A sparse latent class model incorporating response times

Abstract

Diagnostic models (DM) have been widely used to classify respondents' latent attributes in cognitive and non‐cognitive assessments. The integration of response times (RTs) with DM presents additional evidence to understand respondents' problem‐solving behaviours. While recent research has explored using sparse latent class models (SLCM) to infer the latent structure of items based on item responses, the incorporation of RT data within these models remains underexplored. This study extends the SLCM framework to include RT, relaxing the conditional independence assumption between RT and latent attributes given individual speed. This adaptation provides a more flexible framework for jointly modelling RT and item responses. While the proposed model holds promise for applications in educational assessment, this study applied the model to the Fisher Temperament Inventory, yielding findings that provide a novel perspective on utilizing DM with RT in personality assessments. Additionally, a Gibbs sampling algorithm is proposed for parameter estimation. Results from Monte Carlo simulations demonstrate the algorithm's accuracy and efficiency.

1. INTRODUCTION

In recent years, researchers have become increasingly interested in cognitive diagnostic assessments, which can provide fine‐grained information about respondents' educational skills or psychological states. Specifically, sparse latent class model (SLCM; Chen et al., ²⁰²⁰), as an inclusive and comprehensive modelling approach, can fit many existing cognitive diagnostic models (CDM; Tatsuoka, ¹⁹⁹⁵) in either a confirmatory or an exploratory setting. By estimating the underlying item–attribute structure, exploratory SLCM aims to illuminate how each item measures the latent structure in an exploratory fashion (Chen et al., 2020; Culpepper, ²⁰¹⁹; Culpepper & Balamuta, ²⁰²¹; He et al., ²⁰²²).

In testing scenarios, SLCMs can be used to measure response accuracy (RA), and the latent attribute can be either defined as a concrete knowledge point or a more abstract latent construct. In addition to RA, response times (RTs) are often collected as collateral information to gain insights into respondent behaviour. As collecting RT data has become a standard practice in both high‐ and low‐stakes tests, RT is frequently used alongside RA in traditional item response theory (IRT) models to investigate cognitive processes underlying tests (De Boeck & Jeon, 2019). Although some efforts have been made to incorporate RTs within the CDM framework (Qiao & Jiao, 2021; Wang et al., ²⁰²⁰; Wang, Yang, et al., ²⁰¹⁸; Wang, Zhang, et al., ²⁰¹⁸; Zhan et al., ²⁰¹⁸), RT information has largely been overlooked in the development of exploratory CDMs.

In IRT models, the hierarchical RT model is widely used to capture the relationship between latent ability and latent speed (van der Linden, ²⁰⁰⁷). In this framework, RA and RT are modelled separately using different measurement models, while the speed and ability parameters are connected at a higher level through a correlational structure. This approach assumes conditional independence between RT and ability given speed, meaning that all information about ability present in the RT data are fully captured by the speed parameter. Similarly, the conditional independence between RA and speed given ability suggests that all information about speed contained in the RA data are fully accounted for by the ability parameter. The model aims to improve the precision of ability and speed estimation when the correlation between these two parameters is significantly different from zero.

In existing research integrating RA and RT in CDM (De La Torre & Douglas, 2004; Templin, ²⁰⁰⁴; Zhan et al., ²⁰¹⁸), a continuous general latent ability is often constructed within a logit function to account for the variability in binary attributes, allowing it to fit into hierarchical simple structure model discussed above. However, in practice, the correlation between speed and ability may be negligible, limiting the model's ability to extract useful information from RT for estimating ability. Additionally, the simple structure model only considers a person's overall effective speed by aggregating RT across all items, neglecting how high‐ and low‐ability individuals may employ different time allocation strategies across various items (Bolsinova & Tijmstra, 2018). Therefore, exploring models with alternative dependency structures is necessary to better understand how integrating RT models with CDMs could impact the measurement of latent attributes.

Not only is methodological innovation limited, but the potential applications of the joint dependency model are also underexplored. Although Ranger and Ortner (2011). found that incorporating RT into the estimation of personality traits can enhance measurement precision, most existing research on RT and RA primarily focuses on assessing cognitive proficiency, with limited attention given to its application in personality assessment. For personality assessments collected from Likert‐scale, RA is more appropriately described as response ratings (RRs). Existing studies that incorporate RTs in personality assessments predominantly use IRT models with personality traits conceptualized as continuous variables (Ferrando & Lorenzo‐Seva, 2007; Lyu & Bolt, ²⁰²⁴; Ranger, ²⁰¹³). In contrast, studies that treat personality as categorical variables have largely utilized mixture models to capture RA, but without incorporating RT data (Clark et al., 2013; Hallquist & Wright, ²⁰¹⁴; Lenzenweger et al., ²⁰⁰⁷; Wright & Hallquist, ²⁰¹⁴). This highlights a significant gap in the literature regarding exploratory diagnostic modelling approaches that integrate RT data to assess personality trait structures.

To address these limitations, the current study proposes a new approach (sparse latent class model with response time [SLCMRT]) to incorporating RT into exploratory SLCMs. This approach diverges from the traditional hierarchical RT model, which posits a correlational structure between the latent variable and response latency. Instead, the proposed model introduces a cross‐loading structure, where the log‐normal distribution of RT depends on a linear function of both the individual's speed variable and their latent attribute profiles, thereby violating the conditional independence assumption between RT and ability. In addition to a person's overall speed, this structure assumes that the RT for a given item also depends on how the item measures each attribute and whether the individual possesses these attributes. In summary, the model proposes a novel structure to capture item and person dependencies in the joint modelling of RA/RR and RT. The flexibility of SLCM to accommodate multiple CDMs further extends the applicability of the proposed model.

The remainder of this paper is structured as follows: Section 2 introduces the proposed Bayesian formulation for the SLCMRT. Section 3 presents a real data application using the Fisher Temperament Inventory (FTI). Section 4 reports the results of a simulation study based on the empirically estimated parameters. Finally, Section 5 discusses the limitations of our study and outlines potential directions for future research.

2. AN SLCM WITH RT

We begin by introducing a general framework for SLCMRT. Section 2.1 presents the statistical formulation of the model, while Section 2.2 details the Bayesian framework and describes a Gibbs sampling algorithm for parameter estimation.

2.1. Model configurations

Consider a questionnaire comprising $J$ items with $K$ underlying attributes, and we denote the vector of response probabilities for latent class $c$ on item $j$ as ${\mathit{\theta }}_{cj}={\left({\theta }_{cj0},\dots ,{\theta }_{cj,P-1}\right)}^{\prime }$, where ${\theta }_{cjp}$ represents the probability of observing response category $p$ on item $j$ for latent class $c$. It is important to note that the scale can be either dichotomous ($P=2$) or polytomous ($P>2$). Given the vector of response probabilities ${\mathit{\theta }}_{cj}$, the probability of observing an ordinal response $y_j$ is expressed as:

$$\begin{array}{l}\hfill P\left(y_j|{\mathit{\theta }}_{cj},{\mathit{\zeta }}^{\prime }\mathit{v}=c\right)=\sum _{p=0}^{P-1}{\theta }_{cjp}ℐ\left(y_j=p\right),\end{array}$$ (1)

where $y_j\in \{0,\dots ,P-1\}$ and $ℐ$ is an indicator function. We assume the existence of k binary attributes that can explain the $P^J$ observed response patterns. In this framework, a total of $2^K$ latent classes are created. We denote the latent class index $c\in \{0,\dots ,2^K-1\}$, where each latent class is associated with a $K$‐vector of latent binary attributes $\mathit{\zeta }\in {\{0,1\}}^K$. The mapping of $\mathit{\zeta }$ to an integer index $c$ is achieved through the bijection ${\mathit{\zeta }}^{\prime }\mathit{v}=c$, where $\mathit{v}={(2^{K-1},2^{K-2},\dots ,1)}^{\prime }$.

Next, we assume that the membership in class $c$ follows a multinomial distribution with structural parameters ${\pi }_c\in {\pi }_0,\dots ,{\pi }_{2^K-1}$, where ${\pi }_c$ represents the probability of membership in latent class $c$ and ${\sum }_{c=0}^{2^K-1}{\pi }_c=1$. By integrating out the latent class variable $c$, we can express the marginal likelihood of observing a random vector $\mathit{y}=(y_1,\dots ,y_J)$ as:

$$\begin{array}{l}\hfill p(\mathit{y}|\mathbf{\Theta },\mathit{\pi })=\sum _{c=0}^{2^K-1}{\pi }_c\prod _{j=1}^{J}P\left(y_j|{\mathit{\theta }}_{cj},{\mathit{\zeta }}^{\prime }\mathit{v}=c\right),\end{array}$$ (2)

where $\mathbf{\Theta }\in {ℝ}^{J\times 2^K\times P}$ represents the multivariate simplex of the response probability, and $\mathit{\pi }$ denotes the structural parameter vector. In a sample of $N$ respondents, we denote the ordinal responses for respondent $i$ as vector ${\mathit{y}}_i$, where $i=1,\dots ,N$. The likelihood of observing this sample can be expressed as:

$$\begin{array}{l}\hfill p(\mathit{Y}|\mathbf{\Theta },\mathit{\pi })=\prod _{i=1}^{n}p({\mathit{y}}_i|\mathbf{\Theta },\mathit{\pi }),\end{array}$$ (3)

where $\mathit{Y}$ denotes the $N\times J$ response matrix.

Furthermore, to define the ordinal responses, we utilize a cumulative link function $\mathrm{\Psi }(·)$ (Culpepper, 2019). This function allows us to compute the probability of each response category by taking the difference between two adjacent cumulative probabilities. The response probability for latent class $c$ on item $j$ can be expressed as:

$$\begin{array}{ll}\hfill & {\theta }_{cjp}=\mathrm{\Phi }\left({\kappa }_{j,p+1}-{\mu }_{cj}\right)-\mathrm{\Phi }\left({\kappa }_{j,p}-{\mu }_{cj}\right)\end{array}$$ (4)

$$\begin{array}{ll}\hfill & \mathrm{\Phi }\left({\kappa }_{j,p}-{\mu }_{cj}\right)=P\left(y_j\le p|{\mu }_{cj},{\kappa }_{j,p}\right),\end{array}$$ (5)

where $\mathit{\kappa }\in \{{\kappa }_{j,0},\dots ,{\kappa }_{j,P}\}$. To ensure appropriate boundaries for the cumulative probabilities, we set ${\kappa }_{j,0}=-\infty$ and ${\kappa }_{j,P}=\infty$. Consequently, this choice leads to $\mathrm{\Phi }\left({\kappa }_{j,0}-{\mu }_{cj}\right)=0$ and $\mathrm{\Phi }\left({\kappa }_{j,P}-{\mu }_{cj}\right)=1$. In this context, ${\mu }_{cj}$ represents the latent class mean parameter, defined as ${\mu }_{cj}={\mathit{\alpha }}_c^T\mathit{\beta }$. The term $\mathrm{\Phi }({\kappa }_{j,p}-{\mu }_{cj})$ denotes the cumulative probability of observing a response at level $p$ or lower.

Assuming local independence, the response distribution $\mathit{Y}$ given $\mathit{B}$ and $\mathit{\pi }$ can be expressed as follows:

$$\begin{array}{l}\hfill p(\mathit{Y}|\mathit{B},\mathit{\pi })=\prod _{j=1}^{J}p\left({\mathit{y}}_j|{\mathit{\beta }}_j,\mathit{\pi }\right)=\prod _{i=1}^N\sum _{c=0}^{2^K-1}{\pi }_c\prod _{j=1}^{J}p(y_{ij}|{\mathit{\beta }}_j,{\mathit{\alpha }}_{\mathit{i}}={\mathit{\alpha }}_{\mathit{c}}),\end{array}$$ (6)

where $y_{ij}$ represents the response of individual $i$ on item $j$, ${\mathit{\alpha }}_i$ denotes the attribute profile vector for individual $i$ and ${\mathit{\beta }}_j=({\beta }_{j0},\dots ,{\beta }_{j,2^K-1})$ is a vector of item parameters that describes the relationships between attributes and items. In SLCM, the $K$ binary variables are modelled as independent Bernoulli parameter. To identify which main effect and/or interactions of the $K$ variables are relevant to the item, a $2^K$‐dimensional structure vector incorporating all main‐effect and interaction‐effect coefficients is proposed. When we concatenate the $\mathit{\beta }$ by rows, we obtain matrix $\mathit{B}$ with $J$ rows and $2^K$ columns (Chen et al., 2020).

With $K=4$ binary attributes, a respondent's attribute profile $\mathit{\alpha }$ is represented as: $(1,{\alpha }_4,{\alpha }_3,{\alpha }_3{\alpha }_4,{\alpha }_2,{\alpha }_2{\alpha }_4,{\alpha }_2{\alpha }_3,{\alpha }_2{\alpha }_3{\alpha }_4,{\alpha }_1,{\alpha }_1{\alpha }_4,{\alpha }_1{\alpha }_3,{\alpha }_1{\alpha }_3{\alpha }_4,{\alpha }_1{\alpha }_2,{\alpha }_1{\alpha }_2{\alpha }_4,{\alpha }_1{\alpha }_2{\alpha }_3,{\alpha }_1{\alpha }_2{\alpha }_3{\alpha }_4)$. The design vector is composed of three parts: (1) the intercept parameter ‘0000’; (2) main effect parameters ‘0001’, ‘0010’, ‘0100’, ‘1000’ and (3) the interaction parameters ‘0011’, ‘0101’, ‘1001’, ‘0110’, ‘1010’, ‘1100’, ‘0111’, ‘1011’, ‘1101’, ‘1110’, ‘1111’. For example, the coefficient ${\alpha }_1{\alpha }_2{\alpha }_4$ denoted as ‘1101’ represents the interaction effect between attribute 1,2,4. If a respondent possesses these three attributes, this component is set to be 1. Table 1 displays the attribute profile matrix $A$, where the rows represent the $2^4=16$ latent classes. The first column lists the latent class labels, where 1 indicates mastery and 0 indicates non‐mastery of attributes or their interactions. For instance, the latent class ${\zeta }_{15}={(1,1,1,0)}^T$ corresponds to ${\mathit{\alpha }}_{15}=(1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0)$, with the 1s denoting the active attribute components in the design vector. Chen et al. (2020). provides a detailed discussion on how existing CDMs can be incorporated into the SLCM framework using this specification.

TABLE 1.

Example design matrix A of latent classes by the mastery of attributes for $K=4$.

$\mathit{\alpha }$	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111
(0,0,0,0)	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
(0,0,0,1)	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
(0,0,1,0)	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
(0,0,1,1)	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0
(0,1,0,0)	1	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
(0,1,0,1)	1	1	0	0	1	1	0	0	0	0	0	0	0	0	0	0
(0,1,1,0)	1	0	1	0	1	0	1	0	0	0	0	0	0	0	0	0
(0,1,1,1)	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0
(1,0,0,0)	1	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
(1,0,0,1)	1	1	0	0	0	0	0	0	1	1	0	0	0	0	0	0
(1,0,1,0)	1	0	1	0	0	0	0	0	1	0	1	0	0	0	0	0
(1,0,1,1)	1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0
(1,1,0,0)	1	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
(1,1,0,1)	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0	0
(1,1,1,0)	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0
(1,1,1,1)	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1

Compared with the ordinal response data $\mathit{Y}$, the log‐transformed RT $\mathit{T}$ are specified to fit in a lognormal model:

$$\begin{array}{l}\hfill p(\log t_{ij}|{\eta }_j,{\tau }_i,{\mathit{\alpha }}_i,{\mathit{h}}_j,{\sigma }_{(t)j})=N({\mathit{\alpha }}_i^{\prime }{\mathit{h}}_j-{\eta }_j{\tau }_i,{\sigma }_{(t)j}^2).\end{array}$$ (7)

Same as the cross‐loading method introduced by Bolsinova and Tijmstra (2018), we added an item–attribute time coefficient vector $\mathit{h}$, which is multiplied by $\mathit{\alpha }$ in the RT model. The item–attribute time coefficients are represented by vector ${\mathit{h}}_j=(h_{j0},\dots ,h_{j,2^K-1})$, which describes the time‐wise relationships between attributes and items. Here the $h_{j0}$ is the intercept parameter that describes the time intensity of item $j$. When we concatenate the ${\mathit{h}}_j$ by items, we obtain matrix $\mathit{H}$ with $J$ rows and $2^k$ columns. The sparsity of $\mathit{B}$ (i.e. coefficient matrix for RRs) and $\mathit{H}$ (i.e. coefficient matrix for RT) is the same because both matrices share the same underlying structure defined by the sparsity matrix $\mathrm{\Delta }$. The $\mathrm{\Delta }$ matrix ensures that if a coefficient in $\mathit{B}$ is active (i.e. non‐zero), the corresponding entry in $\mathit{H}$ must also be active. This design reflects a unified framework where the attribute‐to‐item relationships for RRs and RTs are modelled in parallel. Using the same sparsity matrix simplifies the interpretation of item–attribute relationships while enabling simultaneous analysis of response behaviours across both dimensions.

The speed residual for respondent $i$ is represented by parameter ${\tau }_i$, while the item time‐discrimination parameter is denoted as ${\eta }_j$. If two respondents with same latent attribute profiles (i.e. ${\mathit{\alpha }}_{i1}={\mathit{\alpha }}_{i2}$) answering the same item, the one with a higher speed parameter (i.e. ${\tau }_1>{\tau }_2$) would take less time. The added term of ${\mathit{\alpha }}_i^{\prime }{\mathit{h}}_j$ relaxes the conditional independence assumption that RT is fully explained by $\tau$. The parameter vector ${\sigma }_{(t)j}^2$ implies the item‐wise RT residual variance. Note that the conditional dependence in Equation (7) indicates that part of the variability in RT can be attributed to respondents' latent attributes, while the remaining variability is influenced by their response speed. Thus, the residual speed parameter (also referred to as the latent speed parameter) captures individual differences in item RT that are not explained by their latent traits.

2.2. Gibbs sampling algorithm

In this section, we introduce a Bayesian model formulation for the SLCMRT. The joint model encompasses two components: a cumulative probit response model for handling response data and a lognormal RT model for handling RT data. The cumulative probit response model is an adaptation of the SLCM designed for binary attributes in the context of ordinal response data (Culpepper, 2019; He et al., ²⁰²²).

The joint dependency model is defined as follows:

$$\begin{array}{ll}\hfill & y_{ij}^{\ast }|{\mathit{\alpha }}_i,{\mathit{\beta }}_j,y_{ij}\sim N({\mathit{\alpha }}_i^{\prime }{\mathit{\beta }}_j,1)I({\kappa }_{y_{ij}}<y_{ij}^{\ast }<{\kappa }_{y_{ij}+1})\end{array}$$ (8)

$$\begin{array}{ll}\hfill & \log t_{ij}|{\eta }_j,{\tau }_i,{\mathit{\alpha }}_i,{\mathit{h}}_j,{\sigma }_{(t)j}\sim N({\mathit{\alpha }}_i^{\prime }{\mathit{h}}_j-{\eta }_j{\tau }_i,{\sigma }_{(t)j}^2).\end{array}$$ (9)

The priors for the elements in the coefficient matrices $\mathit{B}$ and $\mathit{H}$ are defined separately as follows:

$$\begin{array}{ll}\hfill {\beta }_{jc}& |{\delta }_{jc}\sim \left\{\begin{array}{cc}N(0,{\sigma }_{\beta }^2)I({\beta }_{jc}>L_{{\beta }_{jc}})& \text{if}{\delta }_{jc}=1\\ I({\beta }_{jc}=0)& \text{if}{\delta }_{jc}=0\end{array}\right.\end{array}$$ (10)

$$\begin{array}{ll}\hfill h_{jc}& |{\delta }_{jc}\sim \left\{\begin{array}{cc}N(0,{\sigma }_h^2)I(h_{jc}<0)& \text{if}{\delta }_{jc}=1\\ I(h_{jc}=0)& \text{if}{\delta }_{jc}=0\end{array}\right.\end{array}$$ (11)

$$\begin{array}{ll}\hfill {\delta }_{jc}& |\omega \sim \text{Bernoulli}(\omega ),\omega \sim \text{Beta}({\omega }_0,{\omega }_1).\end{array}$$ (12)

The prior distributions for the latent attributes ${\mathit{\alpha }}_i$ and latent speed ${\tau }_i$ are specified as:

$$\begin{array}{ll}\hfill & {\mathit{\alpha }}_i|\mathit{\pi }\sim \text{Multinomial}(\mathit{\pi }),\mathit{\pi }\sim \text{Dirichlet}({\mathit{d}}_0)\end{array}$$ (13)

$$\begin{array}{ll}\hfill & {\tau }_i\sim N(0,1).\end{array}$$ (14)

The priors for the other model parameters are specified as:

$$\begin{array}{ll}\hfill & {\eta }_j\sim N(0,{\sigma }_{\eta }^2)I({\eta }_j>0),\frac{1}{{\sigma }_{\eta }^2}\sim \text{gamma}(a_{\eta },b_{\eta })\end{array}$$ (15)

$$\begin{array}{ll}\hfill & \frac{1}{{\sigma }_{(t)j}^2}\sim \text{gamma}(a_t,b_t)\end{array}$$ (16)

$$\begin{array}{ll}\hfill & \frac{1}{{\sigma }_h^2}\sim \text{gamma}(a_h,b_h).\end{array}$$ (17)

We implemented the Gibbs sampling algorithm described in Culpepper (2019) and Chen et al. (2020) to estimate the model parameters. We summarized the sampling algorithm in Algorithms 1 and 2, where the detailed derivation of the conditional distributions can also be found in the Appendices. The algorithm was implemented using the ‘R’ programming language version 3.6.2, along with the ‘C++’ programming language. We utilized the ‘RcppArmadillo’ and ‘Rcpp’ packages (Eddelbuettel, 2013; Eddelbuettel & Balamuta, ²⁰¹⁸; Eddelbuettel & François, ²⁰¹¹) to enhance computational efficiency and optimize the code execution.

3. EMPIRICAL EXAMPLE: THE FTI

In the following sections, we will begin by presenting an overview of the empirical data, followed by a discussion of the imposed model constraints and the findings derived from our real‐data analyses.

3.1. The FTI data

The FTI is a questionnaire consisting of 56 items that are specifically designed to assess a novel construct of temperament. The FTI identifies four distinct collections of behaviours that are associated with four broad neural systems: dopamine, serotonin, testosterone and oestrogen/oxytocin (Brown et al., 2013). Each of these neural systems corresponds to a specific temperament dimension, capturing different aspects of an individual's temperament as (1) curious/energetic, (2) cautious/social norm compliant, (3) analytical/tough‐minded and (4) prosocial/empathetic.

In this study, we applied the SLCMRT model to the FTI data which can be accessed on the Open Psychometrics website (https://openpsychometrics.org/tests/FTI/). To mitigate the impact of outliers, we select participants whose mean and standard deviation of RTs fall within the 5%–95% quantile range. This subset comprises a sample of $N=4310$ participants and does not include any missing data. The FTI consists of 56 items that assess four broad dimensions of temperament, with each dimension represented by 14 items. Table A1 in the Appendices includes the item index and their respective affiliations with the temperament dimensions: ‘CE’ for ‘curious/energetic’, ‘CS’ for ‘cautious/social norm compliant’, ‘AT’ for ‘analytical/tough‐minded’ and ‘PE’ for ‘prosocial/empathetic’. Participants responded to each item on a Likert scale, with options ranging from 0 for ‘strongly disagree’, 1 for ‘disagree’, 2 for ‘agree’ and 3 for ‘strongly agree’.

In addition to the item responses, the dataset also includes parallel item‐wise RTs data. Figure 1illustrates the distribution of log‐transformed RTs for each item. These figures suggest that the log‐transformed RTs are approximately normally distributed with a slight right‐skewed tendency. Although several useful extensions of the base lognormal RT model have been proposed (Fox & Marianti, 2016; Wang et al., ²⁰¹³), the lognormal RT model was still selected due to its ease of implementation in practice.

FIGURE 1.

Histogram of log‐transformed response time distribution for each item (in milliseconds).

Our empirical analyses are guided by three primary objectives. First, we aim to use this model as an exploratory tool to uncover the item–attribute structure within the personality inventory. Second, we seek to examine whether the inclusion of RT data enhances the estimation of latent attributes and improves the overall model‐data fit for RRs. Third, we investigate whether the latent attributes help explain variability in RT.

3.2. Model constraints and assumptions

Similar to conventional IRT, the model is built on several underlying assumptions: trait invariance, item invariance and local independence. In addition, enforcing identifiability constraints is essential for models that demand high interpretability, as it ensures that distinct sets of parameter values generate unique distributions of observed variables. If a probabilistic model is not identifiable, recovering the true parameter values becomes challenging, even with an infinite number of observations. Recent research has made significant advancements in establishing identifiability conditions for SLCMs, including binary attributes with dichotomous responses (Chen et al., 2020), binary attributes with ordinal responses (Culpepper, 2019), ordinal attributes with ordinal responses (He et al., 2022) and binary attributes with nominal responses (Liu & Culpepper, 2024). For binary attributes and responses, a strict and generic identifiability condition requires the sparse structure matrix $\mathrm{\Delta }$ to follow specific configurations, as detailed in Chen et al. (2020). Our sampling algorithm ensures that the $\mathrm{\Delta }$ always maintains an identifiability format whenever an element in the matrix is sampled.

We also imposed the monotonicity constraints on the $\beta$ parameters. Specifically, in instances where an item measures two dimensions, individuals who exhibit high scores in both dimensions are theorized to demonstrate a greater propensity for agreement with the item, relative to individuals who exhibit a high score in only one dimension. Additionally, individuals with a high score in one dimension are not directly comparable to those with a high score in another. Furthermore, individuals who score low in both dimensions are expected to exhibit the least tendency of agreeing with the item. To satisfy these assumptions, a lower bound has been derived (Chen et al., 2020). Specifically, if the algorithm decides to sample a $\beta$ parameter, it will sample it from a truncated normal distribution, with the lower limit depending on the other $\beta$ coefficients for this item.

Regarding the RT coefficients $h$, we applied negative constraints to the non‐intercept coefficients, indicating that a higher score on an attribute is linked to shorter RTs. In empirical terms, this implies that individuals who express greater agreement with the item description take less time to select their answers. This assumption is given based on a pilot study. We computed the correlation coefficients between the item‐wise averaged RT and item–attribute coefficients $\mathit{\beta }$ obtained from a single SLCM model. The results demonstrated that the $\beta$ coefficients for all specified attributes have a negative relationship with RT.

3.3. The exploratory analysis of the empirical data

As the dataset is conventionally assumed to have four dimensions, we conducted an exploratory analysis using the SLCMRT model with $K=4$. To obtain reliable results, we performed 20 Markov Chain Monte Carlo (MCMC) chains, each consisting of 60,000 iterations with 40,000 burn‐in iterations inside. We selected one chain with the highest log‐likelihood for the posterior inference. Throughout the iterations, we enforced the monotonicity constraints for $\beta$. We also compute the potential scale reduction factor, known as $\widehat{R}$ for each parameter. The results show that all model parameters have converged and mixed well with $\widehat{R}<1.1$. The estimated parameters of $\mathit{B}$, $\mathit{\pi }$, $\mathit{H}$ and $\mathit{\eta }$ are presented in Tables 2 and 3.

TABLE 2.

Estimated model parameters $\mathit{B}$ and $\mathit{\pi }$ of SLCMRT with non‐zero elements.

	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111
CE1	.43					.62
CE2	.56					.46
CE3	.71					.54
CE4	1.03					.69							.17
CE5	.51					.20							.38
CE6	.56	.15				.24				.17
CE7	.73					.64							.19
CE8	.47					.50
CE9	.50					.23							.47
CE10	.73					.57
CE11	.35												.17
CE12	.82					.59	.23			.24
CE13	.32					.30									.17
CE14	.92					.46
CS15	.56										.28
CS16	.78						.12				.34
CS17	.37										.12
CS18	.51		.10								.40
CS19	.54										.26
CS20	.51						.18				.20
CS21	.16										.13
CS22	.54						.17				.32
CS23	.66						.16
CS24	.89										.36
CS25	.40										.32
CS26	.80										.43
CS27	.30										.11
CS28	.57		.14				.13				.13
AT29	.30					.14	.55
AT30	.45				.19			.54
AT31	.64						.43
AT32	.69						.80
AT33	.89					.29	.51			.15
AT34	.51						.66
AT35	.89					.29	.60
AT36	.63						.44
AT37	.53				.19			.51
AT38	.24						.10
AT39	.48						.47									.30
AT40	.58						.39
AT41	.83						.38
AT42	.54						.70
PE43	.72					.14			.81
PE44	.94	.11							.98
PE45	.31									.37
PE46	.63	.46								.35
PE47	.53	.14				.16
PE48	.54								.54	.28
PE49	.68	.48					.15		.28
PE50	.62								1.02
PE51	.71	.40						.17		.33
PE52	.46	.14							.93
PE53	.69	.44								.38
PE54	.58	.26							.22
PE55	.58	.22							.30
PE56	.74								1.07
$\pi$	.05	.10	.05	.10	.04	.03	.16	.10	.02	.11	.05	.05	.02	.08	.02	.02

TABLE 3.

Estimated model parameters $\mathit{H}$ and $\mathit{\eta }$ of SLCMRT with non‐zero elements.

	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111	$\eta$
CE1	8.49					−.18											.27
CE2	8.18					−.13											.25
CE3	8.38					−.12											.23
CE4	8.14					−.22							−.06				.22
CE5	8.07					−.10							−.07				.23
CE6	8.03	−.07				−.06				−.03							.23
CE7	8.12					−.20							−.05				.24
CE8	8.15					−.13											.23
CE9	8.13					−.04							−.17				.24
CE10	8.38					−.18											.24
CE11	8.41												−.05				.24
CE12	8.21					−.19	−.06			−.11							.25
CE13	8.01					−.03									−.06		.21
CE14	8.60					−.13											.24
CS15	8.58										−.16						.27
CS16	8.55						−.02				−.15						.24
CS17	8.66		−.02														.26
CS18	8.17		−.07								−.15						.24
CS19	8.30		−.06								−.07						.24
CS20	8.37						−.06				−.06						.26
CS21	8.30					−.08											.25
CS22	8.25						−.02				−.09						.28
CS23	8.44						−.04										.23
CS24	8.40										−.16						.24
CS25	8.16										−.07						.25
CS26	8.63										−.23						.27
CS27	8.52																.26
CS28	8.58		−.03				−.02				−.03						.25
AT29	8.21					−.04	−.09										.24
AT30	8.15				−.01			−.22									.22
AT31	8.48						−.10										.27
AT32	8.27						−.29										.24
AT33	8.37					−.08	−.23			−.10							.26
AT34	8.51						−.15										.24
AT35	8.10					−.10	−.17										.22
AT36	8.05						−.14										.27
AT37	8.36				−.03			−.18									.27
AT38	8.76																.23
AT39	8.54						−.05									−.11	.25
AT40	8.56						−.06										.24
AT41	8.14						−.15										.23
AT42	8.79						−.17										.26
PE43	8.46					−.03			−.26								.25
PE44	8.51	−.03							−.31								.27
PE45	8.70						−.07			−.05							.27
PE46	8.16	−.21								−.08							.20
PE47	8.22	−.07				−.09											.22
PE48	8.52								−.12	−.07							.27
PE49	8.63	−.15					−.03		−.07								.27
PE50	8.20								−.27								.24
PE51	8.36	−.18						−.03		−.08							.23
PE52	8.27	−.04							−.21								.24
PE53	8.03	−.23								−.09							.22
PE54	8.54	−.07							−.06								.25
PE55	8.80	−.06							−.06								.25
PE56	7.99								−.31								.22

In Table 2, the estimated item–attribute relationships can be visually summarized. In personality assessment, the empirical interpretation of binary attributes is different from that of cognitive assessment. The latter represents the possession of a cognitive ability, whereas the former indicates a high level in the personality dimension; that is, the person has a strong preference for a relevant style of cognition and behaviours. The main effect coefficients, denoted as ‘0001’, ‘0010’, ‘0100’ and ‘1000’, correspond to the dimension ‘CE’, ‘CS’, ‘AT’ and ‘PE’, respectively. From the table, we can see how the model clusters the items compared to their original affiliations. If we classify items into a dimension by identifying the largest coefficient $\beta$ and determining which dimension this coefficient refers to, the categorization can be summarized as follows: The majority of ‘CE’ items primarily load on the ‘CE’ and ‘AT’ dimensions, with Items CE5, CE9 and CE11 cross‐loading on the ‘PE’ and ‘AT’ dimensions. The majority of ‘CS’ items primarily load on the ‘CS’ and ‘PE’ dimension, with Item CS23 cross‐loading on the ‘AT’ and ‘CS’ dimensions. Items labelled as ‘AT’ cross‐load on the ‘AT’ and ‘CS’ dimension, with Items AT30 and AT37 cross‐loading on the ‘AT’, ‘CS’ and ‘CE’ dimensions. The majority of ‘PE’ items load on the ‘PE’ or ‘CE’ dimension or cross‐load on their interaction.

We can also find various item types from Table 2. For example, Item PE50, which states ‘I am very sensitive to people's feelings and needs’, is a simple structure item that loads on the PE (prosocial/empathetic) dimension. Item CE13, which states ‘I have more energy than most people’, can be viewed as a conjunctive item that loads on the interaction of the AT (analytical/tough‐minded) and CE (curious/energetic) dimensions. For this type of item, only respondents who exhibit high scores in both of these attributes are likely to resonate with it. Additionally, item PE55, which states ‘When reading I enjoy when the writer takes a sidetrack to say something beautiful or meaningful’, is a compensatory item that loads on the PE (prosocial/empathetic) and CE (curious/energetic) dimensions. For this type of item, having high scores on either dimension compensates for not having another, so respondents with high scores in either dimension are likely to agree with what the item describes. By analysing the stems of the items, we can identify the ambiguity in these items and propose potential revisions to clarify their intended meaning. It is worth noting that many items in personality scales do not follow a simple structure, and practitioners should be cautioned when interpreting the model scores.

Table 3 displays the item RT coefficient matrix $\mathit{H}$. The smaller the non‐intercept coefficients, individuals with the related attribute are more likely to respond rapidly to the item. Figure 2 presents the estimated mean RTs ${\mathit{\mu }}_H$ for each item across the 16 latent classes, calculated from ${\mathit{\alpha }}_c^{\prime }{\mathit{h}}_j$, where the x‐axis in this subplot corresponds to the same ordering of latent classes as presented in Table 1. This figure illustrates how variations in RTs are influenced by item–attribute relationships and latent class profiles. It highlights that some items demonstrate greater variance explained by latent trait attributes compared to others, such as item CE12 versus CE20. For ordinal responses, the mode response is defined as the response category with the highest posterior probability based on the model estimates. The mode response for each item is presented in Table 4, which highlights how different latent classes exhibit distinct response patterns to items within the model framework.

FIGURE 2.

Latent class response time mean parameter across items.

TABLE 4.

Posterior mode responses of SLCMRT model: ‘1’ for ‘strongly disagree’, ‘2’ for ‘slightly disagree’, ‘3’ for ‘slightly agree’, ‘4’ for ‘strongly agree’.

	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111
CE1	2	2	2	2	2	3	2	3	2	2	2	2	2	3	2	3
CE2	2	2	2	2	2	3	2	3	2	2	2	2	2	3	2	3
CE3	2	2	2	2	2	3	2	3	2	2	2	2	2	3	2	3
CE4	3	3	3	3	3	4	3	4	3	3	3	3	3	4	3	4
CE5	2	2	2	2	2	2	2	2	2	2	2	2	2	3	2	3
CE6	2	2	2	2	2	2	2	2	2	2	2	2	2	3	2	3
CE7	2	2	2	2	2	3	2	3	2	2	2	2	2	3	2	3
CE8	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
CE9	2	2	2	2	2	2	2	2	2	2	2	2	2	3	2	3
CE10	2	2	2	2	2	3	2	3	2	2	2	2	2	3	2	3
CE11	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
CE12	2	2	2	2	2	3	3	3	2	3	2	3	2	3	3	4
CE13	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
CE14	2	2	2	2	2	3	2	3	2	2	2	2	2	3	2	3
CS15	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
CS16	2	2	2	2	2	2	2	2	2	2	3	3	2	2	3	3
CS17	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
CS18	2	2	2	2	2	2	2	2	2	2	3	3	2	2	3	3
CS19	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
CS20	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
CS21	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
CS22	2	2	2	2	2	2	2	2	2	2	2	2	2	2	3	3
CS23	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
CS24	2	2	2	2	2	2	2	2	2	2	3	3	2	2	3	3
CS25	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
CS26	2	2	2	2	2	2	2	2	2	2	3	3	2	2	3	3
CS27	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
CS28	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
AT29	1	1	1	1	1	2	2	2	1	1	1	1	1	2	2	2
AT30	2	2	2	2	2	2	2	3	2	2	2	2	2	2	2	3
AT31	2	2	2	2	2	2	3	3	2	2	2	2	2	2	3	3
AT32	2	2	2	2	2	2	3	3	2	2	2	2	2	2	3	3
AT33	2	2	2	2	2	3	3	4	2	3	2	3	2	3	3	4
AT34	2	2	2	2	2	2	3	3	2	2	2	2	2	2	3	3
AT35	2	2	2	2	2	3	3	4	2	2	2	2	2	3	3	4
AT36	2	2	2	2	2	2	3	3	2	2	2	2	2	2	3	3
AT37	2	2	2	2	2	2	2	3	2	2	2	2	2	2	2	3
AT38	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
AT39	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	3
AT40	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
AT41	2	2	2	2	2	2	3	3	2	2	2	2	2	2	3	3
AT42	2	2	2	2	2	2	3	3	2	2	2	2	2	2	3	3
PE43	2	2	2	2	2	2	2	2	3	3	3	3	3	3	3	3
PE44	2	3	2	3	2	3	2	3	4	4	4	4	4	4	4	4
PE45	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
PE46	2	3	2	3	2	3	2	3	2	3	2	3	2	3	2	3
PE47	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
PE48	2	2	2	2	2	2	2	2	3	3	3	3	3	3	3	3
PE49	2	3	2	3	2	3	2	3	2	3	2	3	2	3	3	3
PE50	2	2	2	2	2	2	2	2	3	3	3	3	3	3	3	3
PE51	2	3	2	3	2	3	2	3	2	3	2	3	2	3	2	3
PE52	2	2	2	2	2	2	2	2	3	3	3	3	3	3	3	3
PE53	2	3	2	3	2	3	2	3	2	3	2	3	2	3	2	3
PE54	2	2	2	2	2	2	2	2	2	3	2	3	2	3	2	3
PE55	2	2	2	2	2	2	2	2	2	3	2	3	2	3	2	3
PE56	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4

3.4. Comparisons of SLCMRT and SLCM

To provide evidence of the model‐fit improvement achieved by incorporating RTs, we compared the SLCM model to the SLCMRT model. However, it is not possible to directly compare their absolute model fit due to the inclusion of additional observed data in SLCMRT. To assess the necessity of including item–attribute RT coefficients in the H matrix, we introduced a competing model called SLCMRT0. In this model, the item–attribute coefficients in the $\mathit{H}$ matrix were removed, and only the intercept $h_0$, representing the item‐level speed coefficients, was retained. The model fit of the competing models was evaluated using the deviance information criterion (DIC), as shown in Table 5. The posterior mean deviance $\overline{D}$ is the mean of the deviance over the posterior distribution. The deviance at the posterior mean $D(\overline{\theta })$ is the deviance computed at the posterior mean parameter values. $p_D$ is the effective number of parameters that measures the complexity of the model, where DIC is calculated as $\overline{D}+p_D$ and $p_D=\overline{D}-D(\overline{\theta })$. Results indicated that the SLCMRT model was favoured over the SLCMRT0 model based on the DIC, despite the SLCMRT model's higher model complexity. This finding suggests that incorporating item–attribute speed coefficients improved the overall model fit.

TABLE 5.

Model fit results in empirical data analysis.

Model	$\overline{D}$	$p_D$	$D(\theta )$	DIC
SLCM	694,064	274	693,790	694,338
SLCMRT0	1,022,606	4240	1,018,366	1,026,846
SLCMRT	1,016,708	4411	1,012,297	1,021,119

To assess whether adding RT effectively improves the model's ability to capture the overall RA data, we performed a posterior predictive model check (PPMC) comparing the SLCMRT and SLCM models. The PPMC in our study is designed to evaluate how well each model recovers the responses by calculating the root mean squared error (RMSE) between the observed and predicted responses in an element‐wise manner. Specifically, the following steps are used to calculate the RMSE for each model:

1. Draw samples of the item parameter matrix $\mathit{B}$ and latent class profiles $\mathit{\alpha }$ from the MCMC samples after burn‐in.

2. Generate model‐based predictive dominance responses using the sampled parameters obtained in Step 1.

3. For each of the $n=1,\dots ,$ 20,000 samples, compute the element‐wise RMSE between the model‐based predictive dominance responses and the observed responses.

4. Compute the 5% and 95% quantiles of the RMSE across the 20,000 samples to assess the range of the RMSE distribution.

The 5% and 95% quantiles of the RMSE for the SLCMRT model fall between .8533 and .8728, while for the SLCM model, they range from 1.0402 to 1.0683. This indicates that incorporating RT into the model improves the overall model‐data fit for RRs.

Additionally, to assess whether the SLCMRT model enhances the predictability of respondent classes, Shannon entropy was calculated for respondent $i$ based on their posterior class probability vector ${\tilde{\mathit{\pi }}}_i$, using the formula $H({\tilde{\mathit{\pi }}}_i)=-{\sum }_{c=1}^{C}P({\tilde{\mathit{\pi }}}_{ic}){\log }_{2}P({\tilde{\mathit{\pi }}}_{ic})$, with ${\log }_2$ denoting the logarithm of base two. Higher entropy values in this context suggest greater unpredictability of outcomes. Figure 3 displays the Shannon entropy values for all respondents (N=4310), comparing the SLCM and SLCMRT models. The posterior probability vectors ${\tilde{\mathit{\pi }}}_i$ were derived by calculating the relative frequency of each class label in MCMC posterior draws. The results show that for approximately $64.5\%$ of the 4310 respondents, the Shannon entropy values calculated from the SLCMRT model are lower than those from the SLCM model. This indicates that the SLCMRT model yields more confident classification results for $64.5\%$ of respondents compared to the SLCM model.

FIGURE 3.

Shannon entropy of the posterior latent class probability estimates from SLCM and SLCMRT ($N=4310$).

4. SIMULATIONS BASED ON EMPIRICAL DATA

To assess the parameter recovery capabilities of the proposed model, we conduct a simulation study using the Bayesian MCMC estimation. The empirical population item parameters ($\mathit{B}$), ($\mathit{H}$) and ($\eta$'s) are sourced from Tables 2 and 3. Latent class profiles ($\mathit{\alpha }$) are generated using the estimated proportion parameter ($\mathit{\pi }$) in Table 2. The population person speed parameters ($\tau$'s) are generated from standard normal distributions. Finally, the ordinal responses (Y) and RTs (T) are randomly generated in accordance with the model's specifications.

In this simulation study, we construct a personality test with 56 items (J), where the model assumes four latent attributes (K), two attribute levels (M) and five response categories (P). We vary the sample size as follows: 1000, 2000, 3000, 4000 and 5000, and each sample size condition is replicated 20 times. For each replicated dataset, we run 10 chains, and each chain consists of a total of 60,000 iterations. We discard the first 40,000 iterations as burn‐in and use the remaining iterations for posterior inference. Among the chains, we select the one with the highest likelihood to perform the posterior inference and assess the recovery accuracy. By conducting this simulation study with various sample sizes, we aim to evaluate how well the proposed model recovers the true parameters under different conditions. On average, the algorithm takes approximately 46 min runtime for a sample size of 5000 on a single core of Apple M2 CPU with 8 GB RAM.

Table 6 presents the recovery accuracy of parameters $\mathrm{\Delta }$ in the simulation study. The first row, labelled as ‘$\mathrm{\Delta }$’, displays the average recovery accuracy across all entries of $\mathrm{\Delta }$. The second row, labelled as ‘$\mathrm{\Delta }=1$’, shows the proportion of entries in $\mathrm{\Delta }$ that are correctly recovered and have a true value of 1. Similarly, the third row, labelled as ‘$\mathrm{\Delta }=0$’, represents the proportion of entries in $\mathrm{\Delta }$ that are correctly recovered and have a true value of 0. The estimation accuracy of parameters $\mathit{B}$ and $\mathit{H}$ is evaluated using the RMSE in Table 7. Additionally, the RMSE for the parameter $\mathit{\pi }$ and $\mathit{\eta }$ is found to be approximately 0 for all sample size conditions.

TABLE 6.

Summary of Monte Carlo simulation recovery rate of $\mathrm{\Delta }$ by $N$ = 1000, 2000, 3000, 4000, 5000.

	1000	2000	3000	4000	5000
$\mathrm{\Delta }$	.931	.942	.950	.956	.968
$\mathrm{\Delta }=1$	.639	.687	.712	.769	.821
$\mathrm{\Delta }=0$	.988	.992	.995	.994	.995

TABLE 7.

Summary of Monte Carlo simulation study RMSE of $\mathit{B}$, $\mathit{H}$ by $N$ = 1000, 2000, 3000, 4000, 5000.

	1000		2000		3000		4000		5000
	H	B	H	B	H	B	H	B	H	B
$\mathrm{\Delta }$	.001	.006	.001	.004	.000	.003	.000	.003	.000	.002
$\mathrm{\Delta }=1$	.005	.033	.004	.024	.003	.021	.002	.017	.001	.014
$\mathrm{\Delta }=0$	.000	.001	.000	.001	.000	.000	.000	.001	.000	.001

The results indicate that the estimation accuracy of both parameters improves as the sample size increases. Notably, considering the number of items in this study, a larger sample size is suggested to achieve a satisfactory recovery of $\mathrm{\Delta }$. These findings from the simulation study provide strong support for the effectiveness of our Gibbs algorithm in successfully recovering the model with the above empirical population parameters.

5. DISCUSSION

The speed at which individuals respond during personality and attitude assessments reflects their decision‐making efficiency in determining the extent to which a statement describes their judgments or behaviours. It is highly probable that respondents work at different speeds across items, especially when the items differ in how strongly they represent the underlying dimensions of personality. As digital‐based personality assessments have become more prevalent, RT information has become more readily available. There is considerable interest in examining how item attributes interact with individual response speed to impact response latency. To address this need, the present study introduces an exploratory SLCMRT model that incorporates a dependency relationship between RT and latent attributes, designed to fit data with ordinal responses and RT.

To showcase a potential application, we apply the SLCMRT model to an empirical dataset of FTI. The results of the analysis demonstrate that integrating RT into SLCM yields a viable model, improving the overall model‐data fit and leading to a modest improvement in the predictability of respondent classification compared to the SLCM model. Additionally, a simulation study was conducted, employing population parameters derived from the empirical dataset and demonstrated satisfactory recovery of the model's parameters.

The proposed model contributes to the existing literature in several important ways. First, from a methodological perspective, it introduces an innovative theory on RT measurement by utilizing a cross‐loading framework that relaxes the conditional independence assumption between RT and latent attributes. Second, this study represents a pioneering effort to incorporate RT in personality assessments, a domain where psychometric properties are generally weaker compared to cognitive measures. The ambiguity in defining personality traits complicates the validation of construct validity across various scales and methodologies. SLCMs have already been instrumental in capturing underlying item–attribute structures. However, if a dependency between RT and latent attributes exists in practice, integrating RTs (SLCMRT) can not only enhance the precision of ability measurement but also reveal hidden latent structures, thereby advancing our understanding of personality assessments. Although the example presented in this paper focuses on non‐cognitive assessment, we emphasize that the model can also be directly applied to cognitive and educational assessments. We encourage future researchers to explore its application to educational assessments, which are widely used with CDMs. Furthermore, we suggest comparing the data‐driven item–attribute structure with expert evaluations to further validate the model's performance from an exploratory perspective.

While the current study makes important contributions, it also has some limitations that can be addressed in future research. First, the algorithm used in the study requires the selection of parallel chains with differing initial values, resulting in a lengthy estimation process. Future research could focus on finding ways to initialize parameter values that approximate the true structure, potentially reducing the time required for estimation. Moreover, the model is prone to overfitting as the number of attributes increases. Therefore, a method for determining the optimal number of attributes (Chen et al., 2023) should be considered as an extension to the current research.

Second, further empirical and theoretical evidence is needed to investigate whether the trait dimensions in Fisher temperament should be treated as discrete or continuous. The enduring debate over categorical or dimensional personality traits can be traced back to the early development of psychiatric nosology (Kendell, 1975). Psychological disorders have traditionally been understood through categorical, dimensional and hybrid approaches. The categorical approach posits a dichotomy in classification, for example, an illness is either present or absent, while the dimensional approach conceptualizes the severity of a disorder as existing on a continuum, ranging from normal to severely ill. A hybrid approach combines elements of both, taking on different forms, such as dimensions that intersect with discrete classes or discrete classes within which dimensions describe individual differences in personality (Avasthi et al., 2014). For instance, Lenzenweger et al. (2007). have suggested using mixture models to represent distinct classes of schizotypic risk, where genuine continuous variation exists within each class.

In psychiatry, it is historically believed that categorical conceptualizations are more useful for clinical assessment and decision‐making, as they align with the practical need to diagnose and treat based on clear, binary criteria (Livesley, 2007). However, some qualitative reviews have concluded that the strongest evidence for continuity is seen in certain types of internalizing psychopathology (e.g. social anxiety and posttraumatic stress disorder), while the strongest evidence for discreteness is found in conditions such as dissociation, antisocial behaviour and schizotypy (Haslam, 2007). This debate has also extended to the general understanding of personality traits (Wright & Hallquist, 2014). Several widely used psychological inventories are based on categorical personality types, including the Myers–Briggs Type Indicator (Briggs‐Mayers et al., 1998). In contrast, dimensional theories, such as the Big Five Traits or the Five‐Factor Model (Goldberg, 2013), provide substantial evidence supporting the continuous nature of personality traits. To date, it is widely recognized that the viability of personality theory depends on its ability to provide a robust and empirically justifiable foundation. A well‐formulated personality structure should be empirically tested using contemporary qualitative and quantitative modelling techniques.

Third, there is a lack of existing literature that compares and summarizes the pros and cons of current psychometric models used for personality assessment. The most popular statistical model with continuous trait is the factor analysis model, which has been frequently used for personality psychologists to analyse Likert‐scale data. This approach relies on several key assumptions, such as the normality of latent trait distributions, linear relationships between items and factors, and homoscedasticity, which are often violated in practice (Flora et al., 2012; Yang & Liang, ²⁰¹³). These dimensional trait models are also called as the variable‐centred approach, as focusing on examining relations among variables. By contrast, the other modelling methods, for instance, cluster analysis, latent class analysis and factor mixture modelling (Hicks et al., 2017), are more appropriate for personality traits that do not always follow a normal distribution. For instance, the sadism, as a psychology disorder trait, typically present in a small subpopulation but entirely absent in others (Morey, 1988). These modelling approaches are defined as person‐centred approaches, which identify subgroups of individuals based on their similarities on set of variables. In the era of artificial intelligence and machine learning, there has been a growing attempt of using clustering methods to analyse personality traits (Bleidorn & Hopwood, 2019; Gerlach et al., ²⁰¹⁸; Stachl et al., ²⁰²⁰). By broadening the conceptualization of traits, psychometric models can integrate advanced techniques from fields such as machine learning and data science, leading to more flexible and data‐driven approaches for understanding and measuring personality.

Fourth, for non‐cognitive psychological constructs, it has been suggested that the response process is better characterized by an unfolding mechanism. Unfolding IRT models relax the assumption of a monotonic item response function (IRF) and instead assume a unimodal IRF. Specifically, the degree of preference a respondent has for an item is inversely related to the Euclidean distance between the respondent's location and the item's location. Ferrando and Lorenzo‐Seva (2007). proposed a method incorporating a person‐item distance measure as a covariate in RT models using personality data. Similarly, Ranger and Ortner (2011). proposed a related theory focused on the likelihood of the chosen response in relation to the content trait level. According to their theory, responses with a higher likelihood (e.g. items where the appropriate response category is clear to the respondent) tend to be given more quickly. Future research could further explore the unfolding relationship with latent class models (De Soete & Heiser, 1993) between latent traits and responses.

Lastly, RT is widely recognized as a natural and intuitive source of data for investigating cognitive processes. Researchers have demonstrated that the unfolding IRT model can be integrated with response style variables using the IRTree model to capture behaviours within a sequential process (He & Kern, 2024). In the context of cognitive measurements, decomposing RT based on hypothesized sequential processes is relatively straightforward (De Boeck & Jeon, 2019). However, in attitudinal and personality assessments, RTs are less clearly understood within a sequential framework. Future research could explore how incorporating RT data might facilitate more robust process inferences.

AUTHOR CONTRIBUTIONS

Siqi He: conceptualization; investigation; funding acquisition; writing – original draft; writing – review and editing; visualization; validation; software; formal analysis; project administration; data curation; supervision; resources; methodology. Steven Andrew Culpepper: supervision; conceptualization; methodology. Jeffrey A. Douglas: supervision; conceptualization; methodology.

CONFLICT OF INTEREST STATEMENT

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

APPENDIX A. FULL CONDITIONAL DISTRIBUTIONS OF GIBBS ALGORITHM

Under the above formulation, the full conditional distributions of each model parameters can be explicitly derived in a closed format, which enables the implementation of a Gibbs sampling estimation algorithm.

For parameter ${\mathit{\alpha }}_i$, we assume a multinomial prior such that ${\mathit{\alpha }}_i|\mathit{\pi }\sim \text{Multinomial}(\mathit{\pi })$. For the two likelihood function that conditioned on ${\mathit{\alpha }}_i$, we reframe the Equation (8) as $p(y_{ij}|{\mathit{\alpha }}_i={\mathit{\alpha }}_c,{\mathit{\beta }}_j):={\theta }_{jc,y_{ij}}$ and Equation (9) as $p(\log t_{ij}|{\eta }_j,{\tau }_i,{\mathit{\alpha }}_i,{\mathit{h}}_j,{\sigma }_{(t)j}^2):={\varphi }_{jc,\log t_{ij}}$, the conditional distribution of ${\mathit{\alpha }}_i$ given the other model parameters follow a multinomial distribution:

$$\begin{array}{ll}\hfill & {\mathit{\alpha }}_i={\mathit{\alpha }}_c|y_{ij},\log t_{ij},{\eta }_j,{\tau }_i,{\mathit{\beta }}_j,{\mathit{h}}_j,{\sigma }_{(t)j}^2\sim \text{Multinomial}(\tilde{\mathit{\pi }})\end{array}$$ (A1)

$$\begin{array}{ll}\hfill & \text{where}\widetilde{{\pi }_c}=\frac{{\pi }_c{\prod }_{j=1}^J{\theta }_{jc,y_{ij}}{\varphi }_{jc,\log t_{ij}}}{{\sum }_{c=0}^C{\pi }_c{\prod }_{j=1}^J{\theta }_{jc,y_{ij}}{\varphi }_{jc,\log t_{ij}}}.\end{array}$$ (A2)

For hyperparameter $\mathit{\pi }$, we assume a Dirichlet prior as $\mathit{\pi }\sim \text{Dirichlet}({\mathit{d}}_0)$; the conditional distribution of ${\pi }_c|\mathit{\alpha }$ is derived as:

$$\begin{array}{ll}\hfill & {\mathit{\alpha }}_i|\mathit{\pi }\sim \text{Multinomial}(\mathit{\pi })\end{array}$$ (A3)

$$\begin{array}{ll}\hfill & p({\pi }_c|{\alpha }_i={\alpha }_c)\sim \prod _{i=1}^N\prod _{c=0}^C{\pi }_c^{I\left({\alpha }_i={\alpha }_c\right)}+d_{oc},\end{array}$$ (A4)

thus, the posterior distribution is also Dirichlet with $p(\mathit{\pi }|\mathit{\alpha })\sim \text{Dirichlet}(\tilde{\mathit{N}})$, where $\widetilde{N_c}={\sum }_{i=1}^{N}I({\mathit{\alpha }}_i={\mathit{\alpha }}_c)+d_{oc}$.

For the item–attribute parameter ${\beta }_{jc}$, with the prior specified as ${\beta }_{jc}|{\sigma }_{\beta }^2\sim N(0,{\sigma }_{\beta }^2)I({\beta }_{jc}>L_{{\beta }_{jc}})$ and likelihood function specified in Equation (8), the conditional distribution of ${\beta }_{jc}$ is a normal distribution:

$$\begin{array}{ll}\hfill {\beta }_{jc}|(y_{ij}^{\ast },{\mathit{\alpha }}_i,{\mathit{\beta }}_{j(c)},{\delta }_{jc}=1)& \sim N({\tilde{\mu }}_{{\beta }_{jc}|\ast },{\tilde{\sigma }}_{{\beta }_c|\ast }^2)I({\beta }_{jc}>L_{{\beta }_{jc}}^{\ast })\end{array}$$ (A5)

$$\begin{array}{ll}\hfill p({\beta }_{jc}=0|{\delta }_{jc}=0)& =1,\end{array}$$ (A6)

where the mean ${\tilde{\mu }}_{{\beta }_{jc}|\ast }$ and variance parameters ${\tilde{\sigma }}_{{\beta }_c|\ast }^2$ are specified as

$$\begin{array}{ll}\hfill & {\tilde{\mu }}_{{\beta }_{jc}|\ast }={\tilde{\sigma }}_{{\beta }_c|\ast }^2\sum _{i=1}^N{\alpha }_{ic}(y_{ij}^{\ast }-{\mathit{\alpha }}_{i(c)}^{\prime }{\mathit{\beta }}_{j(c)})\end{array}$$ (A7)

$$\begin{array}{ll}\hfill & {\tilde{\sigma }}_{{\beta }_c|\ast }^2={\left(\frac{1}{{\sigma }_{\beta }^2}+\sum _{i=1}^N{\alpha }_{ic}^2\right)}^{-1}.\end{array}$$ (A8)

Note here $c$ refers to the index of coefficients; in some cases, when the model is deparameterized, $c$ could be smaller than the number of latent classes. Proof of $L_{{\beta }_{jc}}^{\ast }$ can be found in Chen et al. (2020).

For parameter ${\delta }_{jc}$, we assume a prior as ${\delta }_{jc}|\omega \sim \text{Bernoulli}(\omega )$; with the likelihood function specified in Equations (8) and (9), we can derive the conditional distribution of ${\delta }_{jc}$ as

$$\begin{array}{ll}\hfill & {\delta }_{jc}|(y_{ij}^{\ast },\log t_{ij},{\eta }_j,{\tau }_i,{\mathit{\alpha }}_{i(c)},{\mathit{\beta }}_{j(c)},{\mathit{h}}_{j(c)},{\sigma }_{(t)j}^2)\sim \text{Bernoulli}({\tilde{\omega }}_{jc}),\end{array}$$ (A9)

where

$$\begin{array}{l}{\tilde{\omega }}_{jc}=\frac{2\omega \mathrm{\Phi }{\left(-\frac{L_{{\beta }_{jc}^{\ast }}}{{\sigma }_{\beta }}\right)}^{-1}\mathrm{\Phi }\left(\frac{{\tilde{\mu }}_{{\beta }_{jc}|\ast }-L_{{\beta }_{jc}^{\ast }}}{{\tilde{\sigma }}_{{\beta }_c|\ast }}\right)\mathrm{\Phi }\left(-\frac{{\tilde{\mu }}_{h_{jc}|\ast }}{{\tilde{\sigma }}_{h_{jc}|\ast }}\right)\frac{{\tilde{\sigma }}_{\beta c|\ast }}{{\sigma }_{\beta }}exp\left(\frac{{\tilde{\mu }}_{{\beta }_{jc}|\ast }^2}{2{\tilde{\sigma }}_{{\beta }_c|\ast }^2}\right)\frac{{\tilde{\sigma }}_{hc|\ast }}{{\sigma }_h}exp(\frac{{\tilde{\mu }}_{h_{jc}|\ast }}{{\tilde{\sigma }}_{h_c|\ast }^2})}{2\omega \mathrm{\Phi }{\left(-\frac{L_{{\beta }_{jc}^{\ast }}}{{\sigma }_{\beta }}\right)}^{-1}\mathrm{\Phi }\left(\frac{{\tilde{\mu }}_{{\beta }_{jc}|\ast }-L_{{\beta }_{jc}^{\ast }}}{{\tilde{\sigma }}_{{\beta }_c|\ast }}\right)\mathrm{\Phi }\left(-\frac{{\tilde{\mu }}_{h_{jc}|\ast }}{{\tilde{\sigma }}_{h_{jc}|\ast }}\right)\frac{{\tilde{\sigma }}_{\beta c|\ast }}{{\sigma }_{\beta }}exp\left(\frac{{\tilde{\mu }}_{{\beta }_{jc}|\ast }^2}{2{\tilde{\sigma }}_{{\beta }_c|\ast }^2}\right)\frac{{\tilde{\sigma }}_{hc|\ast }}{{\sigma }_h}exp\left(\frac{{\tilde{\mu }}_{h_{jc}|\ast }^2}{2{\tilde{\sigma }}_{h_c|\ast }^2}\right)+(1-\omega )}.\end{array}$$ (A10)

For parameter $\omega$, we assume a prior distribution of $\omega |{\omega }_0,{\omega }_1\sim \text{Beta}({\omega }_0,{\omega }_1)$. Given that ${\delta }_{jc}\sim \text{Bernoulli}(\omega )$, we have the conditional distribution of

$$\begin{array}{l}\hfill \omega |{\delta }_{jc},{\omega }_0,{\omega }_1\sim {\omega }^{\sum _{j,c}{\delta }_{jc}+{\omega }_0-1}{(1-\omega )}^{\sum _{j,c}(1-{\delta }_{jc})+{\omega }_1-1}.\end{array}$$ (A11)

Therefore, the posterior of $\omega$ follows a distribution of $\text{Beta}({\sum }_{j,c}{\delta }_{jc}+{\omega }_0,{\sum }_{j,c}(1-{\delta }_{jc})+{\omega }_1)$.

For parameter ${\tau }_i$, we assume a normal prior as ${\tau }_i\sim N(0,1)$. With the likelihood function specified in Equation (7), the conditional distribution of ${\tau }_i$ given the response data and the other parameters $\ast$ follows a normal distribution:

$$\begin{array}{l}\hfill {\tau }_i|\log t_{ij},{\eta }_j,{\mathit{\alpha }}_i,{\mathit{h}}_j,{\sigma }_{(t)j}^2\sim N({\tilde{\mu }}_{{\tau }_i|\ast },{\tilde{\sigma }}_{\tau |\ast }^2),\end{array}$$ (A12)

where

$$\begin{array}{ll}\hfill {\tilde{\mu }}_{{\tau }_i|\ast }& ={\tilde{\sigma }}_{\tau |\ast }^2\sum _{j=1}^J\frac{{\eta }_j({\mathit{\alpha }}_i^{\prime }{\mathit{h}}_j-\log t_{ij})}{{\sigma }_{(t)j}^2}\end{array}$$ (A13)

$$\begin{array}{ll}\hfill {\tilde{\sigma }}_{\tau |\ast }^2& ={\left(1+\sum _{j=1}^J\frac{{\eta }_j^2}{{\sigma }_{(t)j}^2}\right)}^{-1}.\end{array}$$ (A14)

For the parameter ${\sigma }_{(t)j}^2$, we assume the precision parameter $\frac{1}{{\sigma }_{(t)j}^2}$ has a gamma distribution prior such that $\frac{1}{{\sigma }_{(t)j}^2}|a_t,b_t\sim \text{gamma}(a_t,b_t)$. With the likelihood function specified in Equation (7), the conditional distribution of $\frac{1}{{\sigma }_{(t)j}^2}$ given the response data and the other parameters is also a gamma distribution:

$$\begin{array}{l}\hfill \frac{1}{{\sigma }_{(t)j}^2}|\log t_{ij},{\eta }_j,{\tau }_i,{\mathit{\alpha }}_i,{\mathit{h}}_j,a_t,b_t\sim \text{gamma}\left(a_t+\frac{N}{2},b_t+\frac{{\sum }_{i=1}^N{(\log t_{ij}+{\eta }_j{\tau }_i-{\mathit{\alpha }}_i^{\prime }{\mathit{h}}_j)}^2}{2}\right).\end{array}$$ (A15)

For parameter ${\eta }_j$, we assume a normal prior as ${\eta }_j|{\sigma }_{\eta }\sim N(0,{\sigma }_{\eta }^2)$. With the likelihood function specified in Equation 7, the conditional distribution given the response data and the other parameters $\ast$ follows a normal distribution:

$$\begin{array}{l}\hfill {\eta }_j|\log t_{ij},{\tau }_i,{\mathit{\alpha }}_i,{\mathit{h}}_j,{\sigma }_{(t)j}^2\sim N({\tilde{\mu }}_{{\eta }_j|\ast },{\tilde{\sigma }}_{{\eta }_j|\ast }^2),\end{array}$$ (A16)

where

$$\begin{array}{ll}\hfill {\tilde{\mu }}_{{\eta }_j|\ast }& ={\tilde{\sigma }}_{{\eta }_j|\ast }^2\frac{{\sum }_{i=1}^N{\tau }_i({\mathit{\alpha }}_i^{\prime }{\mathit{h}}_j-\log t_{ij})}{{\sigma }_{(t)j}^2}\end{array}$$ (A17)

$$\begin{array}{ll}\hfill {\tilde{\sigma }}_{{\eta }_j|\ast }^2& ={\left(\frac{1}{{\sigma }_{\eta }^2}+\frac{{\sum }_{i=1}^N{\tau }_i^2}{{\sigma }_{(t)j}^2}\right)}^{-1}.\end{array}$$ (A18)

For the hyperparameter ${\sigma }_{\eta }^2$, its inversion $\frac{1}{{\sigma }_{\eta }^2}$ as a precision parameter follows a gamma distribution such that $\frac{1}{{\sigma }_{\eta }^2}|a_{\eta },b_{\eta }\sim \text{gamma}(a_{\eta },b_{\eta })$. With the assumed ${\eta }_j|{\sigma }_{\eta }^2\sim N(0,{\sigma }_{\eta }^2)$, the conditional distribution of $\frac{1}{{\sigma }_{\eta }^2}$ is also a gamma distribution:

$$\begin{array}{l}\hfill \frac{1}{{\sigma }_{\eta }^2}|{\eta }_j,a_{\eta },b_{\eta }\sim \text{gamma}\left(a_{\eta }+\frac{J}{2},b_{\eta }+\frac{{\sum }_{j=1}^J{\eta }_j^2}{2}\right).\end{array}$$ (A19)

For the item–attribute speed intensity parameter $h_{jc}$, the likelihood function in Equation 7 can be reframed as

$$\begin{array}{l}\hfill p(\log t_{ij}|{\eta }_j,{\tau }_i,{\mathit{\alpha }}_{i(c)},{\alpha }_{ic},{\mathit{h}}_{j(c)},h_{jc},{\sigma }_{(t)j})=N({\mathit{\alpha }}_{i(c)}^{\prime }{\mathit{h}}_{j(c)}+{\alpha }_{ic}{h}_{jc}-{\eta }_j{\tau }_i,{\sigma }_{(t)j}^2).\end{array}$$ (A20)

With the prior specified as $h_{jc}|{\sigma }_h^2\sim N(0,{\sigma }_h^2)I(h_{jc}<0)$, the conditional distribution of $h_{jc}$ is a normal distribution:

$$\begin{array}{ll}\hfill h_{jc}|(\log t_{ij},{\eta }_j,{\tau }_i,{\mathit{\alpha }}_{i(c)},{\alpha }_{ic},{\mathit{h}}_{j(c)},{\sigma }_h,{\sigma }_{(t)j},{\delta }_{jc}=1)& \sim N({\tilde{\mu }}_{h_{jc}|\ast },{\tilde{\sigma }}_{h_c|\ast }^2)I(h_{jc}<0)\end{array}$$ (A21)

$$\begin{array}{ll}\hfill p(h_{jc}=0|{\delta }_{jc}=0)& =0,\end{array}$$ (A22)

where the mean ${\tilde{\mu }}_{h_{jc}|\ast }$ and variance parameters ${\tilde{\sigma }}_{h_c|\ast }^2$ are specified as

$$\begin{array}{ll}\hfill {\tilde{\mu }}_{h_{jc}|\ast }& ={\tilde{\sigma }}_{h_{jc}|\ast }^2\frac{{\sum }_{i=1}^N{\alpha }_{ic}(\log t_{ij}+{\eta }_j{\tau }_i-{\mathit{\alpha }}_{i(c)}^{\prime }{\mathit{h}}_{j(c)})}{{\sigma }_{(t)j}^2}\end{array}$$ (A23)

$$\begin{array}{ll}\hfill {\tilde{\sigma }}_{h_{jc}|\ast }^2& ={\left(\frac{1}{{\sigma }_h^2}+\frac{{\sum }_{i=1}^N{\alpha }_{ic}^2}{{\sigma }_{(t)j}^2}\right)}^{-1}.\end{array}$$ (A24)

The term ${\mathit{\alpha }}_{i(c)}$ denotes the respondent $i$'s attribute profile with $c_{th}$ element removed; ${\mathit{h}}_{j(c)}$ denotes the item $j$'s time intensity vector with $c_{th}$ element removed; the terms ${\alpha }_{ic}$ and $h_{ic}$ denote the removed elements, respectively.

For parameter ${\sigma }_h^2$, we assume the precision parameter $\frac{1}{{\sigma }_h^2}$ has a gamma distribution prior such that $\frac{1}{{\sigma }_h^2}|a_h,b_h\sim \text{gamma}(a_h,b_h)$. The conditional distribution of $\frac{1}{{\sigma }_h^2}$ given the coefficient matrix ${\mathit{H}}_{(1)}$ is also a gamma distribution:

$$\begin{array}{l}\hfill \frac{1}{{\sigma }_h^2}|{\mathit{H}}_{(1)},a_h,b_h\sim \text{gamma}\left(a_h+\sum _{j(c>1)}{\delta }_{jc},b_h+\sum _{j(c>1)}{h}_{jc}^2\right).\end{array}$$ (A25)

Note that the ${\mathit{H}}_{(1)}$ does not include the intercept column, because we assume the intercept coefficients $h_{j1}$ are active such that ${\delta }_{jc}=1$ holds for any item $j$.

Fisher temperament inventories

TABLE A1.

Fisher temperament inventory (FTI) items.

Item	Statement
CE1	I find unpredictable situations exhilarating.
CE2	I do things on the spur of the moment.
CE3	I get bored when I have to do the same familiar things.
CE4	I have a very wide range of interests.
CE5	I am more optimistic than most people.
CE6	I am more creative than most people.
CE7	I am always looking for new experiences.
CE8	I am always doing new things.
CE9	I am more enthusiastic than most people.
CE10	I am willing to take risks to do what I want to do.
CE11	I get restless if I have to stay home for any length of time.
CE12	My friends would say I am very curious.
CE13	I have more energy than most people.
CE14	On my time off I like to be free to do whatever looks fun.
CS15	I think consistent routines keep life orderly and relaxing.
CS16	I consider (and reconsider) every option thoroughly before making a plan.
CS17	People should behave according to established standards of proper conduct.
CS18	I enjoy planning way ahead.
CS19	In general I think it is important to follow rules.
CS20	Taking care of my possessions is a high priority for me.
CS21	My friends and family would say I have traditional values.
CS22	I tend to be meticulous in my duties.
CS23	I tend to be cautious but not fearful.
CS24	People should behave in ways that are morally correct.
CS25	It is important to respect authority.
CS26	I would rather have loyal friends than interesting friends.
CS27	Long established customs need to be respected and preserved.
CS28	I like to work in a straightforward path towards completing the task.
AT29	I understand complex machines easily.
AT30	I enjoy competitive conversations.
AT31	I am intrigued by rules and patterns that govern systems.
AT32	I am more analytical and logical than most people.
AT33	I pursue intellectual topics thoroughly and regularly.
AT34	I am able to solve problems without letting emotion get in the way.
AT35	I like to figure out how things work.
AT36	I am tough‐minded.
AT37	Debating is a good way to match my wits with others.
AT38	I have no trouble making a choice even when several alternatives seem equally good at first.
AT39	When I buy a new machine (e.g. camera computer or car) I want to know all of its technical features.
AT40	I like to avoid the nuances and say exactly what I mean.
AT41	I think it is important to be direct.
AT42	When making a decision I like to stick to the facts rather than be swayed by people's feelings.
PE43	I like to get to know my friends' deepest needs and feelings.
PE44	I highly value deep emotional intimacy in my relationships.
PE45	Regardless of what is logical I generally listen to my heart when making important decisions.
PE46	I frequently catch myself daydreaming.
PE47	I can change my mind easily.
PE48	After watching an emotional film I often still feel moved by it several hours later.
PE49	I vividly imagine both wonderful and horrible things happening to me.
PE50	I am very sensitive to people's feelings and needs.
PE51	I often find myself getting lost in my thoughts during the day.
PE52	I feel emotions more deeply than most people.
PE53	I have a vivid imagination.
PE54	When I wake up from a vivid dream it takes me a few seconds to return to reality.
PE55	When reading I enjoy when the writer takes a sidetrack to say something beautiful or meaningful.
PE56	I am very empathetic.

He, S. , Culpepper, S. A. , & Douglas, J. A. (2025). A sparse latent class model incorporating response times. British Journal of Mathematical and Statistical Psychology, 78, 757–784. 10.1111/bmsp.12380

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.