# Characterizing the Manifest Probability Distributions of Three Latent Trait Models for Accuracy and Response Time

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## Abstract

In this paper we study the statistical relations between three latent trait models for accuracies and response times: the hierarchical model (HM) of van der Linden (Psychometrika 72(3):287–308, 2007), the signed residual time model (SM) proposed by Maris and van der Maas (Psychometrika 77(4):615–633, 2012), and the drift diffusion model (DM) as proposed by Tuerlinckx and De Boeck (Psychometrika 70(4):629–650, 2005). One important distinction between these models is that the HM and the DM either assume or imply that accuracies and response times are independent given the latent trait variables, while the SM does not. In this paper we investigate the impact of this conditional independence property—or a lack thereof—on the manifest probability distribution for accuracies and response times. We will find that the manifest distributions of the latent trait models share several important features, such as the dependency between accuracy and response time, but we also find important differences, such as in what function of response time is being modeled. Our method for characterizing the manifest probability distributions is related to the Dutch identity (Holland in Psychometrika 55(6):5–18, 1990).

## Keywords

drift diffusion model Dutch identity graphical model hierarchical model item response theory response times signed residual time model conditional independence## 1 Introduction

In this paper we wish to study the statistical relations between three latent trait models for accuracies and response times: the hierarchical model (HM) of van der Linden (2007), the signed residual time model (SM) of Maris and van der Maas (2012), and the drift diffusion model (DM) proposed by Tuerlinckx and De Boeck (2005). These models come from different backgrounds and differ in many respects. A key distinction between the three latent trait models is in the relation they stipulate between accuracy and response time after conditioning on the latent variables. Whereas responses are independent of response times after conditioning on the latent variables in both the DM and the HM, this is not the case for the SM. The conditional independence property has received much attention in the psychometric literature on response time modeling (e.g., Bolsinova, De Boeck, & Tijmstra, 2010; Bolsinova & Maris, 2016; Bolsinova & Tijmstra, 2016; Bolsinova, Tijmstra, & Molenaar, 2017; van der Linden & Glas, 2017).

The way that these and other latent trait models for accuracy and response time are related has been the topic of several publications (e.g., Molenaar, Tuerlinckx, & van der Maas, 2017a; 2015b; van Rijn & Ali, 2015a), but what sets our approach apart from earlier comparison attempts is that we do not work with their latent trait formulations. A serious complication with comparing latent trait models is that they are usually not defined on a common metric or space. As a result, it is unclear what the conditional independence property, for example, says about the distribution of observables, or how one can compare latent variables and their impact on observables across models. The manifest distribution —i.e., the distribution of observables after having integrated out the latent variables—does not suffer from these complications and is easily compared. We therefore work with manifest distributions in this paper.

The comparison of manifest probability distributions crucially depends on having their analytic expressions available to us, but unfortunately, this is not the case for the latent trait models that we study here. To overcome this complication, we reverse-engineer an approach that was originally used by Kac (1968) to find a latent variable expression of a graphical model known now as the Ising model (Ising, 1925). The work of Kac has revealed a broad equivalence between psychometric item response models and network models from statistical physics (Epskamp, Maris, Waldorp, & Borsboom, 2018; Marsman et al., 2018; Marsman, Tanis, Bechger, & Waldorp, 2019). Here we use it to characterize the manifest distribution of latent trait models that are in the exponential family. Another way to express the manifest distributions of latent trait models is the Dutch identity (Holland, 1990). Our approach and the Dutch identity are, of course, very much related, and we will study this relation in detail.

The remainder of this paper is structured as follows: In the next section, we formally introduce the three latent trait models. We will focus on versions of the latent trait models that either use or imply the two-parameter logistic model for the marginal distribution of response accuracies—i.e., the conditional distribution of accuracies given the latent variables after having integrated out the response times. After having introduced the three latent trait models we introduce our approach for characterizing their manifest probability distributions. Here we will also study the relation between our approach and the Dutch identity. We then characterize and analyze the manifest probability distributions that are implied by the three latent trait models. Our paper ends with a discussion of these results.

## 2 Models

*p*response accuracies—\(x_i \in \{0,\,1\}\)—and use \(\mathbf {t}\) to denote a vector of

*p*response times—\(t_i \in \mathbb {R}^+\) for the HM and DM and \(t_i \in (0,\,\text {d}_i)\) for the SM, see below. The two latent variables ability and speed will be denoted with \(\theta \) and \(\eta \), respectively. Finally, we will use “marginal distribution” to refer to the conditional distribution of one type of observable, e.g.,

### 2.1 The Hierarchical Model

To conclude the HM we specify a distribution for ability \(\theta \) and speed \(\eta \). 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.^{1}

### 2.2 The Signed Residual Time Model

*i*. This scoring rule encourages persons to work fast but punishes guessing: Residual time \(\text {d}_i - t_i\) is gained when the response is correct, but is lost when the response is incorrect. Van Rijn and Ali (2017b) 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.

*pseudo*-response times \(\mathbf {t}^*\).

*Pseudo*-response times are obtained from response times through the transformation

*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.,

### 2.3 The Drift Diffusion Model

The DM was introduced by Ratcliff
(1978) 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, 1970). The drift \(\mu \) of the diffusion process determines how fast information is accumulated, the volatility \(\sigma \) determines how noisy the accumulation process is, and the distance between the two boundaries \(\alpha \) 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 \(T_{(er)}\). A commonly used simplification of the DM assumes that the process is unbiased \(z = \frac{1}{2}\alpha \).

## 3 Characterizing Manifest Probabilities of Latent Trait Models

*item response theory (IRT)*model for accuracy and response times in an exponential family form:

### Definition 1

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 (1925) model by Kac (1968, see also Marsman et al., 2018; Epskamp, Maris, Waldorp, & Borsboom, 2018), but a similar construction can also be found in, for instance, Cressie and Holland (1983, Eq. A9) and McCullagh (1994). We can now state our first result.

### Theorem 1

### Theorem 2

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, 1983; Tjur, 1982), 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 \(k(\varvec{\zeta })\) for the practical application of Theorem 1. The following corollary shows how a multivariate normal kernel distribution \(k(\varvec{\zeta })\) can be used to express the manifest distribution in a simple analytic form.

### Corollary 1

*m*and variance \(v^2\). Inserting this identity into the expression of the manifest distribution in Eq. (10) with \(r_1=\sum _is_{1i} = s_{1+}\) and \(r_2=\sum _is_{2i}=s_{2+}\), we end up with the following expression

*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 (1990), Corollary 1 of Ip (2002), and Theorem 2 of Hessen (2012), see also the log-multiplicative association models of Anderson and Vermunt (2000), and Anderson and Yu (2007), and the fused latent and graphical IRT model of Chen, Li, Liu, and Ying (2018).

## 4 The Manifest Probabilities of the Three Latent Trait Models

*h*. What we shall see is that the three latent trait models fundamentally differ in what the random variable \(\mathbf {q}\) is, revealing key differences in the function of response time they are modeling, but also that accuracies and responses times are dependent in the manifest distribution of each latent trait model, except for fringe cases that are uncommon in practice. We will now consider each of the models in turn.

### 4.1 The Hierarchical Model

*a priori*correlation between ability and speed, and \(v_\eta ^2\) the

*a priori*variance of speed.

*a priori*restriction on the variance of the speed parameter \(v_\eta ^2\) and the item-specific precisions \(\phi _i\). 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

*i*is zero; the precision \(\phi _i\) of item

*i*is zero, and/or the

*a priori*variance \(v_\eta ^2\) 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

*p*. One way to write this distribution more succinctly is to express it in terms of the random variables \(y_i = (2x_i-1)\) and residual times \(r_i = \text {d}_i-t_i\), which gives

*p*intercepts \(\mu _i\), and \(\varvec{\Sigma }\) is a symmetric \(p \times p\) matrix of pairwise interactions \(\sigma _{ij}\), 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 \(\varvec{\mu } = \mathbf {r} \odot \varvec{\beta }\) and \(\varvec{\Sigma } = \frac{1}{2}\mathbf {r}\mathbf {r}^\mathsf{T}\). This view of the SM thus provides a novel way for modeling the intercepts and matrix of associations in the Ising model.

*i*given the accuracies and residual times of the remaining items

#### 4.2.2 The Manifest Distribution of Accuracy and *Pseudo*-response Time

*pseudo*-response times, which is of the form

*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 \(p(\mathbf {x},\,\mathbf {t}^*\mid \theta )\) in the exponential family form of Eq. (7) and then apply Corollary 1 using a normal kernel distribution with mean \(m =0\) and variance \(v^2 = 1\) 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

*pseudo*-response times

*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).

*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 \(d_i = 0\) would imply a zero second time limit for item

*i*.

### 4.3 The Drift Diffusion Model

There are two important characteristics that can be observed from the manifest probability distribution of our version of the DM. The first observation is that the association between both accuracies and response times is scaled by the total time \(t_+\) that is spent on the test. This implies smaller associations between accuracies and response times for pupils that take longer to complete the test, and larger associations for pupils that take less time to complete the test. In none of the three other manifest probability distributions that were considered here have we seen an influence of the total time that was spent on the test.

## 5 Discussion

The goal of this paper was the statistical comparison of three latent trait models for accuracy and response time: the hierarchical model (HM) of van der Linden (2007), the signed residual time model (SM) of Maris and van der Maas (2012), and the drift diffusion model (DM) as proposed by Tuerlinckx and De Boeck (2005). Our idea was to work with the manifest distributions of observables that were generated by these latent trait models, as they are more easily compared than their original latent trait formulations. To characterize these manifest distributions we have reverse-engineered an approach by Kac (1968), which inspired a new method for expressing manifest distributions. This method is summarized in our Theorem 1 and Corollary 1 and is related to the Dutch identity (Holland, 1990), which is our Theorem 2 for the response models considered in this paper. Our assumption of a normal kernel density for the latent trait parameters appeared to be closely related to the posterior normality assumption that is often used with the Dutch identity, but more importantly, it has allowed us to characterize the manifest distributions of observables analytically. So what did this formal exercise teach us about the three psychometric models?

The observation that accuracies and response times are dependent in the analyzed manifest distributions is a warm reminder of the fact that integrating over a common cause, or set of correlated common causes, will generate a dependency between observables that are conditionally independent. In fact, the statistical modeling of such manifest dependencies is what latent variable models are made for. Viewed in this way, it hardly seems relevant how the three latent trait models treat these dependencies locally, e.g., assuming conditional independence between observables or not, since these local properties have disappeared in the manifest distribution. Given that the manifest probabilities are all that we can ever learn from our observables, the conditional independence property may be a convenient tool to model dependencies at the latent trait level, but for the three response models it is hardly more than that.

A more sensible division of the three latent trait models appears to be the response time function that is being modeled, as it is here that we find major differences between the three response models. For example, the log-transformed response times are modeled in the HM, the residual response times or *pseudo*-response times are modeled in the SM, and in the DM the response times are modeled directly, although the latter does so in proportion to the total test time. This offers an interesting new view on response models that take response times into account, and one may wonder if there is a way to find out which function tells us the most about the unknown abilities. The manifest distributions in this paper offer one approach to address such questions.

^{2}we observed some interesting properties in their manifest distributions. In the manifest expression for the DM, for example, the associations between accuracies and response times are a function of the total time the pupil has spent on the test, an aspect that is not being modeled in any of the other manifest expressions. This property could be used to inform about the underlying strategies that pupils use, for example. The manifest expression of the SM, on the other hand, provides a new and interesting way to view an old model, the Ising model. The Ising model is an undirected graphical model that is characterized by the following distribution,

*p*-dimensional vector of \((0,\,1)\) or \((-1,\,1)\) variables \(x_i\), \(\varvec{\mu }\) a

*p*-dimensional vector of main effects, and \(\varvec{\Sigma }\) a \(p\times p\) symmetric matrix of pairwise associations between variables. Whereas the pairwise associations are fixed effects in the Ising model, the manifest distribution of the SM indicates one way to model these associations as a random effect.

One famous conjecture from Holland
(1990, p. 11) is that if there are large number of items on a test, and a smooth unidimensional IRT model (for accuracies) is used, the posterior distribution of the latent trait will be approximately normal. This conjecture has inspired several publications on the posterior normality of the latent trait in the context of IRT models for response accuracy (e.g., Chang and Stout, 1993; Chang, 1996; Zhang and Stout, 1997). An interesting conclusion that Holland
(1990) deduced from this conjecture, in combination with the assumption that the log-likelihoods of the *p* items can be approximated using a *p*-variate normal with a rank one covariance matrix, is that the log of the manifest distribution of accuracy is approximately of quadratic form consisting of *p* main effects and a \(p \times p\) matrix of associations that was of rank one. This enticed Holland
(1990) to add a second conjecture that only two parameters can be consistently estimated per item. This idea points to interesting avenues of future research, such as the asymptotic posterior normality of the latent trait in the context of IRT models for response accuracy and response times. If it is reasonable to approximate the posterior of the latent trait (or the log-likelihood function) with a normal distribution, then we can use this approximation in combination with Corollary 1 or Theorem 2 to investigate the complexity of models for response accuracy and response times, and how model complexity is impacted by the conditional independence property of the underlying latent trait model.

The latent variable distribution \(g(\varvec{\zeta })\) has allowed us to express the manifest probability distributions for a large class of latent trait models, but it also generated an unexpected parameter restriction in the manifest distribution of the HM, where we found that the variance of the speed variable \(v_\eta ^2\) needed to be smaller than the smallest log-normal variance \(\phi _i^{-1}\). This parameter restriction follows from omitting the normalizing constants \(\text {Z}_i(\varvec{\zeta })\) of the latent variable model in Eq. (7), which provides prior model structure. When a regular latent variable distribution is used—for example, a normal distribution on \(\eta \)—the model structure that is provided by the normalizing constants \(\text {Z}_i(\varvec{\zeta })\) is integrated instead. Marsman et al. (2018) studied a similar scaling issue of the posterior distribution that results from using the latent variable distribution \(g(\varvec{\zeta })\) in the context of multi-dimensional IRT (see also Marsman et al., 2017). The correspondence that we have found between our normal kernel assumption and the posterior normality assumption with the Dutch identity suggests that similar observations can be made for the prior and posterior of the latent variables in Corollary 1 of Holland (1990), Corollary 1 of Hessen (2012), and Theorem 1 in Ip (2002).

## Footnotes

- 1.
Alternatively, one may choose an item

*i*and constrain \(\beta _i\) and \(\xi _i\) to zero and constrain \(\alpha _i\) to one for this item. - 2.
Observe, however, that for both models the conditional distribution \(p(\mathbf {x} \mid \mathbf {t})\) is an MRF, because they share the Markov property that when the association between the accuracy of an item

*i*and an item*j*is equal to zero, these two variables are independent given the accuracy on the remaining items. Incidentally, for the manifest expressions of both the SM and the DM the associated conditional distribution is another instance of the Ising model.

## Notes

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