1. Bayesian Implementation

Our ABDM lends itself naturally to Bayesian analysis due its functional form. Specifically, assuming the parameterisation introduced above, where the rate of information accumulation for person p answering item i is determined by the person’s ability ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} and the item easiness ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} , $v_{\mathrm{ip}}={\beta }_i+{\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{ip} = \beta _i + \theta _p$$\end{document} , and the volatility varies across items, $s_i={\alpha }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_i = \alpha _i$$\end{document} , the density of the ABDM is:

(6) $\begin{array}{cc}f(t_{\mathrm{ip}},x_{\mathrm{ip}}\mid {\beta }_i,{\theta }_p,{\alpha }_i)=& \pi c^{2}exp\left([x_{\mathrm{ip}}-1/2],\frac{{\beta }_i+{\theta }_p}{c{\alpha }_i},-,t_{\mathrm{ip}},\frac{{({\beta }_i+{\theta }_p)}^2}{2{\alpha }_i^2}\right)\\ & \times \sum _{k=1}^{\infty }\left(k,sin,\left(\frac{\pi k}{2}\right),exp,\left(-,\frac{{\pi }^2{c}^2{k}^2}{2},t_{\mathrm{ip}}\right)\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(t_{ip},x_{ip} \mid \beta _i, \theta _p, \alpha _i) =&\, \pi c^2 \exp {\left( [x_{ip}-1/2]\frac{\beta _i + \theta _p}{c \alpha _i}-t_{ip}\frac{(\beta _i + \theta _p)^2}{2\alpha _i^2} \right) } \nonumber \\&\times \sum _{k=1}^{\infty }\left[ k\sin \left( \frac{\pi k }{2} \right) \exp {\left( -\frac{\pi ^2 c^2 k^2}{2} t_{ip}\right) }\right] \end{aligned}$$\end{document}

and dropping all constants not related to ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} , ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} , or ${\alpha }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _i$$\end{document} and letting ${\tilde{\alpha }}_i={\alpha }_i^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i=\alpha ^{-1}_i$$\end{document} :

$\begin{array}{c}\propto exp\left([x_{\mathrm{ip}}-1/2],{\tilde{\alpha }}_i,\frac{({\beta }_i+{\theta }_p)}{c},-,t_{\mathrm{ip}},{\tilde{\alpha }}_i^2,\frac{{({\beta }_i+{\theta }_p)}^2}{2}\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \propto \exp { \left( [x_{ip}-1/2]\tilde{\alpha }_i\frac{(\beta _i + \theta _p)}{c}-t_{ip}\tilde{\alpha }^2_i\frac{(\beta _i + \theta _p)^2}{2} \right) }. \end{aligned}$$\end{document}

If we assume for now that the value of c is known, then this form of the density suggests a conjugate analysis for the estimation of the remaining parameters. Choosing a multivariate normal distribution as joint prior for the model parameters ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} , ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} , and ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} , and an inverse Wishart prior for the variance–covariance matrix, will yield a multivariate normal posterior distribution for all parameters. Due to the interpretation of the volatility ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} as a standard deviation, only positive values are admissible for this parameter and the prior distribution for ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} needs to be truncated at zero, which complicates the form of the joint posterior distribution considerably. Nevertheless, conjugacy will still hold for the full conditional posterior distributions of ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} , ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} and ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} . Hence, our model can be straightforwardly analysed using Gibbs sampling.

1.1. Random Effects

The likelihood function of our ABDM, Eq. (6), has a simple form that makes it amenable to extension. The terms of the series expression do not depend on any of the model parameters and can therefore be dropped in analyses of the likelihood function, which leaves a simple exponential family model. Moreover, the item and person effects on drift rate, ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} and ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} , and the item effects on the information volatility, ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} , only appear in the first and second powers. Hence, the full-conditional distributions we introduce below take the form of a normal kernel, which allows for simple conjugate analysis and easy sampling-based estimation.

One problem that is regularly of interest in psychometrics is the modelling of unobserved heterogeneity. We show here how our ABDM can be extended to accommodate a random effects structure. Extensions to other statistical problems such as missing data, inclusion of manifest predictor variables, or the addition of autoregressive components are straightforward.

To model heterogeneity in the data, we assume the following population-level distributions:

$\begin{array}{c}\begin{array}{ccc}{\beta }_i\sim N({\mu }_I,{\sigma }_I^2),& {\theta }_p\sim N({\mu }_P,{\sigma }_P^2),& {\tilde{\alpha }}_i\sim N({\mu }_A,{\gamma }_A^2)[0,\infty ).\end{array}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{lll} \beta _i \sim \mathscr {N}(\mu _I, \sigma ^2_I),&\theta _p \sim \mathscr {N}(\mu _P, \sigma ^2_P),&\tilde{\alpha }_i \sim \mathscr {N}(\mu _A, \gamma ^2_A)[0, \infty ). \end{array} \end{aligned}$$\end{document}

Here, $N(\mu ,{\sigma }^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {N}(\mu ,\sigma ^2)$$\end{document} denotes the normal distribution with mean $\mu$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and variance ${\sigma }^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2$$\end{document} . The expression $[0,\infty )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0, \infty )$$\end{document} indicates the support of the truncated distribution. The population-level parameters can be assigned prior distributions in a similar manner:

$\begin{array}{c}\begin{array}{cccc}{\mu }_P\sim N({\nu }_{P0},{\tau }_{P0}^2),& {\sigma }_P^2\sim invG(a_{P0},b_{P0}),& {\mu }_I\sim N({\nu }_{I0},{\tau }_{I0}^2),& {\sigma }_I^2\sim invG(a_{I0},b_{I0}).\\ {\mu }_A\sim N({\nu }_{A0},{\tau }_{A0}^2),& {\gamma }_A^2\sim invG(a_{A0},b_{A0}).& & \end{array}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{llll} \mu _P \sim \mathscr {N}(\nu _{P0}, \tau ^2_{P0}), &{} \sigma ^2_P \sim invG(a_{P0}, b_{P0}), &{} \mu _I \sim \mathscr {N}(\nu _{I0}, \tau ^2_{I0}), &{} \sigma ^2_I \sim invG(a_{I0}, b_{I0}).\\ \mu _A \sim \mathscr {N}(\nu _{A0}, \tau ^2_{A0}), &{} \gamma ^2_A \sim invG(a_{A0}, b_{A0}). &{} &{} \end{array} \end{aligned}$$\end{document}

Here, invG(a, b) denotes the inverse gamma distribution with shape parameter a and scale parameter b.

The full-conditional conjugate posterior distributions for the person and item effects on drift rate $v_{\mathrm{ip}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{ip}$$\end{document} for $N_P$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_P$$\end{document} persons answering $N_I$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_I$$\end{document} items are:

$\begin{array}{c}\begin{array}{cc}{\beta }_i\sim N({\hat{\beta }}_{i,N_P},{\hat{\sigma }}_{i,N_P}^2),& {\theta }_p\sim N({\hat{\theta }}_{p,N_I},{\hat{\sigma }}_{p,N_I}^2),\end{array}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{ll} \beta _i \sim \mathscr {N}(\hat{\beta }_{i,N_P}, \hat{\sigma }^2_{i,N_P}),&\theta _p \sim \mathscr {N}(\hat{\theta }_{p,N_I}, \hat{\sigma }^2_{p,N_I}), \end{array} \end{aligned}$$\end{document}

where

$\begin{array}{c}\begin{array}{cc}{\hat{\sigma }}_{i,N_P}^2={\left({\sum }_{p=1}^{N_P},\frac{t_{\mathrm{ip}}}{{\alpha }_i^2},+,\frac{1}{{\sigma }_I^2}\right)}^{-1},& {\hat{\beta }}_{i,N_P}=\left({\sum }_{p=1}^{N_P},\frac{(x_{\mathrm{ip}}-1/2)}{c{\alpha }_i},-,{\sum }_{p=1}^{N_P},\frac{t_{\mathrm{ip}}{\theta }_p}{{\alpha }_i^2},+,\frac{{\mu }_I}{{\sigma }_I^2}\right){\hat{\sigma }}_{i,N_P}^2\\ {\hat{\sigma }}_{p,N_I}^2={\left({\sum }_{i=1}^{N_I},\frac{t_{\mathrm{ip}}}{{\alpha }_i^2},+,\frac{1}{{\sigma }_P^2}\right)}^{-1},& {\hat{\theta }}_{p,N_I}=\left({\sum }_{i=1}^{N_I},\frac{(x_{\mathrm{ip}}-1/2)}{c{\alpha }_i},-,{\sum }_{i=1}^{N_I},\frac{t_{\mathrm{ip}}{\beta }_i}{{\alpha }_i^2},+,\frac{{\mu }_P}{{\sigma }_P^2}\right){\hat{\sigma }}_{p,N_I}^2.\end{array}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{ll} \hat{\sigma }^2_{i,N_P} = \left( \sum _{p=1}^{N_P} \frac{ t_{ip}}{\alpha _i^2} + \frac{1}{\sigma ^2_I} \right) ^{-1}, &{} \hat{\beta }_{i,N_P} = \left( \sum _{p=1}^{N_P}\frac{ (x_{ip}-1/2)}{c \alpha _i} - \sum _{p=1}^{N_P}\frac{ t_{ip} \theta _p}{\alpha _i^2} + \frac{\mu _{I}}{\sigma ^2_I} \right) \hat{\sigma }^2_{i,N_P} \\ \hat{\sigma }^2_{p,N_I} = \left( \sum _{i=1}^{N_I}\frac{ t_{ip}}{\alpha _i^2} + \frac{1}{\sigma ^2_P} \right) ^{-1}, &{} \hat{\theta }_{p,N_I} = \left( \sum _{i=1}^{N_I} \frac{ (x_{ip}-1/2)}{c \alpha _i} - \sum _{i=1}^{N_I}\frac{ t_{ip} \beta _i}{\alpha _i^2} + \frac{\mu _{P}}{\sigma ^2_P} \right) \hat{\sigma }^2_{p,N_I}. \end{array} \end{aligned}$$\end{document}

Similarly, the full-conditional conjugate posterior distribution for item effects on the reciprocal volatility parameter $s_{\mathrm{ip}}^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^{-1}_{ip}$$\end{document} is:

$\begin{array}{c}{\tilde{\alpha }}_i\sim N({\hat{\alpha }}_{i,N_P},{\hat{\gamma }}_{i,N_P}^2)[0,\infty ),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{\alpha }_i \sim \mathscr {N}(\hat{\alpha }_{i,N_P}, \hat{\gamma }^2_{i,N_P})[0, \infty ), \end{aligned}$$\end{document}

where

$\begin{array}{c}\begin{array}{cc}& {\hat{\gamma }}_{i,N_P}^2={\gamma }_A^2{\left({\sum }_{p=1}^{N_P},t_{\mathrm{ip}},{({\beta }_i+{\theta }_p)}^2,{\gamma }_A^2,+,1\right)}^{-1},\\ & {\hat{\alpha }}_{i,N_P}=\left({\mu }_A,/,{\gamma }_A^2,+,{\sum }_{p=1}^{N_P},(x_{\mathrm{ip}}-1/2),({\beta }_i+{\theta }_p),/,c\right){\hat{\gamma }}_{i,N_P}^2.\end{array}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{ll} &{} \hat{\gamma }^2_{i,N_P} = \gamma ^2_A \left( \sum _{p=1}^{N_P} t_{ip}(\beta _i+\theta _p)^2\gamma ^2_A + 1 \right) ^{-1},\\ &{} \hat{\alpha }_{i,N_P} = \left( \mu _{A}/\gamma ^2_{A} + \sum _{p=1}^{N_P} (x_{ip}-1/2)(\beta _i+\theta _p)/c \right) \hat{\gamma }^2_{i,N_P}. \end{array} \end{aligned}$$\end{document}

The full-conditional posterior distributions for the population-level parameters are:

$\begin{array}{c}\begin{array}{cccc}{\mu }_P& \sim N({\nu }_{P,N_P},{\tau }_{P,N_P}^2),& \text{where}{\nu }_{P,N_P}=\frac{{\tau }_P^2{\sum }_{p=1}^{N_P}{\theta }_p+{\sigma }_P^2{\nu }_{P,0}}{{\tau }_P^2{N}_P+{\sigma }_P^2},& {\tau }_{P,N_P}^2=\frac{{\tau }_P^2{\sigma }_P^2}{{\tau }_P^2{N}_P+{\sigma }_P^2}\\ {\sigma }_P^2& \sim invG(a_{P,N_P},b_{P,N_P}),& \text{where}{a}_{P,N_P}=a_{P,0}+N_P/2,\\ & b_{P,N_P}={\sum }_{p=1}^{N_P}{({\theta }_p-{\mu }_P)}^2/2+b_{P,0}\\ {\mu }_I& \sim N({\nu }_{I,N_I},{\tau }_{I,N_I}^2),& \text{where}{\nu }_{I,N_I}=\frac{{\tau }_I^2{\sum }_{i=1}^{N_I}{\beta }_i+{\sigma }_I^2{\nu }_{I,0}}{{\tau }_I^2{N}_I+{\sigma }_I^2},& {\tau }_{I,N_I}^2=\frac{{\tau }_I^2{\sigma }_I^2}{{\tau }_I^2{N}_I+{\sigma }_I^2}\\ {\sigma }_I^2& \sim invG(a_{I,N_I},b_{I,N_I}),& \text{where}{a}_{I,N_I}=a_{I,0}+N_I/2,\\ & b_{I,N_I}={\sum }_{i=1}^{N_I}{({\beta }_i-{\mu }_I)}^2/2+b_{I,0}\\ {\mu }_A& \sim N({\nu }_{A,N_I},{\tau }_{A,N_I}^2),& \text{where}{\nu }_{A,N_I}=\frac{{\tau }_A^2{\sum }_{i=1}^{N_I}{\tilde{\alpha }}_i+{\gamma }_A^2{\nu }_{A,0}}{{\tau }_A^2{N}_I+{\gamma }_A^2},& {\tau }_{A,N_I}^2=\frac{{\tau }_A^2{\gamma }_A^2}{{\tau }_A^2{N}_I+{\gamma }_A^2}\\ {\gamma }_A^2& \sim invG(a_{A,N_I},b_{A,N_I}),& \text{where}{a}_{A,N_I}=a_{A,0}+N_I/2,\\ & b_{A,N_I}={\sum }_{i=1}^{N_I}{({\tilde{\alpha }}_i-{\mu }_A)}^2/2+b_{A,0}.\end{array}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{llll} \mu _P &{} \sim \mathscr {N}(\nu _{P,N_P}, \tau ^2_{P,N_P}), &{} \text { where } \nu _{P,N_P} = \frac{\tau ^2_P \sum _{p=1}^{N_P} \theta _p + \sigma ^2_P \nu _{P,0}}{\tau ^2_P N_P + \sigma ^2_P}, &{} \tau ^2_{P,N_P} = \frac{\tau ^2_P \sigma ^2_P}{\tau ^2_P N_P + \sigma ^2_P}\\ \sigma ^2_P &{} \sim invG(a_{P,N_P}, b_{P,N_P}), &{} \text { where } a_{P,N_P} = a_{P,0} + N_P/2, \\ &{} b_{P,N_P} = \sum _{p=1}^{N_P} (\theta _p - \mu _P)^2/2 + b_{P,0}\\ \mu _I &{} \sim \mathscr {N}(\nu _{I,N_I}, \tau ^2_{I,N_I}), &{} \text { where } \nu _{I,N_I} = \frac{\tau ^2_I \sum _{i=1}^{N_I} \beta _i + \sigma ^2_I \nu _{I,0}}{\tau ^2_I N_I + \sigma ^2_I}, &{} \tau ^2_{I,N_I} = \frac{\tau ^2_I \sigma ^2_I}{\tau ^2_I N_I + \sigma ^2_I}\\ \sigma ^2_I &{} \sim invG(a_{I,N_I}, b_{I,N_I}), &{} \text { where } a_{I,N_I} = a_{I,0} + N_I/2, \\ &{} b_{I,N_I} = \sum _{i=1}^{N_I} (\beta _i - \mu _I)^2/2 + b_{I,0}\\ \mu _A &{} \sim \mathscr {N}(\nu _{A,N_I}, \tau ^2_{A,N_I}), &{} \text { where } \nu _{A,N_I} = \frac{\tau ^2_A \sum _{i=1}^{N_I} \tilde{\alpha }_i + \gamma ^2_A \nu _{A,0}}{\tau ^2_A N_I + \gamma ^2_A}, &{} \tau ^2_{A,N_I} = \frac{\tau ^2_A \gamma ^2_A}{\tau ^2_A N_I + \gamma ^2_A}\\ \gamma ^2_A &{} \sim invG(a_{A,N_I}, b_{A,N_I}), &{} \text { where } a_{A,N_I} = a_{A,0} + N_I/2,\\ &{} b_{A,N_I} = \sum _{i=1}^{N_I} (\tilde{\alpha }_i - \mu _A)^2/2 + b_{A,0}. \end{array} \end{aligned}$$\end{document}

Let $\overrightarrow{t}=[t_{11},\dots ,t_{\mathrm{IP}}]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {t}=[t_{11}, \ldots , t_{IP}]$$\end{document} be the vector of response times and let $\overrightarrow{x}=[x_{11},\dots ,x_{\mathrm{IP}}]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {x}=[x_{11}, \ldots , x_{IP}]$$\end{document} be the vector of accuracies. Moreover, denote by $\Theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} the set of all item, person, and population-level parameters, then:

$\begin{array}{cc}& f(\overrightarrow{t},\overrightarrow{x}\mid \Theta )f\left(\Theta \right)\propto exp\left(\sum _{i=1}^{N_I},\sum _{p=1}^{N_P},\left({\tilde{\alpha }}_i,[x_{\mathrm{ip}}-1/2],\frac{{\beta }_i+{\theta }_p}{c},-,t_{\mathrm{ip}},{\tilde{\alpha }}_i^2,\frac{{({\beta }_i+{\theta }_p)}^2}{2}\right)\right)\\ & \times exp\left(-,\left(\sum _{i=1}^{N_I},\frac{{({\beta }_i-{\mu }_I)}^2}{2{\sigma }_I^2},+,\sum _{p=1}^{N_P},\frac{{({\theta }_p-{\mu }_P)}^2}{2{\sigma }_P^2},+,\sum _{i=1}^{N_I},\frac{{({\tilde{\alpha }}_i-{\mu }_A)}^2}{2{\gamma }_A^2}\right)\right)\\ & \times exp\left(-,\left(\frac{{({\mu }_I-{\nu }_{I0})}^2}{2{\tau }_{I0}^2},+,\frac{{({\mu }_P-{\nu }_{P0})}^2}{2{\tau }_{P0}^2},+,\frac{{({\mu }_A-{\nu }_{A0})}^2}{2{\tau }_{A0}^2},+,\frac{b_I}{{\sigma }_I^2},+,\frac{b_P}{{\sigma }_P^2},+,\frac{b_A}{{\gamma }_A^2}\right)\right)\\ & \times {\sigma }_I^{-2(a_I+1+N_I/2)}{\sigma }_P^{-2(a_P+1+N_P/2)}{\gamma }_A^{-2(a_A+1+N_I/2)}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&f(\vec {t},\vec {x} \mid \Theta )f(\Theta ) \propto \exp {\left( \sum _{i=1}^{N_I} \sum _{p=1}^{N_P} \left( \tilde{\alpha }_i[x_{ip}-1/2]\frac{\beta _i + \theta _p}{c}-t_{ip}\tilde{\alpha }^2_i\frac{(\beta _i + \theta _p)^2}{2} \right) \right) }\\&\quad \times \exp {\left( -\left( \sum _{i=1}^{N_I} \frac{(\beta _i - \mu _{I})^2}{2\sigma ^2_I} + \sum _{p=1}^{N_P} \frac{(\theta _p - \mu _{P})^2}{2\sigma ^2_P} + \sum _{i=1}^{N_I} \frac{(\tilde{\alpha }_i - \mu _{A})^2}{2\gamma ^2_A}\right) \right) }\\&\quad \times \exp {\left( -\left( \frac{(\mu _{I} - \nu _{I0})^2}{2\tau ^2_{I0}} + \frac{(\mu _{P} - \nu _{P0})^2}{2\tau ^2_{P0}} + \frac{(\mu _{A} - \nu _{A0})^2}{2\tau ^2_{A0}} + \frac{b_I}{\sigma ^2_I} + \frac{b_P}{\sigma ^2_P} + \frac{b_A}{\gamma ^2_A} \right) \right) } \\&\quad \times \sigma ^{-2(a_I+1+N_I/2)}_{I} \sigma ^{-2(a_P+1+N_P/2)}_{P} \gamma ^{-2(a_A+1+N_I/2)}_{A}. \end{aligned}$$\end{document}

This random effects extension of our model is not identified, which we address by fixing two model parameters:

$\begin{array}{c}{\mu }_P=0,{\sigma }_P^2=1.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu _P = 0, \quad \sigma ^2_P = 1. \end{aligned}$$\end{document}

The first constraint addresses the trade-off between the ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} and ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} , where adding a constant to all ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} and subtracting the same constant from all ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} yields the same value for the likelihood as the original values ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} and ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} . By requiring ${\mu }_P=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _P = 0$$\end{document} , we fix the location of the person effects on drift rate. The second constraint addresses the indeterminate scale of the item and person effects, where multiplying all ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} and ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} by a constant is compensated for by multiplying the corresponding variance terms by the same constant.

1.2. Regression Models for Person Effects

Our ABDM allows for the inclusion of regression models for the latent person effects and can also be extended to latent regression of item parameters on manifest covariates. We will illustrate this for the case where the person effects ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} are regressed on a set of covariates. A more complete discussion of latent regression models can be found in, for example, Fox (Reference Fox and Glas2001), Fox and Glas (Reference Fox2006), Mislevy (Reference Mislevy1985), and Zwinderman (Reference Zwinderman1991).

Due to the functional form of our ABDM, the addition of a regression extension is straightforward. The random effects structure for the latent variable of interest can be replaced by a regression equation. For an appropriately chosen prior distribution on the regression coefficients and residual variance, this yields normal–inverse gamma posterior distributions for the regression coefficients and residual variance and leaves the normal–normal conjugacy for the latent variables intact.

To regress the person effect ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} on a set of covariates, we specify the regression model as:

(7) $\begin{array}{c}{\theta }_p={\overrightarrow{\lambda }}^T{\overrightarrow{y}}_p+{ϵ}_p,{ϵ}_p\sim N(0,{\sigma }_{\mathrm{res}}^2),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta _p = \vec {\lambda }^T \vec {y}_p + \epsilon _p, \quad \epsilon _p \sim \mathscr {N}(0, \sigma ^2_{res}), \end{aligned}$$\end{document}

where ${\overrightarrow{y}}_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {y}_p$$\end{document} is a $(K+1)\times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(K+1) \times 1$$\end{document} vector with first entry equal to 1 and the remaining K entries being the manifest covariate values for person p, $\overrightarrow{\lambda }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {\lambda }$$\end{document} is a $(K+1)\times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(K+1) \times 1$$\end{document} vector of regression weights, and ${ϵ}_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _p$$\end{document} is a normally distributed error term with mean 0 and variance ${\sigma }_{\mathrm{res}}^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2_{res}$$\end{document} . If we assign the non-informative prior distribution $[\overrightarrow{\lambda },{\sigma }_{\mathrm{res}}^2]\propto 1/{\sigma }_{\mathrm{res}}^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\vec {\lambda },\sigma ^2_{res}] \propto 1/\sigma ^2_{res}$$\end{document} and replace the random effects term for ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} by the regression model in Eq. (7), our model becomes:

$\begin{array}{cc}f(\overrightarrow{t},\overrightarrow{x}\mid \overrightarrow{\theta },\overrightarrow{\lambda },{\sigma }_{\mathrm{res}}^2)& f\left(\overrightarrow{\lambda }\right)f\left({\sigma }_{\mathrm{res}}^2\right)\propto exp\left(-,\sum _{p=1}^{N_P},{\theta }_p^2,\left(\sum _{i=1}^{N_I},\frac{t_{\mathrm{ip}}{\tilde{\alpha }}_i^2}{2},+,\frac{1}{2{\sigma }_{\mathrm{res}}^2}\right)\right)\\ & \times exp\left(2,\sum _{p=1}^{N_P},{\theta }_p,\left(\sum _{i=1}^{N_I},{\tilde{\alpha }}_i,\frac{[x_{\mathrm{ip}}-1/2]}{2c},-,\sum _{i=1}^{N_I},t_{\mathrm{ip}},{\tilde{\alpha }}_i^2,\frac{{\beta }_i}{2},+,\frac{{\overrightarrow{\lambda }}^T{\overrightarrow{y}}_p}{2{\sigma }_{\mathrm{res}}^2}\right)\right)\\ & \times {\left({\sigma }_{\mathrm{res}}^2\right)}^{-\left(N_p,/,2,+,1\right)}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(\vec {t},\vec {x} \mid \vec {\theta }, \vec {\lambda }, \sigma ^2_{res})&f(\vec {\lambda })f(\sigma ^2_{res}) \propto \exp \left( -\sum _{p=1}^{N_P}\theta ^2_p \left( \sum _{i=1}^{N_I} \frac{t_{ip}\tilde{\alpha }^2_i}{2} + \frac{1}{2\sigma ^2_{res}}\right) \right) \\&\times \exp \left( 2\sum _{p=1}^{N_P} \theta _p \left( \sum _{i=1}^{N_I} \tilde{\alpha }_i\frac{[x_{ip}-1/2]}{2c} - \sum _{i=1}^{N_I} t_{ip}\tilde{\alpha }^2_i\frac{\beta _i}{2} + \frac{\vec {\lambda }^T \vec {y}_p}{2 \sigma ^2_{res}} \right) \right) \\&\times (\sigma ^2_{res})^{-\left( N_p/2+1\right) }. \end{aligned}$$\end{document}

Here, $\overrightarrow{\theta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {\theta }$$\end{document} denotes the $N_p\times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_p \times 1$$\end{document} vector of person effects on drift rate. Note that replacing the expression for the random effect on ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} by the regression model replaces the variance term ${\sigma }_P^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2_P$$\end{document} by the residual variance ${\sigma }_{\mathrm{res}}^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2_{res}$$\end{document} . From the expression above, it can be seen that the full-conditional posterior distributions for $\overrightarrow{\lambda }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {\lambda }$$\end{document} , ${\sigma }_{\mathrm{res}}^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2_{res}$$\end{document} , and ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} have the form (for details on the derivation, see chapter 14 in Gelman et al., Reference Gelman, Carlin, Stern, Dunson, Vehtari and Rubin2013):

$\begin{array}{c}\begin{array}{ccc}\overrightarrow{\lambda }\sim N(\hat{\lambda },V_{\lambda }{\sigma }_{\mathrm{res}}^2),& {\sigma }_{\mathrm{res}}^2\sim invG(\frac{n}{2},\frac{n-K}{2}{\hat{\sigma }}_{\mathrm{res}}^2),& {\theta }_p\sim N({\hat{\theta }}_{p,N_I},{\hat{\sigma }}_{p,N_I}^2),\end{array}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{lll} \vec {\lambda } \sim \mathbf {\mathscr {N}}(\hat{\lambda }, V_{\lambda }\sigma ^2_{res}),&\sigma ^2_{res} \sim invG(\frac{n}{2}, \frac{n-K}{2}\hat{\sigma }^2_{res}),&\theta _p \sim \mathscr {N}(\hat{\theta }_{p,N_I}, \hat{\sigma }^2_{p,N_I}), \end{array} \end{aligned}$$\end{document}

where

Here, Y is the $N_p\times (K+1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_p \times (K+1)$$\end{document} design matrix with all entries of the first column equal to 1 and the remaining columns being the K vectors containing the values of the predictors $Y_1,\dots ,Y_K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_1,\ldots ,Y_K$$\end{document} . Hence, MCMC samples for latent regression in the ABDM can be easily obtained using Gibbs-sampling.

As before, this extended model is unidentified, which we address by fixing two parameters. Firstly, we set ${\lambda }_0=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _0=0$$\end{document} to fix the location of the person effects ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} , which we previously addressed by setting ${\mu }_P=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _P=0$$\end{document} . Secondly, we set ${\sigma }_{\mathrm{res}}^2=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2_{res} = 1$$\end{document} to fix the scale of the person effects ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} , which we previously addressed by setting ${\sigma }_P^2=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2_P=1$$\end{document} .

1.4. Estimation of the Volatility-to-Caution Ratio

As pointed out above, estimation of the volatility-to-caution ratio c requires the evaluation of the full density of the ABDM, including the series expression in Eq. (6). The assumption that c is constant across persons and items means that the estimation problem is one dimensional. Fully Bayesian estimation of c would be computationally intractable. However, simple iterative optimisation can be carried out by estimating the person and item effects ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} , ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} , and ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} for each candidate value $\tilde{c}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{c}$$\end{document} using Gibbs sampling and using the ABDM likelihood in Eq. (6) evaluated at the posterior means as the objective function.

If the objective function has a unique global maximum, efficient adaptive estimation can be achieved by the golden section algorithm (Kiefer, Reference Kiefer1953). If the objective function does not have a unique global maximum, other approaches such as a grid search or adaptive gradient descent methods can be employed. Initial values for the latter type of search algorithm can be obtained from the moment estimator of c. Specifically, if we fix all item and person effects on the ABDM parameters and write $p=P(X=1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=\mathbb {P}(X=1)$$\end{document} for the probability of a correct response, then Eq. (3) gives:

$\begin{array}{c}p=\frac{1}{1+exp\left(-,\frac{v}{\mathrm{cs}}\right)}⟺\frac{v}{\mathrm{cs}}=log\left(\frac{p}{1-p}\right)=:L.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p = \frac{1}{1+\exp {\left( -\frac{v}{cs}\right) }} \iff \frac{v}{cs} = \log \left( \frac{p}{1-p} \right) {=:} L. \end{aligned}$$\end{document}

Writing M for the mean RT and substituting L into the expression in Eq. (2) gives:

$\begin{array}{c}M=\frac{a}{2v}\frac{1-exp\left(-,\frac{\mathrm{va}}{s^2}\right)}{1+exp\left(-,\frac{\mathrm{va}}{s^2}\right)}=\frac{1}{2L{c}^2}\frac{1-exp\left(-,L\right)}{1+exp\left(-,L\right)}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} M = \frac{a}{2v} \frac{1-\exp {\left( -\frac{va}{s^2}\right) }}{1 + \exp {\left( -\frac{va}{s^2}\right) }} = \frac{1}{2Lc^2}\frac{1-\exp {\left( -L\right) }}{1+\exp {\left( -L\right) }}. \end{aligned}$$\end{document}

Solving for c and using the fact that $c>0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c>0$$\end{document} , we get the estimator:

$\begin{array}{c}\hat{c}=\sqrt{\frac{1}{2LM}\frac{1-exp\left(-,L\right)}{1+exp\left(-,L\right)}}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{c} = \sqrt{\frac{1}{2LM}\frac{1-\exp {\left( -L\right) }}{1+\exp {\left( -L\right) }}}. \end{aligned}$$\end{document}

As the joint likelihood of the data conditional on c had a unique global maximum in all our example applications, we used a golden section algorithm to estimate c.

2. Simulation Study

We tested the sampling behaviour of our ABDM in a simulation study. We used the rdiffusion function from the RTdists R package (Singmann et al., Reference Singmann, Scott, Gretton, Heathcote, Voss, Voss and Terry2016) to generate RT and choice data for 120 persons answering 70 items. The generating population-level parameters were set to ${\mu }_I=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _I=1$$\end{document} , ${\sigma }_I^2=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2_I=1$$\end{document} , ${\mu }_P=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _P=0$$\end{document} , ${\sigma }_P^2=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2_P=1$$\end{document} , ${\mu }_A=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{A}=1$$\end{document} , and ${\gamma }_A^2=1/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^2_{A}={1}/{2}$$\end{document} . We truncated the generating distribution for ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} at $1/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1}/{2}$$\end{document} instead of 0 to avoid unrealistically large values of the information volatility in the simulated data. We furthermore set the caution-to-volatility ratio to $c=0.35$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c = 0.35$$\end{document} . Panel A of Fig. 2 shows the joint distribution of mean accuracies and mean RTs for the simulated items. As can be seen, both variables spanned a realistic range of values. Panel B of Fig. 2 shows the marginal distribution of RTs. All RTs fell in the range below 10s. Moreover, the distribution exhibits the skew and long tail that is characteristic of RT data.

Figure 2. Simulation study with 120 persons answering 70 items. a Accuracy and mean RT for simulated items. b Marginal RT distribution over items and persons. c Estimated and true values for the item ( ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} ) and person ( ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} ) effects on the rate or information accumulation, and for the item effects on the reciprocal volatility ( ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} ). d Example MCMC chains for item and person effects after a burn-in period of 2,000 samples.

We implemented our random effects model using custom-written R code in combination with the algorithm for sampling from truncated normal distributions implemented in the truncnorm package (Mersmann et al., Reference Mersmann, Trautmann, Steuer and Bornkamp2018). For the estimation of the volatility-to-caution ratio c, we used the method described in the preceding section. R code for the simulations and model fitting is available on OSF: https://osf.io/fh4wp/. We considered the search space $C=[0.1,1]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=[0.1,1]$$\end{document} . We set the maximum number of iterations for the golden section algorithm to 10, and we terminated the search if the relative change in the log-likelihood between iterations was less than 1%. To estimate the person and item parameters for each candidate value of c, we generated a single chain of 2,000 MCMC samples and discarded the first 1,000 samples as burn-in samples. Once we had obtained an estimate of c, we generated the final set of three chains with 12000 MCMC samples. We discarded the first 2000 samples as burn-in samples. The golden section algorithm yielded an estimate of $\hat{c}=0.343$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{c}=0.343$$\end{document} . Panel C of Fig. 2 shows the posterior mean estimates of the person and item effects ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} , ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} , and ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} . As can be seen, all parameters could be recovered accurately. Panel D of Fig. 2 shows three example chains for the person and item effects. All three parameters, ${\beta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _2$$\end{document} , ${\theta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _2$$\end{document} , and ${\tilde{\alpha }}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_2$$\end{document} , showed good sampling behaviour with relatively small autocorrelations.

Table 1 shows a summary of the potential scale reduction factor $\hat{R}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R}$$\end{document} (Gelman & Rubin, Reference Gelman and Rubin1992). As can be seen, $\hat{R}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R}$$\end{document} was below 1.02 for all population parameters. Moreover, $\hat{R}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R}$$\end{document} did not exceed 1.03 for all chains for all parameters ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} , ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} , and ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} . Taken together, the results of our simulation study show that person and item effects on drift rate and item effects on the diffusion coefficient can be estimated with good efficiency and generating values can be recovered with high accuracy.

Table 1. Convergence results for simulated data.

Quantiles are computed across persons or items.

In a further set of simulations, we explored the effect of a neglected additive non-decision time on the ABDM parameter estimates. As pointed in the introduction, our ABDM assumes that non-decision processes affect the decision time by interrupting decision processes, and changes in non-decision time are therefore reflected by changes in the rate of information processing. However, it might be argued that some quick-paced non-decision processes such as stimulus encoding and response execution do not affect decision processes and result in a small additive non-decision component of the observable response time. To gauge the effect of such a small additive non-decision component on parameter estimates in our ABDM, we generated data from the ABDM and added a small non-decision component to the generated RTs.

As in our first simulation, we generated RT and accuracy data for 120 persons answering 70 items with the same settings for the generating population parameters. We generated data with four different values for the additive non-decision time component, $t_0=0.1,0.5,1,2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0=0.1, 0.5, 1, 2$$\end{document} . Relative to the mean decision time (i.e. the mean response time without the additive non-decision component) of 1.1s, an additive non-decision component of 2s represents an extreme case where non-intrusive non-decision processes triple the observable decision time. All fitting procedures were the same as in the first simulation.

Figure 3 shows how the additive non-decision component affects estimates of the ABDM parameters. Correlations $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} between generating and estimated parameter values are given in the top left corner of each plot. As can be seen, item effects on the rate of information processing ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} (leftmost column) are largely unaffected by $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} ; but only extreme values of $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} lead to a noticeable overestimation of large values of ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} . This is also confirmed by the overall high correlation between generating and estimated parameter values. Person effects on the rate of information processing ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} (second column from the left) are largely unaffected by small values of $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} . Large values of $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} result in an underestimation of large values of ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} and an overestimation of small values of ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} . However, the bias is monotonic, which means that the order of ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} across persons is not affected. Hence, even in the presence of an unrealistically large additive non-decision component, inferences about the relative rate of information processing among test takers are still valid. This is further confirmed by the overall high correlation between generating and estimated parameter values. Item effects on the volatility parameter ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} (third column from the left) are largely unaffected by very small values of $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} . However, for larger values of $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} , the ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} tend to be underestimated and the precision with which the generating value can be recovered decreases. This is also reflected in a sizeable decrease in the correlation between generating and estimated parameter values for large values of $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} . Finally, estimates of the population-level parameters (rightmost column) are largely unaffected by small values of $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} . For extreme values of $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} , ${\mu }_I$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _I$$\end{document} remains largely unaffected, whilst the variance ${\sigma }_I^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2_I$$\end{document} is overestimated, and ${\mu }_A$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _A$$\end{document} and ${\gamma }_A^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^2_A$$\end{document} are considerably underestimated. Estimates of the boundary-to-volatility ratio c are unaffected for small values of $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} but tend to be underestimated for larger values of $t_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} .

Taken together, the results of these simulations show that neglecting realistic, small additive non-decision components in the ABDM does not significantly bias parameter estimates. Large additive non-decision components mainly affect estimates of item volatility. However, these biases only become significant when the additive non-decision component is at least as large as the mean decision time. Substantively, this means that non-intrusive non-decision processes such as stimulus encoding and response execution would need to occupy as much time as decision processes, which is implausible in realistic psychometric settings.

Figure 3. Parameter recovery simulation under additive non-decision time. Plots show the relationship between true and estimated values of the ABDM parameters when the generating model includes an additive non-decision time component. The generating population-level parameters were ${\mu }_I=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _I=1$$\end{document} , ${\sigma }_I^2=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2_I=1$$\end{document} , ${\mu }_A=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{A}=1$$\end{document} , ${\gamma }_A^2=1/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^2_{A}={1}/{2}$$\end{document} , and $c=0.35$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.35$$\end{document} .

3. Application

We gauged the performance of our ABDM in realistic settings by re-analysing data sets used in Van Rijn and Ali (Reference Van Rijn and Ali2017) and Molenaar et al. (Reference Molenaar, Tuerlinckx and van der Maas2015). Since Molenaar et al.’s (Reference Molenaar, Tuerlinckx and van der Maas2015) data set was considerably smaller, it provides little insight into the ABDM’s performance in the large-data contexts for which the model was developed. Therefore, we only report our analysis of Van Rijn and Ali’s (Reference Van Rijn and Ali2017) data here, the analysis of Molenaar et al.’s (Reference Molenaar, Tuerlinckx and van der Maas2015) data can be found in Appendix C. The data set from Van Rijn and Ali (Reference Van Rijn and Ali2017) consists of RT and accuracy data from 4899 first-year students in teachers college who completed 50 spelling items as part of a Dutch language test. Items had a two-alternative forced choice format and were presented in random order. There were no time limits imposed for the completion of individual items, but the duration of the entire language test was limited to 90 min. For our analysis, we only used data from the A version of the test and excluded test takers with incomplete data, who spent less than 3 s per item on average, or had taken (a version of) the test previously.

Figure 4 shows the marginal RT quantiles (0, 0.2, 0.4, 0.6, 0.8, and 1.0) for each of the 50 items and for the first 50 persons, ordered by mean RT. As can be seen, marginal RT distributions for the items (left plot) had a sharp leading edge and a typical long tail. Minimum RTs were generally small compared to the median RT; the ratio of the minimum observed RT to the median RT ranged between 0.11 and 0.29, with a mean of 0.21. Since item-specific additive non-decision components cannot exceed the minimum RT, this means that, in line with the assumptions of our ABDM, any possible additive non-decision components were negligible relative to the median RT. Marginal RT distributions for the persons (right plot) also exhibited the typical skewed shape with a long tail. The leading edge tended to be less steep compared to the marginal distributions for the items. However, due to the small number of items, the exact shape of the marginal distributions is insufficiently constrained by the data. Although the ratio of the minimum observed RT to the median RT across all 4899 participants ranged between 0.10 and 0.80, for 90% of the participants the minimum RT was smaller than half their median RT. This means that any potential person-specific additive non-decision components were small relative to the median RT.

Figure 4. Marginal RT quantiles (0, 0.2, 0.4, 0.6, 0.8, and 1.0) for items (left plot) and 50 persons (right plot). Data are ordered by mean RT. Maximum RTs exceeding 200 s are indicated by asterisks.

We aimed to compare the performance of our ABDM to Van der Linden’s (Reference Van der Linden2009) popular hierarchical model and to Van der Maas et al.’s (Reference Van der Maas, Molenaar, Maris, Kievit and Borsboom2011) Q-diffusion model. We compared our ABDM to the two competitor models in terms of relative and absolute model fit. For the comparison of the relative model fit with the hierarchical model, we fitted both models to the full data set of 4899 persons and 50 items. For the comparison of the absolute model fit, we only used the data of the first 50 persons on all 50 items due to the formidable computational costs of generating data from the full ABDM. For the comparison of the relative as well as the absolute model fit with the Q-diffusion model, we only used the data of the first 200 persons and the first 15 items due to prohibitively long computing times of the Q-diffusion model (see also Van Rijn & Ali, Reference Van Rijn and Ali2017). Moreover, as the DM likelihood function approaches zero quickly for RTs above 10s, we fitted all models to the RTs in tenths of seconds to guarantee numerical stability.

Van der Linden’s (Reference Van der Linden2009) hierarchical model describes log-RTs as normally distributed, where the RT for a person completing one item is described by a person-dependent speed parameter, and item-dependent discrimination and time-intensity parameters. The probability of giving a correct response is described by a normal ogive model with a person-dependent ability parameter and two item-dependent discrimination and difficulty parameters. Dependencies between RTs and accuracies are introduced through multivariate normal population-level distributions for the item and person parameters. Hence, the model as implemented by Fox et al. (Reference Fox, Klotzke and Entink2019) describes a data set with $N_I$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_I$$\end{document} items and $N_P$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_P$$\end{document} persons in terms of $2N_I+5N_P+25$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2N_I+5N_P+25$$\end{document} parameters.

Van der Maas et al.’s (Reference Van der Maas, Molenaar, Maris, Kievit and Borsboom2011) Q-diffusion model describes the joint distribution of RTs and accuracies in terms of the classic DM. The Q-diffusion model describes the rate of information processing and boundary separation as a ratio of a person and an item parameter. Moreover, non-decision time is assumed to be an additive component of the observed RT and is determined by an item parameter. Together with two parameters for the population-level variance of the person parameters, the model describes a data set with $N_I$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_I$$\end{document} items and $N_P$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_P$$\end{document} persons in terms of $3N_I+2N_P+2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3N_I+2N_P+2$$\end{document} parameters.

For the comparison with the hierarchical model, we fitted our ABDM using the routine described earlier. We first estimated the boundary-to-volatility ratio c to using the golden section algorithm, where the algorithm terminated if either a maximum number of ten iterations had been reached or the change in the log-likelihood between iterations was less than 1%. To estimate the person and item parameters for each candidate value of c, we drew 2,000 posterior samples for all other model parameters and discarding the first 1,000 samples as burn-in samples. The golden section estimation of c was completed after seven iterations with a change in the log-likelihood of less than 0.03%. In a second step, we obtained the desired number of posterior samples for the best-fitting value of c. We sampled three chains with 12,000 posterior samples each of which we discarded the first 2,000 samples as burn-in. All chains showed good convergence, with $\hat{R}<1.02$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R} < 1.02$$\end{document} . We used the ddiffusion function implemented in the rtdists R package (Singmann et al., Reference Singmann, Scott, Gretton, Heathcote, Voss, Voss and Terry2016) to compute the ABDM likelihood. We fitted van der Linden’s hierarchical model using the LNIRT function in Fox et al.’s (Reference Fox, Klotzke and Entink2019) LNIRT R package. We obtained three chains with 12,000 posterior samples each of which we discarded the first 2,000 samples as burn-in. All chains showed good convergence, with $\hat{R}<1.02$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{R} < 1.02$$\end{document} .

We assessed relative model fit by means of AIC, BIC and DIC. For the computation of AIC and BIC, we evaluated the conditional model likelihoodsFootnote 1 at the posterior means. We also based our computation of DIC on the conditional likelihood for both models.Footnote 2 Due to the long computing times required for the evaluation of the full ABDM likelihood function, we based the computation of the model selection criteria on the first 1,000 posterior samples after burn-in from a single MCMC chain. Table 2 shows the relative fit of the two models. As can be seen, the log-likelihood was larger for the ABDM than for the hierarchical model and all three model selection criteria indicated that the ABDM also fitted the data better when model complexity was taken into account.

Table 2. Relative model fits.

We assessed the absolute model fit by means of the posterior predictive mean RTs and accuracies for persons and items. Due to the high computational costs of generating samples from the full ABDM, we limited our analysis to the first 50 items and persons in the data set. We generated 500 posterior predictive samples for each item and person. We used the rdiffusion function in the rtdists R package (Singmann et al., Reference Singmann, Scott, Gretton, Heathcote, Voss, Voss and Terry2016) to generate posterior predictive samples for the ABDM, and we used custom-written code to generate posterior predictive samples for the hierarchical model.

Figure 5 shows the comparison of the data and the posterior predictive samples generated by the two models, ordered by the mean RT in the data. As can be seen, the ABDM generated wider posterior predictive intervals for RTs (left column in the left panel) than the hierarchical model (left column in the right panel). Whereas the ABDM’s posterior predictives matched the mean item RTs relatively closely (top left plot in the left panel), the hierarchical model systematically underpredicted the mean item RTs (top left plot in the right panel). The ABDM overpredicted some of the mean person RTs (bottom left plot in the left panel), whereas the hierarchical model matched the mean person RTs closely (bottom left plot in the right panel). Both models matched the mean item (top right plot in each panel) and person accuracies (bottom right plot in each panel) closely, although the ABDM overpredicted some particularly low item accuracies. Taken together, both models fitted the data well, although the hierarchical model showed a systematic overprediction of mean item RTs that was not present in the ABDM.

Figure 5. Posterior predicted mean RTs and accuracies for 50 items and persons. Results are ordered by mean RT. Results for the ABDM are shown in the left panel, results for the hierarchical model are shown in the right panel. Intervals indicate the range between the 0.025 and the 0.975 quantile of the posterior predictive values.

As there are no Bayesian implementations available of the Q-diffusion, we developed a maximum-likelihood fitting method for our ABDM to compare both models on an equal footing. To avoid the computationally costly evaluation of the full ABDM likelihood required for the maximum-likelihood estimation of the boundary-to-volatility ratio c, we fixed c to the value estimated based on the complete data set in our comparison with the hierarchical model.

Joint maximum-likelihood estimates of the ABDM parameters can be obtained by differentiating the logarithm of the likelihood (6) with respect to the person and item effects on the rate of information processing and volatility, which yields:

$\begin{array}{cc}\frac{\partial }{\partial {\beta }_i}log\left(f,(,\overrightarrow{t},,,\overrightarrow{x},\mid ,\Theta ,)\right)& =\sum _{p=1}^{N_P}\frac{x_{\mathrm{ip}}-1/2}{c}{\tilde{\alpha }}_i-t_{\mathrm{ip}}({\beta }_i+{\theta }_p){\tilde{\alpha }}_i^2\\ \frac{\partial }{\partial {\theta }_p}log\left(f,(,\overrightarrow{t},,,\overrightarrow{x},\mid ,\Theta ,)\right)& =\sum _{i=1}^{N_I}\frac{x_{\mathrm{ip}}-1/2}{c}{\tilde{\alpha }}_i-t_{\mathrm{ip}}({\beta }_i+{\theta }_p){\tilde{\alpha }}_i^2\\ \frac{\partial }{\partial {\tilde{\alpha }}_i}log\left(f,(,\overrightarrow{t},,,\overrightarrow{x},\mid ,\Theta ,)\right)& =\sum _{p=1}^{N_P}\frac{x_{\mathrm{ip}}-1/2}{c}({\beta }_i+{\theta }_p)-t_{\mathrm{ip}}{({\beta }_i+{\theta }_p)}^2{\tilde{\alpha }}_i.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial }{\partial \beta _i} \log \left( f(\vec {t},\vec {x} \mid \Theta ) \right)&= \sum _{p=1}^{N_P} \frac{x_{ip}-1/2}{c}\tilde{\alpha }_i - t_{ip}(\beta _i + \theta _p)\tilde{\alpha }^2_i \\ \frac{\partial }{\partial \theta _p} \log \left( f(\vec {t},\vec {x} \mid \Theta ) \right)&= \sum _{i=1}^{N_I} \frac{x_{ip}-1/2}{c}\tilde{\alpha }_i - t_{ip}(\beta _i + \theta _p)\tilde{\alpha }^2_i \\ \frac{\partial }{\partial \tilde{\alpha }_i} \log \left( f(\vec {t},\vec {x} \mid \Theta ) \right)&= \sum _{p=1}^{N_P} \frac{x_{ip}-1/2}{c}(\beta _i + \theta _p) - t_{ip}(\beta _i + \theta _p)^2\tilde{\alpha }_i. \end{aligned}$$\end{document}

Equating to 0 and solving for ${\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i$$\end{document} , ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} , and ${\tilde{\alpha }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\alpha }_i$$\end{document} yields for the estimators:

$\begin{array}{cc}{\hat{\beta }}_i& =\frac{{\sum }_{p=1}^{N_P}(x_{\mathrm{ip}}-1/2)-c{t}_{\mathrm{ip}}{\theta }_p{\tilde{\alpha }}_i}{c{\sum }_{p=1}^{N_P}{t}_{\mathrm{ip}}{\tilde{\alpha }}_i}\\ {\hat{\theta }}_p& =\frac{{\sum }_{i=1}^{N_I}(x_{\mathrm{ip}}-1/2)-c{t}_{\mathrm{ip}}{\beta }_i{\tilde{\alpha }}_i}{c{\sum }_{p=1}^{N_P}{t}_{\mathrm{ip}}{\tilde{\alpha }}_i}\\ {\hat{\tilde{\alpha }}}_i& =\frac{{\sum }_{p=1}^{N_P}(x_{\mathrm{ip}}-1/2)}{c{\sum }_{p=1}^{N_P}{t}_{\mathrm{ip}}({\beta }_i+{\theta }_p)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{\beta }_i&= \frac{\sum _{p=1}^{N_P} (x_{ip}-1/2) - ct_{ip}\theta _p \tilde{\alpha }_i}{c\sum _{p=1}^{N_P} t_{ip} \tilde{\alpha }_i} \\ \hat{\theta }_p&= \frac{\sum _{i=1}^{N_I} (x_{ip}-1/2) - ct_{ip}\beta _i \tilde{\alpha }_i}{c\sum _{p=1}^{N_P} t_{ip} \tilde{\alpha }_i} \\ \hat{\tilde{\alpha }}_i&= \frac{\sum _{p=1}^{N_P} (x_{ip}-1/2)}{c\sum _{p=1}^{N_P} t_{ip} (\beta _i + \theta _p)}, \end{aligned}$$\end{document}

which can be solved by iterative computation. To identify the model, we imposed the same constraints on the ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} as we imposed on the population mean and variance in our Bayesian implementation, fixing the mean and variance of the ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} to $\overline{\theta }=1/N_P{\sum }_{p=1}^{N_P}{\theta }_p=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{\theta } = {1}/{N_P}\sum _{p=1}^{N_P}\theta _p = 0$$\end{document} and ${\sigma }^2=1/N_P{\sum }_{p=1}^{N_P}{({\theta }_p-\overline{\theta })}^2=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2 = {1}/{N_P}\sum _{p=1}^{N_P}(\theta _p-\bar{\theta })^2 = 1$$\end{document} . Note that the remaining population-level parameters, ${\mu }_I,{\sigma }_I^2,{\mu }_A$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _I, \sigma ^2_I, \mu _A$$\end{document} , and ${\gamma }_A^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^2_A$$\end{document} , are not estimated in the JML approach. We obtained maximum-likelihood estimates of the Q-diffusion parameters using the diffIRT function in the diffIRT R package (Molenaar, Reference Molenaar2015). We fitted both models to RTs divided by 10 to guarantee numerical stability when evaluating the model likelihood. One reviewer pointed out that the selection of the 15 items might have affected the outcomes of the model comparisons. We therefore repeated the model comparisons with a different set of 15 items. As the results of both analyses agreed closely, we only report the results for the first set of 15 items. The analysis for the second set of 15 items is reported in Appendix D.

We assessed relative model fit by means of AIC and BIC. Table 3 shows the relative fit of the two models. As can be seen, the log-likelihood was smaller for the ABDM than for the Q-diffusion model. AIC preferred the Q-diffusion model over the ABDM, whereas BIC preferred the ABDM over the Q-diffusion model. This difference between the two criteria is due to the higher penalty for model complexity imposed by BIC, which thus avoids selecting overly complex models at small sample sizes. Moreover, the difference in AIC between the two models was considerably smaller than the difference in BIC. These results suggest that the ABDM accounted as well for the data as the Q-diffusion model when model complexity was taken into account.

Table 3. Relative model fits.

We assessed absolute model fit through a cross-validation approach. To this end, we split up the data in two ways. Firstly, we divided the data into five folds of 40 persons each, removed one fold from the data and fitted both models to the remaining data (person-based fit). Secondly, we divided the data into five folds of three items each, removed one fold from the data and fitted both models to the remaining data (item-based fit). We used the results of the person-based fit to compute the item parameters for the item fold removed in the item-based fit, and we used the results of the item-based fit to compute the person parameters for the person-fold removed in the person-based fit. We subsequently combined the person and item parameters obtained in the two steps and used expressions (2) and (3) to predict RTs and accuracies for the person-by-item combinations contained in the removed folds. This was repeated for all 25 person-by-item folds.

Figure 6 shows the deviation between observed and predicted RTs (top row of plots) and accuracies (bottom row of plots) for the ABDM (left column of plots) and the Q-diffusion model (right column of plots). Each column of dots shows the results for the 40 persons in one person-fold, each cluster of five columns of dots shows the results for all five person folds on one item, and each group of three items shows the results for all five person folds in one item fold. As can be seen, the prediction errors for RTs for the ABDM clustered symmetrically around 0, with a few large positive outliers. This means that the ABDM predicted RTs were largely unbiased, except for very long RTs, in which case the ABDM tended to underpredict observed RTs. For the Q-diffusion model, prediction errors also clustered symmetrically around 0, except for the third item fold, for which the model systematically underpredicted the observed RTs. The spread of the prediction error for the Q-diffusion model was comparable to that for the ABDM. Finally, the Q-diffusion model showed a less pronounced tendency to underpredict long RTs than the ABDM; the Q-diffusion model equally often overpredicted short RTs.

The results for the predicted accuracy are again similar between the two models. For the ABDM prediction, errors tend to be positive, which indicates a tendency to underpredict accuracies. Nevertheless, most prediction errors are close to 0, which means that the ABDM generally captured the probability of a correct response well. The Q-diffusion model showed a similar tendency to underpredict accuracies but, in general, predicted the probability of a correct response well, with most prediction errors being slightly larger than 0. The spread of the prediction errors is also comparable to those observed for the ABDM. However, for the third item fold, the Q-diffusion model showed a systematic underprediction of accuracies. Taken together, both models provided a comparable absolute fit to the data.

Figure 6. Out-of-sample predictions in fivefold cross-validation for the ABDM and the Q-diffusion model. Predictions are based on data from 200 persons and 15 items. Results for the ABDM are shown on the left results for the Q-diffusion model are shown on the right. Each column of dots shows the results for one person fold, each cluster of five columns shows the results for all five person folds in one item, and each group of three items shows the results for all five person folds on one item fold.

To sum up, our comparison with the hierarchical model showed that our ABDM provides a better relative fit and a comparable absolute fit to the data, with a considerably smaller number of parameters. Our comparison with the Q-diffusion model further showed that the ABDM provides a comparable relative and absolute fit to the data. This latter result indicates that our ABDM does not lose much descriptive power compared to an instantiation of the classic DM, but requires fewer parameters ( $N_I+N_P+4$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_I+N_P+4$$\end{document} , to be precise).

3.1. Regression Model

As a final step in the assessment of our ABDM, we tested whether the estimated person effects on drift rate behave in accordance with their substantive interpretation. Van Rijn and Ali’s (Reference Van Rijn and Ali2017) data set included information on test takers’ prior education. If individual differences in the person effects on drift rate ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} do indeed reflect differences in the speed of information processing, one would expect that ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} is larger for test takers who completed a higher level of prior education. To test this substantive hypothesis, we extended our ABDM with a regression model that described the ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} in terms of the dummy-coded prior education levels, with MBO as the reference level.

Prior education was categorised as MBO, HAVO, or VWO, which reflect increasing levels in the Dutch education system. We excluded the data of 348 test takers from this analysis as there were either no data available on their prior education or their prior education did not fall into one of the three categories. As a first analysis step, we fitted the ABDM without the regression component to the data. The left panel in Fig. 7 shows the posterior means of ${\theta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _p$$\end{document} for the three education levels. As can be seen, the estimated person effects were smallest for test takers with MBO as their prior education and largest for test takers with a VWO education. As a second step, we fitted the ABDM with the regression extension to the data. The right panel in Fig. 7 shows the equal-tailed 95% credible intervals for the regression coefficients. In line with the hypothesis, the regression coefficient for the difference between HAVO and MBO was positive and the credible interval did not include 0. The difference between VWO and MBO was considerably larger than the difference between HAVO and MBO, and the credible interval again did not include 0. Taken together, these results are in line with the substantive hypothesis that higher education levels are associated with faster information processes, which is represented by the person effects on drift rate.

Figure 7. Relationship between person effect on drift rate and prior education. Posterior means from the model without regression component are shown on the left. Data are jittered for improved visibility. Horizontal lines indicate the group means. Posterior means (dots) and 95% equal-tailed credible interval (lines) for the regression coefficients are shown on the right.