A Partially Confirmatory Approach to the Multidimensional Item Response Theory with the Bayesian Lasso

Abstract

For test development in the setting of multidimensional item response theory, the exploratory and confirmatory approaches lie on two ends of a continuum in terms of the loading and residual structures. Inspired by the recent development of the Bayesian Lasso (least absolute shrinkage and selection operator), this research proposes a partially confirmatory approach to estimate both structures using Bayesian regression and a covariance Lasso within a unified framework. The Bayesian hierarchical formulation is implemented using Markov chain Monte Carlo estimation, and the shrinkage parameters are estimated simultaneously. The proposed approach with different model variants and constraints was found to be flexible in addressing loading selection and local dependence. Both simulated and real-life data were analyzed to evaluate the performance of the proposed model across different situations.

Appendices

Derivation of Conditional Distributions

Equation (2) can be rewritten as $y_{ij} ={{\varvec{\uplambda }}}_{j}^\mathrm{T} {\varvec{\uptheta }}_{i} +\varvec{\upmu }_{j} +\varvec{\upvarepsilon }_{ij}$. After rearranging the loading vector for item j, the trait vector is ${\varvec{\uptheta }}_{i} =\left( {{\begin{array}{*{20}c} {{\varvec{\uptheta }}_{i}^{\prime } } \\ {{\varvec{\uptheta }}_{i}^{\prime \prime } } \\ \end{array} }} \right)$, where the $K' \times 1$ vector ${\varvec{\uptheta }}_{i}^{\prime }$ and $K'' \times 1$ vector ${\varvec{\uptheta }}_{i}^{\prime \prime }$ are associated with the unspecified loadings and zero/specified loadings, respectively. Similarly, one can denote the rearranged trait matrix as $\varvec{\Theta }=\left( {{\begin{array}{*{20}c} \varvec{\Theta }^{\prime } \\ \varvec{\Theta }^{\prime \prime } \\ \end{array} }} \right) =\left( {{\begin{array}{*{20}c} {{\varvec{\uptheta }}_{1}^{\prime } , \ldots {\varvec{\uptheta }}_{i}^{\prime } , \ldots {\varvec{\uptheta }}_{N}^{\prime } } \\ {{\varvec{\uptheta }}_{1}^{\prime \prime } , \ldots {\varvec{\uptheta }}_{i}^{\prime \prime } , \ldots {\varvec{\uptheta }}_{N}^{\prime \prime } } \\ \end{array} }} \right)$. Denote y$_{i(-j)}$ as the subvector of y$_{i}$ with the jth element deleted. The conditional distribution for ${\varvec{\uplambda }}_{j}^{\prime }$ can be obtained as:

$$\begin{aligned}&\left. {p({\varvec{\uplambda }}_{j}^{\prime } } \right| \mathbf{Z }, {\varvec{\Lambda }}_{-j} , {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Psi }}, \varvec{\delta } _{j} )=\left. {p({\varvec{\uplambda }}_{j}^{\prime } } \right| \mathbf{Y }_{j} , \mathbf{Y }_{-j} , {\varvec{\Theta }}, {\varvec{\Lambda }}_{-j} , {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Psi }}, \varvec{\delta }_{j} )\\&\quad \propto p(\mathbf{Y }_{j} \vert \mathbf{Y }_{-j} , {\varvec{\Theta }}, {\varvec{\uplambda }}_{j}^{\prime } , {\varvec{\Lambda }}_{-j} , {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Psi }})p({\varvec{\uplambda }}_{j}^{\prime } ) \\&\quad =\mathop \prod \limits _{i=1}^{N} p(y_{ij} \vert \mathbf{y }_{i(-j)} , {\varvec{\uptheta }}_{i} , {\varvec{\uplambda }}_{j}^{\prime } , {\varvec{\Lambda }}_{-j} , {\varvec{\upmu }}, {\varvec{\Psi }})p({\varvec{\uplambda }}_{j}^{\prime } ) \\&\quad \propto \mathop \prod \limits _{i=1}^{N} \exp \left\{ \frac{1}{2}\left[ {y_{ij} -\mu _{j} -({\varvec{\uplambda }}_{j}^{\prime } )^\mathrm{T}{\varvec{\uptheta }}_{i}^{\prime } -{\varvec{\uppsi }}_{j}^\mathrm{T} {\varvec{\uppsi }}_{-jj}^{-1} (\mathbf{y }_{i(-j)} -{\varvec{\upmu }}_{-j} -{\varvec{\Lambda }}_{-j} {\varvec{\uptheta }}_{i} )} \right] ^{\text{ T }}\times \right. \\&\quad \left. (\psi _{jj} -{\varvec{\uppsi }}_{j}^\mathrm{T} {\varvec{\Psi }}_{-jj}^{-1} {\varvec{\uppsi }}_{j} )^{-1}\left[ {y_{ij} -\mu _{j} -({\varvec{\uplambda }}_{j}^{\prime } )^\mathrm{T}{\varvec{\uptheta }}_{i}^{\prime } -{\varvec{\uppsi }}_{j}^\mathrm{T} {\varvec{\Psi }}_{-jj}^{-1} (\mathbf{y }_{i(-j)} -{\varvec{\upmu }}_{-j} -{\varvec{\Lambda }}_{-j} {\varvec{\uptheta }}_{i} )} \right] \right\} \times p({\varvec{\uplambda }}_{j}^{\prime } ). \end{aligned}$$

With $\psi _{jj}^{*} =\psi _{jj} -{\varvec{\uppsi }}_{j}^\mathrm{T} {\varvec{\Psi }}_{-jj}^{-1} {\varvec{\uppsi }}_{j}$, $\mathbf{Y }_{j}^{*} =\mathbf{Y }_{j} -{\varvec{\uppsi }}_{j}^\mathrm{T} {\varvec{\Psi }}_{-jj}^{-1} \mathbf{Y }_{-j}$ or $y_{ij}^{*} =y_{ij} -{\varvec{\uppsi }}_{j}^\mathrm{T} {\varvec{\Psi }}_{-jj}^{-1} \mathbf{y }_{i(-j)}$, and ${\varvec{\upmu }}_{j}^{*} =\mu _{j} -{\varvec{\uppsi }}_{j}^\mathrm{T} {\varvec{\Psi }}_{-jj}^{-1} ({\varvec{\upmu }}_{-j} +{\varvec{\Lambda }}_{-j} {\varvec{\Theta }})$ or $\mu _{ij}^{*} =\mu _{j} -{\varvec{\uppsi }}_{j}^\mathrm{T} {\varvec{\Psi }}_{-jj}^{-1} ({\varvec{\upmu }}_{-j} +{\varvec{\Lambda }}_{-j} {\varvec{\uptheta }}_{i} )$, it becomes:

$$\begin{aligned}&\mathop \prod \limits _{i=1}^{N} \exp \left\{ {-\frac{1}{2}(y_{ij}^{*} -\mu _{ij}^{*} -({\varvec{\uplambda }}_{j}^{\prime } )^\mathrm{T}{\varvec{\uptheta }}_{i}^{\prime } )^{\text{ T }}\psi _{jj}^{*-1} (y_{ij}^{*} -\mu _{ij}^{*} -({\varvec{\uplambda }}_{j}^{\prime } )^\mathrm{T}{\varvec{\uptheta }}_{i}^{\prime } )} \right\} \times p({\varvec{\uplambda }}_{j}^{\prime } )\\&\quad =\exp \left\{ -\frac{1}{2}\psi _{jj}^{*-1} (\mathbf{Y }_{j}^{*} -{\varvec{\upmu }}_{j}^{*} -({\varvec{\uplambda }}_{j}^{\prime } )^\mathrm{T}{\varvec{\Theta }}^{\prime } )^{\text{ T }}(\mathbf{Y }_{j}^{*} -{\varvec{\upmu }}_{j}^{*} -({\varvec{\uplambda }}_{j}^{\prime } )^\mathrm{T}{\varvec{\Theta }}^{\prime } )\right\} \times p({\varvec{\uplambda }}_{j}^{\prime } ). \end{aligned}$$

With ${\varvec{\uplambda }}_{j}^{\prime } \sim {N}(\mathbf{0 }, \mathbf{D }_{\tau _{j} } )$, one has:

$$\begin{aligned}&\left. {{\varvec{\uplambda }}_{j}^{\prime } } \right| \mathbf{Z }, {\varvec{\Lambda }}_{-j} , {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Psi }}, \updelta _{j} \sim N[(\psi _{jj}^{*-1} ({\varvec{\Theta }}^{\prime } \text{( }{\varvec{\Theta }}^{\prime } )^\mathrm{T})+\mathbf{D }_{\tau _{j} }^{-1} \text{) }^{-1}\psi _{jj}^{*-1} {\varvec{\Theta }}^{\prime } (\mathbf{Y }_{j}^{*} -{\varvec{\upmu }}_{j}^{*} ), \\&\quad (\psi _{jj}^{*-1} ({\varvec{\Theta }}^{\prime } \text{( }{\varvec{\Theta }}^{\prime } )^\mathrm{T})+\mathbf{D }_{\tau _{j} }^{-1} \text{) }^{-1}]. \end{aligned}$$

The conditional distribution for $\tau _{jk}$ can be expressed as:

$$\begin{aligned}&p(\tau _{jk}^{2} \left| \mathbf{Y }, {\varvec{\Lambda }}, {\varvec{\Theta }}, {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Psi }})\propto p(\lambda _{jk} \right| \tau _{jk}^{2} )p(\tau _{jk}^{2} )\\&\quad \propto (\tau _{jk}^{2} )^{-\frac{1}{2}}\exp \left\{ {-\frac{1}{2\tau _{jk}^{2} }\lambda _{jk}^{2} } \right\} \exp \left\{ {-\frac{\delta _{j}^{2} }{2}\tau _{jk}^{2} } \right\} .\\&\quad \Rightarrow p\left( \left. {\frac{1}{\tau _{jk}^{2} }} \right| \mathbf{Y }, {\varvec{\Lambda }}, {\varvec{\Theta }}, {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Psi }}\right) \propto \text {Inv-Gaussian}\left( \sqrt{\frac{\delta _{j}^{2} }{\lambda _{jk}^{2} }} , \delta _{j}^{2} \right) . \end{aligned}$$

The conditional distribution for $\delta _{j}$ can be expressed as:

$$\begin{aligned}&p(\delta _{j}^{2} \left| \mathbf{Y }, {\varvec{\Lambda }}, {\varvec{\Theta }}, {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Psi }})\propto p(\tau _{jk}^{2} \right| \delta _{j}^{2} )p(\delta _{j}^{2} ) \\&\quad \propto \frac{\delta _{j}^{2} }{2}\exp \left\{ {-\frac{\delta _{j}^{2} }{2}\tau _{jk}^{2} } \right\} \times (\delta _{j}^{2} )^{\alpha _{j} -1}\exp \left\{ {-\beta _{j} (\delta _{j}^{2} )} \right\} \\&\quad \propto (\delta _{j}^{2} )^{(\alpha _{j} +1)-1}\exp \left\{ {-(\frac{\tau _{jk}^{2} }{2}+\beta _{j} )(\delta _{j}^{2} )} \right\} \\&\quad \propto Gamma(\alpha _{j} +1, \frac{\tau _{jk}^{2} }{2}+\beta _{j} ). \end{aligned}$$

Similarly, with ${\varvec{\uplambda }}_{j}^{"} \sim {N}(\lambda _{0j} , \mathbf{H }_{0j} )$, the conditional distribution for ${\varvec{\uplambda }}_{j}^{"}$ can be obtained as:

$$\begin{aligned}&\left. {{\varvec{\uplambda }}_{j}^{"} } \right| \mathbf{Z }, {\varvec{\Lambda }}_{-j} , {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Psi }}\sim N[(\psi _{jj}^{*-1} ({\varvec{\Theta }}^{\prime } \text{( }{\varvec{\Theta }}^{\prime } )^\mathrm{T})+\mathbf{H }_{0j}^{-1} \text{) }^{-1}\psi _{jj}^{*-1} {\varvec{\Theta }}^{\prime } ((\mathbf{Y }_{j}^{*} -{\varvec{\upmu }}_{j}^{*} )+\uplambda _{0j} \mathbf{H }_{0j}^{-1} ), \\&\quad (\psi _{jj}^{*-1} ({\varvec{\Theta }}^{\prime } \text{( }{\varvec{\Theta }}^{\prime } )^\mathrm{T})+\mathbf{H }_{0j}^{-1} \text{) }^{-1}]. \end{aligned}$$

The conditional distributions $p(\varvec{\uptheta }_{i}|\mathbf{Y} , \varvec{\Lambda }, \varvec{\upmu }, \varvec{\Phi }, \varvec{\Psi })$ and $\left. p(\varvec{\upmu } \right| \mathbf{Z }, {\varvec{\Lambda }}, {\varvec{\Phi }}, {\varvec{\Psi }})$ are similar to those presented in Lee (2007, pp. 146–147), as:

$$\begin{aligned}&\left. {{\varvec{\uptheta }}_{i} } \right| \mathbf{X }, \mathbf{Y }, {\varvec{\Lambda }}, {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Psi }}\sim N[({\varvec{\Phi }}^{-1}+{\varvec{\Lambda }}^{\text{ T }}{\varvec{\Psi }}^{-1}{\varvec{\Lambda }})^{-1}{\varvec{\Lambda }}^{\text{ T }}{\varvec{\Psi }}^{-1}(\mathbf{y }_{i} -{\varvec{\upmu }}), ({\varvec{\Phi }}^{-1}+{\varvec{\Lambda }}^{\text{ T }}{\varvec{\Psi }}^{-1}{\varvec{\Lambda }})^{-1}],\\&\quad \left. {{\varvec{\upmu }}} \right| \mathbf{Z }, {\varvec{\Lambda }}, {\varvec{\Phi }}, {\varvec{\Psi }}\sim N[(\mathbf{H }_{\mu 0}^{-1} +N{\varvec{\Psi }}^{-1}\text{) }^{-1}({\varvec{\Psi }}^{-1}\mathbf{V }+{\varvec{\upmu }}_{0} \mathbf{H }_{\mu 0}^{-1} ), (\mathbf{H }_{\mu 0}^{-1} +N{\varvec{\Psi }}^{-1}\text{) }^{-1}], \end{aligned}$$

where $\mathbf{V }=\sum \nolimits _{i=1}^N {(\mathbf{y }_{i} -{\varvec{\Lambda } \varvec{\uptheta }}_{i} )}$.

For PCIRM-LI, $\varvec{\Psi } =\text {diag}(\psi _{jj})$ is modeled as a diagonal matrix. The conditional distribution for $\psi _{jj}$ can be expressed as: $\left. {p(\psi _{jj}^{-1} } \right| \mathbf{Z }, {\varvec{\Lambda }}, {\varvec{\upmu }}, {\varvec{\Phi }})\propto p(\mathbf{y }_{j} \vert \mathbf{y }_{-j} , {\varvec{\Lambda }}, {\varvec{\Theta }}, {\varvec{\upmu }}, {\varvec{\Phi }})p(\psi _{jj}^{-1} )$. It can be further simplified as:

$$\begin{aligned} \left. {p(\psi _{jj} } \right| \mathbf{Z }, {\varvec{\Lambda }}, {\varvec{\upmu }}, {\varvec{\Phi }})\propto \text {Inv-Gamma}(\alpha _{0j} +\frac{N}{2}-1, \beta _{0j} +\frac{1}{2}\mathop \sum \limits _{i=1}^{N} [y_{ij} -\mu _{j} -({\varvec{\uplambda }}_{j}^{\prime \prime } \text{) }^\mathrm{T}{\varvec{\uptheta }}_{i}^{\prime \prime } ]^{2}). \end{aligned}$$

Block Gibbs Sampler to Draw $\Sigma$ and $\Psi$ with Local Dependence

The conditional distribution $\left. p({\varvec{\Sigma }} \right| \mathbf{Y }, {\varvec{\upmu }}, {\varvec{\Theta }}, {\varvec{\Lambda }}, {\varvec{\Phi }}, {\varvec{\uptau }}_{s} , \delta _{s} )$ can be decomposed as follows:

$$\begin{aligned}&\left. p({\varvec{\Sigma }} \right| \mathbf{Y }, {\varvec{\upmu }}, {\varvec{\Theta }}, {\varvec{\Lambda }}, {\varvec{\Phi }}, {\varvec{\uptau }}_{s} , \delta _{s} )\propto \left. p(\mathbf{Y } \right| {\varvec{\Sigma }}, {\varvec{\upmu }}, {\varvec{\Theta }}, {\varvec{\Lambda }}, {\varvec{\Phi }})p({\varvec{\uptau }}_{s} , \delta _{s} )\\&\quad \propto \left| {{\varvec{\Sigma }}} \right| ^{N/2}\exp \left[ {-\text {tr}\left( -\frac{1}{2}\mathbf{S \varvec{\Sigma }}\right) } \right] \mathop \prod \limits _{i<j} \exp \left( {-\frac{\sigma _{ij}^{2} }{2\tau _{ij} }} \right) \times \mathop \prod \limits _{j=1}^{J} \exp \left( {-\frac{\delta _{s} \sigma _{jj} }{2}} \right) \text {I}({\varvec{\Sigma }}>0) \end{aligned}$$

where ${\varvec{\uptau }}_{s} =(\tau _{ij} )_{i<j}$ is the vector of the latent scale parameters, and

$$\begin{aligned} \mathbf{S }=\sum _{i=1}^N {({\mathbf{y}}_{i} -{\varvec{\upmu }}-{\varvec{\Lambda \theta }}_{i} \text{) }(\mathbf{y }_{i} -{\varvec{\upmu }}-{\varvec{\Lambda \theta }}_{i} \text{) }^{\mathrm{T}}}. \end{aligned}$$

For $j = 1, \ldots , J$ and without loss of generality, one can partition and rearrange the columns of $\varvec{\Sigma }$ and S as follows:

$$\begin{aligned} {\varvec{\Sigma }}=\left( {{\begin{array}{*{20}c} {{\varvec{\Sigma }}_{-jj} } &{} {{\varvec{\upsigma }}_{j} } \\ {{\varvec{\upsigma }}_{j}^\mathrm{T} } &{} {\sigma _{jj} } \\ \end{array} }} \right) , \quad \mathbf{S }=\left( {{\begin{array}{*{20}c} {\mathbf{S }_{-jj} } &{} {\mathbf{s }_{j} } \\ {\mathbf{s }_{j}^\mathrm{T} } &{} {s_{jj} } \\ \end{array} }} \right) \end{aligned}$$

where $\sigma _{jj}$ is the jth diagonal element of $\varvec{\Sigma }$, $\varvec{\sigma }_{j}=\left( \sigma _{j1}, {\ldots , }\sigma _{j,j-1}, \sigma _{j,j+1}, \ldots , \sigma _{jJ} \right) ^{\mathrm {T}}$is the vector of all off-diagonal elements of the jth column, and $\varvec{\Sigma }_{-jj}$ is the ($J \quad -$ 1) $\times$ ($J \quad -$ 1) matrix resulting from deleting the jth row and jth column of $\varvec{\Sigma }$. Similar, $s_{-jj}$ is the jth diagonal element of S, s$_{j}$ is the vector of all-diagonal elements of the jth column of $\mathbf{S }$, and $\mathbf{S} _{-jj}$ is the matrix with thejth row and jth column of S deleted. Then we have:

$$\begin{aligned}&p\left( {\varvec{\sigma }_{j}, \sigma _{jj}} \vert {\varvec{\Sigma }}_{-jj}\mathrm {,} \mathbf{Y} , \varvec{\Theta }, \varvec{\upmu }, \varvec{\Lambda }, \varvec{\Phi }, \varvec{\uptau }_{s}, \delta _{s}\right) \\&\quad {\propto \left( \sigma _{jj}-\varvec{\upsigma }_{j}^{\mathrm {T}}\varvec{\Sigma }_{-jj}^{-1}\varvec{\upsigma }_{j} \right) }^{\frac{N}{2}}\times \mathrm {exp}\left\{ -\frac{1}{2}[\varvec{\upsigma }_{j}^{\mathrm {T}}{} \mathbf{M} _{\tau }^{\mathrm {T}}\varvec{\upsigma }_{j}+2\mathbf {s}_{j}^{\mathrm {T}}\varvec{\upsigma }_{j}+\left( s_{jj}+\delta _{s} \right) \sigma _{jj}]\right\} \end{aligned}$$

where $\mathbf{M} _{\tau }$ is the diagonal matrix with diagonal elements $\tau _{j1}, \ldots , \tau _{j,j-1}, \tau _{j,j+1}, \ldots , \tau _{jJ}$.

Let $\varvec{\beta }=\varvec{\upsigma }_{j}$ and $\varvec{\gamma }=\sigma _{jj}-\varvec{\upsigma }_{j}^{\mathrm {T}}\varvec{\Sigma }_{-jj}^{-1}\varvec{\upsigma }_{j}$. It can be shown that:

$$\begin{aligned}&p\left( \varvec{\beta } \vert {\varvec{\Sigma }}_{-jj}, \mathbf{Y} , \varvec{\Theta }, \varvec{\upmu }, \varvec{\uplambda }, \varvec{\Phi }, \varvec{\uptau }_{s}, \delta _{s}\right) \\&\quad \propto N\left( -\left[ \left( s_{jj}+\delta _{s} \right) \varvec{\Sigma }_{-jj}^{-1}+\mathbf{M} _{\tau }^{-1} \right] ^{-1}\mathbf {s}_{j}, \left[ \left( s_{jj}+\delta _{s} \right) \varvec{\Sigma }_{-jj}^{-1}+\mathbf{M} _{\tau }^{-1} \right] ^{-1}\right) , \\&\quad p\left( \varvec{\gamma } \vert {\varvec{\Sigma }}_{-jj}, \mathbf{Y} , \varvec{\Omega }, \varvec{\upmu }, \varvec{\Lambda }, \varvec{\Phi }, \varvec{\uptau }_{s}, \delta _{s}\right) \propto Gamma\left( \frac{N}{2}+1, \frac{s_{jj}+\delta _{s}}{2}\right) . \end{aligned}$$

After drawing from the above conditional distributions, we can obtain $\varvec{\sigma }_{j}=\varvec{\beta }$, $\varvec{\sigma }_{j}^{\mathrm {T}}=\varvec{\beta }^{\mathrm {T}}$ and $\sigma _{jj}=\varvec{\gamma }+\varvec{\sigma }_{j}^{\mathrm {T}}\varvec{\Sigma }_{-jj}^{-1}\varvec{\sigma }_{j}$, then the jth column and row of $\varvec{\Sigma }$ can be updated one at a time. At the end, ${\varvec{\Psi }}={\varvec{\Sigma }}^{-1}$is computed.

The conditional distribution for ${\varvec{\uptau }}_{s} =(\tau _{ij} )_{i<j}$ can be expressed as:

$$\begin{aligned}&p({\varvec{\uptau }}_{s} \left| \mathbf{Y }, {\varvec{\Lambda }}, {\varvec{\Theta }}, {\varvec{\upmu }}, \varvec{\Phi }, {\varvec{\Sigma }}, \delta _{s} )\propto p({\varvec{\Sigma }} \right| {\varvec{\uptau }}_{s} , \delta _{s} )p({\varvec{\uptau }}_{s} \vert \delta _{s} )\\&\quad \propto \prod \limits _{i<J} {\tau _{ij}^{-\frac{1}{2}} \exp \left\{ {-\frac{\sigma _{ij}^{2} +\tau _{ij}^{2} \delta _{s}^{2} }{2\tau _{ij} }} \right\} }. \end{aligned}$$

It can be shown that for $i < j$,

$$\begin{aligned} p\left( \left. {\frac{1}{\tau _{ij} }} \right| \mathbf{Y }, {\varvec{\Lambda }}, {\varvec{\Theta }}, {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Sigma }}, \delta _{s} \right) \propto \text {Inv-Gaussian}\left( \sqrt{\frac{\delta _{s}^{2} }{\sigma _{ij}^{2} }} , \delta _{s}^{2} \right) . \end{aligned}$$

The conditional distribution for $\delta _{s}$ can be expressed as:

$$\begin{aligned}&p(\delta _{s} \left| {\mathbf{Y }, {\varvec{\Lambda }}, {\varvec{\Theta }}, {\varvec{\upmu }}, {\varvec{\Phi }}, {\varvec{\Sigma }}, {\varvec{\uptau }})\propto p({\varvec{\uptau }}} \right| \delta _{s} )p(\delta _{s} )\\&\quad \propto Gamma\left( \alpha _{s} +\frac{J\text{( }J+1\text{) }}{2}, \beta _{s} +\frac{1}{2}\sum \nolimits _{i=1}^J {\sum \nolimits _{j=1}^J {\left| {\sigma _{ij} } \right| } } \right) . \end{aligned}$$

Details of Nonzero Loading Estimates

See Tables 12, 13, 14 and 15.

Table 12 Nonzero loading estimates for PCIRM-LI and PCIRM under local independence

Table 13 Nonzero loading estimates for BSEMs and GLFM-DR under local independence

Table 14 Nonzero loading estimates for PCIRM-LI, PCIRM, and CIRM-LD under local dependence

Table 15 Nonzero loading estimates for BSEMs and GLFM-DR under local dependence

The Revised Eysenck Personality Questionnaire Short Scale

No.	Content	T
1	Would you take drugs which may have strange or dangerous effects?	P
2	Do you prefer to go your own way rather than act by the rules?	P
3	Do you think marriage is old fashioned and should be done away with?	P
4	Do you think people spend too much time safeguarding their future with savings and insurance?	P
5	Would you like other people to be afraid of you?	P
6(R)	Do you take much notice of what people think?	P
7(R)	Would being in debt worry you?	P
8(R)	Do good manners and cleanliness matter much to you?	P
9(R)	Do you enjoy co-operating with others?	P
10(R)	Does it worry you if you know there are mistakes in your work?	P
11(R)	Do you try not to be rude to people?	P
12(R)	Is it better to follow society’s rules than go your own way?	P
13	Are you a talkative person?	E
14	Are you rather lively?	E
15	Can you usually let yourself go and enjoy yourself at a lively party?	E
16	Do you enjoy meeting new people?	E
17	Do you usually take the initiative in making new friends?	E
18	Can you easily get some life into a rather dull party?	E
19	Do you like mixing with people?	E
20	Can you get a party going?	E
21	Do you like plenty of bustle and excitement around you?	E
22	Do other people think of you as being very lively?	E
23(R)	Do you tend to keep in the background on social occasions?	E
24(R)	Are you mostly quiet when you are with other people?	E
25	Does your mood often go up and down?	N
26	Do you ever feel ‘just miserable’ for no reason?	N
27	Are you an irritable person?	N
28	Are your feelings easily hurt?	N
29	Do you often feel ‘fed-up’?	N
30	Are you often troubled about feelings of guilt?	N
31	Would you call yourself a nervous person?	N
32	Are you a worrier?	N
33	Would you call yourself tense or ‘highly-strung’?	N
34	Do you worry too long after an embarrassing experience?	N
35	Do you suffer from ‘nerves’?	N
36	Do you often feel lonely?	N

Note. Item responses were dichotomous (i.e., Yes/No); T $=$ targeted trait; P $=$ psychoticism; E $=$ extraversion; N $=$ neuroticism; negatively worded items were marked by “R.”