2.1. The Hierarchical Model

The HM, as proposed by van der Linden (Reference van der Linden2007), is a general statistical framework for modeling accuracies and responses times that is based on the idea that there are two latent traits at work; ability $\theta$ \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} governs the response accuracy distribution and speed $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} the response time distribution. Importantly, the response accuracy distribution is assumed to be independent of speed $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} given ability $\theta$ \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 response time distribution is assumed to be independent of ability $\theta$ \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} given speed $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} , and it is also assumed that accuracies and response times are independent given the full set of latent traits, i.e.,

$\begin{array}{c}p(x,t\mid \theta ,\eta )=p(x\mid \theta )p(t\mid \eta ).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x} ,\,\mathbf {t} \mid \theta ,\,\eta ) = p(\mathbf {x} \mid \theta ) \, p(\mathbf {t} \mid \eta ). \end{aligned}$$\end{document}

This setup provides a plug-and-play framework for modeling accuracy and response times: The measurement model for ability $\theta$ \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} —marginal distribution of accuracies $p(x\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x} \mid \theta )$$\end{document} —can be chosen independently of the measurement model for speed $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} —marginal distribution of response times $p(t\mid \eta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {t} \mid \eta )$$\end{document} . The HM is concluded with a model for the two latent traits.

Different measurement models for ability $\theta$ \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} have been used in the literature. For example, van der Linden (Reference van der Linden2007) used normal ogive models, Bolsinova, De Boeck, and Tijmstra (Reference Bolsinova, De Boeck and Tijmstra2017) used their logistic counterparts, while Zhan, Jiao, and Liao (Reference Zhan, Jiao and Liao2018) used cognitive diagnosis models instead. Here, we use the two-parameter logistic model,

(1) $\begin{array}{c}p(x\mid \theta )=\prod _{i=1}^{p}p(x_i\mid \theta )=\prod _{i=1}^p\frac{exp\left(x_i,,{\alpha }_i,(\theta +{\beta }_i)\right)}{1+exp\left({\alpha }_i,(\theta +{\beta }_i)\right)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x} \mid \theta ) = \prod _{i=1}^p p(x_i \mid \theta ) = \prod _{i=1}^p \frac{\exp \left( x_i\, \alpha _i(\theta + \beta _i)\right) }{1+\exp \left( \alpha _i(\theta + \beta _i)\right) }, \end{aligned}$$\end{document}

where ${\alpha }_i$ \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} is an item discrimination parameter and ${\beta }_i$ \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} is an item easiness parameter. There are two reasons for using the two-parameter logistic model here. Firstly, the marginal distribution of accuracies $p(x\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x} \mid \theta )$$\end{document} that is implied by the versions of the SM and DM that are used here is also a two-parameter logistic model. Secondly, the two-parameter logistic model is a member of the exponential family of distributions, which will be convenient for expressing the manifest probability distribution $p(x,t)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x} ,\,\mathbf {t})$$\end{document} .

Different measurement models for speed $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} have also been used in the literature. For example, van der Linden (Reference van der Linden2006) used a log-normal distribution, Fox, Klein Entink, and van der Linden (Reference Fox, Klein Entink and van der Linden2007) used a linear factor model for log-transformed response times, and Klein Entink, van der Linden, and Fox (Reference Klein Entink, van der Linden and Fox2009) used models based on Box–Cox transformations of response times. In this paper we will use the log-normal distribution,

(2) $\begin{array}{c}p(t\mid \eta )=\prod _{i=1}^{p}p(t_i\mid \eta )=\prod _{i=1}^p\sqrt{\frac{{\varphi }_i}{2\pi }}\frac{1}{t_i}exp\left(-,\frac{1}{2},,{\varphi }_i,,{\left(ln,\left(t_i\right),+,\eta ,-,{\xi }_i\right)}^2\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {t} \mid \eta ) = \prod _{i=1}^p p(t_i \mid \eta ) = \prod _{i=1}^p \sqrt{\frac{\phi _i}{2\pi }}\,\frac{1}{t_i}\, \exp \left( -\frac{1}{2}\, \phi _i\, \left( \ln (t_i) + \eta - \xi _i\right) ^2\right) , \end{aligned}$$\end{document}

where ${\xi }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _i$$\end{document} is an item time intensity parameter and ${\varphi }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _i$$\end{document} an item precision parameter. The item precision ${\varphi }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _i$$\end{document} is the response time analogue of the item discrimination in the two-parameter logistic model; larger values of ${\varphi }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _i$$\end{document} imply that speed explains a larger portion of the log-time variance. The log-normal distribution is a common choice for the measurement model for speed $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} in the HM framework and is also a member of the exponential family of distributions.

To conclude the HM we specify a distribution for ability $\theta$ \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} and speed $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} . Typically, a bivariate normal distribution is used in which ability and speed are correlated. To identify the model, however, the means of the bivariate normal need to be constrained to zero and the marginal variance of ability needs to be constrained to one.Footnote ¹

2.2. The Signed Residual Time Model

The SM has been proposed by Maris and van der Maas (Reference Maris and van der Maas2012) as a measurement model for ability $\theta$ \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} in the context of tests with item-level time limits. The model was specifically designed for tests that use the following scoring rule,

$\begin{array}{c}\text{s}=\sum _{i=1}^p{\text{s}}_i=(2x_i-1)({\text{d}}_i-t_i),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {s} = \sum _{i=1}^p\text {s}_i=(2x_i- 1 )(\text {d}_i-t_i), \end{aligned}$$\end{document}

where ${\text{d}}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {d}_i$$\end{document} is the time limit for item i. This scoring rule encourages persons to work fast but punishes guessing: Residual time ${\text{d}}_i-t_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {d}_i - t_i$$\end{document} is gained when the response is correct, but is lost when the response is incorrect. Van Rijn and Ali (Reference van Rijn and Ali2017b) demonstrated that the SM is also appropriate for applications where these time limits are not specified a priori, but “estimated” from the observed response time distributions.

The SM specifies the following distribution for accuracy $x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {x}$$\end{document} and response times $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {t}$$\end{document} :

(3) $\begin{array}{c}p(x,t\mid \theta )=\prod _{i=1}^p(\theta +{\beta }_i)\frac{exp\left((2x_i-1),({\text{d}}_i-t_i),(\theta +{\beta }_i)\right)}{exp\left({\text{d}}_i(\theta +{\beta }_i)\right)-exp(-{\text{d}}_i(\theta +{\beta }_i))},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\,\mathbf {t}\mid \theta ) = \prod _{i=1}^p (\theta +\beta _i) \frac{\exp \left( (2x_i-1)(\text {d}_i-t_i)(\theta + \beta _i)\right) }{\exp (\text {d}_i(\theta +\beta _i))-\exp (-\text {d}_i(\theta +\beta _i))}, \end{aligned}$$\end{document}

where ${\beta }_i$ \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} is an item easiness parameter. Observe that the SM is an exponential family model and that the scoring rule $\text{s}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {s}$$\end{document} is the sufficient statistic for ability $\theta$ \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 SM has been generalized by van Rijn and Ali (Reference van Rijn and Ali2017b) allowing the items to differ in their discriminative power even when the time limits are the same across items. In this paper we will use the standard version of the SM.

Whereas the HM characterizes the joint distribution of accuracies and response times by specific choices of the marginals $p(x\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x} \mid \theta )$$\end{document} and $p(t\mid \eta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {t} \mid \eta )$$\end{document} , the SM directly specifies a joint distribution for accuracies and response times $p(x,t\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x},\,\mathbf {t} \mid \theta )$$\end{document} . By integrating out the response times we obtain the marginal distribution for accuracies $p(x\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x} \mid \theta )$$\end{document} . Maris and van der Maas (Reference Maris and van der Maas2012) show that this marginal distribution is the two-parameter logistic model in Eq. (1), where the item discrimination ${\alpha }_i$ \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} is equal to the item time limit ${\text{d}}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {d}_i$$\end{document} . In a similar way, we obtain the marginal distribution of response times $p(t\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {t} \mid \theta )$$\end{document} by summing out the accuracies,

$\begin{array}{c}p(t\mid \theta )=\prod _{i=1}^p(\theta +{\beta }_i)\frac{exp\left(({\text{d}}_i-t_i)(\theta +{\beta }_i)\right)+exp(-({\text{d}}_i-t_i)(\theta +{\beta }_i))}{exp\left({\text{d}}_i(\theta +{\beta }_i)\right)-exp(-{\text{d}}_i(\theta +{\beta }_i))}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {t} \mid \theta ) = \prod _{i=1}^p (\theta + \beta _i)\frac{\exp ((\text {d}_i - t_i)(\theta + \beta _i))+\exp (-(\text {d}_i-t_i)(\theta +\beta _i))}{\exp (\text {d}_i(\theta +\beta _i))-\exp (-\text {d}_i(\theta +\beta _i))}. \end{aligned}$$\end{document}

An alternative specification of the SM is in terms of accuracies and what Maris and van der Maas (Reference Maris and van der Maas2012, p. 624) refer to as pseudo-response times $t^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {t}^*$$\end{document} . Pseudo-response times are obtained from response times through the transformation

(4) $\begin{array}{c}{t}_i^{\ast }=\left(\begin{array}{cc}{t}_i& \text{if}{x}_i=1,\\ {\text{d}}_i-t_i& \text{if}{x}_i=0.\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} t^*_i = {\left\{ \begin{array}{ll} t_i &{} \text { if }x_i = 1,\\ \text {d}_i - t_i &{} \text { if }x_i = 0. \end{array}\right. } \end{aligned}$$\end{document}

This transformation from response times to pseudo-response times is one-to-one, so that no information is lost. One convenient feature of using pseudo-response times instead of response times is that the pseudo-response times and accuracies are (conditionally) independent in the SM, i.e.,

$\begin{array}{c}p(x,t^{\ast }\mid \theta )=p(x\mid \theta )p(t^{\ast }\mid \theta ),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\,\mathbf {t}^*\mid \theta ) = p(\mathbf {x}\mid \theta )p(\mathbf {t}^*\mid \theta ), \end{aligned}$$\end{document}

where the marginal distribution $p(t^{\ast }\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {t}^*\mid \theta )$$\end{document} is equal to,

(5) $\begin{array}{c}p(t^{\ast }\mid \theta )=\prod _{i=1}^p(\theta +{\beta }_i)\frac{exp\left(-,t_i^{\ast },\left(\theta ,+,{\beta }_i\right)\right)}{1-exp\left(-,{\text{d}}_i,\left(\theta ,+,{\beta }_i\right)\right)}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {t}^*\mid \theta ) = \prod _{i=1}^p (\theta +\beta _i)\,\frac{\exp \left( -t_i^*\left( \theta +\beta _i\right) \right) }{1- \exp \left( -\text {d}_i\left( \theta +\beta _i\right) \right) }. \end{aligned}$$\end{document}

2.3. The Drift Diffusion Model

The DM was introduced by Ratcliff (Reference Ratcliff1978) as a model for two-choice experiments. In the DM, evidence for either choice accumulates over time until a decision boundary is reached. One way to characterize this evidence accumulation process is in terms of a Wiener process with constant drift and volatility, and absorbing upper and lower boundaries (Cox & Miller, Reference Cox and Miller1970). The drift $\mu$ \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} of the diffusion process determines how fast information is accumulated, the volatility $\sigma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} determines how noisy the accumulation process is, and the distance between the two boundaries $\alpha$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} determines how much evidence needs to be accumulated before a choice is made. The process has two additional parameters: a bias parameter z that indicates the distance from the starting point to the lower boundary, and the non-decision time $T_{\left(er\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{(er)}$$\end{document} . A commonly used simplification of the DM assumes that the process is unbiased $z=\frac{1}{2}\alpha$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z = \frac{1}{2}\alpha $$\end{document} .

The DM has been extended to model differences between persons and tasks. For example, Tuerlinckx and De Boeck (Reference Tuerlinckx and De Boeck2005) proposed to decompose the drift $\mu$ \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} of the accumulation process into a person and an item part —i.e., $\mu =\theta +{\beta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu = \theta + \beta _i$$\end{document} —and to treat the distance between the boundaries as an item characteristic —i.e., $\alpha ={\alpha }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\alpha _i$$\end{document} . The person component $\theta$ \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} in the drift specification carries the interpretation of an ability in item response theory models, as a higher value of $\theta$ \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} implies an increased probability of choosing the correct alternative. To identify the DM, Tuerlinckx and De Boeck (Reference Tuerlinckx and De Boeck2005) fixed the volatility $\sigma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} to one. The joint distribution of decision times and the chosen alternatives —i.e., response accuracies if the upper and the lower boundaries correspond to the correct and incorrect responses—is then equal to:

(6) $\begin{array}{cc}p(x,t\mid \theta )& =\prod _{i=1}^p\frac{\pi }{{\alpha }_i^2}exp\left(\frac{1}{2},{\alpha }_i,\left(2,x_i,-,1\right),(\theta +{\beta }_i),-,\frac{1}{2},\left(t_i,-,T_{\left(er\right)}\right),{(\theta +{\beta }_i)}^2\right)\\ & \times \sum _{n=1}^{\infty }sin\left(\frac{1}{2},\pi ,n\right)exp\left(-,\frac{1}{2{\alpha }_i^2},{\pi }^2,n^2,\left(t_i,-,T_{\left(er\right)}\right)\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\,\mathbf {t}\mid \theta )&= \prod _{i = 1}^p \frac{\pi }{\alpha _i^2}\exp \left( \frac{1}{2}\alpha _i\left( 2x_i-1\right) (\theta + \beta _i) - \frac{1}{2}\left( t_i-T_{(er)}\right) (\theta + \beta _i)^2\right) \nonumber \\&\quad \times \sum _{n = 1}^{\infty } \sin \left( \frac{1}{2}\pi n\right) \exp \left( -\frac{1}{2\alpha _i^2}\pi ^2n^2\left( t_i-T_{(er)}\right) \right) . \end{aligned}$$\end{document}

Both the SM and the DM directly specify a joint distribution of accuracies and decision times (response times) that is based on one latent trait, ability $\theta$ \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} . In contrast to the SM, however, accuracies and response times are independent given ability in the (unbiased) DM. The marginal $p(x\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x} \mid \theta )$$\end{document} is the two-parameter logistic model in Eq. (1), where the discrimination parameter equals the distance between the boundaries in the diffusion process, and the easiness parameter is an item effect on drift of the diffusion process. The marginal $p(t\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {t} \mid \theta )$$\end{document} is equal to

$\begin{array}{cc}p(t\mid \theta )& =\prod _{i=1}^p\frac{2\pi }{{\alpha }_i^2}cosh\left(\frac{1}{2},{\alpha }_i,(\theta +{\beta }_i)\right)exp\left(-,\frac{1}{2},\left(t_i,-,T_i^{\left(er\right)}\right),{(\theta +{\beta }_i)}^2\right)\\ & \times \sum _{n=1}^{\infty }sin\left(\frac{1}{2},\pi ,n\right)exp\left(-,\frac{1}{2{\alpha }_i^2},{\pi }^2,n^2,\left(t_i,-,T_i^{\left(er\right)}\right)\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {t} \mid \theta )&= \prod _{i = 1}^p \frac{2\pi }{\alpha _i^2} \cosh \left( \frac{1}{2}\alpha _i(\theta + \beta _i)\right) \exp \left( -\frac{1}{2}\left( t_i-T_{i}^{(er)}\right) (\theta + \beta _i)^2\right) \\&\quad \times \sum _{n = 1}^{\infty } \sin \left( \frac{1}{2}\pi n\right) \exp \left( -\frac{1}{2\alpha _i^2}\pi ^2n^2\left( t_i-T_{i}^{(er)}\right) \right) . \end{aligned}$$\end{document}

Even though the marginal distribution $p(x\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x} \mid \theta )$$\end{document} is a member of the exponential family, neither the marginal distribution $p(t\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {t}\mid \theta )$$\end{document} nor the joint distribution $p(x,t\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x},\,\mathbf {t}\mid \theta )$$\end{document} is a member of the exponential family. The primary reason that the latter cannot be written in exponential family form is because it implies a statistic ${\text{s}}_1={\text{s}}_1(x,t)={\sum }_i{\alpha }_i{x}_i-{\sum }_i{\beta }_i{t}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {s}_1 = \text {s}_1(x,\,t) = \sum _i\alpha _i\,x_i -\sum _{i}\beta _i\,t_i$$\end{document} that is sufficient for $\theta$ \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} and another statistic ${\text{s}}_2={\text{s}}_2\left(t\right)={\sum }_i{t}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {s}_2 = \text {s}_2(t) = \sum _it_i$$\end{document} that is sufficient for $-\frac{1}{2}{\theta }^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\frac{1}{2}\theta ^2$$\end{document} . If we express the latter as $\eta =\eta \left(\theta \right)=-\frac{1}{2}{\theta }^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta = \eta (\theta ) = -\frac{1}{2}\theta ^2$$\end{document} , we end up with an exponential family model subject to constraints on the parameters $\theta$ \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} and $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} : $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} is functionally related to ability $\theta$ \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} . This is known as a curved exponential family model (Efron, Reference Efron1975, Reference Efron1978).

Definition 1

For models $p(x,t\mid \zeta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x},\,\mathbf {t} \mid \varvec{\zeta })$$\end{document} that are of the form of Eq. (7) or Eq. (8) we may define the latent variable distribution

(9) $\begin{array}{c}g\left(\zeta \right)=\frac{1}{\text{Z}}\prod _{i=1}^p{\text{Z}}_i\left(\zeta \right)k\left(\zeta \right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g(\varvec{\zeta }) =\frac{1}{\text {Z}} \prod _{i=1}^p \text {Z}_i(\varvec{\zeta })\, k(\varvec{\zeta }), \end{aligned}$$\end{document}

where $k\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(\varvec{\zeta })$$\end{document} is a kernel density and $\text{Z}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Z}$$\end{document} is the normalizing constant of $g\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\varvec{\zeta })$$\end{document} . For every kernel distribution $k\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(\varvec{\zeta })$$\end{document} for which the normalizing constant $\text{Z}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Z}$$\end{document} is finite, i.e.,

$\begin{array}{c}0<\text{Z}={\int }_{{\Omega }_{\zeta }}\prod _{i=1}^p{\text{Z}}_i\left(\zeta \right)k\left(\zeta \right)\text{d}\zeta <\infty ,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0< \text {Z} = \int _{\Omega _{\varvec{\zeta }}}\prod _{i=1}^p \text {Z}_i(\varvec{\zeta }) \, k(\varvec{\zeta }) \,\text {d}\varvec{\zeta } < \infty , \end{aligned}$$\end{document}

where ${\Omega }_{\zeta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\varvec{\zeta }}$$\end{document} is the support of $\zeta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\zeta }$$\end{document} , $g\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\varvec{\zeta })$$\end{document} is a valid probability distribution.

The distribution in Definition 1 was inspired by the latent trait distribution that has been introduced with the latent variable expression of a graphical model from physics known as the Ising (Reference Ising1925) model by Kac (Reference Kac1968, see also Marsman et al., Reference Marsman, Borsboom, Kruis, Epskamp, van Bork and Waldorp2018; Epskamp, Maris, Waldorp, & Borsboom, Reference Epskamp, Maris, Waldorp, Borsboom, Irwing, Hughes and Booth2018), but a similar construction can also be found in, for instance, Cressie and Holland (Reference Cressie and Holland1983, Eq. A9) and McCullagh (Reference McCullagh1994). We can now state our first result.

Theorem 1

When $p(x,t\mid \zeta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x} ,\,\mathbf {t}\mid \varvec{\zeta })$$\end{document} is of the form of Eq. (7) and the latent variable distribution $g\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\varvec{\zeta })$$\end{document} in Eq. (9) is a valid probability distribution, then the manifest distribution $p(x,t)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x},\,\mathbf {t})$$\end{document} is given by

$\begin{array}{c}p(x,t)=\frac{1}{\text{Z}}\prod _{i=1}^p{\text{b}}_{i}E\left(\prod _{i=1}^p,exp,\left(s_i^T,\zeta \right)\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\,\mathbf {t}) = \frac{1}{\text {Z}} \prod _{i=1}^p\text {b}_i\,\mathbb {E}\left( \prod _{i=1}^p\exp \left( \mathbf {s}_i^\mathsf{T}\varvec{\zeta }\right) \right) , \end{aligned}$$\end{document}

where $\text{Z}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Z}$$\end{document} is a normalizing constant and the expectation is an integral with respect to the kernel density $k\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(\varvec{\zeta })$$\end{document} .

We omit the simple proof of Theorem 1, which requires one to fill in the definitions of the models in Eqs. (7) and (9), and then integrate out the latent variable $\zeta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\zeta }$$\end{document} . In a similar way, the manifest distribution for latent trait models that are of the form of Eq. (8) can be expressed as

(10) $\begin{array}{c}p(x,t)=\frac{1}{\text{Z}}\prod _{i=1}^p{\text{b}}_{i}E\left(\prod _{i=1}^p,exp,\left(s_{i1},\theta ,-,s_{i2},\frac{1}{2},{\theta }^2\right)\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\,\mathbf {t}) = \frac{1}{\text {Z}} \prod _{i=1}^p\text {b}_i\,\mathbb {E}\left( \prod _{i=1}^p\exp \left( s_{i1}\theta - s_{i2}\frac{1}{2}\theta ^2\right) \right) . \end{aligned}$$\end{document}

Theorem 1 shows that for any latent trait model of the form of Eq. (7), combined with a latent variable distribution of the form of Eq. (9), the manifest distribution can be characterized in terms of the base measures ${\text{b}}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {b}_i$$\end{document} and the moment generating function of the kernel distribution $k\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(\varvec{\zeta })$$\end{document} . This is similar to Holland’s (Reference Holland1990) Dutch identity, which was initially formulated for locally independent binary response models by Holland (Reference Holland1990) and extended to a locally dependent response model by Ip (Reference Ip2002) and a polytomous response model by Hessen (Reference Hessen2012). The following theorem gives an extension of the Dutch identity for response models that are of the exponential family form of Eq. (7). Its proof is in the Appendix.

Theorem 2

(The Dutch identity) Suppose that $p(x,t)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x},\,\mathbf {t})$$\end{document} is of the form

$\begin{array}{c}p(x,t)={\int }_{{\Omega }_{\zeta }}p(x,t\mid \zeta )f\left(\zeta \right)\text{d}\zeta ,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\,\mathbf {t}) = \int _{\Omega _\zeta } p(\mathbf {x},\,\mathbf {t} \mid \varvec{\zeta })\, f(\varvec{\zeta }) \text { d}\varvec{\zeta }, \end{aligned}$$\end{document}

where ${\Omega }_{\zeta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\varvec{\zeta }}$$\end{document} is the support of $\zeta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\zeta }$$\end{document} , and $p(x,t\mid \zeta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x},\,\mathbf {t} \mid \varvec{\zeta })$$\end{document} is of the form of Eq. (7). Then for any vector $y\in {\Omega }_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {y} \in \Omega _{\mathbf {x}}$$\end{document} and $w\in {\Omega }_t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {w} \in \Omega _{\mathbf {t}}$$\end{document} , where ${\Omega }_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\mathbf {x}}$$\end{document} and ${\Omega }_t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\mathbf {t}}$$\end{document} denote the support of $x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {x}$$\end{document} and $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {t}$$\end{document} , respectively, we have

$\begin{array}{c}\frac{p(x,t)}{p(y,w)}=\prod _{i=1}^p\frac{{\text{b}}_i(x_i,t_i)}{{\text{b}}_i(y_i,w_i)}E\left(\left(\prod _{i=1}^p,exp,\left({\left(s_i,(x_i,t_i),-,s_i,(y_i,w_i)\right)}^T,\zeta \right),\right),,S,=,\sum _i,s_i,(y_i,w_i)\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{p(\mathbf {x},\,\mathbf {t})}{p(\mathbf {y},\,\mathbf {w})} = \prod _{i=1}^p \frac{\text {b}_i(x_i,\,t_i)}{\text {b}_i(y_i,\,w_i)} \mathbb {E}\left( \left. \prod _{i=1}^p \exp \left( \left[ \mathbf {s}_i(x_i,\,t_i) - \mathbf {s}_i(y_i,\,w_i)\right] ^\mathsf{T} \varvec{\zeta }\right) \,\right| \, \mathbf {S} = \sum _i\mathbf {s}_i(y_i,\,w_i)\right) , \end{aligned}$$\end{document}

where the expectation is an integral with respect to the posterior density $f(\theta \mid S={\sum }_i{s}_i(y_i,w_i))$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta \mid \mathbf {S} = \sum _i\mathbf {s}_i(y_i,\,w_i))$$\end{document} .

It is easy to verify that whenever $f\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\varvec{\zeta })$$\end{document} is of the form of $g\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\varvec{\zeta })$$\end{document} in Eq. (9), the identity in Theorem 2 reduces to

$\begin{array}{c}\frac{p(x,t)}{p(y,w)}=\prod _{i=1}^p\frac{{\text{b}}_i(x_i,t_i)}{{\text{b}}_i(y_i,w_i)}\times \frac{E\left({\prod }_{i=1}^p,exp,\left(s_i,{(x_i,t_i)}^T,\zeta \right)\right)}{E\left({\prod }_{i=1}^p,exp,\left(s_i,{(y_i,w_i)}^T,\zeta \right)\right)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{p(\mathbf {x},\,\mathbf {t})}{p(\mathbf {y},\,\mathbf {w})} = \prod _{i=1}^p \frac{\text {b}_i(x_i,\,t_i) }{\text {b}_i(y_i,\,w_i)} \times \frac{\mathbb {E}\left( \prod _{i=1}^p\exp \left( \mathbf {s}_i(x_i,\,t_i)^\mathsf{T} \varvec{\zeta }\right) \right) }{\mathbb {E}\left( \prod _{i=1}^p\exp \left( \mathbf {s}_i(y_i, w_i)^\mathsf{T} \varvec{\zeta }\right) \right) }, \end{aligned}$$\end{document}

which was to be expected from Theorem 1. Observe that in this case the expectations imply integrating with respect to the kernel density $k\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(\varvec{\zeta })$$\end{document} that is used to define $g\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\varvec{\zeta })$$\end{document} in Eq. (9).

Both Theorems 1 and 2 characterize the manifest distribution in terms of a moment generating function, and for their practical application it is important to find a convenient form for this moment generating function. The Dutch identity, for example, has provided a general analytic solution for the (extended) Rasch model (Cressie & Holland, Reference Cressie and Holland1983; Tjur, Reference Tjur1982), but to come to an analytic expression for other latent trait models an assumption has to be made about the posterior distribution of the latent variable. In a similar way, we have to choose a kernel $k\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(\varvec{\zeta })$$\end{document} for the practical application of Theorem 1. The following corollary shows how a multivariate normal kernel distribution $k\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(\varvec{\zeta })$$\end{document} can be used to express the manifest distribution in a simple analytic form.

Corollary 1

If $p(x,t\mid \zeta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x},\,\mathbf {t}\mid \varvec{\zeta })$$\end{document} is of the form in Eq. (7), and the latent variable distribution $g\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\varvec{\zeta })$$\end{document} is of the form in Eq. (9) with a multivariate normal kernel $k\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(\varvec{\zeta })$$\end{document} having a mean vector $m$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {m}$$\end{document} and covariance matrix $V$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {V}$$\end{document} , then (i)

(11) $\begin{array}{c}p(x,t)=\frac{1}{\text{Z}}exp\left(\sum _{i=1}^p,ln,{\text{b}}_i,+,{\left(\sum _{i=1}^p,s_i\right)}^T,m,+,\frac{1}{2},{\left(\sum _{i=1}^p,s_i\right)}^T,V,\left(\sum _{i=1}^p,s_i\right)\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\,\mathbf {t}) = \frac{1}{\text {Z}} \exp \left( \sum _{i=1}^p \ln \text {b}_i + \left[ \sum _{i=1}^p \mathbf {s}_i\right] ^\mathsf{T}\mathbf {m} + \frac{1}{2}\left[ \sum _{i=1}^p \mathbf {s}_i\right] ^\mathsf{T}\mathbf {V}\left[ \sum _{i=1}^p \mathbf {s}_i\right] \right) , \end{aligned}$$\end{document}

where $\text{Z}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Z}$$\end{document} is a normalizing constant, and (ii) the posterior distribution $g(\zeta \mid s_1,\cdots ,s_p)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\varvec{\zeta } \mid \mathbf {s}_1,\,\dots ,\,\mathbf {s}_p)$$\end{document} is multivariate normal with mean vector $m+V\left({\sum }_{i=1}^p,s_i\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {m} + \mathbf {V}\left[ \sum _{i=1}^p\mathbf {s}_i\right] $$\end{document} and covariance matrix $V$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {V}$$\end{document} .

We will omit the proof of Corollary 1, which requires one to insert in Theorem 1 the moment generating function of the multivariate normal distribution, i.e.,

$\begin{array}{c}E\left(exp,\left(r^T,\zeta \right)\right)=exp\left(r^T,m,+,\frac{1}{2},r^T,V,r\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}\left( \exp \left( \mathbf {r}^\mathsf{T}\varvec{\zeta }\right) \right) = \exp \left( \mathbf {r}^\mathsf{T}\mathbf {m} + \frac{1}{2}\mathbf {r}^\mathsf{T}\mathbf {V}\mathbf {r}\right) , \end{aligned}$$\end{document}

using $r={\sum }_i{s}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {r} = \sum _i\mathbf {s}_i$$\end{document} as the interpolating parameter vector. Since we cannot make use of this moment generating function for the curved exponential family models in Eq. (8), the normal kernel does not result in the same simple form for these models. We can, however, make use of the following identity

$\begin{array}{c}E\left(exp,\left(r_1,\theta ,-,r_2,\frac{1}{2},{\theta }^2\right)\right)=\sqrt{\frac{v^2}{r_2{v}^2+1}}exp\left(\frac{r_1^2{v}^2+2r_{1}m-r_2{m}^2}{2r_2{v}^2+2}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbb {E}\left( \exp \left( r_{1}\theta - r_{2}\frac{1}{2}\theta ^2\right) \right) = \sqrt{\frac{v^2}{r_{2}\,v^2+1}}\, \exp \left( \frac{r_{1}^2\,v^2 + 2r_{1}\,m - r_{2}m^2}{2r_{2}\,v^2 + 2}\right) , \end{aligned}$$\end{document}

where the expectation is taken with respect to a normal distribution with mean m and variance $v^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v^2$$\end{document} . Inserting this identity into the expression of the manifest distribution in Eq. (10) with $r_1={\sum }_i{s}_{1i}=s_{1+}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_1=\sum _is_{1i} = s_{1+}$$\end{document} and $r_2={\sum }_i{s}_{2i}=s_{2+}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_2=\sum _is_{2i}=s_{2+}$$\end{document} , we end up with the following expression

(12) $\begin{array}{c}p(x,t)=\frac{1}{\text{Z}}\prod _{i=1}^p{\text{b}}_i\sqrt{\frac{v^2}{s_{2+}{v}^2+1}}exp\left(\frac{s_{1+}^2{v}^2+2s_{1+}m-s_{2+}{m}^2}{2s_{2+}{v}^2+2}\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\,\mathbf {t}) = \frac{1}{\text {Z}} \prod _{i=1}^p\text {b}_i\,\sqrt{\frac{v^2}{s_{2+}\,v^2+1}}\, \exp \left( \frac{s_{1+}^2\,v^2 + 2s_{1+}\,m - s_{2+}m^2}{2s_{2+}\,v^2 + 2}\right) . \end{aligned}$$\end{document}

This is a relatively simple analytic expression when we set m to zero. We will use Corollary 1 and Eq. (12) to characterize the manifest probability distributions of the three latent trait models.

Corollary 1 mirrors the results for assuming posterior normality of the latent trait in combination with the Dutch identity as evidenced in Corollary 1 of Holland (Reference Holland1990), Corollary 1 of Ip (Reference Ip2002), and Theorem 2 of Hessen (Reference Hessen2012), see also the log-multiplicative association models of Anderson and Vermunt (Reference Anderson and Vermunt2000), and Anderson and Yu (Reference Anderson and Yu2007), and the fused latent and graphical IRT model of Chen, Li, Liu, and Ying (Reference Chen, Li, Liu and Ying2018).

There appears to be a deeper connection between the prior assumption that leads to our Corollary 1 and the posterior assumption that leads to Corollary 1 of Holland (Reference Holland1990). What we know is that the latent variable distribution in Eq. (9) with a normal kernel is one way to ensure posterior normality of the latent variables for any latent trait model of the form of Eq. (7). To see why this is the case, consider the posterior with a prior distribution $f\left(\zeta \right)=g\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\varvec{\zeta }) = g(\varvec{\zeta })$$\end{document} of the form of Eq. (9),

$\begin{array}{cc}g(\zeta \mid s)& \propto \prod _{i=1}^p\frac{1}{{\text{Z}}_i\left(\zeta \right)}{e}^{\left(s_i,{(y_i,w_i)}^T,\zeta \right)}g\left(\zeta \right)\\ & \propto \prod _{i=1}^p\frac{1}{{\text{Z}}_i\left(\zeta \right)}{e}^{\left(s_i,{(y_i,w_i)}^T,\zeta \right)}\prod _{i=1}^p{\text{Z}}_i\left(\zeta \right)k\left(\zeta \right)\\ & =exp\left(\left(\sum _{i=1}^p,s_i,{(y_i,w_i)}^T\right),\zeta \right)k\left(\zeta \right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g(\varvec{\zeta } \mid \mathbf {s})&\propto \prod _{i=1}^p\frac{1}{\text {Z}_i(\varvec{\zeta })} \,e^{\left[ \mathbf {s}_i(y_i,\,w_i)^\mathsf{T} \varvec{\zeta }\right] } g(\varvec{\zeta }) \\&\propto \prod _{i=1}^p\frac{1}{\text {Z}_i(\varvec{\zeta })} \,e^{\left[ \mathbf {s}_i(y_i,\,w_i)^\mathsf{T} \varvec{\zeta }\right] } \prod _{i=1}^p\text {Z}_i(\varvec{\zeta })\, k(\varvec{\zeta })\\&= \exp \left( \left[ \sum _{i=1}^p\mathbf {s}_i(y_i,\,w_i)^\mathsf{T}\right] \varvec{\zeta }\right) \, k(\varvec{\zeta }), \end{aligned}$$\end{document}

which is proportional to a multivariate normal distribution if and only if the kernel $k\left(\zeta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(\varvec{\zeta })$$\end{document} is a multivariate normal distribution. The reverse need not be true since there exists at least one counterexample: Suppose that $s_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {s}_i$$\end{document} follows a multivariate normal distribution with a mean equal to $\zeta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\zeta }$$\end{document} (or some linear function of $\zeta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\zeta }$$\end{document} ), such that $p(s_i\mid \zeta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {s}_i \mid \varvec{\zeta })$$\end{document} can be written in the form of Eq. (7), then a normal prior distribution for $\zeta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\zeta }$$\end{document} , which is not of the form of Eq. (9), also ensures posterior normality of the latent variables. It may well be the case that this is the only exception to the general correspondence between the prior assumption that leads to our Corollary 1 and the posterior assumption that leads to Corollary 1 of Holland (Reference Holland1990).

4.1. The Hierarchical Model

The version of the HM that is considered here is of the form

$\begin{array}{c}p\left(x,,,,t,\mid ,\zeta ,=,{(\theta ,\eta )}^T\right)g\left(\zeta ,=,{(\theta ,\eta )}^T\right)=p(x\mid \theta )p(t\mid \eta )g(\theta ,\eta ),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p\left( \mathbf {x},\,\mathbf {t} \mid \varvec{\zeta } = (\theta ,\, \eta )^\mathsf{T}\right) \,g\left( \varvec{\zeta } = (\theta ,\, \eta )^\mathsf{T}\right) = p(\mathbf {x}\mid \theta )\, p(\mathbf {t} \mid \eta )\, g(\theta ,\,\eta ), \end{aligned}$$\end{document}

where the marginal $p(x\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x} \mid \theta )$$\end{document} is the two-parameter logistic model introduced in Eq. (1), the marginal $p(t\mid \eta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {t} \mid \eta )$$\end{document} is the log-normal distribution introduced in Eq. (2), and $g(\theta ,\eta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\theta ,\,\eta )$$\end{document} is of the form of Eq. (9) using a bivariate normal kernel distribution $k(\theta ,\eta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(\theta ,\,\eta )$$\end{document} for ability $\theta$ \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} and speed $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} , using a mean vector $m=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {m} = \mathbf {0}$$\end{document} and covariance matrix

$\begin{array}{c}V=\left(\begin{array}{cc}1& \rho v_{\eta }\\ \rho v_{\eta }& v_{\eta }^2\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbf {V} = \begin{pmatrix} 1 &{} \rho \, v_\eta \\ \rho \, v_\eta &{} v_\eta ^2 \end{pmatrix}, \end{aligned}$$\end{document}

where $\rho$ \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} denotes the a priori correlation between ability and speed, and $v_{\eta }^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_\eta ^2$$\end{document} the a priori variance of speed.

To come to an expression for the manifest probability distribution of this version of the HM we first rewrite the conditional distribution of accuracies and response times $p(x,t\mid \theta ,\eta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x},\,\mathbf {t} \mid \theta ,\,\eta )$$\end{document} to fit the form of Eq. (7). To this aim, we introduce the statistic

$\begin{array}{c}{s}_i=s_i(x_i,t_i)=\left(\begin{array}{c}{x}_i{\alpha }_i\\ -ln\left(t_i\right)\frac{1}{2}{\varphi }_i\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbf {s}_i = \mathbf {s}_i(x_i,\,t_i) = \begin{pmatrix} x_i\alpha _i \\ -\ln (t_i)\,\frac{1}{2}\phi _i \end{pmatrix}, \end{aligned}$$\end{document}

the base measures

$\begin{array}{c}{\text{b}}_i={\text{b}}_i(x_i,t_i)=exp\left(x_i,{\alpha }_i,{\beta }_i,-,ln,{\left(t_i\right)}^2,,\frac{1}{2},{\varphi }_i,-,ln,\left(t_i\right),,(1-{\varphi }_i{\xi }_i)\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {b}_i = \text {b}_i(x_i,\,t_i) = \exp \left( x_i\alpha _i\beta _i - \ln (t_i)^2\,\frac{1}{2}\phi _i - \ln (t_i) \, (1-\phi _i\xi _i)\right) , \end{aligned}$$\end{document}

and the normalizing constants

$\begin{array}{cc}{\text{Z}}_i(\theta ,\eta )& =\frac{\sqrt{2\pi }}{\sqrt{{\varphi }_i}}exp\left(\frac{1}{2},,{\varphi }_i,,{(\eta -{\xi }_i)}^2\right)\left(1,+,exp,\left({\alpha }_i,(\theta +{\beta }_i)\right)\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {Z}_i(\theta ,\,\eta )&= \frac{\sqrt{2\pi }}{\sqrt{\phi _i}}\,\exp \left( \frac{1}{2}\,\phi _i\,(\eta - \xi _i)^2\right) \,\left\{ 1 + \exp \left( \alpha _i(\theta +\beta _i)\right) \right\} . \end{aligned}$$\end{document}

Having expressed the conditional distribution of accuracies and response times in the form of Eq. (7), we can now apply Corollary 1 to obtain the manifest distribution. It is convenient to characterize this manifest distribution in terms of log-transformed response times $u_i=ln\left(t_i\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_i = \ln (t_i)$$\end{document} instead of response times. The manifest probability distribution of accuracies and log-transformed response times that results is equal to

$\begin{array}{cc}p(x,u)& =\frac{1}{\text{Z}}exp\left({\left(\begin{array}{c}x\\ u\end{array}\right)}^T,\left(\begin{array}{c}\alpha \odot \beta \\ \varphi \odot \xi \end{array}\right),+,\frac{1}{2},{\left(\begin{array}{c}x\\ u\end{array}\right)}^T,\left(\begin{array}{cc}\alpha {\alpha }^T& -\rho v_{\eta }\alpha {\varphi }^T\\ -\rho v_{\eta }\varphi {\alpha }^T& v_{\eta }^2\varphi {\varphi }^T\end{array}\right),\left(\begin{array}{c}x\\ u\end{array}\right)\right)\\ & \prod _{i=1}^{p}exp\left(-,\frac{1}{2},{\varphi }_i,,u_i^2\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\,\mathbf {u})&= \frac{1}{\text {Z}} \exp \left( \begin{pmatrix}\mathbf {x} \\ \mathbf {u}\end{pmatrix}^\mathsf{T} \begin{pmatrix}\varvec{\alpha }\odot \varvec{\beta } \\ \varvec{\phi } \odot \varvec{\xi } \end{pmatrix} + \frac{1}{2} \begin{pmatrix} \mathbf {x}\\ \mathbf {u} \end{pmatrix}^\mathsf{T} \begin{pmatrix} \varvec{\alpha } \, \varvec{\alpha }^\mathsf{T} &{} - \rho \, v_\eta \, \varvec{\alpha }\, \varvec{\phi }^\mathsf{T}\\ - \rho \, v_\eta \, \varvec{\phi }\, \varvec{\alpha }^\mathsf{T} &{} v_\eta ^2\,\varvec{\phi }\,\varvec{\phi }^\mathsf{T} \end{pmatrix} \begin{pmatrix} \mathbf {x}\\ \mathbf {u} \end{pmatrix}\right) \\&\quad \prod _{i=1}^p \exp \left( -\frac{1}{2}\phi _i \,u_i^2\right) , \end{aligned}$$\end{document}

where $\odot$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\odot $$\end{document} refers to Hadamard product. Observe that this manifest distribution is of the form of Eq. (13) for a random variable $q={(x^T,u^T)}^T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {q} = (\mathbf {x}^\mathsf{T}, \mathbf {u}^\mathsf{T})^\mathsf{T}$$\end{document} , with intercepts

$\begin{array}{c}\mu =\left(\begin{array}{c}\alpha \odot \beta \\ \varphi \odot \xi \end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{\mu } = \begin{pmatrix}\varvec{\alpha }\odot \varvec{\beta }\\ \varvec{\phi } \odot \varvec{\xi } \end{pmatrix}, \end{aligned}$$\end{document}

a rank two matrix of pairwise interactions

$\begin{array}{c}\Sigma =\frac{1}{2}\left(\begin{array}{cc}\alpha {\alpha }^T& -\rho v_{\eta }\alpha {\varphi }^T\\ -\rho v_{\eta }\varphi {\alpha }^T& v_{\eta }^2\varphi {\varphi }^T\end{array}\right)=\frac{1}{2}\left(\begin{array}{c}\alpha \\ v_{\eta }\varphi \end{array}\right)\left(\begin{array}{cc}1& -\rho \\ -\rho & 1\end{array}\right){\left(\begin{array}{c}\alpha \\ v_{\eta }\varphi \end{array}\right)}^T,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{\Sigma } = \frac{1}{2} \begin{pmatrix} \varvec{\alpha } \, \varvec{\alpha }^\mathsf{T} &{} - \rho \, v_\eta \, \varvec{\alpha }\, \varvec{\phi }^\mathsf{T}\\ - \rho \, v_\eta \, \varvec{\phi }\, \varvec{\alpha }^\mathsf{T} &{} v_\eta ^2\,\varvec{\phi }\,\varvec{\phi }^\mathsf{T} \end{pmatrix} = \frac{1}{2} \begin{pmatrix} \varvec{\alpha } \\ v_\eta \, \varvec{\phi } \end{pmatrix} \begin{pmatrix} 1 &{} -\rho \\ -\rho &{} 1 \end{pmatrix} \begin{pmatrix} \varvec{\alpha } \\ v_\eta \, \varvec{\phi } \end{pmatrix}^\mathsf{T}, \end{aligned}$$\end{document}

and a normal base measure

$\begin{array}{c}\omega \left(u\right)=\prod _{i=1}^p{\omega }_i\left(u_i\right)=\prod _{i=1}^{p}exp\left(-,\frac{1}{2},{\varphi }_i,,u_i^2\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \omega (\mathbf {u}) = \prod _{i=1}^p \omega _i(u_i) = \prod _{i=1}^p \exp \left( -\frac{1}{2}\phi _i \,u_i^2\right) , \end{aligned}$$\end{document}

with precisions ${\varphi }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _i$$\end{document} . In this model the associations, or pairwise interactions, between accuracies and log-response times, are of the opposite sign of the correlation $\rho$ \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 the two latent variables of the HM: Faster responses correspond to correct answers when $\rho >0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho > 0$$\end{document} ; slower responses correspond to correct answers when $\rho <0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho < 0$$\end{document} .

There are two important characteristics that can be observed from the manifest probability distribution of our version of the HM. Firstly, the base measure $\omega \left(u\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (\mathbf {u})$$\end{document} that is used here stipulates an a priori restriction on the variance of the speed parameter $v_{\eta }^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_\eta ^2$$\end{document} and the item-specific precisions ${\varphi }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _i$$\end{document} . To see this, observe that for this base measure the manifest distribution is a proper probability distribution—i.e., integrates to one—if and only if

$\begin{array}{c}{v}_{\eta }^2<\underset{i}{min}\left(\frac{1}{{\varphi }_i}\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} v_\eta ^2 < \min _i\left( \frac{1}{\phi _i}\right) . \end{aligned}$$\end{document}

Since the variance of the $log\left(t_i\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log (t_i)$$\end{document} is equal to

$\begin{array}{c}\text{Var}(log\left(t_i\right))=E\left(\text{Var}(log\left(t_i\right)\mid \eta )\right)+\text{Var}\left(E(log\left(t_i\right)\mid \eta )\right)=\frac{1}{{\varphi }_i}+v_{\eta }^2,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {Var}(\log (t_i)) = \mathbb {E}(\text {Var}(\log (t_i) \mid \eta )) +\text {Var}(\mathbb {E}(\log (t_i) \mid \eta )) = \frac{1}{\phi _i} + v_\eta ^2, \end{aligned}$$\end{document}

this restriction on $v_{\eta }^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_\eta ^2$$\end{document} and ${\varphi }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _i$$\end{document} implies that the speed variable can account for less than 50% of the total variance of the $log\left(t_i\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log (t_i)$$\end{document} .

A second important characteristic that can be observed from the manifest probability distribution of our version of the HM concerns the associations between accuracies and log-response times, which are encoded in the matrix of pairwise interactions $\Sigma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma }$$\end{document} . First, note that when the interaction between an accuracy $x_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i$$\end{document} and log-transformed response time $u_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_i$$\end{document} in the manifest distribution is equal to zero, these variables are independent conditional upon the remaining accuracies and log-transformed response times:

Thus, $x_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i$$\end{document} and $u_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_i$$\end{document} are conditionally independent whenever one of the following conditions apply: The correlation $\rho$ \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 speed and ability is zero; the discrimination ${\alpha }_i$ \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} of item i is zero; the precision ${\varphi }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _i$$\end{document} of item i is zero, and/or the a priori variance $v_{\eta }^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_\eta ^2$$\end{document} of the speed variable is zero. The only non-trivial condition that leads to conditional independence between accuracy and response time is the a priori independence of ability and speed in the HM. But this entails the extreme case in which all of the accuracies are independent of all of the response times, which is unlikely to occur in psychometric practice.

4.2. The Signed Residual Time Model

There are two versions of the SM that are considered here. The first version of the SM stipulates a distribution of accuracies and residual response times, and the second version of the SM stipulates a distribution of accuracies and pseudo-response times. We will first characterize the manifest probability distribution of accuracies and residual times and revert to pseudo-response times after that.

4.2.1. The Manifest Distribution of Accuracy and Residual Response Time

The SM was introduced in Eq. () and can be expressed in the exponential family form of Eq. (7) with statistics

$\begin{array}{c}{\text{s}}_i={\text{s}}_i(x_i,t_i)=(2x_i-1)({\text{d}}_i-t_i),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {s}_i = \text {s}_i(x_i,\,t_i) = (2x_i-1)(\text {d}_i - t_i), \end{aligned}$$\end{document}

base measures

$\begin{array}{c}{\text{b}}_i={\text{b}}_i(x_i,t_i)=exp\left((2x_i-1),({\text{d}}_i-t_i),{\beta }_i\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {b}_i = \text {b}_i(x_i,\,t_i) = \exp \left( (2x_i-1)(\text {d}_i-t_i)\beta _i\right) , \end{aligned}$$\end{document}

and normalizing constants

$\begin{array}{c}{\text{Z}}_i\left(\theta \right)=\frac{exp\left({\text{d}}_i,(\theta +{\beta }_i)\right)-exp\left(-,{\text{d}}_i,(\theta +{\beta }_i)\right)}{\theta +{\beta }_i}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {Z}_i(\theta ) = \frac{\exp \left( \text {d}_i(\theta + \beta _i)\right) - \exp \left( -\text {d}_i(\theta +\beta _i)\right) }{\theta + \beta _i}. \end{aligned}$$\end{document}

Having expressed the conditional distribution of accuracies and residual times of the SM in the form of Eq. (7) we can now use Corollary 1 to obtain their manifest distribution. Assuming a normal kernel with mean $m=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m = 0$$\end{document} and variance $v^2=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v^2 = 1$$\end{document} , Corollary 1 leads us to the following manifest distribution,

$\begin{array}{cc}p(x,t)=& \frac{1}{\text{Z}}exp\left({\left((2x-1_p),\odot ,(d-t)\right)}^T,\beta ,+,\frac{1}{2},{\left(\left(2,x,-,1_p\right),\odot ,\left(d,-,t\right)\right)}^T,\left(\left(2,x,-,1_p\right)\right)\right)\\ & \left(\left(\odot ,\left(d,-,t\right),)\right)\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\, \mathbf {t})= & {} \frac{1}{\text {Z}}\exp \left( \left[ (2\mathbf {x}-\mathbf {1}_p) \odot (\mathbf {d} - \mathbf {t})\right] ^\mathsf{T} \varvec{\beta } + \frac{1}{2}\left( \left( 2\mathbf {x} - \mathbf {1}_p\right) \odot \left( \mathbf {d} - \mathbf {t}\right) \right) ^\mathsf{T}\left( \left( 2\mathbf {x} - \mathbf {1}_p\right) \right. \right. \\&\quad \left. \left. \odot \left( \mathbf {d} - \mathbf {t}\right) \big )\right. \phantom {\left( \left[ (2\mathbf {x}-\mathbf {1}_p) \odot (\mathbf {d} - \mathbf {t})\right] ^\mathsf{T} \varvec{\beta } + \frac{1}{2}\left( \left( 2\mathbf {x} - \mathbf {1}_p\right) \odot \left( \mathbf {d} - \mathbf {t}\right) \right) ^\mathsf{T}\left( \left( 2\mathbf {x} - \mathbf {1}_p\right) \right. \right. }\right) , \end{aligned}$$\end{document}

where $1_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {1}_p$$\end{document} is the unit vector of length p. One way to write this distribution more succinctly is to express it in terms of the random variables $y_i=(2x_i-1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_i = (2x_i-1)$$\end{document} and residual times $r_i={\text{d}}_i-t_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_i = \text {d}_i-t_i$$\end{document} , which gives

(15) $\begin{array}{c}p(y,r)=\frac{1}{\text{Z}}exp\left({\left(y,\odot ,r\right)}^T,\beta ,+,\frac{1}{2},{\left(y,\odot ,r\right)}^T,\left(y,\odot ,r\right)\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {y},\,\mathbf {r}) = \frac{1}{\text {Z}}\exp \left( \left( \mathbf {y}\odot \mathbf {r}\right) ^\mathsf{T} \varvec{\beta } + \frac{1}{2}\left( \mathbf {y}\odot \mathbf {r}\right) ^\mathsf{T}\left( \mathbf {y}\odot \mathbf {r}\right) \right) . \end{aligned}$$\end{document}

This is of the form of the manifest distribution in Eq. (13) for the random variable $q=y\odot r$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {q} = \mathbf {y}\odot \mathbf {r}$$\end{document} , with intercepts $\mu =\beta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\mu } = \varvec{\beta }$$\end{document} , a rank one matrix of pairwise interactions $\Sigma =\frac{1}{2}{1}_p{1}_p^T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma } = \frac{1}{2}\mathbf {1}_p\mathbf {1}_p^\mathsf{T}$$\end{document} , and a uniform base measure $\omega \left(q\right)=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (\mathbf {q}) = 1$$\end{document} . In this model, larger residual times (faster responses) are associated with an increased probability that a person responds accurately to easy items ( ${\beta }_i>0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i > 0$$\end{document} ) and a decreased probability to respond accurately to difficult items ( ${\beta }_i<0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _i < 0$$\end{document} ). The association between the response accuracies of different items is positive and increases with increasing residual response times (faster responses).

There are two important characteristics that can be observed from the manifest probability distribution of the SM. One characteristic appears when we view the manifest distribution as a distribution for the random variables $y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {y}$$\end{document} , as this closely resembles a graphical model from physics that is known as the Ising model (Lenz, Reference Lenz1920; Ising, Reference Ising1925). The Ising model is characterized by the following probability distribution over realizations of $y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {y}$$\end{document}

$\begin{array}{c}p\left(y\right)=\frac{1}{\text{Z}}exp\left(y^T,\mu ,+,y^T,\Sigma ,,y\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {y}) = \frac{1}{\text {Z}}\exp \left( \mathbf {y}^\mathsf{T}\varvec{\mu } + \mathbf {y} ^\mathsf{T}\varvec{\Sigma } \,\mathbf {y}\right) , \end{aligned}$$\end{document}

where $\mu$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\mu }$$\end{document} denotes a vector of p intercepts ${\mu }_i$ \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} , and $\Sigma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma }$$\end{document} is a symmetric $p\times p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times p$$\end{document} matrix of pairwise interactions ${\sigma }_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{ij}$$\end{document} , similar to our Eq. (13). However, where the intercepts and interactions are fixed effects in the Ising network model, they are random effects here, with $\mu =r\odot \beta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\mu } = \mathbf {r} \odot \varvec{\beta }$$\end{document} and $\Sigma =\frac{1}{2}r{r}^T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma } = \frac{1}{2}\mathbf {r}\mathbf {r}^\mathsf{T}$$\end{document} . This view of the SM thus provides a novel way for modeling the intercepts and matrix of associations in the Ising model.

A second important characteristic that can be observed from the manifest probability distribution of the SM concerns the association between accuracies and residual response times. Whereas we have found that for the manifest distribution of the HM that accuracy can be conditionally independent of response time in a non-trivial manner, at least in theory, this is not the case with the SM. This can be observed, for example, from the conditional distribution of accuracy and residual response time for an item i given the accuracies and residual times of the remaining items

$\begin{array}{c}p\left(y_i,,,,r_i,\mid ,y^{\left(i\right)},,,,t^{\left(i\right)}\right)=\frac{exp\left(y_i,r_i,\left({\beta }_i,+,{\sum }_{j\ne i},y_j,r_j\right)\right)}{\frac{exp\left({\beta }_i,+,{\sum }_{j\ne i},y_j,r_j\right)-exp\left(-,{\beta }_i,-,{\sum }_{j\ne i},y_j,r_j\right)}{{\beta }_i+{\sum }_{j\ne i}{y}_j{r}_j}},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p\left( y_i,\,r_i \mid \mathbf {y}^{(i)},\,\mathbf {t}^{(i)}\right) = \frac{\exp \left( y_ir_i\left[ \beta _i + \sum _{j \ne i} y_jr_j\right] \right) }{\frac{\exp \left( \beta _i + \sum _{j \ne i} y_jr_j\right) - \exp \left( -\beta _i - \sum _{j \ne i} y_jr_j\right) }{\beta _i + \sum _{j \ne i} y_jr_j}}, \end{aligned}$$\end{document}

from which it is clear that there are no 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} (or $y^{\left(i\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {y}^{(i)}$$\end{document} and $r^{\left(i\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {r}^{(i)}$$\end{document} ) that render $y_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_i$$\end{document} and $r_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_i$$\end{document} (conditionally) independent.

4.2.2. The Manifest Distribution of Accuracy and Pseudo-response Time

An alternative formulation of the SM is in terms of accuracies and pseudo-response times, which is of the form

$\begin{array}{c}p(x,t^{\ast }\mid \theta )=p(x\mid \theta )p(t^{\ast }\mid \theta ),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p(\mathbf {x},\,\mathbf {t}^*\mid \theta ) = p(\mathbf {x}\mid \theta )\,p(\mathbf {t}^*\mid \theta ), \end{aligned}$$\end{document}

where the marginal $p(x\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x} \mid \theta )$$\end{document} is the two-parameter logistic model in Eq. (1), and the marginal $p(t^{\ast }\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {t}^*\mid \theta )$$\end{document} is given in Eq. (5).

To characterize the manifest distribution of accuracies and pseudo-response times for this version of the SM we can take two approaches. Firstly, we may express the conditional distribution of accuracies and pseudo-response times $p(x,t^{\ast }\mid \theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\mathbf {x},\,\mathbf {t}^*\mid \theta )$$\end{document} in the exponential family form of Eq. (7) and then apply Corollary 1 using a normal kernel distribution with mean $m=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m =0$$\end{document} and variance $v^2=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v^2 = 1$$\end{document} to this conditional distribution. Alternatively, we may rewrite the sufficient statistic for residual response times in the manifest distribution in Eq. (15) through the relation

$\begin{array}{c}{\text{s}}_i^{\ast }=x_i{\text{d}}_i-t_i^{\ast }=\left(\begin{array}{cc}{\text{d}}_i-t_i& \text{if}{x}_i=1\\ -({\text{d}}_i-t_i)& \text{if}{x}_i=0\end{array}\right)⟺(2x_i-1)({\text{d}}_i-t_i)=y_i{r}_i={\text{s}}_i.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {s}_i^*= x_i\text {d}_i - t_i^*= {\left\{ \begin{array}{ll} \text {d}_i - t_i &{} \text { if }x_i = 1\\ - (\text {d}_i-t_i) &{} \text { if }x_i = 0 \end{array}\right. } \Longleftrightarrow (2x_i-1)(\text {d}_i - t_i) = y_ir_i = \text {s}_i. \end{aligned}$$\end{document}

Both approaches lead to the following manifest distribution of accuracies and pseudo-response times

$\begin{array}{c}p(x,t^{\ast })=\frac{1}{\text{Z}}exp\left({\left(\begin{array}{c}x\\ t^{\ast }\end{array}\right)}^T,\left(\begin{array}{c}d\odot \beta \\ -1_p\odot \beta \end{array}\right),+\frac{1}{2},{\left(\begin{array}{c}x\\ t^{\ast }\end{array}\right)}^T,\left(\begin{array}{c}d\\ -1_p\end{array}\right),{\left(\begin{array}{c}d\\ -1_p\end{array}\right)}^T,\left(\begin{array}{c}x\\ t^{\ast }\end{array}\right)\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {p(\mathbf {x},\,\mathbf {t}^*)} = {\frac{1}{\text {Z}} \exp }\left( \begin{pmatrix} {\mathbf {x}}\\ {\mathbf {t}^*} \end{pmatrix}^{\mathsf{T}}\begin{pmatrix}{\mathbf {d}\odot \varvec{\beta }}\\ {-\mathbf {1}_p \odot \varvec{\beta }} \end{pmatrix} {+ \frac{1}{2}}\begin{pmatrix} {\mathbf {x}}\\ {\mathbf {t}^*} \end{pmatrix}^{\mathsf{T}} \begin{pmatrix} \mathbf {d}\\ -\mathbf {1}_p \end{pmatrix} \begin{pmatrix} \mathbf {d}\\ -\mathbf {1}_p \end{pmatrix}^\mathsf{T} \begin{pmatrix} {\mathbf {x}}\\ {\mathbf {t}^*} \end{pmatrix} \right) , \end{aligned}$$\end{document}

which is of the form of the manifest distribution in Eq. (13) for the random variable $q={(x^T,t^{\ast T})}^T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {q} = (\mathbf {x}^\mathsf{T},\,\mathbf {t}^{*\mathsf T})^\mathsf{T}$$\end{document} , with intercepts

$\begin{array}{c}\mu =\left(\begin{array}{c}d\odot \beta \\ -1_p\odot \beta \end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{\mu } = \begin{pmatrix} \mathbf {d}\odot \varvec{\beta }\\ -\mathbf {1}_p\odot \varvec{\beta } \end{pmatrix}, \end{aligned}$$\end{document}

a rank one matrix of pairwise interactions

$\begin{array}{c}\Sigma =\frac{1}{2}\left(\begin{array}{c}d\\ -1_p\end{array}\right){\left(\begin{array}{c}d\\ -1_p\end{array}\right)}^T,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{\Sigma } = \frac{1}{2}\begin{pmatrix} \mathbf {d}\\ -\mathbf {1}_p \end{pmatrix} \begin{pmatrix} \mathbf {d}\\ -\mathbf {1}_p \end{pmatrix}^\mathsf{T}, \end{aligned}$$\end{document}

and a uniform base measure $\omega \left(q\right)\propto 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (\mathbf {q}) \propto 1$$\end{document} . Observe that for this model both the associations between accuracies and the associations between pseudo-response times are positive, yet the associations between accuracies and pseudo-response times are negative: Correct responses are associated with smaller pseudo-time values (faster response times); incorrect responses are associated with larger pseudo-time values (slower response times).

The manifest distribution of accuracies and pseudo-response times of the SM is of the same form as the manifest distribution of accuracies and the log-transformed response times of the HM, and thus they share certain characteristics. For example, both models share the following Markov property: When the association between an accuracy and a response time for an item i is equal to zero, then this implies that these two variables are independent conditional upon the remaining accuracies and response times,

However, it is immediately clear that this association is never zero in practice, since $d_i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i = 0$$\end{document} would imply a zero second time limit for item i.

4.3. The Drift Diffusion Model

The version of the DM that is considered here—which was introduced in Eq. (6)—can be expressed in the form of Eq. (8), with statistics

$\begin{array}{c}{s}_i(x_i,t_i)=\left(\begin{array}{c}{\alpha }_i{x}_i-{\beta }_i{t}_i\\ t_i\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbf {s}_i(x_i,\,t_i) = \begin{pmatrix} \alpha _i\,x_i - \beta _i\,t_i\\ t_i \end{pmatrix}, \end{aligned}$$\end{document}

base measures

$\begin{array}{c}{\text{b}}_i={\text{b}}_i(x_i,t_i)=exp\left({\alpha }_i,,{\beta }_i,,x_i,-,\frac{1}{2},,{\beta }_i^2,,t_i\right)\sum _{n=1}^{\infty }sin\left(\frac{1}{2},\pi ,n\right)exp\left(-,\frac{1}{2{\alpha }_i^2},{\pi }^2,n^2,\left(t_i,-,T_{\left(er\right)}\right)\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {b}_i = \text {b}_i(x_i,\,t_i) = \exp \left( \alpha _i\,\beta _i\,x_i - \frac{1}{2}\,\beta _i^2\,t_i\right) \sum _{n = 1}^{\infty } \sin \left( \frac{1}{2}\pi n\right) \exp \left( -\frac{1}{2\alpha _i^2}\pi ^2n^2\left( t_i-T_{(er)}\right) \right) , \end{aligned}$$\end{document}

and normalizing constants

$\begin{array}{c}{\text{Z}}_i\left(\theta \right)=\frac{{\alpha }_i^2}{\pi }exp\left(\frac{1}{2},{\alpha }_i,(\theta +{\beta }_i),-,\frac{1}{2},T_{\left(er\right)},{(\theta +{\beta }_i)}^2\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {Z}_i(\theta ) = \frac{\alpha _i^2 }{ \pi } \exp \left( \frac{1}{2}\alpha _i(\theta +\beta _i)-\frac{1}{2}T_{(er)}(\theta + \beta _i)^2\right) . \end{aligned}$$\end{document}

Having expressed the DM in the form of Eq. (8), we may now use Eq. (12) to express its manifest distribution. Assuming a latent trait distribution $g\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(\theta )$$\end{document} of the form of Eq. (9) with a normal kernel with a mean $m=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m = 0$$\end{document} and variance $v^2=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v^2 = 1$$\end{document} , Eq. (12) leads us to the following manifest distribution

$\begin{array}{c}p(x,t)=\frac{1}{\text{Z}}exp\left({\left(\begin{array}{c}x\\ t\end{array}\right)}^T,\left(\begin{array}{c}\alpha \odot \beta \\ -\frac{1}{2}\beta \odot \beta \end{array}\right),+\frac{1}{2t_{+}+2},{\left(\begin{array}{c}x\\ t\end{array}\right)}^T,\left(\begin{array}{c}\alpha \\ -\beta \end{array}\right),{\left(\begin{array}{c}\alpha \\ -\beta \end{array}\right)}^T,\left(\begin{array}{c}x\\ t\end{array}\right)\right)\omega \left(t\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {p(\mathbf {x},\,\mathbf {t}) = \frac{1}{\text {Z}} \exp }\left( \begin{pmatrix}{\mathbf {x}}\\ {\mathbf {t}}\end{pmatrix}^{\mathsf{T}} \begin{pmatrix} {\varvec{\alpha } \odot \varvec{\beta }}\\ {-\frac{1}{2}\varvec{\beta } \odot \varvec{\beta }} \end{pmatrix} {+ \frac{1}{2t_+ + 2} }\begin{pmatrix}{\mathbf {x}}\\ {\mathbf {t}}\end{pmatrix}^{\mathsf{T}} \begin{pmatrix}{\varvec{\alpha }}\\ {-\varvec{\beta }} \end{pmatrix} \begin{pmatrix}{\varvec{\alpha }}\\ {-\varvec{\beta }}\end{pmatrix}^{\mathsf{T}} \begin{pmatrix}{\mathbf {x}}\\ {\mathbf {t}}\end{pmatrix}\right) \, {\omega (\mathbf {t})} \end{aligned}$$\end{document}

where $t_{+}={\sum }_i{t}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_+ = \sum _it_i$$\end{document} , and $\omega \left(t\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (\mathbf {t})$$\end{document} is a base measure,