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Correction for Item Response Theory Latent Trait Measurement Error in Linear Mixed Effects Models

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Abstract

When latent variables are used as outcomes in regression analysis, a common approach that is used to solve the ignored measurement error issue is to take a multilevel perspective on item response modeling (IRT). Although recent computational advancement allows efficient and accurate estimation of multilevel IRT models, we argue that a two-stage divide-and-conquer strategy still has its unique advantages. Within the two-stage framework, three methods that take into account heteroscedastic measurement errors of the dependent variable in stage II analysis are introduced; they are the closed-form marginal MLE, the expectation maximization algorithm, and the moment estimation method. They are compared to the naïve two-stage estimation and the one-stage MCMC estimation. A simulation study is conducted to compare the five methods in terms of model parameter recovery and their standard error estimation. The pros and cons of each method are also discussed to provide guidelines for practitioners. Finally, a real data example is given to illustrate the applications of various methods using the National Educational Longitudinal Survey data (NELS 88).

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Notes

  1. The T-score is a standardized score, which was in fact a transformation of an IRT \(\theta \) score.

  2. Originally, \({\varvec{\Sigma }}_{u}\) needs to be constrained to be nonnegative definite. However, this is not a box constraint that “optim” function can handle. We therefore impose constraints on the variance and correlation terms.

  3. We used the list-wise deletion because we wanted to create a complete data set for illustration. Our intention was to evaluate the performance of different methods without possible interference of missing data. Because we used the NELS provided item parameters and because our structural model is simple, the possible bias introduced by list-wise deletion may be ignored.

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Acknowledgements

This project is supported by IES R305D160010 and NSF SES-1659328

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Authors and Affiliations

  1. Measurement and Statistics, College of Education, University of Washington, 312E Miller Hall, Box 353600, Seattle, WA, 98195-3600, USA

    Chun Wang

  2. University of Michigan, Ann Arbor, USA

    Gongjun Xu

  3. Northeast Normal University, Changchun, China

    Xue Zhang

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  1. Chun Wang
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Appendices

Appendix A: Closed-form Marginal Likelihood

In this appendix, we provide detailed derivations for the closed-form marginal likelihood for a general model where the design matrices for the fixed and random effects in the latent growth curve model are different, i.e., Eq. (3) in the paper is updated as

$$\begin{aligned} \varvec{\theta }_i=\varvec{X}_i\varvec{\beta }+\varvec{Z}_i\varvec{u}_i+\varvec{e}_i. \end{aligned}$$
(1)

The subscript in \(\varvec{X}_i\) and \(\varvec{Z}_i\) indicates the model allows for unbalanced design.

Given (1) and the measurement error model, the marginal likelihood of the structural parameters, \(L(\beta , \varvec{\Sigma }_u, \sigma ^2)\), is proportional to

$$\begin{aligned}&\prod _i \int |\sigma ^2\varvec{I}_i|^{-1/2} \hbox {e}^{-\frac{1}{2\sigma ^2}(\varvec{\theta }_i-\varvec{X}_i\varvec{\beta }-\varvec{Z}_i\varvec{u}_i)^t (\varvec{\theta }_i-\varvec{X}_i\varvec{\beta }-\varvec{Z}_i\varvec{u}_i)} \hbox {e}^{-\frac{1}{2}(\varvec{\theta }^*_i-\varvec{\theta }_i)^t\varvec{\Sigma }_{\theta _i}^{-1}(\varvec{\theta }^*_i-\varvec{\theta }_i)} |\varvec{\Sigma }_u|^{-1/2} \hbox {e}^{-\frac{1}{2}\varvec{u}_i^t\varvec{\Sigma }_u^{-1}\varvec{u}_i} \mathrm d \theta _i \mathrm d u_i \\&\quad \propto \prod _i \int |\sigma ^2\varvec{\Sigma }_{\theta _i}^{-1}+\varvec{I}_i|^{-1/2} |\varvec{\Sigma }_u|^{-1/2} \hbox {e}^{-\frac{1}{2\sigma ^2}||\varvec{X}_i\varvec{\beta }+\varvec{Z}_i\varvec{u}_i||^2 +\frac{1}{2}||(\varvec{\Sigma }_{\varvec{\theta }_i}^{-1}+\sigma ^{-2}\varvec{I}_i)^{-1/2} \{\varvec{\Sigma }_{\varvec{\theta }_i}^{-1}\varvec{\theta }^*_i+\sigma ^{-2}(\varvec{X}_i\varvec{\beta }+\varvec{Z}_i\varvec{u}_i)\}||^2}\\&\qquad -\frac{1}{2}\varvec{u}_i^t\varvec{\Sigma }_u^{-1}\varvec{u}_i \mathrm d u_i, \end{aligned}$$

where \(|\cdot |\) denote the determinant of a matrix and \(||\theta ||^2 = \theta ^t\theta \).

Observing the coefficient of the squared term of \(\varvec{u}_i\) in the power of e is

$$\begin{aligned}&-\frac{1}{2}\sigma ^{-2}\varvec{Z}_i^t\varvec{Z}_i + \frac{1}{2}\sigma ^{-4}\varvec{Z}_i^t(\varvec{\Sigma }_{\theta _i}^{-1}+\sigma ^{-2}\varvec{I}_i)^{-1}\varvec{Z}_i -\frac{1}{2}\varvec{\Sigma }_u^{-1} \\&\quad = -\frac{1}{2}\left( \varvec{\Sigma }_u^{-1} + \sigma ^{-2}\varvec{Z}_i^t\varvec{Z}_i - \sigma ^{-4} \varvec{Z}_i^t(\sigma ^2\varvec{\Sigma }_{\theta _i}^{-1}+\varvec{I}_i)^{-1}\varvec{Z}_i \right) \end{aligned}$$

and the coefficient of \(u_i\) in the power of e is

$$\begin{aligned}&-\frac{1}{2}2\sigma ^{-2}\varvec{Z}_i^t(\varvec{X}_i\varvec{\beta }) + \frac{1}{2}2\varvec{Z}_i^t(\varvec{\Sigma }_{\theta _i}^{-1}+\sigma ^{-2}\varvec{I}_i)^{-1}(\varvec{\Sigma }_{\theta _i}^{-1}\varvec{\theta }^*_i + \sigma ^{-2}\varvec{X}_i\varvec{\beta }) \cdot \sigma ^{-2} \\&\quad = -\left( \sigma ^{-2}\varvec{Z}_i^t(\varvec{X}_i\varvec{\beta }) - \varvec{Z}_i^t(\sigma ^2\varvec{\Sigma }_{\theta _i}^{-1}+\varvec{I}_i)^{-1} (\varvec{\Sigma }_{\theta _i}^{-1}\varvec{\theta }^*_i + \sigma ^{-2}\varvec{X}_i\varvec{\beta })\right) \end{aligned}$$

Thus,

$$\begin{aligned} l(\beta , \varvec{\Sigma }_u, \sigma ^2) \propto&-n\log |\varvec{\Sigma }_u| - \frac{N}{\sigma ^2}||\varvec{X}_i\varvec{\beta }||^2 + \sum _{i=1}^N \left( \log |\varvec{\Sigma }_{u,i}^*| - \log |\sigma ^2\varvec{\Sigma }_{\theta _i}^{-1}+\varvec{I}_i| \right) \\&+ \sum _{i=1}^N \left( ||(\varvec{\Sigma }_{\theta _i}^{-1}+\sigma ^{-2}\varvec{I}_i)^{-1/2} \{\varvec{\Sigma }_{\theta _i}^{-1}{\theta }^*_i+\sigma ^{-2}\varvec{X}_i\varvec{\beta }\}||^2 + ||(\varvec{\Sigma }_{u,i}^*)^{1/2}\mu _{u,i}^*||^2 \right) \end{aligned}$$

where

$$\begin{aligned}&(\varvec{\Sigma }_{u,i}^*)^{-1}=\varvec{\Sigma }_u^{-1} + \sigma ^{-2}\varvec{Z}_i^t\varvec{Z}_i - \sigma ^{-4}\varvec{Z}_i^t(\sigma ^2\varvec{\Sigma }_{\theta _i}^{-1}+\varvec{I}_i)^{-1}\varvec{Z}_i\\&\varvec{\mu }_{u,i}^* = \sigma ^{-2}\varvec{Z}_i^t(\varvec{X}_i\varvec{\beta }) - \varvec{Z}_i^t(\sigma ^2\varvec{\Sigma }_{\theta _i}^{-1}+\varvec{I}_i)^{-1}(\varvec{\Sigma }_{\theta _i}^{-1}\varvec{\theta }^*_i+\sigma ^{-2}\varvec{X}_i\varvec{\beta }). \end{aligned}$$

For multivariate models, let \(\varvec{\theta }_i=(\theta _{i11},\ldots ,\theta _{i1T},\ldots ,\theta _{iD1},\ldots ,\theta _{iDT})^t\), a \(n_iD \times 1\) vector where \(n_i\) denotes the number of measurement waves for person i. Then in the general form, the associative latent growth curve model still takes the same form as in (1), but \(\varvec{X}_i\) becomes a \(n_iD \times Dp\) design matrix, and \(\varvec{\beta } = (\beta _{01}, \beta _{02},\ldots , \beta _{0D}, \beta _{11}, \beta _{12},\ldots , \beta _{1d},\ldots , \beta _{(p-1)1},\ldots , \beta _{(p-1)D})^t\) is a \(Dp \times 1\) vector. \(\varvec{Z}_i\) is \(n_iD \times Dk\) design matrix, assuming there are k random effects. In our simulation setting, \(n_i=4\), \(p=k\); hence, \(\varvec{Z}\) takes the same form as \(\varvec{X}\).

\(\varvec{u}_i\) is a \(Dk \times 1\) vector. The covariance matrix of \(\varvec{u}_i\) is \(\varvec{\Sigma }_u\). \(e_i = (e_{i11},\ldots ,e_{i1T},\ldots ,e_{iD1},\ldots ,e_{iDT})^t\) is a \(n_iD \times 1\) vector of residuals. The covariance matrix of \(e_i\), \(\varvec{\Sigma }\), is a diagonal block matrix. It has the structure of \(\begin{pmatrix} \varvec{\Sigma } &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad \varvec{\Sigma } &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \varvec{\Sigma }\\ \end{pmatrix}_{n_iD \times n_iD}\) where \(\Sigma _{D\times D} = diag((\sigma _1)^2, (\sigma _2)^2,\ldots , (\sigma _D)^2)\) and \(\Sigma \) has \(n_i\) such diagonal blocks. Then the marginal likelihood of model parameters is:

$$\begin{aligned}&L(\varvec{\beta }, \varvec{\Sigma }_u, \varvec{\Sigma }) \\&\quad \propto \prod _i \int |\varvec{\Sigma }|^{-1/2} \hbox {e}^{-\frac{1}{2} (\varvec{\theta }_i-\varvec{X}_i\varvec{\beta }-\varvec{Z}_i\varvec{u}_i)^t \varvec{\Sigma }^{-1}(\varvec{\theta }_i-\varvec{X}_i\beta -\varvec{Z}_i\varvec{u}_i)}\\&\qquad |\varvec{\Sigma }_{\varvec{\theta }_i}|^{-1/2} \hbox {e}^{-\frac{1}{2}(\varvec{\theta }^*_i-\varvec{\theta }_i)^t\varvec{\Sigma }_{\varvec{\theta }_i}^{-1}(\varvec{\theta }^*_i-\varvec{\theta }_i)} |\varvec{\Sigma }_u|^{-1/2} \hbox {e}^{-\frac{1}{2}\varvec{u}_i^t\varvec{\Sigma }_u^{-1}\varvec{u}_i} \mathrm d \varvec{\theta }_i \mathrm d \varvec{u}_i\\&\quad \propto \prod _i \int |\varvec{\Sigma }|^{-1/2} |\varvec{\Sigma }^{-1}+\varvec{\Sigma }_{\varvec{\theta }_i}^{-1}|^{-1/2}\\&\qquad \times \hbox {e}^{-\frac{1}{2}((\varvec{X}_i\varvec{\beta }+\varvec{Z}_i\varvec{u}_i)^t \varvec{\Sigma }^{-1} (\varvec{X}_i\varvec{\beta }+\varvec{Z}_i\varvec{u}_i) + \frac{1}{2} || (\varvec{\Sigma }^{-1}+\varvec{\Sigma }_{\varvec{\theta }_i}^{-1})^{-1/2} \{\varvec{\Sigma }_{\varvec{\theta }_i}^{-1}\varvec{\theta }^*_i + \varvec{\Sigma }^{-1}(\varvec{X}_i\varvec{\beta }+\varvec{Z}_i\varvec{u}_i)\} ||^2})\\&\qquad \times |\varvec{\Sigma }_u|^{-1/2} \hbox {e}^{-\frac{1}{2}\varvec{u}_i^t\varvec{\Sigma }_u^{-1}u_i} \mathrm d u_i\\&\quad \propto \prod _i |\varvec{\Sigma }|^{-1/2}|\varvec{\Sigma }^{-1}+\varvec{\Sigma }_{\varvec{\theta }_i}^{-1}|^{-1/2} |\varvec{\Sigma }_u|^{-1/2} |\varvec{\Sigma }_{u,i}^*|^{1/2}\\&\qquad \times \hbox {e}^{-\frac{1}{2}(\varvec{X}_i\varvec{\beta })^t\varvec{\Sigma }^{-1}(\varvec{X}_i\varvec{\beta }) + \frac{1}{2}|| (\varvec{\Sigma }^{-1}+\varvec{\Sigma }_{\varvec{\theta }_i}^{-1})^{-1/2} \{\varvec{\Sigma }_{\varvec{\theta }_i}^{-1}\varvec{\theta }^*_i + \varvec{\Sigma }^{-1}(\varvec{X}_i\varvec{\beta })\} ||^2 + \frac{1}{2}|| (\varvec{\Sigma }_{u,i}^*)^{1/2} \varvec{\mu }_{u,i}^* ||^2} \end{aligned}$$

where

$$\begin{aligned} (\varvec{\Sigma }_{u,i}^*)^{-1}= & {} \varvec{\Sigma }_u^{-1} + \varvec{Z}_i^t\varvec{\Sigma }^{-1}\varvec{Z}_i - \varvec{Z}_i^t\varvec{\Sigma }^{-1} (\varvec{\Sigma }^{-1} + \varvec{\Sigma }_{\varvec{\theta }_i}^{-1})^{-1} \varvec{\Sigma }^{-1} \varvec{Z}_i\\ \varvec{\mu }_{u,i}^*= & {} \varvec{Z}_i^t \varvec{\Sigma }^{-1} (\varvec{X}_i\varvec{\beta })^t - \varvec{Z}_i^t(\varvec{\Sigma }^{-1}+ \varvec{\Sigma }_{\varvec{\theta }_i}^{-1})^{-1} \varvec{\Sigma }^{-1} \{\varvec{\Sigma }_{\varvec{\theta }_i}^{-1}\varvec{\theta }^*_i+ \varvec{\Sigma }^{-1}\varvec{X}_i\varvec{\beta }\}. \end{aligned}$$

Appendix B: Computational Details of the EM Standard Error (MIRT)

The important component of computing the standard error is the complete data Fisher information matrix \({\varvec{I}}_{c} ({\hat{\varvec{\psi }}})\). Below, we present the specific forms of these components for the MIRT models. Results from the UIRT models can be considered as a special case. Assuming a Monte Carlo sampling version of the EM algorithm is used, i.e., Eq. (19), we have,

$$\begin{aligned} \frac{\partial ^{2}E(\varvec{\psi }\vert {\hat{\varvec{\psi }}}^{(m)})}{\partial \varvec{\beta }^{2}}= & {} -N{\varvec{X}}_{i}^{t} ( {\hat{{\sigma }}^{-2}\mathrm{\mathbf{I}}_{DT} } ){\varvec{X}}_{i}, \end{aligned}$$
(B1)
$$\begin{aligned} \frac{\partial ^{2}E(\varvec{\psi }\vert {\hat{\varvec{\psi }}}^{(m)})}{\partial ( {\sigma ^{2}} )^{2}}= & {} \frac{1}{2Q}\sum \limits _{i=1}^N \sum \limits _{q=1}^Q [ T{\hat{\sigma }^{ - 4}} - 2{{[ {{{({\varvec{\theta }}_i^q)}^t} - {{\varvec{X}}_i}\hat{{\varvec{\beta }}} - Z{\varvec{u}}_i^q} ]}^t}( {{{\hat{\sigma } }^{ - 6}}{{\mathbf{I}}_{DT}}} )\nonumber \\&[ {{{({\varvec{\theta }}_i^q)}^t} - {{\varvec{X}}_i}\hat{{\varvec{\beta }}} - Z{\varvec{u}}_i^q}] ], \end{aligned}$$
(B2)
$$\begin{aligned} \frac{\partial ^{2}E(\varvec{\psi }\vert {\hat{\varvec{\psi }}}^{(m)})}{\partial \varvec{\beta }\partial \sigma ^{2}}= & {} -\frac{1}{Q}\sum \limits _{i=1}^N {\sum \limits _{q=1}^Q {[ {(\varvec{\theta }_{i}^{q} )^{t}-{\varvec{X}}_{i} {\hat{\varvec{\beta }}}-Z{\varvec{u}}_{i}^{q} } ]^{t}( {\hat{{\sigma }}^{-4}\mathrm{\mathbf{I}}_{DT} } ){\varvec{X}}_{i} } }, \end{aligned}$$
(B3)

To obtain the second derivatives with respect to the elements in the covariance matrix

$$\begin{aligned} \frac{\partial ^{2}E(\varvec{\varphi }\vert \hat{{\varvec{\varphi }} }^{(m)})}{\partial x_{p}\partial x_{q}}= & {} \frac{1}{Q}\sum \limits _{i=1}^N \sum \limits _{q=1}^Q \left\{ -\frac{1}{2}\hbox {tr}\left[ \left( \hat{\Sigma }_{u}^{-1}\frac{\partial \hat{\Sigma }_{u}}{\partial x_{q}}\hat{\Sigma }_{u}^{-1} \right) \frac{\partial \hat{\Sigma }_{u}}{\partial x_{p}}\right] \right. \nonumber \\&\left. -\frac{1}{2}{({\varvec{u}}_{i}^{q})}^{t}\left[ \hat{\Sigma }_{u}^{-1}\left( \frac{\partial \hat{\Sigma }_{u}}{\partial x_{p}}\hat{\Sigma }_{u}^{-1}\frac{\partial \hat{\Sigma }_{u}}{\partial x_{q}}+\frac{\partial \hat{\Sigma }_{u}}{\partial x_{q}}\hat{\Sigma }_{u}^{-1}\frac{\partial \hat{\Sigma }_{u}}{\partial x_{p}} \right) \hat{\Sigma }_{u}^{-1} \right] {\varvec{u}}_{i}^{q} \right\} , \end{aligned}$$
(B4)

where \(x_{p}\) and \(x_{q}\) are the two elements in the covariance matrix \(\hat{\Sigma }_{u}\). For instance, using UIRT set up as an example, if taking the second derivative of log-likelihood with respect to \(\tau _{00}\), we would set \(x_{p}=x_{q}=\tau _{00}\) in (B4), and then \(\frac{\partial \hat{\Sigma }_{u}}{\partial \tau _{00}}=\left[ {\begin{array}{c@{\quad }c} 1 &{} 0\\ 0 &{} 0\\ \end{array} } \right] \). The parameters in (B1)–(B4) are final estimates upon convergence. Hence, the Fisher information matrix for the complete data has the following form as

$$\begin{aligned} {\varvec{I}}\mathrm {=-}\left[ {\begin{array}{c@{\quad }c@{\quad }c} \frac{\partial ^{2}E(\varvec{\varphi }\vert \hat{{\varvec{\varphi }} }^{(m)})}{\partial \varvec{\beta }^{2}} &{} \frac{\partial ^{2}E(\varvec{\varphi }\vert \hat{{\varvec{\varphi }} }^{(m)})}{\partial \varvec{\beta }\partial \sigma ^{2}} &{} 0\\ \frac{\partial ^{2}E(\varvec{\varphi }\vert \hat{{\varvec{\varphi }} }^{(m)})}{\partial \varvec{\beta }\partial \sigma ^{2}} &{} \frac{\partial ^{2}E(\varvec{\varphi }\vert \hat{{\varvec{\varphi }} }^{(m)})}{\partial {{\varvec{(}}\sigma ^{2}{\varvec{)}}}^{2}} &{} 0\\ 0 &{} 0 &{} \frac{\partial ^{2}E(\varvec{\varphi }\vert \hat{{\varvec{\varphi }} }^{(m)})}{\partial {(\Sigma _{{\varvec{u}}})}^{2}}\\ \end{array} } \right] . \end{aligned}$$

The information matrix can also be obtained similarly if a closed-form conditional expectation is obtained.

Appendix C: The MCMC Algorithm

The Metropolis–Hastings algorithm within Gibbs sampler is used. For the ease of exposition below, we can rewrite the linear mixed model in (3) as follows:

$$\begin{aligned} \begin{aligned} \theta _{it}&=\pi _{0i} +\pi _{1i} \left( {t-1} \right) +e_{it} \\ \pi _{0i}&=\beta _{00} +u_{0i} \\ \pi _{1i}&=\beta _{01} +u_{1i} \end{aligned}, \end{aligned}$$
(C1)

The conjugate priors are selected whenever available. Below is an outline of the sampling schemes. At the \((m+1)^{\mathrm {th}}\) iteration, we have

Step 1: Sample \({{{\varvec{\uptheta }}}}_{i}^{*} \sim N({{{\varvec{\uptheta }} }}_{i}^{(m)},\sigma _{\theta }^{2} )\) and \(\mathrm{\mathbf{u}}_{i} \sim U( {0,1} )\), and set \(\theta _{it}^{(m+1)} =\theta _{it}^{*} \) when

$$\begin{aligned} \mathrm{\mathbf{u}}_{i} <\frac{P( {{{{\varvec{\uptheta }} }}_{i}^{*} | {{\varvec{X}}_{i} {{{\varvec{\uppi }} }}_{i}^{(m)},\sigma ^{2(m)}} } )L( {{\varvec{Y}}_{i} | {{{{\varvec{\uptheta }} }}_{i}^{*} } } )}{P( {{ {{\varvec{\uptheta }} }}_{i}^{(m)} | {{\varvec{X}}_{i} {{{\varvec{\uppi }} }}_{i}^{(m)},\sigma ^{2(m)}} } )L( {{\varvec{Y}}_{i} | {{{ {\varvec{\uptheta }} }}_{i}^{(m)} } } )}, \end{aligned}$$
(C2)

where \(\varvec{\pi }_{i}={(\pi _{0i},\pi _{1i})}^{t},\)\(L({\varvec{Y}}_{i} \vert \theta _{it}^{*} )\) is the likelihood obtained from item responses, and \(P({{{\varvec{\uptheta }} }}_{i}^{*} \vert {\mathbf{X}{\varvec{\uppi }} }_{i}^{(m)},\sigma ^{2(m)})\) is the normal density with a mean of \({\varvec{X}}_{i} {{{\varvec{\uppi }} }}_{i}^{(m)} \) and variance of \(\sigma ^{2(m)}\).

Step 2: Sample \(\pi _{i}^{(m+1)} \) from the multivariate normal distribution with covariance

$$\begin{aligned} \Sigma _{{{{\varvec{\uppi }} }}_{i} \left| {{{{\varvec{\uptheta }}}}_{i} ,\sigma ^{2},} \right. \mathrm{\mathbf{X}},{{{\varvec{\upbeta }}}},{{\varvec{\Sigma }}}_{u} } =\left( {\sigma ^{-2(m)}{\varvec{X}}_{i}^{t} {\varvec{X}}_{i} +{{\varvec{\Sigma }}}_{u}^{-1(m)} } \right) ^{-1}, \end{aligned}$$
(C3)

and mean

$$\begin{aligned} {{\varvec{\upmu } }}_{{{{\varvec{\uppi }}}}_{i} | {{{{\varvec{\uptheta }}}}_{i} ,\sigma ^{2},{\mathbf {X}},{\varvec{\upbeta }},{{\varvec{\Sigma }}}_{u} } } =\Sigma _{{\varvec{\uppi }}_{i} | {{\varvec{\uptheta }}_{i} ,\sigma ^{2},{\mathbf {X}},{\varvec{\upbeta }},{{\varvec{\Sigma }}}_{u} } } \cdot ( {{{\varvec{\Sigma }}}_{u}^{-1(m)} {\varvec{\upbeta }}^{(m)}+\sigma ^{-2(m)}{\mathbf {X}}^{t}{\varvec{\uptheta }}_{i}^{(m+1)} } ). \end{aligned}$$
(C4)

Step 3: Sample \(\sigma ^{2(m+1)}\) from the full condition distribution

$$\begin{aligned} \sigma ^{2(m+1)}\sim \hbox {Inv-Gamma}\left( {\alpha _{0} +\frac{TN}{2},\alpha _{1} +\frac{1}{2}\sum \limits _i {( {{\varvec{\uptheta }}_{i}^{(m+1)} -{\varvec{X}}_{i} {\varvec{\uppi }}_{i}^{(m+1)} } )^{t}( {{\varvec{\uptheta }}_{i}^{(m+1)} -{\varvec{X}}_{i} {\varvec{\uppi }}_{i}^{(m+1)} } )} } \right) ,\nonumber \\ \end{aligned}$$
(C5)

where the prior distribution of \(\sigma ^{2}\) is Inv-Gamma (\(\alpha _{0},\alpha _{1})\). We selected \(\alpha _{0}=.0001\) and \(\alpha _{1}=1\) (Congdon, 2001) as the hyper-parameters of a non-information prior distribution.

Step 4: Sample \({\varvec{\upbeta }}^{(m+1)}\) and \({{\varvec{\Sigma }}}_{u}^{(m+1)} \) from the normal inverse Wishart distribution with parameters \(( {{{\varvec{\upmu } }}_{{\varvec{\upbeta }}| {{\varvec{\uppi }}} }^{(m+1)},\kappa _{n},{{\varvec{\Sigma }}}_{{\mathbf {u}}| {{\varvec{\uppi }}} }^{(m+1)},\nu _{n} } )\). Here

$$\begin{aligned}&\mu _{{{\varvec{\upbeta }}|\pi } }^{(m+1)} =\frac{\kappa _{0} }{\kappa _{0} +N}\mu _{{\varvec{\upbeta }},0} +\frac{N}{\kappa _{0} +N}{{\bar{\varvec{\uppi }}}}^{(m+1)}, \end{aligned}$$
(C6)
$$\begin{aligned}&{{\varvec{\Sigma }}}_{u| \pi }^{(m+1)} \nonumber \\&\quad =\Sigma _{u,0} +\sum \limits _i {( {{\varvec{\uppi }}_{i}^{(m+1)} -{\varvec{\bar{{\uppi }}}}^{(m+1)}} )} ( {{\varvec{\uppi }}_{i}^{(m+1)} -{{\bar{\varvec{\uppi }}}}^{(m+1)}} )^{t}\nonumber \\&\qquad +\frac{\kappa _{0} }{\kappa _{0} +n}( {{{\bar{\varvec{\uppi }}}}^{(m+1)}-{{\varvec{\upmu } }}_{{\varvec{\upbeta }},0} } )( {{{\bar{\varvec{\uppi }}}}^{(m+1)}-{{\varvec{\upmu } }}_{{\varvec{\upbeta }},0} } )^{t},\nonumber \\&\kappa _{n} =\kappa _{0} +N,\nonumber \\&\nu _{n} =\nu _{0} +N, \end{aligned}$$
(C7)

where \({{\bar{\varvec{\uppi }}}}^{(m+1)}=( {\sum \nolimits _i {{\pi _{0i}^{(m+1)} } / {N},} \sum \nolimits _i {{\pi _{1i}^{(m+1)} } / N} })^{t}\). And the prior distribution is a normal inverse Wishart distribution with parameters \(({{{\varvec{\upmu } }}_{{\varvec{\upbeta }},0},\kappa _{0},{{\varvec{\Sigma }}}_{u,0},\nu _{0} })\). Regarding the hyper-parameters, \({\varvec{\upmu }}_{{\beta ,0}}\) is a 2-by-1 zero vector, \(\kappa _{{0}}=0\), \({{\varvec{\Sigma }}}_{u,0} \) is a 2-by-2 identity matrix, and \(\nu _{{0}}=-1\), yielding a non-informative prior (Murphy, 2007).

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Wang, C., Xu, G. & Zhang, X. Correction for Item Response Theory Latent Trait Measurement Error in Linear Mixed Effects Models. Psychometrika 84, 673–700 (2019). https://doi.org/10.1007/s11336-019-09672-7

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