2.1 Multidimensional Latent Space Item Response Models

Throughout the manuscript, $Y_{pi}$ represents an item response of respondent p ( $p = 1, \dots , P$ ) to item i ( $i = 1, \dots , I$ ). To derive extensions of the LSIRM, we start from the multidimensional IRT models with a logit link function (although a probit function or other choices can be used) such as the Rasch model and the two-parameter logistic model (2PLM), and integrate them with a latent space. First with the 2PLM, an extension of the LSIRM which we call the Multidimensional Latent Space 2PLM (MLS2PLM) can be expressed as follows:

(3) $$ \begin{align} \begin{aligned} logit(P(Y_{pi}=1 | \boldsymbol{\theta}_p, \boldsymbol{a}_i, b_i, \boldsymbol{\xi}_p, \boldsymbol{\zeta}_i)) & = \boldsymbol{a}_i^T \boldsymbol{\theta}_p + b_i - \gamma \cdot d(\boldsymbol{\xi}_p, \boldsymbol{\zeta}_i). \end{aligned} \end{align} $$

The logit of the response accuracy is modeled as a function of several parameters such as 1) $\boldsymbol {\theta }_p$ , a D-dimensional vector of latent abilities for person p, 2) $\boldsymbol {a}_i$ , a D-dimensional vector of discrimination parameters for item i, and 3) $b_i$ , an intercept for item i (usually interpreted as an overall easiness or an overall negative difficulty parameter for ability tests and as a symptom threshold parameter in clinical tests). These correspond to the multidimensional 2PLM. We restrict our interest to the cases where each item measures a single ability only, as in between-item multidimensional IRT models (Adams et al., Reference Adams, Wilson and Wang1997; Rijmen & De Boeck, Reference Rijmen and De Boeck2005). Thus, the dimensionality concerned here corresponds to that in CFA models with a simple structure and no cross loadings. Accordingly, for each item, the vector $\boldsymbol {a}_i$ has only one non-zero element. For example, if item i measures the dth latent ability, then $a_{id}$ is non-zero and freely estimated whereas all other $a_{id'}$ ( $d' \neq d$ ) are set to zero. In this case, the first term on the right side of Equation (3) reduces to $\boldsymbol {a}_i^T \boldsymbol {\theta }_p = a_{id} \theta _{dp}$ . For simplicity, we sometimes denote $a_{id} = a_{i}$ .

Another difference between the proposed MLS2PLM and the standard 2PLM is in the last term $-\gamma \cdot d(\boldsymbol {\xi }_p, \boldsymbol {\zeta }_i)$ on the right side of Equation (3). The MLS2PLM assumes that persons and items can be mapped onto a shared K-dimensional metric space as in previous latent space modeling of psychometric data. Parameters $\boldsymbol {\xi }_p \in \mathcal {R}^{K}$ and $\boldsymbol {\zeta }_i \in \mathcal {R}^{K}$ represent latent positions/coordinates of person p and item i, respectively. On the latent space, distances between persons, between items, and between persons and items can be computed. The function $d(\cdot , \cdot )$ determines how these distances are computed. In this article, we use the Euclidean distance and set $K=2$ for visualization so that $d(\boldsymbol {\xi }_p, \boldsymbol {\zeta }_i) = ||\boldsymbol {\xi }_p - \boldsymbol {\zeta }_i||_2 = \sqrt {\sum _{k=1}^{2}(\xi _{pk} - \zeta _{ik})^2}$ , as done in many previous latent space models (Handcock et al., Reference Handcock, Raftery and Tantrum2007; Hoff et al., Reference Hoff, Raftery and Handcock2002; Jeon et al., Reference Jeon, Jin, Schweinberger and Baugh2021; Kang & Jeon, Reference Kang and Jeon2024; Kang et al., Reference Kang, Jeon and Partchev2023; Smith et al., Reference Smith, Asta and Calder2019). Distances are multiplied by the distance tuning parameter $\gamma $ , which adjusts the scale differences between distances and linear predictor in the logit link function, and then included in Equation (3).

With the distance effect, the latent space can be utilized to extract valuable information to further understand the interactions between respondents and items as well as derive customized diagnoses and feedback for them. The distance effect is assumed to decrease logit accuracy. Thus, the larger the distance between person p and item i is, the more likely the person produces the incorrect response to the item. By visualizing the latent positions of persons and items in a single figure, one can obtain an interaction map with which person–item dynamics unexplained by latent abilities and item parameters can be studied. This map reveals that even respondents with similar levels of latent abilities can produce considerably different response profiles as the same item can have higher or lower probabilities of being solved by these respondents (Jeon et al., Reference Jeon, Jin, Schweinberger and Baugh2021; Kang et al., Reference Kang, Jeon and Partchev2023). If items are from clinical scales, the interaction map can show that persons with similar overall clinical states might exhibit different symptom profiles as the same item can have higher or lower probabilities of being endorsed (Kang and Jeon, Reference Kang and Jeonunder review). Person–item distances can be computed for any person–item pair to formally quantify these kinds of observations. Then, more detailed diagnoses and treatments can be made for a specific person based on the distances from this person to the items.

Because distance has a negative relationship with response probability, we can instead define and rely on a similarity measure to ease straightforward interpretations of person–item interactions. To this end, we introduce a negative exponential decay function defined as follows:

(4) $$ \begin{align} s(\boldsymbol{\xi}_p, \boldsymbol{\zeta}_i) = \exp(-\gamma \cdot d(\boldsymbol{\xi}_p, \boldsymbol{\zeta}_i)). \end{align} $$

This transformed measure $s(\boldsymbol {\xi }_p, \boldsymbol {\zeta }_i)$ has a positive relationship with response probability. Specifically, a larger similarity indicates a smaller distance implying a relatively higher response probability, whereas a smaller similarity indicates a larger distance implying a relatively lower response probability. Also, Equation (4) maps positive-valued distances onto $[0,1]$ , producing scaled measures of person–item interactions that ease comparisons. It is worth noting that this transformation has been widely used in mathematical models in psychology, such as the SIMPLE (termed for scale-independent memory, perception, and learning; Brown et al., Reference Brown, Neath and Chater2007) model of memory and Generalized Context Model (GCM; Nosofsky, Reference Nosofsky1986) of Categorization, as well as in clustering (e.g., Ng et al., Reference Ng, Jordan, Weiss, Dietterich, Becker and Ghahramani2001). Also, it can be derived from Equation (3) that

$$ \begin{align*} \begin{aligned} \frac{\pi_{pi}}{1-\pi_{pi}} & = \exp(\boldsymbol{a}_i^T \boldsymbol{\theta}_p + b_i) \cdot \exp(- \gamma \cdot d(\boldsymbol{\xi}_p, \boldsymbol{\zeta}_i)), \\ \end{aligned} \end{align*} $$

where $\pi _{pi} = P(Y_{pi}=1 | \boldsymbol {\theta }_p, \boldsymbol {a}_i, b_i, \boldsymbol {\xi }_p, \boldsymbol {\zeta }_i)$ . This equation implies that the transformed measure in Equation 4 represents the decreases in the probability of $Y_{pi} = 1$ relative to that of $Y_{pi}=0$ (i.e., odds) due to person–item interactions unexplained by the main model parameters.

It should be noted that the latent space in the proposed models is not the space of factors (latent variables or abilities). As Equation (3) shows, the model estimates factor scores $\boldsymbol {\theta }_p$ and latent positions $\boldsymbol {\xi }_p$ and $\boldsymbol {\zeta }_i$ simultaneously. The factor score for respondent p provides the overall level of this respondent regarding the ability (or any psychological construct) being measured by items. Once estimated, this factor score does not depend on and vary across items, meaning that the factor score captures the global characteristics of the respondent. Beyond this information, the person–item distances $d(\boldsymbol {\xi }_p, \boldsymbol {\zeta }_i)$ of the same respondent p to different items capture person–item interactions that vary across items. In this way, the distance effects account for item-specific variations and reactivities of the same respondent that are not captured by the factor score. These person–item interactions can be utilized to provide detailed information, diagnosis, and/or feedback for this respondent.

For more detailed descriptions of how to use the latent space models and their resulting interaction map for practical purposes, we refer the readers to some previous literature regarding the integration of latent space and psychometric models (e.g., Jeon et al., Reference Jeon, Jin, Schweinberger and Baugh2021; Kang & Jeon, Reference Kang and Jeon2024; Kang et al., Reference Kang, Jeon and Partchev2023, see also Section 4.3. of the current article for examples of utilizing CD and latent positions). Below in this article, we focus on illustrating the nested models of the MLS2PLM, the relativity of CD with simulation-based studies, empirical applications, and theoretical discussions. All these compare estimated latent spaces from the MLS2PLM and its reduced models described in the following section.

2.2 Related models

Among the proposed MLSIRMs, the most complex MLS2PLM serves as a main framework for us to illustrate the relativity of CD because, with appropriate parameter constraints, it reduces to simpler models with larger expected CD. Primarily we focus on two constraints: (1) no item discrimination parameter, i.e., $a_i = 1$ for all items ( $i = 1, \dots , I$ ) and (2) unidimensionality $\theta _{1p} = \theta _{2p} = ... = \theta _{Dp} = \theta _{p}$ for all persons ( $p = 1, \dots , P$ ).

Applying the first constraint, the MLS2PLM reduces to the multidimensional Rasch model with a latent space integrated. We will refer to this model as the Multidimensional Latent Space Rasch Model (MLSRM). Similarly applying the second constraint, the MLS2PLM reduces to the Unidimensional Latent Space 2PLM (ULS2PLM). Lastly, with both constraints applied, the MLS2PLM is simplified to the LSIRM (Jeon et al., Reference Jeon, Jin, Schweinberger and Baugh2021):

(5) $$ \begin{align} \begin{aligned} logit(P(Y_{pi}=1 | \theta_p, b_i, \boldsymbol{\xi}_p, \boldsymbol{\zeta}_i)) & = \theta_p + b_i - \gamma \cdot d(\boldsymbol{\xi}_p, \boldsymbol{\zeta}_i). \end{aligned} \end{align} $$

In this article, we will call this model the Unidimensional Latent Space Rasch Model (ULSRM), just for consistency with the other models. Table 1 provides a taxonomy of the model abbreviations for future reference.

Table 1

Taxonomy of the proposed multidimensional latent space item response models

Note: The table provides the abbreviations of the model names. See the main text for the full names.

It would be worth noting that another constraint of $\gamma = 0$ can reduce the models described above to traditional IRT models. Without latent spaces, these models assume CI and are not able to capture CD, beyond what can be explained by person and item parameters. Leaving CD unexplained may produce unwanted influences on the parameter estimation when CD is substantial. In this article, we demonstrate a potential bias in estimates due to CD by comparing the MLS2PLM, against its reduced version with $\gamma = 0$ constraint and the CI assumption, which we will call the Multidimensional IRTl (MIRM) hereafter. Note that this is just the traditional multidimensional IRT model, and we focus on the multidimensional 2PLM in the following comparisons.

2.3 Inference

Most of the previous latent space modeling approaches have exploited Bayesian methods to estimate the model parameters. Also for the proposed models in this article, we developed a Stan (Stan Development Team, Reference Team2024) program, which utilizes the Hamiltonian Monte Carlo method for model estimation. The Stan code to fit the most complex MLS2PLM can be found in Section S1 of the Supplementary Material.

Samples from the joint posterior distribution can be obtained with the following specifications of the prior distributions as our recommendations:

(6) $$ \begin{align} \begin{aligned} \boldsymbol{\theta}_p & \sim MVN(\boldsymbol{0}, \boldsymbol{\Phi}), \ \ p = 1, \cdots, P, \\ \log(a_i) &\sim N(\mu_a, \sigma_a^2), \ \ i = 1, \cdots, I, \\ b_i &\sim N(\mu_b, \sigma_b^2), \ \ i = 1, \cdots, I, \\ \boldsymbol{\Phi} &\sim LKJ(1), \\ \boldsymbol{\xi}_p &\sim MVN_K (\boldsymbol{0}, \boldsymbol{I}_K), \ \ p = 1, \cdots, P, \\ \boldsymbol{\zeta}_i &\sim MVN_K (\boldsymbol{0}, \boldsymbol{I}_K), \ \ i = 1, \cdots, I, \\ \log(\gamma) &\sim N(\mu_{\gamma}, \sigma_{\gamma}). \end{aligned} \end{align} $$

Here $MVN(\boldsymbol {\mu }, \boldsymbol {\Sigma })$ is a multivariate normal distribution with mean vector $\boldsymbol {\mu }$ and covariance matrix $\boldsymbol {\Sigma }$ , with an appropriate dimension, $N(\mu , \sigma ^2)$ is a normal distribution with mean $\mu $ and standard deviation (SD) $\sigma $ , and $LKJ(s)$ is a Lewandowski–Kurowicka–Joe distribution for a correlation matrix (Lewandowski et al., Reference Lewandowski, Kurowicka and Joe2009) with a shape parameter s. To establish the identifiability of the model, the mean vector of the latent variables $\boldsymbol {\theta }_p$ is fixed to the zero vector $\boldsymbol {0}$ and means and SDs of the latent positions $\boldsymbol {\xi }_p$ and $\boldsymbol {\zeta }_i$ are set to $0$ and $1$ , respectively. Hyperparameters can be given appropriate hyperprior distributions or specific values. For item parameters, hyperpriors can be given as, e.g., $N(0, 1^2)$ for $\mu _a$ , $N(0, 5^2)$ for $\mu _b$ and $Half$ - $Cauchy(5)$ for $\sigma _a^2$ and $\sigma ^2_b$ . Note that a prior on $\log {a_i}$ allows positive discrimination parameters only, implying that the probability of endorsing an item or giving the correct response increases as the latent trait/ability increases. This choice is based on the assumption that all items measure the target constructs appropriately (Baker, Reference Baker1985) and reverse-keyed items are also reverse-coded before the main analysis. If a more general application is required, a normal prior can be imposed on $a_i$ instead of $\log {a}_i$ . For the log-transformed distance tuning parameter, $\log (\gamma )$ , $\mu _\gamma = 0.5$ and $\sigma _\gamma = 1$ can be used as in the previous latent space approaches (Jeon et al., Reference Jeon, Jin, Schweinberger and Baugh2021; Kang et al., Reference Kang, Jeon and Partchev2023), with which $\gamma $ on its raw scale has mean of $2.718$ and SD of $3.563$ .

On the latent space, the Euclidean distance function $d(\cdot , \cdot )$ exhibits translational, rotational, and reflectional invariance with respect to the latent positions. Consequently, different configurations of these positions can yield identical distance values for all respondents and items. This issue can be resolved using matching methods commonly employed in multidimensional scaling. For the proposed models, we use Procrustes matching (e.g., Borg & Gorenen, Reference Borg and Gorenen2005, Chapter 20). After obtaining posterior samples of all model parameters, we apply this method to each posterior sample of the latent positions. First, the posterior sample of model parameters with the highest log posterior density should be identified. The latent positions within this posterior sample can serve as the reference set. Then, configurations of latent positions in all other posterior samples can be aligned with this reference set while preserving the distances of all respondent-item pairs. Once the matching procedure is complete, the convergence of Bayesian chains can be assessed, and the posterior samples can be explored for further inferences. For practical applications of the Procrustes matching in the context of latent space modeling of psychometric data, we refer readers to the R package prolsirm, which is developed based on the R package MCMCpack (Martin et al., Reference Martin, Quinn and Park2011).

2.4 Statistical test of CD using a slab-and-spike prior

If the main effect parameters such as latent variables and item parameters are insufficient to account for variations in item responses and the residual variations imply some person–item interactions, a latent space in the proposed models can account for some of these residual variations and yield useful information for diagnoses and evaluations. In contrast, in some cases, a model may already be equipped with sufficient main effect parameters to describe data variations, not requiring additional model-based mechanisms to capture CD (i.e., achieving CI). However, the distance effect assumed in the proposed models would always attempt to capture CD if a simple normal prior is given to γ as in Equation (6).

To address this concern, a regularization method with the slab-and-spike prior (Ishwaran & Rao, Reference Ishwaran and Rao2005; Mitchell & Beauchamp, Reference Mitchell and Beauchamp1988) can be given to the distance tuning parameter, as follows:

(7) $$ \begin{align} \begin{aligned} \log(\gamma) \sim (1-\delta)\cdot N_{spike}(\mu_{\gamma 0}, \sigma_{\gamma 0}^2) + \delta \cdot N_{slab}(\mu_{\gamma 1}, \sigma_{\gamma 1}^2). \end{aligned} \end{align} $$

Following the previous applications of this prior to the latent space models (Jeon et al., Reference Jeon, Jin, Schweinberger and Baugh2021; Kang et al., Reference Kang, Jeon and Partchev2023), we use $\mu _{\gamma 0} = -5$ , $\mu _{\gamma 1} = 0.5$ , and $\sigma _{\gamma 0} = \sigma _{\gamma 1} = 1$ , which lead $\gamma | \delta = 0$ (spike) to have a distribution with mean of $0.011$ , mode of $0.002$ , and SD of $0.015$ and $\gamma | \delta = 1$ (slab) to have mean of $2.718$ , mode of $0.607$ , and SD of $3.563$ . Thus, if $\delta = 0$ is selected, the prior effectively shrinks $\gamma $ to zero, removing the distance effect on the logit accuracy from the model. In contrast, if $\delta = 1$ is selected, latent positions and distance effects can be well estimated without large shrinkage toward zero.

One complication in using the slab-and-spike prior with Stan is that the program does not support sampling of a discrete parameter like $\delta $ . As an alternative, $\delta $ can be marginalized to produce a mixture distribution of $\log (\gamma )$ , with $\omega = P(\delta =1)$ and $1-\omega $ as choice proportions of the two components of the slab-and-spike prior. In this case, $\omega $ can be given a prior such as $Beta(1,1)$ for estimation. Also, the posterior inclusion probability (PIP) of $\delta $ can be obtained as

(8) $$ \begin{align} \begin{aligned} P(\delta=1 \ | \ \gamma, \omega, \mu_{\gamma 0}, \sigma_{\gamma 0}, \mu_{\gamma 1}, \sigma_{\gamma 1}) = \frac{\omega \cdot f(\gamma \ | \ \mu_{\gamma 1}, \sigma_{\gamma 1}^2)}{(1-\omega) \cdot f(\gamma \ | \ \mu_{\gamma 0}, \sigma_{\gamma 0}^2) + \omega \cdot f(\gamma \ | \ \mu_{\gamma 1}, \sigma_{\gamma 1}^2)}, \end{aligned} \end{align} $$

where $f(\cdot \ |\ \mu , \sigma ^2)$ is a density function of $N(\mu , \sigma ^2).$

For latent space modeling of psychometric data, simulation-based studies have shown that the slab-and-spike prior can correctly detect significant CD and reject ignorable residual variations (Jeon et al., Reference Jeon, Jin, Schweinberger and Baugh2021; Kang and Jeon, Reference Kang and Jeonunder review; Kang et al., Reference Kang, Jeon and Partchev2023). Thus, in this article, we employ this prior in our simulation studies and empirical illustrations of examining the relativity in CD.

2.5 Model complexity and relativity in CD

As discussed in the Introduction, psychometric models utilize latent variables and item parameters to account for person and item effects within the overall data variations. Also, by combining these main model parameters (e.g., the product of latent variables and item discrimination parameters), the models may capture some systematic person–item interactions. The remaining residual variations may imply CD and furthermore, person–item interactions that cannot be fully explained by simply combining the main effect parameters. These are the primary focus of the latent space approach.

According to this decomposition of data variations, model complexity plays a crucial role in balancing systematic and residual variations, providing some predictions for the MLSIRMs and their nested unidimensional models. Latent space models with fewer main effect parameters are more likely to detect larger CD due to reduced systematic explanations, whereas those with more main parameters would identify reduced CD or even reject it. Among the four models to compare, it is anticipated that the MLS2PLM will generally exhibit the smallest extent of CD, while the simplest ULSRM will show the largest extent. Comparing the MLSRM with the ULS2PLM has some complexity, as their main model parameters are not nested and can account for different types of main effects. Hence, the result of this comparison would be context-dependent and may vary by the appropriate dimensionality of the measurement data, item properties related to the slope of the item characteristic curve, etc.

However, this prediction is not without exceptions. Despite the similarity between the decomposition of $SST$ in linear regression and that of data variations by psychometric models, increases or decreases in model complexity of psychometric models do not always correspond to reductions or enlargements of residual variations and CD, due to several complications. This complexity arises because psychometric models often face greater uncertainty in parameter estimation compared to linear regression models. In contrast to $SSR$ and $SSE$ in linear regression, which have closed-form solutions and monotone relationships with the number of predictor variables, systematic and residual variations in psychometric models lack such simplicity and are linked to model complexity in much more intricate ways. Furthermore, there might be various sources of data variations that cannot be captured by specific person or item parameters. For the models in our examination, if there are unknown data variations that cannot be explained by adding multidimensional factors or item slope parameters, previous predictions about the extent of CD may no longer apply.

Note that the discussion above concerns possibilities rather than established findings. To explore and investigate these possibilities, we proceed with simulation studies and empirical examples using the proposed MLSIRMs and their simplified variations. These models will serve as our primary tools to examine the relative nature of CD.

3.1 Parameter recovery

3.1.2 Results

To assess parameter recovery, we obtained the point estimates of the model parameters from the posterior chains and computed their mean squared error (MSE), bias (evaluated with the absolute difference between estimates and true values), and standard error (SE). The calculation was done for each parameter but averaged across items (for $a_i$ , $b_i$ , $\boldsymbol {\zeta }_i$ ), across persons (for $\boldsymbol {\xi }_p$ ), across persons and factors (for $\boldsymbol {\theta }_{p}$ ) and across matrix elements (for $\boldsymbol {\Phi }$ ) to summarize results. For latent positions, their values were also averaged across $K=2$ dimensions.

Table 2 presents the recovery results with $\phi _{jl} = 0.30$ . The other recovery results (with $\phi _{jl} = 0.00, 0.75$ ) can be found in Section S2 of the Supplementary Material. The results show that the MLS2PLM can recover its parameters reasonably well. Statistics of person parameters ( $\boldsymbol {\theta }_{p}$ and $\boldsymbol {\xi }_p$ ) were a bit large when the number of items is small (e.g., $I_d = 8$ ), but these values were comparable to previously reported MSE values of unidimensional $\theta _p$ in the parameter recovery study of the 2PLM without CD (0.4–0.7; Hulin et al., Reference Hulin, Lissak and Drasgow1982; Natesan et al., Reference Natesan, Nandakumar, Minka and Rubright2016; Stone, Reference Stone1992). Also, recovery showed expected improvements as the number of items increased. Estimation of $\boldsymbol {\xi }_p$ became more accurate with a larger D ( $D=4$ ), because the same latent positions can be constrained with more items ( $I = I_d \times D$ ). However, for $\boldsymbol {\theta }_p$ , a larger D did not improve the estimation because it introduced more factors to estimate. In fact, what was important for $\boldsymbol {\theta }_p$ was the number of items per factor $I_d$ , not the number of total items I.

Table 2

Parameter recovery results of the multidimensional latent space two-parameter logistic model

The values of MSE, bias, and SE for the other parameters were reasonable and exhibited the anticipated effects due to the simulation conditions. For example, item parameters were generally estimated more accurately with a larger number of persons, and person parameters improved with more items. Notably, although having more items means more item parameters to be estimated, all item parameters ( $a_i$ , $b_i$ , and $\boldsymbol {\zeta }_i$ ) were estimated more accurately with larger $I_d$ and I. This would be because the additional items provided better constraints on the person parameters, which in turn, improved the calibration of item parameters.

It is important to note that this simulation study examined parameter recovery under the effect of CD. The results show that the main model parameters to capture person and item effects are not much affected by CD when a model with a latent space is employed to account for unexplained data variations implied by CD. If a model is not equipped with a component to capture CD, its effect can propagate to the recovery of the main model parameters in an unwanted way. In the next simulation study, we perform similar parameter recovery but compare the MLS2PLM and the MIRM to illustrate this point and further demonstrate the advantages of incorporating a latent space in modeling measurement data.

3.2 Impact of ignoring CD

In the second simulation study, we aimed to demonstrate the advantages of implementing a latent space for parameter recovery. To this end, we fitted the MIRM, which has the same main parameters as the MLS2PLM but does not employ a latent space, to the synthetic datasets used in the first simulation study. To balance computational efficiency with the main objectives of the study, we chose four conditions with $P = 500, 1000$ , $I_d = 8, 16$ , and $D = 2$ (i.e., the total number of items were $I = 16, 32$ ). The MIRM was fitted to the synthetic datasets using the same Bayesian estimation method as the MLS2PLM except that latent positions and distance tuning parameters were excluded. Consequently, the MIRM could not account for any residual dependencies, particularly person–item interactions that cannot be explained by the product of item discrimination parameters and latent abilities. The key question we sought to address was how much the person and item parameters of MIRM would be influenced by not accounting for the distance effect and CD considered in the data-generation process.

We compared the MLS2PLM and the MIRM based on MSE, Bias, and SE of the estimates. Figure 1 shows the results. The left, middle, and right panels present results for MSE, Bias, and SE, respectively, as denoted on top of the upper panels. The upper and lower panels present results for $a_i$ and $\boldsymbol {\theta }_p$ , respectively, as denoted at the top-left side of the left-most panels. In each panel, the blue circles with solid connecting lines and the red triangles with dashed connecting lines represent results from the MLS2PLM and the MIRM, respectively. Each panel has four points for each model, representing the computed values of the measure across four conditions as shown as $(P, I)$ on the x-axis.

Figure 1

Comparison of parameter estimation.

As shown in the figure, we restricted the comparison to the two parameters $a_i$ and $\boldsymbol {\theta }_p$ and excluded $b_i$ . This was because $b_i$ estimates from the MIRM are not comparable to the data-generating $b_i$ values used for the MLS2PLM. Due to the negative distance effect, $b_i$ of the MLS2PLM is typically much larger than those in the MIRM, unless the CD is very small. Simply put, overall sizes of $b_i - \gamma \cdot d(\boldsymbol {\xi }_p, \boldsymbol {\zeta }_i)$ correspond to those of $b_i$ in the MIRM, but the former vary across persons, making comparisons with the latter and evaluations of bias infeasible.

In general, the results show that the MLS2PLM outperformed the MIRM. Estimates from the MLS2PLM had larger SEs due to the need to estimate more parameters and higher model complexity. However, they exhibited greater reductions in bias, resulting in considerable reductions in MSEs. Importantly, the impact of CD did not decrease with increasing data size; instead, it turned out from the simulation study that CD exerted larger influences on the MIRM estimates as the data size grew.

The results in Figure 1 can be investigated from an extended perspective based on model predictions. This is because the influences of CD on parameter estimates can be propagated to predictions generated by the models. To demonstrate that the proposed MLS2PLM can yield better predictive accuracy, we performed posterior predictive checking (PPC) with the MLS2PLM and the MIRM, computed predicted item-wise and person-wise response proportions, and examined which model produced better predictions by comparing predictions against data-based item-wise and person-wise proportions. To save space, we present the results in our Supplementary Material (see Section S3), which show that capturing CD with the integrated latent space approach can improve model predictions as well.

The comparison presented here might seem unfair given that the MLS2PLM was used as the true data-generating model. However, the MLS2PLM can still perform better than the MIRM as long as data imply any form of CD. This is because this latent space model can at least partially account for CD, whereas the traditional model cannot. Moreover, even if the data-generating process adheres to the CI assumption and implies no CD at all, the MLS2PLM can reduce to the MIRM with the slab-and-spike priorFootnote ³.

Overall, the results have a clear implication that ignoring CD can distort parameter estimation in psychometric models to a great extent, potentially leading to incorrect inferences and conclusions. Using an integrated psychometric model with a latent space can facilitate more robust parameter estimation.

3.3 Relativity in the extent of CD

After establishing the appropriate parameter recovery of the MLS2PLM and demonstrating the utility of implementing a latent space in the context of parameter estimation, we proceeded to the third simulation study to illustrate the relative nature of CD. We again used the synthetic datasets generated in the first simulation study but limited our analysis to the four conditions used in the second study. The previously obtained results of the MLS2PLM were compared to the results from two reduced models, the ULS2PLM and the MLSRM, both fitted to the same datasets. Comparing the MLS2PLM to the ULS2PLM allows us to investigate the effect of underspecifing the number of factors. Similarly, comparison against the MLSRM reveals the impact of dropping the item discrimination parameters (i.e., fixing all $a_i$ ’s to 1). The difference between the models may yield not only changes in the extent of CD but also substantive and qualitative differences in unexplained interactions (e.g., patterns of latent positions). However, for this simulation, we focus on the effect of the main model parameters on the extent of CD only, because the patterns of interactions can vary across repetitions, making it hard to consistently compare them for all three models and simulation repetitions.

Both competing models would not account for certain sources of person–item interactions, potentially producing increases in the extent of CD. To compare these effects quantitatively, we examined the $\gamma $ estimates ( $\hat {\gamma }$ ) from the three models. Generally, we anticipate that models with constraints on latent variables and main item parameters would exhibit larger CD because they lose some of their capacity to systematically explain data variations. Consequently, overestimated $\gamma $ values are expected in most cases. However, exceptions may occur where $\hat {\gamma }$ is larger in a more complex model due to randomness in estimation. This outcome contrasts with what is typically expected (and can be mathematically proven) in linear regression analysis.

Comparison of the models solely based on $\hat {\gamma }$ would need justification. The parameter $\gamma $ is simply a tuning parameter to adjust the scale difference between the Euclidean distance and the other model parameters. Typically, comparing $\hat {\gamma }$ could not be meaningful in practical cases because its value can vary due to factors other than the extent of CD, such as differences in the underlying structure of latent positions, link functions, etc. For example, if one model detects largely deviant clusters of items or persons while the other model yields randomly distributed latent positions, it could be argued that the former model exhibits much larger CD even if its $\hat {\gamma }$ is smaller. In such cases, sizes of $\hat {\gamma }$ do not correctly represent the extent of CD. However, in the current simulation, data-generating values of latent positions were randomly generated from the standard multivariate normal distribution. Also, all models in comparison use the same logit link function. These constraints in simulation design help observe and evaluate changes in the extent of CD based on $\hat {\gamma }$ across three latent space models.

The comparison results are presented in Figures 2 and 3. In each figure, there are four panels of scatter plots corresponding to four conditions of P and I, as shown at the bottom-right side of each panel. In each panel, $\hat {\gamma }$ values from the MLS2PLM are plotted on the x-axis against those from the two competing models on the y-axis. The purple triangles in Figure 2 and the green circles in Figure 3 represent comparisons against the ULS2PLM and the MLSRM, respectively, as denoted by the y-axis labels. The diagonal line in each panel indicates the points at which $\hat {\gamma }$ values from the MLS2PLM and the other competing models are equivalent. Hence, circles and triangles distributed on the top-left side of the diagonal line indicate that the simpler competing models detect larger CD and produce larger $\hat {\gamma }$ than the MLS2PLM. If this is the case, increases in CD can be attributed to the parameter constraints imposed on the simpler models and their resulting decreases in systematic explanations of data variations.

Figure 2

Comparison of estimated distance tuning parameters: MLS2PLM vs ULS2PLM, i.e., the effect of underspecifying the number of factors.

Figure 3

Comparison of estimated distance tuning parameters: MLS2PLM vs MLSRM, i.e., the effect of dropping the item discrimination parameters.

The results correspond to the anticipated relativity in CD as described in the introduction. The estimates $\hat {\gamma }$ from the MLS2PLM were generally distributed around its data-generating value $1.5$ and as the data size increased, its estimation precision improved. The estimates from the competing models were mostly larger than those from the MLS2PLM. Specifically, the ULS2PLM that underspecified the number of factors produced larger CD than the MLS2PLM in almost all cases, with only two exceptions when $P=500$ and $I=32$ .

The MLSRM, which constrains all item discriminations to be 1, yielded a similar effect of increasing CD. However, there were some exceptions. Specifically, when $P = 500$ , the MLSRM produced smaller $\hat {\gamma }$ values than the MLS2PLM in $12$ and $18$ repetitions (out of $50$ ) in the conditions with $I = 16$ and $I = 32$ , respectively. When $P = 1000$ , overestimated $\hat {\gamma }$ values from the MLSRM were more consistently observed, with only one exception in the condition with $I = 32$ . Also, across both conditions of P, the MLSRM estimates of $\hat {\gamma }$ tended to be closer to those from the MLS2PLM when $I = 32$ than when $I=16$ (i.e., the green circles were closer to the diagonal lines when $I=32$ ). These findings align with expectations, considering the estimation issue with the item discrimination parameters. With larger numbers of persons (P), the MLS2PLM can more precisely estimate $a_i$ and detect the extent of CD. Consequently, there were fewer cases in which the MLS2PLM yielded larger error in estimating $\hat {\gamma }$ and the MLSRM produced smaller $\hat {\gamma }$ values. As the number of items (I) increases, more item discrimination parameters need to be estimated, which can lead to less precise estimates of $\hat {a}_i$ as well as less precisely estimated extents of CD. This could be associated with the observation that the MLSRM estimated smaller $\hat {\gamma }$ values than the MLS2PLM in more repetitions when $I = 32$ , compared to when $I = 16$ .

It might be tempting to compare the two results and examine the differences between the two constraints on CD. However, we do not pursue this here, as the actual differences can vary considerably depending on the number of underlying dimensions, the degree of misspecification, the distribution of true item discriminations, etc. A more thorough and comprehensive simulation study would be required to address this, which is beyond the scope of the current article.

4.1 Example 1: IRDT dataset

For the IRDT dataset, all four models selected the slab part of the slab-and-spike prior with the PIP $> 0.999$ , implying substantial CD. Figure 4 shows the four latent spaces from four different models, as indicated at the top-left side of the panels. All panels have the same ranges for the x- and y-axes for comparison across the models. In each panel, the gray dots represent respondents and the colored numbers represent items. Items measuring the same factor (e.g., items 1–8) were given the same color code (e.g., dark cyan).

Figure 4

Estimated latent spaces for the IRDT dataset.

The results from the ULSRM (top-left) showed seven clusters of items, corresponding to the presumed seven-factor structure. As the model had a unidimensional latent ability that was supposed to influence all items, the differences between items measuring different factors remained unexplained by the main model parameters. These residuals were captured as the item clusters on the latent space, which corresponded to the specific factors used in the previous application of the bi-factor model (Golino & Epskamp, Reference Golino and Epskamp2017). Some item clusters were close to each other, reflecting high correlations between their items beyond what can be explained by the common latent ability. Putting it all together, the latent space showed that the primary source of CD detected by the model would be the item clusters (i.e., unspecified factors). This finding generally replicated the result presented in Kang & Jeon (Reference Kang and Jeon2024). When item discrimination parameters were incorporated (top-right), some item clusters merged, for instance, items 41–48 (lime) and 49–56 (green). This implies that some inter-cluster item correlations can be largely accounted for by item discrimination parameters, leading the corresponding clusters to seemingly merge. However, this did not entirely remove the clustering patterns of items.

The latent spaces produced by the multidimensional models exhibited an important distinction. Both MLSRM and MLS2PLM chose the slab part of the slab-and-spike prior, even after controlling for the effect of the seven underlying factors. In other words, the models detected substantial CD unaccounted for by correlations between those factors, and thus, the primary source of CD from the multidimensional models is not the underlying factors and their corresponding person effects. Due to this difference, configurations of the resulting latent spaces considerably changed. Most importantly, all items were intermixed regardless of which factor they were supposed to measure. This pattern was also consistent with the interpretation that variations due to the seven developmental stages of IRDT were not the main source of CD. From simple visual inspection, it seemed that incorporating item discrimination parameters or not did not yield noticeable differences in estimated latent spaces. Their differences may be revealed by more thorough investigations with quantification of interactions based on distances. An example of this with inter-item distances is provided in our Supplementary Material (Section S6.2).

4.1.1 Reduction in the extent of CD: IRDT

The estimated value of the distance tuning parameter was $2.662$ in the simplest ULSRM and changed to 2.877 (ULS2PLM), 0.969 (MLSRM), and 0.883 (MLS2PLM). The estimates did not always decrease as the model complexity with regard to the main effect parameters increased, but except for the comparison between ULSRM and ULS2PLM, the pattern was consistent with our anticipation and the simulation results. Notably, when the number of factors and the factor structure were correctly specified in the multidimensional model, the estimates were reduced to a large degree.

The spread of latent person and item positions showed similar reductions. For instance, the estimated positions from the unidimensional models are less spread than those from the multidimensional models in Figure 4. The left section of Table 3 presents the $SD$ s of the estimated latent positions, quantifying this pattern in the spread. In general, $SD$ decreased as the model complexity increased, except that the MLS2PLM yielded slightly larger $SD$ s than the MLSRM. Potentially, this could be compensated by the reduction in $\hat {\gamma }$ .

Table 3

Statistics related to Latent Positions. Left section: IRDT, right section: ADHD

Note: CI: $95\%$ Credible intervals of $\gamma $ . SD: Standard deviations of estimated latent positions for persons ( $\hat {\xi }_{1p}, \hat {\xi }_{2p}$ ) and items ( $\hat {\zeta }_{1p}, \hat {\zeta }_{2p}$ ).

Motivated by the findings described above, we took a deeper look into the reduction in CD based on the estimated interaction terms $\hat {\gamma }\cdot d(\hat {\boldsymbol {\xi }}_p, \hat {\boldsymbol {\zeta }}_i)$ . We first computed all person–item distances, then averaged them first over persons and subsequently over items, resulting in person-wise and item-wise distance effects. This process was repeated for each model, providing each person and item with four distance estimates corresponding to the four models under examination. Figure 5 illustrates the changes in the distance effects, with person-wise estimates on the left panel and item-wise estimates on the right panel. In each panel, the x-axis lists the four models in the order of their model complexity, and the y-axis represents the averaged distance effects. Each dashed line corresponds to a single person or a single item. We used the same color codes as the latent spaces in Figure 4. Additionally, for clearer visualization, we randomly selected $10\%$ (approximately $180$ ) of the total sample for the person-wise estimates.

Figure 5

Reduction in the estimated distance effects $\hat {\gamma } \cdot d(\hat {\boldsymbol {\xi }_p}, \hat {\boldsymbol {\zeta }_i})$ as a function of model complexity: The IRDT dataset. Left: Person-wise average distance effects. Right: Item-wise average distance effects.

For both persons and items, the distance effects generally exhibited decreasing trends as a function of the number of person and item parameters incorporated into the model. The differences stemming from item discrimination parameters were weak and somewhat inconsistent, but the differences related to the dimensionality of factors were salient. Also, when the number of factors and the factor structure were adequately specified, the impact of item discrimination parameters became more consistent despite its small size. Overall, this result aligns with our expectations regarding the relativity of CD.

4.2 Example 2: ADHD dataset

We analyzed the ADHD dataset as similarly as we did for the IRDT dataset to examine the relativity in CD. The primary goal was to replicate the findings from the IRDT dataset. Figure 6 shows the four latent spaces from the four models fitted to the ADHD dataset. The associated inter-item distance matrices are provided in the Supplementary Material (Section S6.2). All models yielded that the extent of CD was substantial with the PIPs $> .999$ . In each panel, the gray numbers indicate respondents and words represent abbreviated symptoms used in the measurement. Section S6.1 of the Supplementary Material gives the full list of symptoms with their abbreviations, which can also be accessed from Silk et al. (Reference Silk, Malpas, Beare, Efron, Anderson, Hazell and Sciberras2019, p. 4).

Figure 6

Estimated latent spaces for the ADHD dataset.

The figure suggests implications similar to those from Figure 4. The estimated latent spaces from the unidimensional models in the top panels showed greater variations across both persons and items. Also, items could be separated into two groups, e.g., by the dark gray dashed diagonal line added for reference. Unlike the results from the IRDT dataset, the clustering of items measuring the same factor was less distinct. This difference might be due to the ADHD dataset containing only two factors, compared to seven in the IRDT dataset. The two ADHD factors would generate relatively weaker data variations, making themselves less prominent sources of CD. Nevertheless, the two groups of symptoms remain distinguishable.

In contrast, the items in the latent spaces from the multidimensional models did not exhibit the same distinction. Specifically, the items measuring the hyperactive symptoms (red) were split into two groups, one distributed above the items for the inattentive symptoms and the other distributed below. This pattern was commonly observed in both multidimensional models. However, when item discrimination parameters were added, persons and items gathered more closely with each other.

4.2.1 Reduction in the extent of CD: ADHD

As with the IRDT dataset, we performed further analyses to quantify the findings from the estimated latent spaces and look into the details of the reduction in CD. To this end, the estimated distance tuning parameter $\hat {\gamma }$ and $SD$ s of the estimated latent positions were calculated and presented in the right section of Table 3. The estimates for $\hat {\gamma }$ decreased when the number of factors was correctly specified, but not as a function of incorporating item discrimination parameters. Instead, the $SD$ s of latent positions mostly decreased as more main parameters were employed in the model, indicating a reduction in CD.

Figure 7 provides a more thorough look at the decreases in the distance effects, person-wisely (left panel) and item-wisely (right panel). As in Figure 5, each dashed line corresponds to a single person or a single item. Replicating the findings from the IRDT dataset, persons and items showed generally decreasing extent of CD as a function of model complexity. The differences between unidimensional and multidimensional models were relatively small compared to those found in the IRDT dataset, which can be attributed to the fewer number of factors. However, rise and drop in the size of distance effects due to the factor dimensionality were consistently observed across most persons and items. Changes due to item discrimination were also consistent. Taken together, the four models applied to the ADHD dataset produced results similar to those observed in our first empirical examples, providing comparable implications on the relative nature of CD.

Figure 7

Reduction in the estimated distance effects $\hat {\gamma } \cdot d(\hat {\boldsymbol {\xi }_p}, \hat {\boldsymbol {\zeta }_i})$ as a function of model complexity: The ADHD dataset. Left: Person-wise average distance effects. Right: Item-wise average distance effects.

4.3 Qualitative differences in latent spaces across models

The primary reason we discussed and examined the relative nature of CD was its substantive implications in practical data analysis, particularly in studying unexplained person–item interactions and their implied individual differences, beyond the changes in the extent of CD. To illustrate these points, we revisit our empirical examples again. With the IRDT dataset, we show that a largely different item network can emerge on the latent space depending on which person and item parameters are incorporated in a model. With the ADHD dataset, we demonstrate that choices of main model parameters can lead to different configurations of latent positions, which in turn lead to different evaluations and diagnoses for respondents derived from CD.

4.3.1 Changes in item networks due to main model parameters

To examine qualitative differences in the configurations of latent spaces and the patterns due to CD, Figure 8 presents two latent spaces of the IRDT dataset, obtained from ULSRM and MLS2PLM (left panels). These latent spaces were previously shown in Figure 4. However, unlike Figure 4, which uses the same x- and y-axis ranges for all estimated latent spaces to compare the extent of CD, Figure 8 removes this constraint, allowing each latent space to have its own axis ranges. As a result, the latent space of the MLS2PLM is zoomed in as its positions are more densely clustered compared to those in ULSRM. Also presented in Figure 8 are the inter-item distance matrices (right panels). For this matrix-like visualization of item networks, distances between items in the latent spaces were computed and color-coded according to the legend on the right side of Figure 8.

Figure 8

Estimated latent spaces for the IRDT dataset for the ULSRM and the MLS2PLM and their associated inter-item distance matrices.

As described in Section 4.1, the ULSRM yielded the latent space (the top-left panel of Figure 8) in which item clusters emerged, each of which corresponds to the seven developmental stages considered in IRDT. This could be attributed to the unidimensional latent variable employed in the ULSRM which was not able to sufficiently capture variations due to factor differences. The inter-item matrix plot (the top-right panel of Figure 8) also confirms this pattern, showing a strong block-diagonal structure.

In contrast, the MLS2PLM yielded largely different configurations of latent positions. Most of all, there were seemingly no item clusters associated with the factor structure of IRDT, as judged by the item positions on the latent space (the bottom-left panel of Figure 8) and their inter-item distances (the bottom-right panel). This was because the variations across item clusters due to factor differences were effectively captured by the multidimensional latent variables. However, CD from the remaining variations still indicated substantial person–item interactions, as detected by the slab-and-spike prior with PIP $> .999$ . This means that there were other sources of interactions not accounted for by the main model parameters of the MLS2PLM.

The latent space from the MLS2PLM allows more sophisticated analyses of unexplained interactions. Focusing on the inter-item interactions, a new interaction map reveals some heterogeneity between items measuring the same factor. The most clear pattern can be found from items 17–24 (color-coded as dark-blue), which exhibited two small groups, one on the bottom side (items 17–20) and the other on the top side of the latent space. The inter-item distance matrix also clearly showed this pattern, as highlighted by the thick dark-blue square. This represents that, even though these items were designed to assess the analogy at a specific developmental stage, they exhibited distinct item characteristics. A similar pattern can be found from items 41–48 (lime) as items 44–48 were located roughly around (0.5, $-$ 1.0), item 43 at the origin, and items 41 and 42 at the top side of the latent space. Also for items 25–32 (red), items 25 and 26 exhibited a unique association as they were located far left side of the latent space whereas the other items 27–32 were distributed roughly around (0.5, 0.5).

This investigation can be extended to person–person and person–item relationships to show substantive differences between different models due to the relativity in CD. For instance, as can be seen from the top-left panel of Figure 8, any single person would be judged to have similar distances to items 25–32 (red) by the ULSRM. In contrast, the bottom-left panel of the same figure shows that distances from the same person would vary to a large degree between items 25–26 and the other red items in the result by the MLS2PLM. In this way, different model structures and the relevant relativity in CD can produce largely distinct interpretations of CD and interactions between persons and items.

4.3.2 Utilizing CD for personalized diagnoses: Dual importance of explained and unexplained data variations

We revisit the ADHD dataset to continue our illustration of the relative nature of CD and its consequences in substantive interpretations of person–item interactions. This time, we focus on possible differences in deriving personalized diagnoses and evaluations for different respondents. To this end, we selected two respondents, with ID numbers 94 and 120 (hereafter referred to as P94 and P120), for illustration.

The two histograms on the left side of Figure 9 display the distributions of the estimated latent traits $\hat {\theta }_{1}$ (inattentiveness) and $\hat {\theta }_{2}$ (hyperactiveness), representing the main systematic person effects. Within each histogram, two vertical lines indicate the locations of the latent scores for P94 and P120. As these lines show, the two selected respondents had nearly identical latent scores, meaning that their overall levels of inattentive and hyperactive symptoms were very similar.

Figure 9

Factor score histograms, latent space, and individual symptom profiles from the MLS2PLM applied to the ADHD dataset.

The top-right panel of Figure 9 shows the latent space estimated from the MLS2PLM. This is the same as the one in Figure 6 but now estimated latent positions of P94 and P120 are highlighted with the enlarged ID numbers and the colored dashed circles surrounding them. Below the latent space, two bar charts for P94 (left) and P120 (right) are presented. These bar charts present the symptom similarity (i.e., the negative exponential transformation of a distance using Equation (4)) profiles of each respondent with different items. An individual bar corresponds to the similarity of the selected respondents to the item shown on the bar. The nine bluish bars on the top and the other nine reddish bars on the bottom correspond to inattentive and hyperactive symptoms, respectively. Also, darker colors indicate higher similarities (closer distances and stronger symptoms) while lighter colors represent lower similarities (farther distances and weaker symptoms). To facilitate comparison, items are ordered according to their similarities to P94 on the latent space estimated from the MLS2PLM.

Despite having nearly identical factor scores, the two respondents were located farther away from each other on the estimated latent space, particularly along the y-axis with $\hat {\xi }_{2,94} = 0.717$ and $\hat {\xi }_{2,120} = -0.707$ . As a result, they had largely different profiles of person–item distances. For instance, P94 was closer to hyperactive symptoms on the top side of the latent space, such as interrupt, turn, and talks and accordingly showed higher similarities with them. In contrast, P120 was generally distant from the items, with the closest items being listen (inattentive) and seat (hyperactive). As higher similarities (i.e., smaller distances) are associated with higher probabilities of endorsing items, these differences between the two selected respondents suggest that their specific ADHD symptoms were notably distinct, despite their similar factor scores. This heterogeneity, uncaptured by estimated factor scores, can be further investigated with the latent space as illustrated and utilized to provide personalized diagnoses and feedback for different respondents.

The individual symptom profiles discussed thus far were based on the MLS2PLM. However, the relative nature of CD suggests that using a different model can considerably alter the results. To illustrate this, we examined the result from the ULSRM again. Figure 10 presents the relevant latent space as well as the individual symptom profiles of the two selected respondents, derived from the ULSRM. The single-factor scores of these two respondents were still very similar, with $-$ 0.157 for P94 and $-$ 0.172 for P120.

Figure 10

Individual symptom profiles from the ULSRM applied to the ADHD dataset.

Comparing the ULSRM results from those in Figure 9, it can be noticed that the individual profiles changed considerably. Three key differences emerged: (1) similarities were generally lower (i.e., larger distances) in the ULSRM results (see the range of the x-axis of the profiles), (2) latent positions of P94 and P120 were notably closer in the ULSRM results, leading to relatively similar profiles, and (3) both respondents have very low similarities to hyperactive symptoms, unlike those in the MLS2PLM results. Additionally, as can be seen by comparing changes in the distance profiles of P94 and P120, differences between models would also vary across respondents. Other respondents may show larger or smaller changes across models depending on their estimated latent positions. All these differences ultimately can lead to different diagnoses and evaluations based solely on the choice of model.

The individual symptom profiles underscore the value of studying CD, as they provide valuable insights into person–item interactions, individual differences, and personalized assessment of individuals (as well as items, although not shown). Taking this further, comparing the profiles from the two models (MLS2PLM and ULSRM) leads us back to the importance of considering the relative nature of CD. Models with varying complexities (due to the main model parameters) yield different levels of systematic explanations, producing different extents and configurations of CD, and consequently, different diagnoses, evaluations, and feedback. Also, some aspects of CD may reflect systematic variations that can be accounted for by suitable and interpretable main effect parameters (e.g., specifying factors with an appropriate dimension). Therefore, selecting a model with appropriate main effect parameters is crucial to simultaneously study the overall states of individuals (e.g., by factor scores) and person-wise specificities (e.g., by latent positions and distance effects).