A Nonparametric Multidimensional Latent Class IRT Model in a Bayesian Framework

Abstract

We propose a nonparametric item response theory model for dichotomously-scored items in a Bayesian framework. The model is based on a latent class (LC) formulation, and it is multidimensional, with dimensions corresponding to a partition of the items in homogenous groups that are specified on the basis of inequality constraints among the conditional success probabilities given the latent class. Moreover, an innovative system of prior distributions is proposed following the encompassing approach, in which the largest model is the unconstrained LC model. A reversible-jump type algorithm is described for sampling from the joint posterior distribution of the model parameters of the encompassing model. By suitably post-processing its output, we then make inference on the number of dimensions (i.e., number of groups of items measuring the same latent trait) and we cluster items according to the dimensions when unidimensionality is violated. The approach is illustrated by two examples on simulated data and two applications based on educational and quality-of-life data.

Appendix

1.1 Prior Probabilities of $\mathcal{P}$ and s

In this section, we show how to calculate the prior probability of any partition $\mathcal{P}$ and the prior probability of the number s of groups, conditionally on k, under independent uniform priors on the success probabilities $\lambda _{jc}$. Under more general priors of type (6), analytical derivation of the priors for $\mathcal{P}$ and s becomes prohibitive and resorting to simulations is unavoidable.

We start by calculating the conditional prior probability of a single partition $\mathcal{P}$. For this aim, it is convenient to associate to each item j the ranking vector ${\varvec{q}}_j=(q_{j1},\ldots ,q_{jk})$, in which $q_{jc}$ represents the rank of $\lambda _{jc}$ after ordering $\lambda _{j1},\ldots ,\lambda _{jk}$ in increasing order. Let ${\varvec{Q}}=({\varvec{q}}{'}_{1},\ldots ,{\varvec{q}}{'}_{r})$ be the $k\times r$ matrix of the vectors ${\varvec{q}}_j$, arranged by columns. Each vector ${\varvec{q}}_{j_1}$ can have one out of k! possible configurations, independently from any other ${\varvec{q}}_{j_2}$, for $j_1\ne j_2$. Thus, there are $(k!)^r$ possible configurations of the matrix ${\varvec{Q}}$, all having the same probability under a uniform prior on the success probabilities $\lambda _{jc}$. Notice that ${\varvec{Q}}$ is a function of $\varvec{\Lambda }$.

The number of matrices ${\varvec{Q}}$ determining the same partition $\mathcal{P}$ is

$$\begin{aligned} k!(k!-1)\cdots (k!-s+1), \end{aligned}$$

where $s=|\mathcal{P}|$. This is easily proved considering that the ranking vector of the first item in the first group can have any of the k! possible configurations, which consequently determines the configuration for the ranking vectors of the other items in the same group; the ranking vector of the first item in the second group can have any possible configuration but that of the items in the first group, that is, $k!-1$ possible configurations. By iterating this process, we conclude that the ranking vector of the first item in the s-th group can have any possible configuration but those of the items in the previous groups, that is, $k!-s+1$ possible configurations. Therefore, the probability of any partition $\mathcal{P}$, conditionally on k is given by

$$\begin{aligned} p(\mathcal{P}|k)=\frac{k!(k!-1)\cdots (k!-s+1)}{(k!)^r}, \end{aligned}$$

(15)

where the denominator corresponds to the number of possible configurations of ${\varvec{Q}}$.

We now consider the prior conditional probability of the number s of groups, that is, p(s|k). Since any partition into s groups has the same probability, given in (15), to obtain p(s|k) we simply have to count how many partitions into s groups can be obtained and multiply this number by the probability of the single partition. The number of ways in which a set of r items can be partitioned into s nonempty groups is known as the Stirling number of the second kind (Graham et al., 1988), which is equal to

with

Thus, the conditional probability of s can be obtained as

where $p(\mathcal{P}|k)$ is the probability of a partition $\mathcal{P}$ such that $|\mathcal{P}|=s$, which is given in (15).

1.2 The RJ-MCMC Algorithm

In this section we provide a detailed description of the RJ-MCMC algorithm implemented in this paper. We first illustrate the fixed-dimension moves and then present the changing dimension moves for updating the number of latent classes.

1.2.1 Fixed-Dimension Moves

Model parameters are updated according to the following steps:

• Update class weights. The full conditional distribution of $\varvec{\pi }$ is a Dirichlet distribution with parameters $(1+n_{1},\ldots ,1+ n_{k})$, where $n_c = \#\{i:z_{ic}=1\}$. The vector $\varvec{\pi }$ is then updated by means of Gibbs sampling, drawing a new vector from such a distribution.

• Update latent class variables. The latent class allocation variables $z_{ic}$, $i=1,\ldots ,n$, $c=1,\ldots ,k$, can be updated by means of Gibbs sampling, drawing them independently from their full conditional

  $$\begin{aligned} p(z_{ic}=1|\cdots ) = \frac{\pi _c \prod _{j=1}^r (1-\lambda _{jc})^{1-y_{ij}}\lambda _{jc}^{y_{ij}}}{\sum _{c=1}^k \pi _c \prod _{j=1}^r (1-\lambda _{jc})^{1-y_{ij}}\lambda _{jc}^{y_{ij}}}, \end{aligned}$$

  where “$\cdots$” denotes “all other parameters and data.”

• Update the success probabilities. In order to update the success probabilities, we build a proposal based on independent zero-centered normal increments of $\mathrm{logit}\,(\lambda _{jc})$, separately for each $j=1,\ldots ,r$ and $c=1,\ldots ,k$. The candidate $\lambda _{jc}^{\star }$ is accepted with probability equal to $\min (1,p_{\lambda _{jc}^\star })$, where

  $$\begin{aligned} p_{\lambda _{jc}^\star }=\prod _{i=1}^n&\{ (\lambda _{jc}^\star /\lambda _{jc})^{y_{ij}}[(1-\lambda _{jc}^\star )/(1-\lambda _{jc})]^{(1-y_{ij})}\}^{z_{ic}}\nonumber \\&\times (\lambda _{jc}^\star /\lambda _{jc})^{(\alpha _c-1)}[(1-\lambda _{jc}^\star )/(1-\lambda _{jc})]^{(\beta _c-1)}\nonumber \\&\times (\lambda _{jc}^\star )(1-\lambda _{jc}^\star )/[(\lambda _{jc})(1-\lambda _{jc})]. \end{aligned}$$

  (16)

  The first line on the right side is the likelihood ratio, while the second line corresponds to the ratio between the prior densities. The ratio between the proposal densities cancels out, apart from the Jacobian of the logit transformation, given in the third line of (16).

1.2.2 Changing Dimension Moves for Updating k

• Split/merge move. For the combine proposal, we pick a class at random among $2,\ldots ,k$, with probability $1/(k-1)$, and denote it with $c_2$. Then, we draw another class at random among $1,\ldots ,c_2-1$, with probability $1/(c_2-1)$, and denote it with $c_1$. Classes $c_1$ and $c_2$ are then merged into a new class $c^\prime$, decreasing k by 1, with the merged class $c^\prime$ occupying the place $c_2-1$, once the place $c_1$ has been deleted. We then create new values for $\pi _{c^\prime }$ and $\lambda _{jc^\prime }$, for $j=1,\ldots ,r$, and we reallocate all those observations for which $z_{ic_1}=1$ or $z_{ic_2}=1$ to the merged class $c^\prime$. A new vector of weights is created by letting

  $$\begin{aligned} \pi _{c^\prime }=\pi _{c_1}+\pi _{c_2}. \end{aligned}$$

  The new parameters $\lambda _{jc^\prime }$ are created in such a way that $\lambda _{jc^\prime }=\lambda _{jc_2}$, while the $\lambda _{jc_1}$ are simply deleted for all j. The split proposal begins by choosing a class at random among $1,\ldots ,k$, say $c^\prime$, with probability 1 / k and splitting it into two new classes, labeled $c_1$ and $c_2$, augmenting k by 1. Let $c_2$ take the place $c^\prime +1$ and $c_1$ be inserted in a place preceding $c_2$ and chosen at random with probability $1/c^\prime$. We then need to create new values for $\pi _{c_1},\pi _{c_2}$ and for $\lambda _{jc_1},\lambda _{jc_2}$, $j=1,\ldots ,r$, and reallocate all those observations for which $z_{ic^\prime }=1$, while the hyper-parameters $\alpha _c$ and $\beta _c$ are simply recalculated as $\alpha _c^\star = vc+1$ and $\beta _c^\star =v(k^\star +1-c)+1$, for $c=1,\ldots ,k^\star$, with $k^\star =k+1$. Let us start by splitting the weight $\pi _{c^\prime }$ into $\pi _{c_1}$ and $\pi _{c_2}$ in such a way that $\pi _{c_1}+\pi _{c_2}=\pi _{c^\prime }$. We accomplish this by generating a random value $u\sim \hbox {Beta}\left( u_1,u_2\right)$, where $u_1$ and $u_2$ are the parameters of the Beta density, and setting

  $$\begin{aligned} \pi _{c_1}=\pi _{c^\prime }u \quad \hbox {and}\quad \pi _{c_2}=\pi _{c^\prime }(1-u). \end{aligned}$$

  The new vector of weights is denoted by $\varvec{\pi }^\star$. We then split the parameters $\lambda _{jc^\prime }$ into $\lambda _{jc_1}$ and $\lambda _{jc_2}$ for all j. We accomplish this by sampling a vector ${\varvec{w}}=(w_1,\ldots ,w_r)$ from the prior distribution of $\lambda _{jc_1}$, that is, $w_j\sim \hbox {Beta}(\alpha ^\star _{c_1},\beta _{c_1}^\star )$, for $j=1,\ldots ,r$, and thus setting

  $$\begin{aligned} \lambda _{jc_1}=w_j \quad \hbox {and}\quad \lambda _{jc_2}=\lambda _{jc^\prime }, \quad j=1,\ldots ,r. \end{aligned}$$

  The new matrix of success probabilities is denoted by $\varvec{\Lambda }^\star$. Finally, we reallocate all those observations for which $z_{ic^\prime }=1$ between $c_1$ and $c_2$ in a way analogous to the standard Gibbs allocation move and we let ${\varvec{Z}}^\star$ denote the new allocation matrix. According to the RJ framework, the acceptance probability for the split move is min$(1,p_k)$, where

  $$\begin{aligned} p_k= & {} \prod _{i=1}^n\prod _{j=1}^r \frac{[ (\lambda _{jc_1}^\star )^{y_{ij}} (1-\lambda _{jc_1}^\star )^{(1-y_{ij})}]^{z_{ic_1}^\star } [ (\lambda _{jc_2}^\star )^{y_{ij}} (1-\lambda _{jc_2}^\star )^{(1-y_{ij})}]^{z_{ic_2}^\star }}{[ (\lambda _{jc^\prime })^{y_{ij}} (1-\lambda _{jc^\prime })^{(1-y_{ij})}]^{z_{ic^\prime }}}\nonumber \\&\times \frac{p(k^\star )}{p(k)}\times \frac{\mathcal {D}(\pi _1^\star ,\ldots ,\pi _{k^\star }^\star )}{\mathcal {D}(\pi _1,\ldots ,\pi _{k})}\times \frac{(\pi _{c_1}^\star )^{\sum _{i=1}^n z_{ic_1}^\star }(\pi _{c_2}^\star )^{\sum _{i=1}^n z_{ic_2}^\star }}{(\pi _{c^\prime })^{\sum _{i=1}^n z_{ic^\prime }}} \nonumber \\&\times \frac{\prod _{j=1}^r \prod _{c=1}^{k^\star }\mathcal {B}_{\alpha _c^\star ,\beta _c^\star }(\lambda _{jc}^\star )}{\prod _{j=1}^r \prod _{c=1}^{k} \mathcal {B}_{\alpha _c,\beta _c}(\lambda _{jc})}\times \frac{d_{k^\star }}{b_k p_{\hbox {alloc}}\mathcal {B}_{u_1,u_2}(u)\prod _{j=1}^r \mathcal {B}_{\alpha _{c_1}^\star ,\beta _{c_1}^\star }(w_j)}\times \pi _{c^\prime }.\qquad \end{aligned}$$

  (17)

  The first two lines in (17) represent, respectively, the likelihood and the priors ratio, with $\mathcal {D}$ being the Dirichlet density with parameters all equal to 1, and $\mathcal {B}$ being the Beta density with parameters specified in the subscript. In the third line, the first term represents the proposal ratio, with $p_{\hbox {alloc}}$ being the probability of the particular allocation made in the split move, and the second term is the Jacobian of the transformation from $(\pi _{c^\prime }, \lambda _{1 c^\prime },\ldots , \lambda _{r c^\prime }, u,w_{1}, \ldots , w_{r})$ to $(\pi _{c_1}^\star , \lambda _{1 c_2}^\star ,\ldots , \lambda _{r c_2}^\star , \pi _{c_2}^\star , \lambda _{1 c_1}^\star ,\ldots , \lambda _{r c_1}^\star )$. The combine move is accepted with probability $\min (1,p_k^{-1})$, with some obvious substitutions in the expression for $p_k$.

• Birth/death move. We first make a random choice between birth and death, using the same probabilities $b_k$ and $d_k$ as above. For a birth, we pick a position at random among $1,\ldots ,k^\star$, with probability $1/k^\star$ for the place to be occupied by the new class $c^\prime$. Then, a weight and a vector of success probabilities for the new class are drawn using

  $$\begin{aligned} \pi _{c^\prime }\sim \hbox {Beta}(1,k), \quad \lambda _{jc^\prime }\sim \hbox {Beta}(\alpha ^\star _{c^\prime },\beta _{c^\prime }^\star ), \quad j=1,\ldots ,r, \end{aligned}$$

  where the hyper-parameters $\varvec{\alpha }$ and $\varvec{\beta }$ are recalculated as in the split move, leading to $\varvec{\alpha }^\star$ and $\varvec{\beta }^\star$, respectively. To “make space” for the new class, the existing weights are rescaled, so that all weights sum to 1, using $\pi _c^\star = \pi _c(1 - \pi _{c^\prime })$. The weights and the success probabilities proposed under the birth move are denoted by $\varvec{\pi }^\star$ and $\varvec{\Lambda }^\star$, respectively. The allocation variables remain unchanged, and no subjects are allocated to the new component, that is, $z_{ic^\prime }=0$, $i=1,\ldots ,n$. The increased allocation matrix is indicated as ${\varvec{Z}}^\star$. For a death, a random choice is made between any existing empty components; the chosen component is deleted and the remaining weights are rescaled to sum to 1. No other changes are proposed to the variables and, in particular, the allocations are unaltered. The acceptance probabilities for birth and death are $\min (1, p_k)$ and $\min (1, p_k^{-1})$, respectively, where

  $$\begin{aligned} p_k= & {} \frac{p(k^\star )}{p(k)}\times \frac{\mathcal {D}(\pi _1^\star ,\ldots ,\pi _{k^\star }^\star )}{\mathcal {D}(\pi _1,\ldots ,\pi _{k})}\times \frac{\prod _{c=1}^{k^\star }(\pi _c^\star )^{\sum _{i=1}^n z_{ic}^\star }}{\prod _{c=1}^k \pi _c^{\sum _{i=1}^n z_{ic}}} \times \frac{\prod _{j=1}^r\prod _{c=1}^{k^\star }\mathcal {B}_{\alpha _c^\star ,\beta _c^\star }(\lambda _{jc}^\star )}{\prod _{j=1}^r \prod _{c=1}^{k} \mathcal {B}_{\alpha _c,\beta _c}(\lambda _{jc})}\nonumber \\&\times \frac{d_{k^\star }k^\star }{b_k (k_0+1) \mathcal {B}_{1,k}(\pi _{c^\prime }^\star )\prod _{j=1}^r \mathcal {B}_{\alpha _{c_1}^\star ,\beta _{c_1}^\star }(\lambda _{jc^\prime }^\star )}\times (1-\pi _{c^\prime }^\star ), \end{aligned}$$

  (18)

  with $k_0$ being the number of empty classes, before the birth. In equation (18), the first line is the prior ratio, and the second line contains the proposal ratio and the Jacobian; the likelihood ratio is equal to 1.