J}}_3=\{2K+1,\ldots ,J\}$. Then, the marginal distribution of response $\varvec{Y}$ can be represented as a three-way array $\varvec{T}$ decomposing $\varvec{Y}$ into three parts:

$$\begin{aligned} \begin{aligned} \varvec{T}_{(\varvec{y}^{{\mathbb {J}}_1}, \varvec{y}^{{\mathbb {J}}_2},\varvec{y}^{{\mathbb {J}}_3})}&= P(\varvec{Y}^{{\mathbb {J}}_1}=\varvec{y}^{{\mathbb {J}}_1}, \varvec{Y}^{{\mathbb {J}}_2}=\varvec{y}^{{\mathbb {J}}_2},\varvec{Y}^{{\mathbb {J}}_3}=\varvec{y}^{{\mathbb {J}}_3}\mid \varvec{\pi }, \varvec{B})\\&=\sum _{\varvec{\alpha }} \pi _{\varvec{\alpha }} P(\varvec{Y}^{{\mathbb {J}}_1}=\varvec{y}^{{\mathbb {J}}_1}, \varvec{Y}^{{\mathbb {J}}_2}=\varvec{y}^{{\mathbb {J}}_2},\varvec{Y}^{{\mathbb {J}}_3}=\varvec{y}^{{\mathbb {J}}_3}\mid \varvec{B},\varvec{\alpha })\\&=\sum _{\varvec{\alpha }} \pi _{\varvec{\alpha }} P(\varvec{Y}^{{\mathbb {J}}_1}=\varvec{y}^{{\mathbb {J}}_1}\mid \varvec{B},\varvec{\alpha }) P(\varvec{Y}^{{\mathbb {J}}_2}=\varvec{y}^{{\mathbb {J}}_2}\mid \varvec{B},\varvec{\alpha }) P(\varvec{Y}^{{\mathbb {J}}_3}=\varvec{y}^{{\mathbb {J}}_3}\mid \varvec{B},\varvec{\alpha }). \end{aligned} \end{aligned}$$

(A1)

Let $\varvec{T}_1$, $\varvec{T}_2$, $\varvec{T}_3$ represent distributions of $\varvec{Y}^{{\mathbb {J}}_1}$, $\varvec{Y}^{{\mathbb {J}}_2}$, $\varvec{Y}^{{\mathbb {J}}_3}$ given values of attribute profile $\varvec{\alpha }$. Then, the identifiability is equivalent to the uniqueness of the decomposition of the following tensor (Kruskal, 1977)

$$\begin{aligned} \varvec{T} =\sum _{\varvec{\alpha }} \pi _{\varvec{\alpha }} \varvec{T}_{1,\varvec{\alpha }} \otimes \varvec{T}_{2,\varvec{\alpha }} \otimes \varvec{T}_{3,\varvec{\alpha }} =\sum _{\varvec{\alpha }} \tilde{\varvec{T}}_{1, \varvec{\alpha }} \otimes \varvec{T}_{2,\varvec{\alpha }} \otimes \varvec{T}_{3,\varvec{\alpha }}, \end{aligned}$$

(A2)

where $\varvec{T}_{1, \varvec{\alpha }}$, $\varvec{T}_{2, \varvec{\alpha }}$, $\varvec{T}_{3, \varvec{\alpha }}$ are the $\varvec{\alpha }$-th column of $\varvec{T}_1$, $\varvec{T}_2$, $\varvec{T}_3$, and $\tilde{\varvec{T}}_{1, \varvec{\alpha }}= \pi _{\varvec{\alpha }} \varvec{T}_{1, \varvec{\alpha }}$.

We apply the following theorem shown in Kruskal (1977) for our proof.

Theorem 3

(Kruskal, 1977) If

$$\begin{aligned} rank_K(\tilde{\varvec{T}_1})+rank_K(\varvec{T}_2)+rank_K(\varvec{T}_3)\ge 2 \cdot 2^K +2, \end{aligned}$$

(A3)

then the tensor decomposition of $\varvec{T}$ is unique up to simultaneous permutation and rescaling of the columns.

We have $rank_K(\tilde{\varvec{T}_1})=rank_K(\varvec{T}_1)$ since $\varvec{\pi }$ has positive entries. Moreover, $\varvec{T}_1$, $\varvec{T}_2$ and $\varvec{T}_3$ are all stochastic matrices with column sum 1, so the decomposition of the tensor $\varvec{T}$ is unique up to permutations of columns if (A3) in Theorem 3 is satisfied, which implies model identifiability.

If items have M response levels, we will be able to construct $\varvec{T}_1$, $\varvec{T}_2$ and $\varvec{T}_3$ such that $\varvec{T}_1$, $\varvec{T}_2$ are $M^K\times 2^K$, and $\varvec{T}_3$ is $M^{J-2K}\times 2^K$. So for the first set of K items, the probability matrix $\varvec{T}_1$ is

$$\begin{aligned} \varvec{T}_1=\left[ \bigotimes _{j=1}^K \varvec{\theta }_{j0},\ldots ,\bigotimes _{j=1}^K \varvec{\theta }_{j,2^{K}-1} \right] . \end{aligned}$$

We multiply $\varvec{T}_1$ by a collapsing matrix $\varvec{A}$ that makes $\underset{2^K\times M^K}{\varvec{A}}\underset{M^K\times 2^K}{\varvec{T}_1}$ a square matrix of size $2^K \times 2^K$.

Definition 5

Matrix $\varvec{A}_j$, with size $2\times M$, is a collapsing matrix for item j. For the response probability vector $\theta _{jc}$, $\varvec{A}_j\theta _{jc}$ only gives probabilities of selecting a given response level or not in class c.

Following is an example of matrix $\varvec{A}_j$.

Example 5

Consider $M_j= 3$ and we are collapsing on option 1. Let

$$\begin{aligned} \varvec{A}_j = \begin{bmatrix} 1&{}0&{}1\\ 0&{}1&{}0\\ \end{bmatrix}, \end{aligned}$$

and the response probability vector $\theta _{jc}=(\theta _{jc,0},\theta _{jc,1},\theta _{jc,2})$ for item j in class c. Then, we have

$$\begin{aligned} \varvec{A}_j\theta _{jc}= \begin{bmatrix} 1&{}0&{}1\\ 0&{}1&{}0\\ \end{bmatrix} \begin{bmatrix} 1-\theta _{jc,1}-\theta _{jc,2}\\ \theta _{jc,1}\\ \theta _{jc,2}\\ \end{bmatrix} = \begin{bmatrix} 1-\theta _{jc,1}\\ \theta _{jc,1}\\ \end{bmatrix}. \end{aligned}$$

Let $\varvec{A}=\bigotimes _{j=1}^K \varvec{A}_j$, where $\varvec{A}_j$ is defined in Definition 5. Compared with $\varvec{T}_1$, $\varvec{A} \varvec{T}_1$ is a collapsed version of $\varvec{T}_1$ which only contains probabilities of selecting a given response level or not by latent class membership. So we can write $\varvec{A} \varvec{T}_1$ as

$$\begin{aligned} \varvec{A} \varvec{T}_1&=\bigotimes _{j=1}^K\varvec{A}_j \left[ \bigotimes _{j=1}^K \varvec{\theta }_{j0},\ldots ,\bigotimes _{j=1}^K \varvec{\theta }_{j,2^K-1}\right] \end{aligned}$$

(A4)

$$\begin{aligned}&=\left[ \bigotimes _{j=1}^K\varvec{A}_j \bigotimes _{j=1}^K \varvec{\theta }_{j0},\ldots ,\bigotimes _{j=1}^K\varvec{A}_j \bigotimes _{j=1}^K \varvec{\theta }_{j,2^K-1}\right] \end{aligned}$$

(A5)

$$\begin{aligned}&=\left[ \bigotimes _{j=1}^K (\varvec{A}_j\varvec{\theta }_{j0}),\ldots ,\bigotimes _{j=1}^K (\varvec{A}_j\varvec{\theta }_{j,2^{K}-1}) \right] . \end{aligned}$$

(A6)

Proposition 1

If $\varvec{\Delta }^1$ follows a simple structure shown in Definition 1 and Remark 2, then we have

$$\begin{aligned} \varvec{A}\varvec{T}_1 = \bigotimes _{j=1}^K(\varvec{p}_{j0},\varvec{p}_{j1}) \end{aligned}$$

(A7)

where $\varvec{p}_{j0}=\varvec{A}_{j}\varvec{\theta }_{j0}$ and $\varvec{p}_{j1}=\varvec{A}_{j} \varvec{\theta }_{j,\varvec{e}_j^\top \varvec{v}}$.

Proof

Simple structure implies that we can then write $\varvec{A}\varvec{T}_1$ as a block matrix:

$$\begin{aligned} \varvec{A}\varvec{T}_1&=\left( \varvec{p}_{10}\otimes \left( \bigotimes _{j>1}\varvec{A}_{j}\varvec{\theta }_{j0}\right) ,\dots ,\varvec{p}_{10}\otimes \left( \bigotimes _{j>1}\varvec{A}_{j}\varvec{\theta }_{j,2^{K-1}-1}\right) ,\right. \nonumber \\&\left. \varvec{p}_{11}\otimes \left( \bigotimes _{j>1}\varvec{A}_{j}\varvec{\theta }_{j,2^{K-1}}\right) ,\dots ,\varvec{p}_{11}\otimes \left( \bigotimes _{j>1}\varvec{A}_{j}\varvec{\theta }_{j,2^K-1}\right) \right) \nonumber \\&=\left( \varvec{p}_{10}\otimes \varvec{T}_{(1)0},\varvec{p}_{11} \otimes \varvec{T}_{(1)1}\right) . \end{aligned}$$

(A8)

We next show that simple structure of item 1 implies that,

$$\begin{aligned} \bigotimes _{j>1}\varvec{A}_j\varvec{\Theta }_{j0}=\bigotimes _{j>1}\varvec{A}_j\varvec{\Theta }_{j,2^{K-1}}, \dots , \bigotimes _{j>1}\varvec{A}_j\varvec{\Theta }_{j,2^{K-1}-1}=\bigotimes _{j>1}\varvec{A}_j\varvec{\Theta }_{j,2^{K}-1}. \end{aligned}$$

Notice that items $j>1$ are unrelated to attribute one. Let $\varvec{\alpha }_{(1)}=(\alpha _2,\dots ,\alpha _K)^\top$ denote the response pattern on attributes two through K. Simple structure of item 1 implies that classes with $\varvec{\alpha }=(0,\varvec{\alpha }_{(1)})^\top$ and $\varvec{\alpha }=(1,\varvec{\alpha }_{(1)})^\top$ will have identical response probabilities on items $j>1$. Stated differently, classes $(0,\varvec{\alpha }_{(1)})\varvec{v}=c_0$ and $(1,\varvec{\alpha }_{(1)})\varvec{v}=c_0+2^{K-1}$ have equivalent response probabilities on the remaining $j>1$ items so that $\bigotimes _{j>1}\varvec{A}_j\varvec{\Theta }_{j,c_0}=\bigotimes _{j>1}\varvec{A}_j\varvec{\Theta }_{j,c_0+2^{K-1}}$ for all $c_0\in \{0,\dots ,2^{K-1}-1\}$. Consequently, $\varvec{T}_{(1)0}=\varvec{T}_{(1)1}=\varvec{T}_{(1)}$ and properties of the Kronecker product imply that

$$\begin{aligned} \varvec{A}\varvec{T}_1=\left( \varvec{p}_{10}\otimes \varvec{T}_{(1)},\varvec{p}_{11} \otimes \varvec{T}_{(1)}\right) = \left( \varvec{p}_{10},\varvec{p}_{11}\right) \otimes \varvec{T}_{(1)} \end{aligned}$$

(A9)

where

$$\begin{aligned} \varvec{T}_{(1)} = \left( \bigotimes _{j>1}\varvec{A}_j\varvec{\Theta }_{j0},\dots ,\bigotimes _{j>1}\varvec{A}_j\varvec{\Theta }_{j,2^{K-1}-1} \right) . \end{aligned}$$

(A10)

Item two is also simple structure and repeating the aforementioned steps on $\varvec{T}_{(1)}$ implies that

$$\begin{aligned} \varvec{A}\varvec{T}_1=\left( \varvec{p}_{10},\varvec{p}_{11}\right) \otimes \left( \varvec{p}_{20},\varvec{p}_{21}\right) \otimes \varvec{T}_{(1:2)} \end{aligned}$$

(A11)

where

$$\begin{aligned} \varvec{T}_{(1:2)} = \left( \bigotimes _{j>2}\varvec{A}_j\varvec{\Theta }_{j0},\dots ,\bigotimes _{j>2}\varvec{A}_j\varvec{\Theta }_{j,2^{K-2}-1} \right) . \end{aligned}$$

(A12)

Consequently, simple structure for the remaining $j\in \{3,\dots ,K\}$ items implies that

$$\begin{aligned} \varvec{A}\varvec{T}_1 = \bigotimes _{j=1}^K(\varvec{p}_{j0},\varvec{p}_{j1}). \end{aligned}$$

(A13)

$\square$

Remark 8

Note that properties of the Kronecker product and simple structure in $\varvec{T}_1$ imply that $\varvec{A}\varvec{T}_1$ has rank $2^K$ if $\varvec{p}_{j0}$ and $\varvec{p}_{j1}$ are linearly independent for all j. According to Definition 1, there is at least one $\beta _{jjm}\ne 0$ for $m\in \{1,\dots ,M_j-1\}$, which implies that $\varvec{p}_{j0}\ne \varvec{p}_{j1}$.

Proposition 2

$rank(\varvec{A}^i\varvec{T}_i)=2^K$ if and only if $rank(\varvec{T}_i)=2^K$, $i=1,2$.

Proof

By Sylvester’s rank inequality (Matsaglia & Styan, 1974), we have

$$\begin{aligned} rank(\varvec{A}^i)+rank(\varvec{T}_i)-2^K \le rank(\varvec{A}^i\varvec{T}_i)\le \min \{rank(\varvec{A}^i),rank(\varvec{T}_i)\}. \end{aligned}$$

Given $rank(\varvec{A}_j^i)=2$, and the property of the rank of a Kronecker product, we have $rank(\varvec{A}^i)=\prod _{j=1}^{K}rank(\varvec{A}_j^i)=2^K$. Then, we get

$$\begin{aligned} rank(\varvec{T}_i) \le rank(\varvec{A}^i\varvec{T}_i)\le \min \{2^K,rank(\varvec{T}_i)\}, \end{aligned}$$

which implies that $rank(\varvec{A}^i\varvec{T}_i)=2^K$ if and only if $rank(\varvec{T}_i)=2^K$. $\square$

1.2 Proof of Theorem 1

By Proposition 2, it suffices to show that $rank(\varvec{A}^1\varvec{T}_1)=rank(\varvec{A}^2\varvec{T}_2)=2^K$ and $rank_K(\varvec{T}_3)\ge 2$, where $\varvec{A}^1=\bigotimes _{j=1}^K \varvec{A}_j^1$ and $\varvec{A}^2=\bigotimes _{j=1}^K \varvec{A}_j^2$ are collapsing matrices introduced in Definition 5. According to condition (A1) in Theorem 1, both $\varvec{\Delta }^1$ and $\varvec{\Delta }^2$ satisfy the simple structure, then by Proposition 1 and remark 8 we have $rank(\varvec{A}^1\varvec{T}_1)=rank(\varvec{A}^2\varvec{T}_2)=2^K$. For $\varvec{T}_3$, since each element is nonnegative and each column sums to 1, then given condition (A2), for any two different classes c and $c^\prime$, there must exist one item j such that $\theta _{jcm}\ne \theta _{j^\prime m}$, so that $rank_K(\varvec{T}_3)\ge 2$. By Theorem 3, the model is strictly identified.

1.3 Proof of Theorem 2

According to the tripartition of items set ${\mathbb {J}}$, we can decompose $\varvec{\Delta }$ into $\varvec{\Delta }^1$, $\varvec{\Delta }^2$, $\varvec{\Delta }^\prime$ corresponding to ${\mathbb {J}}_1$, ${\mathbb {J}}_2$, ${\mathbb {J}}_3$, respectively. Similarly, we can decompose parameter space $\Omega (\varvec{B})$ into three parts, $\Omega _{\varvec{\Delta }}(\varvec{B})=\Omega _{\varvec{\Delta }^1}\times \Omega _{\varvec{\Delta }^2}\times \Omega _{\varvec{\Delta }'}$. Therefore, to prove Theorem 2, it suffices to show that under conditions (B1) and (B2), $rank_K(\varvec{T}_1)=rank_K(\varvec{T}_2)=2^K$ and $rank_K(\varvec{T}_3)\ge 2$ hold almost everywhere in $\Omega _{\varvec{\Delta }^1}$, $\Omega _{\varvec{\Delta }^2}$, $\Omega _{\varvec{\Delta }^\prime }$, respectively. Then, by Theorem 3, the identifiability holds almost everywhere in $\Omega _{\varvec{\Delta }}(\varvec{B})$.

Based on Theorem 3 and Proposition 2, we first show that $rank(\varvec{A}^i \varvec{T}_i)=2^K$ holds almost everywhere in $\Omega _{\varvec{\Delta }^i}$, $i=1,2$, given $\varvec{\Delta }$ satisfying the structure shown in Theorem 2. Let

$$\begin{aligned} f_{i}(\varvec{B})=det(\varvec{A}^i\varvec{T}_i):\ \Omega _{\varvec{\Delta }^i}\rightarrow {\mathbb {R}} \end{aligned}$$

(A14)

denote the determinant of matrix $\varvec{A}^i\varvec{T}_i$, where $\varvec{\Delta }^i$ satisfies condition (B1).

Proposition 3

$f_{i}(\varvec{B})$ is a real analytic function of $\varvec{B}$.

Proof

$f_{i}(\varvec{B})$ is a composition function shown as below.

$$\begin{aligned} f_{i}(\varvec{B})=det(\varvec{A}^i\varvec{T}_i)=g(\varvec{\theta }_{1,0},\ldots ,\varvec{\theta }_{K,2^K-1}), \end{aligned}$$

where $\varvec{\theta }_{jc}=(\theta _{jc0},\ldots ,\theta _{jc,M_j-1})^\top$ and $\theta _{jcm}=\frac{\exp \left( \varvec{\alpha }_c^\top \varvec{\beta }_{jm} \right) }{\sum _{m'=0}^{M_j-1}\exp \left( \varvec{\alpha }_c^\top \varvec{\beta }_{jm'} \right) }=\dfrac{1}{1+\sum _{m'\ne m}\exp \left( \varvec{\alpha }_c^\top \varvec{\beta }_{jm'}-\varvec{\alpha }_c^\top \varvec{\beta }_{jm} \right) }$.

$\theta _{jcm}$ is an analytic function because exponential functions are positive analytic functions, and $g(\varvec{\theta }_{1,0},\ldots ,\varvec{\theta }_{K,2^K-1})$ is also a real analytic function of $(\varvec{\theta }_{1,0},\ldots ,\varvec{\theta }_{K,2^K-1})$ given that it is a polynomial. Therefore, we know that $f_{i}(\varvec{B})$ is a real analytic function of $\varvec{B}$, given the fact that the composition of real analytic functions is a real analytic function. $\square$

Next we introduce the following lemma, which shows that the zero set of a real analytic function has Lebesgue measure zero if the function is not constantly equal to zero.

Lemma 1

(Mityagin, 2020; Dang, 2015) Let $f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}$ be a real analytic function on ${\mathbb {R}}^d$. If f is not identically zero, then its zero set $\{\varvec{x}\in {\mathbb {R}}^d: f(\varvec{x})=0\}$ has Lebesgue measure zero.

Proposition 4

If $\varvec{\Delta }^i$ satisfies the structure shown in Theorem 2, then there exists some $\varvec{B}\in \Omega _{\varvec{\Delta }^i}$, such that $f_i(\varvec{B})\ne 0$, $i=1,2$.

Proof

As shown in condition B1 of Theorem 2, assume that for $j=1,\ldots ,K$, $\Delta _j^1$ and $\Delta _{j}^2$ satisfy the following structure:

$$\begin{aligned} \varvec{\Delta }_j= \begin{bmatrix} 0&{}0&{}\cdots &{}0&{}0&{}0&{}\cdots &{}0\\ 1&{}0&{}\cdots &{}0&{}\delta _{jj1}=1&{}0&{}\cdots &{}0\\ 1&{}0&{}\cdots &{}0&{}0&{}0&{}\cdots &{}0\\ \vdots &{}\vdots &{}&{}\vdots &{}\vdots &{}\vdots &{}&{}\vdots \\ 1&{}0&{}\cdots &{}0&{}0&{}0&{}\cdots &{}0\\ \end{bmatrix}_{M_j \times P}. \end{aligned}$$

(A15)

Then, we have

$$\begin{aligned} \varvec{p}_{j0}=\left( 1-\frac{\exp \left( \beta _{j01} \right) }{\exp \left( \beta _{j01} \right) + h_j},\frac{\exp \left( \beta _{j01} \right) }{\exp \left( \beta _{j01} \right) + h_j}\right) ^\top ,\\ \varvec{p}_{j1}=\left( 1-\frac{\exp \left( \beta _{j01}+\beta _{jj1} \right) }{\exp \left( \beta _{j01}+\beta _{jj1} \right) + h_j},\frac{\exp \left( \beta _{j01}+\beta _{jj1} \right) }{\exp \left( \beta _{j01}+\beta _{jj1} \right) + h_j}\right) ^\top , \end{aligned}$$

where

$$\begin{aligned} h_j=\sum _{m'\ne 1}\exp \left( \beta _{j0m^\prime } \right) . \end{aligned}$$

As shown in Definition 1, $\Delta _j^1$ and $\Delta _{j}^2$ satisfy simple structure, and $\varvec{p}_{j0}$ and $\varvec{p}_{j1}$ are linearly independent for $j=1,\ldots ,K$ due to $\beta _{jk1}\ne 0$. Therefore, according to Proposition 1 and Remark 8, we have $rank(\varvec{A}^i\varvec{T}_i)=\prod _{j=1}^{K}rank(\varvec{p}_{j0},\varvec{p}_{j1})=2^K$, which implies that $f_i(\varvec{B})\ne 0$, $i=1,2$. $\square$

Let $S_i$ denote the zero set of function $f_i(\varvec{B})$:

$$\begin{aligned} S_{i}=\{\varvec{B}\in \Omega _{\varvec{\Delta }^i}: f_{i}(\varvec{B})=det(\varvec{A}^i\varvec{T}_i)=0\}, \end{aligned}$$

then by Lemma 1 we can conclude that $S_{i}$ is a measure zero set with respect to $\Omega _{\varvec{\Delta }^i}$ provided $\varvec{\Delta }^i$ satisfies condition (B1).

For condition (B2), we need to show that $rank_K(\varvec{T}_3)\ge 2$ holds almost everywhere in $\varvec{\Omega }_{\varvec{\Delta }^\prime }$. Note $rank_K(\varvec{T}_3)\ge 2$ implies that for any two different attribute profiles $\varvec{\alpha }_c$, $\varvec{\alpha }_{c^\prime }$, there always exist one $j^*>2K$, such that $\theta _{j^*cm}\ne \theta _{j^*c^\prime m}$ for some choice m with $0<m<M_{j^*}$. Note that the $\varvec{\alpha }_c$-th and $\varvec{\alpha }_{c^\prime }$-th columns of $\varvec{T}_3$ are $\varvec{T}_{3,\varvec{\alpha }_c}=\bigotimes _{j>2K}\varvec{\theta }_{j\varvec{\alpha }_c}$ and $\varvec{T}_{3,\varvec{\alpha }_{c^\prime }}=\bigotimes _{j>2K}\varvec{\theta }_{j\varvec{\alpha }_{c^\prime }}$. Then, under condition B2, we have $\varvec{\theta }_{j^*\varvec{\alpha }_c}\ne \varvec{\theta }_{j^*\varvec{\alpha }_{c^\prime }}$, which implies $\varvec{T}_{3,\varvec{\alpha }_c} \ne \varvec{T}_{3,\varvec{\alpha }_{c^\prime }}$. Therefore, $\varvec{T}_{3,\varvec{\alpha }_c} = \varvec{T}_{3,\varvec{\alpha }_{c^\prime }}$ holds only when $\beta _{j^*mk}=0$ for some k and choice m with $\delta _{j^*mk}=1$, which are of Lebesgue measure zero within $\varvec{\Omega }_{\varvec{\Delta }^\prime }$. That proves $rank_K(\varvec{T}_3)\ge 2$ almost everywhere within $\varvec{\Omega }_{\varvec{\Delta }^\prime }$.

Therefore, the inequality A3 shown in Theorem 3 holds almost everywhere in $\Omega _{\varvec{\Delta }}(\varvec{B})$.

Appendix B Posterior Inference

The goal of this section is to describe our strategy for inferring the nominal RLCM parameters. We first discuss the conditional likelihood and apply the Polya-gamma identity to augment the likelihood function. An important feature to note is that we collapse the conditional likelihood, which has the advantage of requiring fewer draws of Polya-gamma augmented variables. Then, we show the derivation of full conditional distributions for a Gibbs sampling algorithm.

1.1 Conditional Likelihood and Polya Gamma Data Augmentation

Let $i=1,\dots ,n$ index individuals so that $y_{ij}$ denotes the observed response for individual i to item j and $\varvec{y}_i=(y_{i1},\dots ,y_{iJ})^\top$ is a J-vector of responses for individual i. Let $\varvec{B}$ denote all of the item parameters. The conditional distribution of a sample of n responses is,

$$\begin{aligned} p(\varvec{y}_{1:n}|\varvec{\alpha }_{1:n},\varvec{B}) = \prod _{i=1}^n p(\varvec{y}_{i}|\varvec{\alpha }_i,\varvec{B})=\prod _{i=1}^n\prod _{j=1}^J \prod _{m=0}^{M_j-1}\left( \frac{\exp \left( \varvec{a}_i^\top \varvec{\beta }_{jm} \right) }{\sum _{m'=0}^{M_j-1}\exp \left( \varvec{a}_i^\top \varvec{\beta }_{jm'} \right) }\right) ^{\mathbb {1}(y_{ij}=m)}. \end{aligned}$$

(B1)

where $\varvec{y}_i$ is the response vector for individual i, $\varvec{\alpha }_i$ is individual i’s attribute profile, $\varvec{y}_{1:n}=(\varvec{y}_{1},\dots ,\varvec{y}_{n})^\top$ is a $n\times J$ matrix of responses, and $\varvec{\alpha }_{1:n}=(\varvec{\alpha }_1,\dots ,\varvec{\alpha }_n)^\top$ denotes all attribute profiles. An important feature to note about the conditional likelihood is that we can aggregate terms in the product for individuals who reside in the same class and select the same response options on items. That is, after switching the order of the i and j products and substituting $\varvec{a}_i = \varvec{a}_{\varvec{\alpha }_i^\top \varvec{v}}$ we can aggregate Eq. B1 as

$$\begin{aligned} p(\varvec{y}_{1:n}|\varvec{\alpha }_{1:n},\varvec{B}) = \prod _{j=1}^J \prod _{c=0}^{2^K-1} \frac{\prod _{m=0}^{M_j-1} \left[ \exp \left( \varvec{a}_c^\top \varvec{\beta }_{jm} \right) \right] ^{n_{jcm}}}{\left[ \sum _{m'=0}^{M_j-1}\exp \left( \varvec{a}_c^\top \varvec{\beta }_{jm'} \right) \right] ^{n_c}} \end{aligned}$$

(B2)

where $n_{jcm} = \sum _{i=1}^n \mathbb {1}(y_{ij}=m) \mathbb {1}(\varvec{\alpha }_i^\top \varvec{v}=c)$ indicates the number of individuals within class c that select option m on item j and $n_c= \sum _{i=1}^n \mathbb {1}(\varvec{\alpha }_i^\top \varvec{v}=c)$ is the number of individuals residing in class c.

The form of Eq. B2 enables us to adopt the Polya-Gamma data augmentation strategy for models involving logistic functions (Polson et al., 2013). In particular, Polson et al. (2013) reported the following identity relating the logistic function with an integral for a Polya-Gamma (PG) random variable,

$$\begin{aligned} \frac{(e^{\psi })^a}{\left( 1+e^\psi \right) ^b} = 2^{-b}e^{\kappa \psi }\int _0^\infty e^{-w \psi ^2/2}p(w)dw,\; a\in {\mathbb {R}},\; b>0 \end{aligned}$$

(B3)

where $\kappa =a-b/2$ and $w\sim \text {PG}(b,0)$. Equation B3 provides a data augmentation strategy for the random variable $\psi$ that is conjugate with normal priors. For instance, see Polson et al. (2013) for a summary of a Gibbs sampling algorithm to infer the posterior distribution of logistic regression model parameters using Markov chain Monte Carlo (MCMC). Additionally, the PG strategy was also used for Bayesian estimation of the two parameter logistic item response theory model (Jiang & Templin, 2019) and binary diagnostic models (Balamuta & Culpepper, 2022).

We apply the PG identity in Eq. B3 by rewriting the portion of the conditional likelihood in Eq. B2 that corresponds with item j in a logistic format. Specifically, as noted by Holmes and Held and Polson et al., the full conditional distribution for $\varvec{\beta }_{jm}$ can be written as,

$$\begin{aligned} p(\varvec{\beta }_{jm}|\varvec{y}_{1:n,j},\varvec{\alpha }_{1:n},\varvec{B}_{j(m)})&=p(\varvec{\beta }_{jm})p(\varvec{y}_{1:n,j}|\varvec{B}_j)\nonumber \\&\propto p(\varvec{\beta }_{jm}) \prod _{c=0}^{2^K-1} \left[ \frac{\exp \left( \eta _{jcm} \right) }{1+\exp \left( \eta _{jcm} \right) }\right] ^{n_{jcm}} \left[ \frac{1}{1+\exp \left( \eta _{ijm} \right) }\right] ^{n_c-n_{jcm}}. \end{aligned}$$

(B4)

where $\varvec{y}_{1:n,j}=(y_{1j},\dots ,y_{nj})$, $\varvec{B}_{j(m)}=(\varvec{\beta }_{j0},\dots ,\varvec{\beta }_{j,m-1},\varvec{\beta }_{j,m+1},\dots ,\varvec{\beta }_{j,M_j-1})$ excludes coefficients for response m on j, and

$$\begin{aligned} \eta _{jcm}&=\varvec{a}_c^\top \varvec{\beta }_{jm}-C_{jcm}\end{aligned}$$

(B5)

$$\begin{aligned} C_{jcm}&=\ln \left( \sum _{m'\ne m} \exp (\varvec{a}_c^\top \varvec{\beta }_{jm'})\right) . \end{aligned}$$

(B6)

1.2 Full Conditional Distributions

1.2.1 $\delta _{jpm}$ and $\gamma$

The full conditional distribution for $\delta _{jpm}$ is

$$\begin{aligned} p(\delta _{jpm}|\beta _{jpm},\gamma ,\sigma ^2_{\beta })\propto p(\beta _{jpm}|\delta _{jpm},\sigma ^2_{\beta })p(\delta _{jpm}|\gamma ) \end{aligned}$$

(B7)

which is

$$\begin{aligned} \delta _{jpm}|\beta _{jpm},\gamma \sim \text {Bernoulli}\left( \tilde{\gamma }_{jpm}\right) \end{aligned}$$

(B8)

where

$$\begin{aligned} \tilde{\gamma }_{jpm}=\frac{\gamma p(\beta _{jpm}|\delta _{jpm}=1,\sigma ^2_{\beta })}{\gamma p(\beta _{jpm}|\delta _{jpm}=1,\sigma ^2_{\beta })+(1-\gamma ) p(\beta _{jpm}|\delta _{jpm}=0,\sigma ^2_{\beta })} \end{aligned}$$

(B9)

The full conditional distribution for $\gamma$ is

$$\begin{aligned} p(\gamma |\varvec{\Delta })\propto p(\varvec{\Delta }|\gamma )p(\gamma )=\left( \prod _{j=1}^J\prod _{p=1}^{P-1} \prod _{m=1}^{M_j-1} p(\delta _{jpm}|\gamma )\right) p(\gamma ) \end{aligned}$$

(B10)

so

$$\begin{aligned} \gamma |\varvec{\Delta }\sim \text {Beta}\left( \sum _{j=1}^J\sum _{p=1}^{P-1} \sum _{m=1}^{M_j-1} \delta _{jpm}+a,J(P-1)(M-1)-\sum _{j=1}^J\sum _{p=1}^{P-1} \sum _{m=1}^{M_j-1} \delta _{jpm}+b\right) \nonumber \\ \end{aligned}$$

(B11)

1.2.2 $\sigma _{\beta }^2$

The full conditional distribution for $\sigma _{\beta }^2$ is

$$\begin{aligned} \sigma _{\beta }^2 \mid \varvec{B}, \varvec{\Delta }\sim IGamma\left( \alpha _{\sigma }+\dfrac{1}{2}\sum _{j=1}^J P(M_j-1),\beta _{\sigma }+\dfrac{1}{2}\sum _{j=1}^J\sum _{p=0}^{P-1} \sum _{m=1}^{M_j-1}\beta _{jpm}^2(D(1-\delta _{jpm})+\delta _{jpm})\right) \nonumber \\ \end{aligned}$$

(B12)

1.2.3 $\varvec{\alpha }_i$ and $\varvec{\pi }$

We update $\varvec{\alpha }$ while integrating $\varvec{\pi }$ out

$$\begin{aligned} p(\varvec{\alpha }_1,\ldots ,\varvec{\alpha }_N)&= \int p(\varvec{\alpha }_1,\ldots ,\varvec{\alpha }_N\mid \varvec{\pi })p(\varvec{\pi })d\varvec{\pi }\nonumber \\&=\dfrac{1}{B(\varvec{d}_0)} \int \left( \prod _{c=0}^{2^K-1} \pi _{c}^{n_{c}+d_{0,c}-1}\right) \textrm{d} \varvec{\pi }\nonumber \\&=\dfrac{B(\varvec{n} + \varvec{d}_0)}{B(\varvec{d}_0)}. \end{aligned}$$

(B13)

Then, the full conditional prior distribution for $\varvec{\alpha }_i$ is

$$\begin{aligned} p(\varvec{\alpha }_i^{\top } \varvec{v}=c\mid \varvec{\alpha }_{(i)})&= \dfrac{p(\varvec{\alpha }_1,\ldots ,\varvec{\alpha }_N)}{p(\varvec{\alpha }_1,\ldots ,\varvec{\alpha }_{i-1},\varvec{\alpha }_{i+1},\ldots ,\varvec{\alpha }_N)}\nonumber \\&= \frac{n_{c(i)}+1}{n-1+2^K}, \end{aligned}$$

(B14)

where $n_{c(i)}$ represents the number of individuals other than i that have attribute profile $\varvec{\alpha }_c$. Full conditional distribution of $\varvec{\alpha }_i$ given $\varvec{y}_{1:n}$ and $\varvec{\alpha }_{(i)}$ is

$$\begin{aligned} p(\varvec{\alpha }_i^{\top } \varvec{v}=c\mid \varvec{\alpha }_{(i)},\varvec{y}_{1:n},\varvec{B}^{(t-1)})&\propto p(\varvec{\alpha }_i^{\top } \varvec{v}=c\mid \varvec{\alpha }_{(i)})p(\varvec{y}_i\mid \varvec{\alpha }_i^{\top } \varvec{v}=c,\varvec{B}^{(t-1)})\nonumber \\&\propto (n_{c(i)}+1)p(\varvec{y}_i\mid \varvec{\alpha }_i^{\top } \varvec{v}=c,\varvec{B}^{(t-1)}), \end{aligned}$$

(B15)

we update $\varvec{\alpha }_i$ sequentially with weight proportional to $(n_{c(i)}+1)p(\varvec{y}_i\mid \varvec{\alpha }_i^{\top } \varvec{v}=c,\varvec{B}^{(t-1)})$. For $\varvec{\pi }$, we have

$$\begin{aligned} \varvec{\pi }\mid \varvec{\alpha }_{1:n} \sim Dirichlet(\varvec{n}+\varvec{d}_0), \end{aligned}$$

(B16)

where $\varvec{n}=(\varvec{n}_0,\ldots ,\varvec{n}_{2^K-1})$.

1.2.4 Item Parameters, $\varvec{\beta }_{jm}$, and Augmented Parameters, $w_{jcm}$

Applying the PG identity in Eq. B3 to Eq. B4 yields,

$$\begin{aligned} p(\varvec{\beta }_{jm}|\varvec{y}_{1:n,j},\varvec{A},\varvec{B}_{j(m)},\varvec{\delta }_{jm},\varvec{w}_{jm})&\propto p(\varvec{\beta }_{jm}|\varvec{\delta }_{jm}) \prod _{c=0}^{2^K-1} \frac{\left[ \exp \left( \eta _{jcm} \right) \right] ^{n_{jcm}}}{\left[ 1+\exp \left( \eta _{jcm} \right) \right] ^{n_c}}\nonumber \\&\propto p(\varvec{\beta }_{jm}|\varvec{\delta }_{jm}) \prod _{c=0}^{2^K-1} \exp \left( {\tilde{y}}_{jcm}\eta _{jcm}-\frac{w_{jcm}\eta _{jcm}^2}{2}\right) , \end{aligned}$$

(B17)

where ${{\tilde{y}}}_{jcm}=n_{jcm}-n_c/2$ and $w_{jm}$ is a $2^K$ vector with element c defined as a PG random variable $w_{jcm}$ with full conditional distribution of $w_{jcm}|\varvec{A},\varvec{B}_j\sim \text {PG}(n_c,\eta _{jcm})$. We see,

$$\begin{aligned}&p(\varvec{\beta }_{jm}|\varvec{y}_{1:n,j},\varvec{A},\varvec{B}_{j(m)},\varvec{\delta }_{jm},\varvec{w}_{jm})\propto p(\varvec{\beta }_{jm}\mid \varvec{\delta }_{jm}) \prod _{c=0}^{2^K-1} \exp \left\{ -\frac{\omega _{jcm}}{2} \left( \frac{{{\tilde{y}}}_{jcm}}{\omega _{jcm}}-\eta _{jcm}\right) ^2\right\} \nonumber \\&\quad = p(\varvec{\beta }_{jm}\mid \varvec{\delta }_{jm}) \prod _{c=0}^{2^K-1} \exp \left\{ -\frac{\omega _{jcm}}{2} \left( z_{jcm}-\varvec{a}_c^\top \varvec{\beta }_{jm}\right) ^2\right\} \nonumber \\&\quad = p(\varvec{\beta }_{jm}\mid \varvec{\delta }_{jm}) \exp \left\{ -\frac{1}{2}\left( \varvec{z}_{jm}-{\textbf {A}}\varvec{\beta }_{jm}\right) ^\top \varvec{\Omega }_{jm}\left( \varvec{z}_{jm}-{\textbf {A}}\varvec{\beta }_{jm}\right) \right\} \end{aligned}$$

(B18)

where ${{\textbf {A}}}$ is a $2^K\times P$ design matrix and

$$\begin{aligned} z_{jcm}&=\frac{{{\tilde{y}}}_{jcm}}{\omega _{jcm}}+C_{jcm}\end{aligned}$$

(B19)

$$\begin{aligned} \varvec{z}_{jm}&=(z_{0jm},\dots ,z_{2^K-1,jm})^\top . \end{aligned}$$

(B20)

Since the prior distribution of $\varvec{\beta }_{jm}$ given $\varvec{\delta }_{jm}$ is

$$\begin{aligned} \varvec{\beta }_{jm}\mid \varvec{\delta }_{jm} \sim {\mathcal {N}} (0,\varvec{\Sigma }_{jm}), \end{aligned}$$

(B21)

where $\varvec{\Sigma }_{jm}=\sigma _\beta ^2 \text {diag}(v_{jm0},\dots ,v_{jm,2^K-1})$ and $v_{jmp}=\delta _{jmp}+(1-\delta _{jmp})/D$. Then, adding the prior term from the exponent for $\varvec{\beta }_{jm}$ yields the posterior distribution

$$\begin{aligned} p(\varvec{\beta }_{jm}|\varvec{y}_{1:n,j},\varvec{A},\varvec{B}_{j(m)},\varvec{\delta }_{jm},\varvec{w}_{jm})&\propto p(\varvec{\beta }_{jm}\mid \varvec{\delta }_{jm}) \prod _{c=0}^{2^K-1} \exp \left\{ -\frac{\omega _{jcm}}{2} \left( \frac{{{\tilde{y}}}_{jcm}}{\omega _{jcm}}-\eta _{jcm}\right) ^2\right\} \nonumber \\&\propto \exp \left\{ -\frac{1}{2}\left( \varvec{\beta }_{jm}^{\top }\varvec{\Sigma }_{jm}^{-1} \varvec{\beta }_{jm}+\left( \varvec{z}_{jm}-{{\textbf {A}}}\varvec{\beta }_{jm}\right) ^\top \varvec{\Omega }_{jm}\left( \varvec{z}_{jm}-{{\textbf {A}}}\varvec{\beta }_{jm}\right) \right) \right\} \nonumber \\&\propto \exp \left\{ -\frac{1}{2} \left( \varvec{\beta }_{jm}-\varvec{\mu }_{jm}\right) ^\top {\textbf{V}}_{jm} \left( \varvec{\beta }_{jm}-\varvec{\mu }_{jm}\right) \right\} . \end{aligned}$$

(B22)

Therefore, the full conditional distribution of $\varvec{\beta }_{jm}$ is

$$\begin{aligned} \varvec{\beta }_{jm}&\mid \varvec{Y}_{1:n,j},\varvec{A},\varvec{B}_{j(m)},\varvec{\delta }_{jm},\varvec{w}_{jm}\sim {\mathcal {N}}_{2^K}\left( \varvec{\mu }_{jm},{\textbf{V}}_{jm}\right) \end{aligned}$$

(B23)

$$\begin{aligned} {\textbf{V}}_{jm}&= \varvec{A}^\top \varvec{\Omega }_{jm}\varvec{A}+\varvec{\Sigma }_{jm}^{-1}\end{aligned}$$

(B24)

$$\begin{aligned} \varvec{\mu }_{jm}&={\textbf{V}}_{jm}^{-1} \varvec{A}^\top \varvec{\Omega }_{jm}\varvec{z}_{jm}={\textbf{V}}_{jm}^{-1} \varvec{A}^\top \left( \tilde{\varvec{y}}_{jm}+\varvec{\Omega }_{jm}\varvec{C}_{jm}\right) \end{aligned}$$

(B25)

where $\varvec{\Sigma }_{jm}=\sigma _\beta ^2 \text {diag}(v_{jm0},\dots ,v_{jmP})$ and $v_{jmp}=\delta _{jmp}+(1-\delta _{jmp})/D$. Note we use the second equality in Eq. B25 to avoid numerical issues associated with dividing by a $\omega _{jcm}$ that is zero.

Similarly, if we instead sample one coefficient at a time, we need the full conditional distribution of $\beta _{jmp}$ as follows:

$$\begin{aligned} p(\beta _{jmp}|\varvec{y}_{1:n,j},\varvec{A},\varvec{B}_{j(m)},\varvec{q}_j,\varvec{w}_{jm})&\propto p(\beta _{jmp}) \prod _{c=0}^{2^K-1} \exp \left\{ -\frac{\omega _{jcm}}{2} \left( \frac{{{\tilde{y}}}_{jcm}}{\omega _{jcm}}-\eta _{jcm}\right) ^2\right\} \nonumber \\&= p(\beta _{jmp}) \exp \left\{ -\frac{1}{2}\left( \varvec{z}_{jm}-{{\textbf {A}}}\varvec{\beta }_{jm}\right) ^\top \varvec{\Omega }_{jm}\left( \varvec{z}_{jm}-{{\textbf {A}}}\varvec{\beta }_{jm}\right) \right\} \nonumber \\&\propto \exp \left\{ -\frac{\beta _{jmp}^2}{2\varvec{\Sigma }_{jmp}} \right\} \exp \left\{ -\frac{1}{2} \left( \tilde{\varvec{z}}_{jm}-\varvec{A}_p\beta _{jmp}\right) ^\top \varvec{\Omega }_{jm}\left( \tilde{\varvec{z}}_{jm}-\varvec{A}_p\beta _{jmp}\right) \right\} \nonumber \\&\propto \left( \varvec{\beta }_{jm}-\varvec{\mu }_{jm}\right) ^\top {\textbf{V}}_{jm} \left( \varvec{\beta }_{jm}-\varvec{\mu }_{jm}\right) . \end{aligned}$$

(B26)

Therefore, the full conditional distribution for $\beta _{jmp}$ given $\varvec{Y}_{1:n,j}$, $\varvec{A}$, $\varvec{\beta }_{jm(p)}$, and $\varvec{B}_{j(m)}$ is

$$\begin{aligned} \beta _{jmp}\mid \varvec{Y}_{1:n,j},\delta _{jmp},\varvec{A}, \varvec{\beta }_{jm(p)},\varvec{B}_{j(m)}\sim \mathcal N(\mu _{jmp},\sigma _{jmp}^2), \end{aligned}$$

(B27)

$$\begin{aligned} \sigma _{jmp}^2&=\frac{1}{\varvec{A}_p^\top \varvec{\Omega }_{jm}\varvec{A}_p + 1/\sigma ^2_\beta v_{jmp}},\end{aligned}$$

(B28)

$$\begin{aligned} \mu _{jmp}&=\sigma _{jmp}^2 \varvec{A}_p^\top \varvec{\Omega }_{jm} \tilde{\varvec{z}}_{jm} = \sigma _{jmp}^2 \varvec{A}_p^\top \left( \tilde{\varvec{y}}_{jm} + \varvec{\Omega }_{jm} \varvec{C}_{jm} - \varvec{\Omega }_{jm} {\textbf{A}}_{(p)}\varvec{\beta }_{jm(p)} \right) , \end{aligned}$$

(B29)

$$\begin{aligned} \tilde{\varvec{z}}_{jm}&=\varvec{z}_{jm}-{{\textbf {A}}}_{(p)}\varvec{\beta }_{jm(p)}=\varvec{z}_{jm} - {{\textbf {A}}}\varvec{\beta }_{jm}+\varvec{A}_p\beta _{jmp}, \end{aligned}$$

(B30)

where $\varvec{A}_p$ is column p of ${{\textbf {A}}}$ and ${{\textbf {A}}}_{(p)}$ excludes column p of ${{\textbf {A}}}$. Note computation of the conditional mean and variance requires, ${{\textbf {A}}}_{p}^\top \varvec{\Omega }_{jm}{{\textbf {A}}}_{p}=\sum _{c=0}^{2^K-1} \text{ A}_{pc}\omega _{jcm}$.

Appendix C Starting Values of $\varvec{B}$

In this appendix, we will show the starting value generation steps of coefficients in $\varvec{B}$.

• 1. Perform a $k-$means clustering (MacQueen, 1967) on the observed responses.

  • (a) First, define the binary response array $\varvec{Y}^b$ with size $N\times J\times M$, where $Y_{ijm}^{b}=\varvec{I}\left\{ Y_{ij}=m \right\}$. Then, reshape the three dimensional array $\varvec{Y}^b$ into a matrix $\varvec{Y}^*$ with size $N\times JM$, through combining columns of every slice of $\varvec{Y}^b$.

  • (b) Partition the $JM-$dimensional vectors corresponding to the N respondents $Y^*=(y_1^*,\ldots ,y_N^*)^\top$ into $2^K$ distinct groups with $n_{c^\prime }$ respondents per group.

  • (c) Initialize the category response probabilities described in Eq. 1 such that $\theta _{j^\prime c^\prime }\in \varvec{\Theta }_{JM\times 2^K}$ is the $j^\prime$-th element of the cluster center for group $c^\prime$.

• 2. Assuming K factors, perform an exploratory factor analysis (EFA) on the slices of observed responses array $\varvec{Y}^b_m$, $m=1,\ldots ,M-1$.

  • (a) Generate factor scores for the $i = 1,\ldots ,N$ respondents across the $j = 1,\ldots ,J$ items and $k = 1,\ldots ,K$ attributes, $\tilde{\theta }_{ik}^m$.

  • (b) Compute within-cluster averages of the factor scores $\tilde{\theta }_{c^{^\prime }k}^m = \dfrac{1}{n_c^{^\prime }}\sum _{i^{c^\prime }=1}^{n_{c^\prime }} \tilde{\theta }_{i^{c^\prime } k}^m$ for each of the $2^K$ groups.

  • (c) Dichotomize the within-cluster factor score averages into pseudo-attributes as $\tilde{\alpha }_{c^{^\prime }k}^m = I\left( \tilde{\theta }_{c^{^\prime }k}^m>0 \right)$.

• 3. Define the pseudo-attribute profiles $\tilde{\varvec{\alpha }}_{c^{^\prime }}^m = (\tilde{\alpha }_{c^{^\prime }1}^m,\ldots ,\tilde{\alpha }_{c^{^\prime }K}^m)^\top$ in terms of the binary-integer bijection $(\tilde{\varvec{\alpha }}_{c^{^\prime }}^m)^\top \varvec{v} = c$.

• 4. Based on the bijection integers in step 3, swap the label of latent class of the initialized category response probabilities matrix got in step 1.

• 5. For $m=1,2,\ldots ,M-1$, initialize the m-th slice of $\varvec{B}$ as follows:

  • (a) Define the category response probability as $Logit^{-1}(\theta _{jcm}) = \varvec{\alpha }_c^\top \varvec{\beta }_{jm}^{(0)}\in \varvec{M}^{(0)}_{\alpha \beta :J\times 2^K}$.

  • (b) Calculate $\varvec{B}_m^{(0)} = \varvec{M}^{(0)}_{\alpha \beta }\varvec{A}(\varvec{A}^\top \varvec{A})^{-1}$, where $\varvec{A}$ is the $2^K \times P$ design matrix.

Table 6 Summary of simulation performance for RLCM with dense $\varvec{\Delta }$ and $\varvec{Q}$

Appendix D Simulation Results for Dense $\varvec{\Delta }$

The unknown denser true $\varvec{\Delta }$ and true $\varvec{Q}$ matrices for each option are shown as follows (columns in $\varvec{\Delta }$ follow the same order as the design vector shown in Eq. 3):

• $\varvec{\Delta }$ cube with $K=2$

  $$\begin{aligned} \varvec{\Delta }_{m=1}= \left( \begin{array}{llll} 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 \end{array}\right) , \quad \varvec{\Delta }_{m=2}=\left( \begin{array}{llll} 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \end{array}\right) , \quad \varvec{\Delta }_{m=3}=\left( \begin{array}{llll} 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 \end{array}\right) \end{aligned}$$

  (B31)

• $\varvec{\Delta }$ cube with $K=3$

  $$\begin{aligned} \varvec{\Delta }_{m=1}= \left( \begin{array}{llllllll} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \end{array}\right) , \varvec{\Delta }_{m=2}= \left( \begin{array}{llllllll} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \end{array}\right) , \varvec{\Delta }_{m=3}= \left( \begin{array}{llllllll} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \end{array}\right) \end{aligned}$$

  (B32)

• $\varvec{Q}$ matrices with $K=2$

  $$\begin{aligned} {\varvec{Q}}_{m=1}= \left( \begin{array}{ll} 0 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 1 \\ 1 &{} 1 \\ 1 &{} 1 \end{array}\right) , \quad {\varvec{Q}}_{m=2}=\left( \begin{array}{ll} 0 &{} 1 \\ 1 &{} 0 \\ 1 &{} 1 \\ 1 &{} 0 \\ 1 &{} 0 \\ 1 &{} 1 \\ 0 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 1 \\ 0 &{} 1 \\ 1 &{} 1 \\ 0 &{} 1 \\ 1 &{} 0 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \end{array}\right) , \quad {\varvec{Q}}_{m=3}=\left( \begin{array}{ll} 1 &{} 0 \\ 0 &{} 1 \\ 0 &{} 1 \\ 1 &{} 0 \\ 1 &{} 0 \\ 1 &{} 0 \\ 1 &{} 0 \\ 0 &{} 1 \\ 0 &{} 1 \\ 0 &{} 1 \\ 1 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 1 \\ 0 &{} 1 \\ 1 &{} 0 \\ 0 &{} 1 \\ 1 &{} 1 \end{array}\right) \end{aligned}$$

  (B33)

• $\varvec{Q}$ matrices with $K=3$

  $$\begin{aligned} {\varvec{Q}}_{m=1}= \left( \begin{array}{lll} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 \end{array}\right) , \quad {\varvec{Q}}_{m=2}=\left( \begin{array}{lll} 1 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \end{array}\right) , \quad {\varvec{Q}}_{m=3}=\left( \begin{array}{lll} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \end{array}\right) \end{aligned}$$

  (B34)

Simulation results are shown in Table 6.