3.1.1 ULD and trait dependence as multi-dimensionality of the p-process

LD due to unmodeled constructs has been named as ULD (Chen & Thissen, Reference Chen and Thissen1997) or as trait dependence (Marais & Andrich, Reference Marais and Andrich2008) and is modeled by assuming a multi-dimensional SRF $P(X|\Theta )=\pi (K|\Theta )$ . In the bivariate case, if GLI is imposed to factorize the SRF in the power set case, one obtains LI, that is, $\pi (K_i, K_{i'}|\Theta )=\pi (K_i|\Theta )\pi (K_{i'}|\Theta )$ . A logit-link function with a suitable choice of kernel reparameterizes the marginal IRFs to, for example, Rasch or 2PL models. If under LI, one assumes, for instance, ${\Theta = \{\theta , \{u_t\}_{t\in T}\},}$ where T is an index set for additional dimensions, due, for instance, to different testlets, one obtains the random effect testlet model (see, e.g., Bradlow et al., Reference Bradlow, Wainer and Wang1999) given by

(25) $$ \begin{align} P(X_{i} =1|\theta, u_t) = \pi(K_i=1|\theta, u_t) = \frac{e^{\theta-b_{i}-u_t}}{1+e^{\theta-b_{i}-u_t}}. \end{align} $$

Models like (25) belong to the general family of bifactor models (see, e.g., Gibbons & Hedeker, Reference Gibbons and Hedeker1992). More in general, any factor model strategy or approach to LD in which additional factors are specified to account for LD violations (see, e.g., Yen, Reference Yen1993) can be subsumed within a multi-dimensional SRF $\pi (K|\Theta )$ .

3.1.2 Response dependence: Factorization of the p-process via the chain rule of probability

RD (Marais & Andrich, Reference Marais and Andrich2008) aims at capturing Table A by assuming that solving item $q_{i'}$ is conditional to the resolution of item $q_i$ . The SRF is factorized using the chain rule of probability into the product $P(X_i, X_{i'}|\theta ) = P(X_i|\theta )P(X_{i'}|X_i, \theta )$ and a logit-link reparameterization is applied to the marginal and conditional IRFs $P(X_i|\theta )$ and $P(X_{i'}|X_i,\theta )$ with kernels $f(X_{i}, \theta )=\theta -b_{i}$ and $f(X_{i'}, \theta )=\theta -b_{i'}-(1-2X_i)d$ which changes the overall difficulty of the $i'$ -th item by either adding or subtracting to $b_{i'}$ a coefficient $d> 0$ conditionally on the solution of the first item. One then obtains

(26) $$ \begin{align} P(X_{i} =x_i|\theta) = \frac{e^{x_{i}(\theta-b_{i})}}{1+e^{\theta-b_{i}}} \quad\text{and}\quad P(X_{i'} = x_{i'}|X_i=x_i, \theta) = \frac{e^{x_{i'}[\theta-b_{i'}-(1-2x_i)d]}}{1+e^{\theta-b_{i'}-(1-2x_i)d}}. \end{align} $$

3.1.3 Order and combination dependency: Reparameterization of the p-process to divide-by-total models

Hoskens and De Boeck (Reference Hoskens and De Boeck1997) distinguished between OD and CD. The former is asymmetric and associated with a sequence of items. The latter is symmetric and implies an interaction between the items. Both are modeled by adding constant parameters to the log-odds of the response patterns. OD (CD) captures Table A (B). These forms are similar to, yet different from, RD and SLD (see below). Both OD and CD follow from a direct reparameterization of the SRF by means of a log-link function. This yields a multinomial/softmax form

(27) $$ \begin{align} P(X_i=x_i, X_{i'}=x_{i'} |\theta, \Gamma) = \frac{e^{f(x, \theta,\Gamma )}}{\sum\limits_{y\in\{0,1\}^2} e^{f(y,\theta, \Gamma)}} \end{align} $$

for the joint distribution of the outcomes $x=\{x_i,x_{i'}\}\in \{0,1\}^2$ , with a choice of kernels

(28) $$ \begin{align} f(x, \theta,\Gamma ) = & \begin{cases} x_i(a\theta-b_i)+x_{i'}(a\theta-b_{i'})-x_{i}(-1)^{1-x_{i'}}b_{ii'} & \text{for OD}\\ x_i(a\theta-b_i)+x_{i'}(a\theta-b_{i'})-x_{i'}x_ib_{ii'} & \text{for CD,} \end{cases} \end{align} $$

in which the interaction parameter $b_{ii'}$ captures the dependence effect. The parameter a is a constant ability weight. A graphical comparison of OD and CD with LI Rasch models and RD is available in the Supplementary Material. Both OD and RD model Table A but make different assumptions: OD is a divide-by-total reparameterization of the SRF $\pi (K|\theta )$ , whereas RD factorizes the SRF via the chain rule of probability (and is thus more akin to a sequential model). Similarly, both CD and SLD model Table B but make different assumptions: CD is a divide-by-total model reparameterization of the SRF, while SLD is a mixture of tables using deterministic LD (see below). Finally, since Bell et al. (Reference Bell, Pattison and Withers1988) provided evidence that LD can be a function of student ability, Hoskens and De Boeck (Reference Hoskens and De Boeck1997) modeled “dimension-dependent” CD and OD by replacing the term $b_{ii'}$ with $b_{ii'}-a\theta $ in the kernels of (27). These are sub-cases of the approach given in the next section.

3.1.4 Locally dependent latent trait models: Reparameterization of the p-process to divide-by-total models

Generalization of Eq. (27) to arbitrary tables was provided by Ip (Reference Ip2002). The resulting models are known as locally dependent latent trait models or generalized log-linear models and are members of the exponential family with canonical functional parameters $\omega _i(\theta )$ for the main effects, $\omega _{ii'}(\theta )$ for the second-order interaction effects, and so on up to $\omega _{1\dots |Q|}(\theta )$ for the interaction of all items, that is,

(29) $$ \begin{align} \log{P(X=x|\theta, \Gamma)} = \sum_{i=1}^{|Q|}x_i\omega_i(\theta) + \sum_{i<j=1}^{|Q|}x_ix_j\omega_{ij}(\theta)+\cdots + \prod_{i=1}^{|Q|}x_i\omega_{1\dots |Q|}(\theta)-k\omega(\theta), \end{align} $$

where $k\in \mathbb {R}$ and $\omega (\theta )$ is a normalization term. In the bivariate case, Eq. (29) yields model (27) with a kernel $f(x, \theta ,\Gamma ) = x_i\omega _i(\theta , \Gamma _i)+x_{i'}\omega _{i'}(\theta , \Gamma _{i'})+x_{i'}x_i\omega _{ii'}(\theta , \Gamma _{ii'})$ . In order to verify that Eq. (29) is a log-link reparameterization of the SRF (23) when $\mathcal {K}= 2^Q$ , one needs to set $\lambda _{\emptyset }(\theta )=-k\omega (\theta )$ , $\lambda ^{\{q_i\}}_K(\theta )=K_i\omega _i(\theta )$ , and $\lambda ^{\{q_i, q_{i'}\}}_K(\theta )=K_iK_{i'}\omega _{ii'}(\theta )$ with $K_i$ such that $K_i=1$ if $q_i\in K$ , zero otherwise, and $K_i=X_i$ since there is no g-process so that $P(X|\theta )=\pi (K|\theta )$ . Given Eq. (29), LI is obtained by setting the main effects to the marginals $\omega _i(\theta )=P(X_i=1|\theta )$ and by setting $\omega _{ii'}(\theta )=0$ for the interaction. Ip (Reference Ip2002) discusses both non-reproducible and semi-reproducible kernels. Non-reproducible means that the item parameters in the marginals do not have the same interpretation that they would have in, for example, a Rasch or a Birnbaum model (see, e.g., Fitzmaurice et al., Reference Fitzmaurice, Laird and Rotnitzky1993; Ip, Reference Ip2002). In the non-reproducible case fall the kernels needed to obtain OD and CD since the calculation of the marginal IRFs like yield models that are different from a Rasch or a Birnbaum model. In the semi-reproducible case, one can set ${\omega _i(\theta )=P(X_i=1|\theta )}$ as in the LI case while allowing for non-null interaction terms like $\omega _{ii'}(\theta )$ . The generalization of Eq. (29) to a multi-dimensional vector $\Theta $ of latent traits is straightforward. If the functional parameters $\omega $ with order greater than or equal to two are set to null, one obtains the LKS model (24) in the case $\mathcal {K}=2^Q$ . If the power set is replaced by a chain and the elements $q_i$ are interpreted as item thresholds, locally dependent latent trait models are thus equivalent to divide-by-total polytomous IRT models.

3.1.5 Other assumptions: Weakened forms of LI, Bahadur’s representation, and copula functions

The conceptualization of Table B as positive covariance leads to two generalizations of LI: local non-negative dependence (LND; Holland, Reference Holland1981) and weak LI in expectation (WLIE; Ip, Reference Ip2001). LND arises from the observation that if items are positively correlated, then frequencies increase in the main diagonal and decrease in the secondary diagonal, thus replacing LI with the boundary conditions

(30) $$ \begin{align} \begin{aligned} P(X_i=1, X_{i'}=1|\theta) & \geq P(X_i=1|\theta)P(X_{i'}=1|\theta)\\ P(X_i=0, X_{i'}=0|\theta) & \geq P(X_i=0|\theta)P(X_{i'}=0|\theta)\\ P(X_i=1, X_{i'}=0|\theta) & \leq P(X_i=1|\theta)P(X_{i'}=0|\theta), \end{aligned} \end{align} $$

which by Eq. (3) are equivalent to assert that the covariance is non-negative. LND thus provides boundary conditions to the SRF $P(X|\theta )=\pi (K|\theta )$ in the form of inequality constraints. In alternative, Ip (Reference Ip2000, Reference Ip2001) applied Bahadur’s representation of a joint distribution to generalize the LI assumption. In the bivariate case, the approach yields a reproducible SRF of the form

(31) $$ \begin{align} \begin{aligned} P(X_i, X_{i'}|\theta) & = P(X_i|\theta)P(X_{i'}|\theta)\left[ 1+ \rho_{ii'}(\theta)Z_i(\theta)Z_{i'}(\theta)\right]\\ \text{with}\quad & Z_i(\theta) = \frac{X_i-P(X_i=1|\theta)}{\sqrt{P(X_i=1|\theta)P(X_i=0|\theta)}}\quad\text{and}\quad \rho_{ii'}(\theta) = E[Z_iZ_{i'}|\theta], \end{aligned} \end{align} $$

in which the correlation function $\rho _{ii'}(\theta )$ captures the LD in Table B. Generalization to higher orders adds terms of the form $\rho _{I}(\theta )\prod _{i\in I}Z_i$ with $\rho _{I}(\theta )=E[\prod _{i\in I}Z_i|\theta ]$ for I a set of indices. The approach encompasses a wide range of conditions weaker than strong LI (1), which is obtained if $\rho _{I}(\theta )=0$ at every order, and stronger than weak LI (3), which is obtained if only $\rho _{ii'}(\theta )$ to zero. WLIE is obtained if $E_{\theta }[\rho _{ii'}(\theta )]=0$ .

Reproducible marginals can also be obtained via copula functions, that is, functions that link multi-variate distributions to their margins (see, e.g., Braeken et al., Reference Braeken, Kuppens, De Boeck and Tuerlinckx2013; Braeken, Reference Braeken2011; Braeken et al., Reference Braeken, Tuerlinckx and De Boeck2007). As in factor analysis for categorical variables, latent response variables $X_i^* = \theta -b_i+\epsilon _i$ are assumed such that $X_i=1$ if $X_i^*>0,$ that is, $\epsilon _i> b_i-\theta $ , zero otherwise. The IRF $P(X_i=1|\theta )=\pi (K_i=1|\theta )$ can be obtained by the marginal cumulative distribution of the residuals $\epsilon _i$ by observing that

$$ \begin{align*} P(X_i = 0|\theta) = P(X_i^* <0) = P(\epsilon_i < b_i-\theta)= F_{\epsilon_{i}}(b_{i}-\theta). \end{align*} $$

By applying a copula function $C: [0,1]^2\rightarrow [0,1],$ the SRF $P(X|\theta )=\pi (K|\theta )$ for a $2\times 2$ table is

(32) $$ \begin{align} \begin{aligned} P(X_i=0, X_{i'}=0|\theta) & = C(F_{\epsilon_{i}}(b_{i}-\theta), F_{\epsilon_{i'}}(b_{i'}-\theta))\\ P(X_i=1, X_{i'}=0|\theta) & = F_{\epsilon_{i'}}(b_{i'}-\theta)-C(F_{\epsilon_{i}}(b_{i}-\theta), F_{\epsilon_{i'}}(b_{i'}-\theta))\\ P(X_i=0, X_{i'}=1|\theta) & = F_{\epsilon_i}(b_i-\theta)-C(F_{\epsilon_{i}}(b_{i}-\theta), F_{\epsilon_{i'}}(b_{i'}-\theta))\\ P(X_i=1, X_{i'}=1|\theta) & = 1-F_{\epsilon_{i}}(b_{i}-\theta)-F_{\epsilon_{i'}}(b_{i'}-\theta)+C(F_{\epsilon_{i}}(b_{i}-\theta), F_{\epsilon_{i'}}(b_{i'}-\theta)) \end{aligned} \end{align} $$

so that the marginal IRFs are reproducible. As the copula functions lie in the region of $[0,1]^2$ established by the Fréchet–Hoeffding bounds, it is interesting to consider the upper bound to monotone increasing positive dependence given by $C(x,y)=\min {(x,y)}$ . In such a case, one retrieves Table A if $b_{i'}>b_i$ and Table B if $b_i=b_{i'}$ , so this boundary case can be considered a case of deterministic LD (see the next section). For a discussion of the same limits in the context of conditional covariance curves, see Douglas et al. (Reference Douglas, Kim, Habing and Gao1998).

3.2.1 Boundary mixture copulas and SLD: Single g-process models

Boundary mixture copulas (Braeken, Reference Braeken2011) and SLD (Chen & Thissen, Reference Chen and Thissen1997) are obtained assuming a competence-dependent g-process and no p-process. Let $\mathcal {S}=\{\{2^Q\},\{\mathcal {K}\}\}$ collect the nominal latent classes expressing the belonging of individuals to either a power set $(Q,2^Q,P_{2^Q})$ or an arbitrary structure $(Q, \mathcal {K}, P_{\mathcal {K}})$ , respectively, modeling LI and LD. Let $(Q,2^Q,P_t)$ model the responses, then the likelihood of the response patterns is a mixture of competence-dependent g-processes, that is,

(33) $$ \begin{align} P_t(X|\theta) & = \sum_{F\in\mathcal{S}} P(X|F,\theta)\pi(F) = P(X|\{2^Q\},\theta)\pi(\{2^Q\})+P(X|\{\mathcal{K}\},\theta)\pi(\{\mathcal{K}\})\nonumber\\ & = (1-\delta_t) P_{2^Q}(X|\theta)+\delta_t P_{\mathcal{K}}(X|\theta), \end{align} $$

in which in the last passage, we have simply set $\pi (\{2^Q\}) = 1-\delta _t$ , $\pi (\mathcal {K})=\delta _t$ , $P_{2^Q}(X|\theta )= P(X|\{2^Q\},\theta )$ , and $P_{\mathcal {K}}(X|\theta ) = P(X|\mathcal {K},\theta )$ so that $\delta _t\in [0,1]$ is the mixture parameter representing the probability of being in the structure $\mathcal {K}$ rather than in the power set. It can be shown that SRF (33) encompasses both the boundary mixture copula approach and SLD. Let us first consider mixture copula models, which assume a mixture of two distributions, respectively, modeling LI and LD, that is, one has the SRF

(34) $$ \begin{align} P_t(X_i,X_{i'}|\theta) = (1-\delta_t)P(X_i|\theta)P(X_{i'}|\theta)+\delta_t P(X_i, X_{i'}|\theta), \end{align} $$

where $P(X_i,X_{i'}|\theta )$ is the SRF given by Eq. (32), and $\delta _t\in [0,1]$ is a mixture parameter. The SRF (33) encompasses (34) since the latter is the case $\mathcal {K}=2^Q$ . Boundary mixture copulas are obtained for the upper bound of maximal positive association $C(x,y)=\min {(x,y)}$ , so that (34) is a mixture of a power set and of the structure $\mathcal {K}_A$ if $b_{i'}> b_i$ , or $\mathcal {K}_B$ , if $b_i=b_{i'}$ . As Eq. (33) considers arbitrary SRFs, it is also more general.

Moving to SLD, it is modeled by assuming that with probability $1-\pi _{LD}$ weak LI holds, while $X_{i'} = X_i$ holds with probability $\pi _{LD}$ regardless of $\theta $ . More in detail, SLD is often defined by

(35) $$ \begin{align} \begin{aligned} \text{with prob. } 1-\pi_{LD}: \quad X_{i'} = \begin{cases} 1 & \text{with}\quad P(X_{i'}=1|\theta) \\ 0 & \text{with}\quad P(X_{i'}=0|\theta) \end{cases}, & \quad \text{with prob. } \pi_{LD}: \quad X_{i'} = \begin{cases} 1 & \text{for}\quad X_i=1 \\ 0 & \text{for}\quad X_i=0 \end{cases} \end{aligned} \end{align} $$

and thus implies a mixture of an incomplete Table B, occurring with probability $\pi _{LD}$ and a complete one, occurring with probability $1-\pi _{LD}$ and expected to obey LI. It is convenient to rewrite (35) as

(36) $$ \begin{align} P^*(X_{i'}=1|\theta) = (1-\pi_{LD})P(X_{i'}=1|\theta) + \pi_{LD}P^*(X_{i} =1|\theta), \end{align} $$

in which $P^*$ (P) represents a probability under LD (LI). Eq. (36) is the symmetric case of the approach to LD suggested by Ackerman and Spray (Reference Ackerman and Spray1987), a proof is given in Appendix A. It can then be shown that Eq. (36) is the marginal distribution of item $q_{i'}$ according to Eq. (33) for the structure $(Q,\mathcal {K}_B,P_B)$ , that is,

$$ \begin{align*} P_t(X_{i'}=1|\theta) = P_t(X_i = 0, X_{i'}=1|\theta)+ P_t(X_i=1, X_{i'}=1|\theta) = (1-\delta_t) P_{2^Q}(X_{i'}=1|\theta) + \delta_t P_{B}(X_i = 1|\theta), \end{align*} $$

it is sufficient to rename $\delta _t$ as $\pi _{LD}$ and to properly identify the probabilities.

3.2.2 Two-process KST-IRT models with competence-independent g-process

Under deterministic LD, prohibited response patterns can be obtained via a g-process $(\mathcal {K}, 2^Q)$ . In order to discuss approaches to LD based on the KST-IRT models suggested by Noventa et al. (Reference Noventa, Spoto, Heller and Kelava2019), we impose a competence-independent g-process. The probabilities $P(X|K)$ capture in the cases of Tables A and B the g-processes $(\mathcal {K}_A, 2^Q)$ and $(\mathcal {K}_B, 2^Q)$ . Considerations on identifiability and parameters interpretation are given later. We argue that an advantage of the deterministic LD approaches over both LI-based and probabilistic LD approaches is that extreme or inflated values of the difficulty or discrimination parameters are not needed to obtain small values of $p_{01}$ in Table A and of $p_{01}$ and $p_{10}$ in Table B since slipping and guessing parameters provide said proportions. To illustrate such a point, comparisons of contingency tables obtained using the different methods are provided for both Tables A and B. The fact that deterministic LD requires less extreme values of the parameters appears to be in line with the result that some types of deterministic LD like SLD are more easily detected than probabilistic LD like ULD. They are indeed modeled via different mechanisms: the former is captured by the structure, the latter by the SRFs.

Table A: Let $Q=\{q_1,q_2\}$ with $q_1$ a prerequisite to $q_2$ (e.g., “ $2+3=?$ ” and “ $\frac {1}{2}+\frac {1}{3}=?$ ”). Application of the $\Theta $ -BLIM (19) to Table A yields

(37) $$ \begin{align} \begin{bmatrix} P(X_1 = 0, X_{2} = 0|\theta)\\ P(X_1 = 0, X_{2} = 1|\theta)\\ P(X_1 = 1, X_{2} = 0|\theta)\\ P(X_1 = 1, X_{2} = 1|\theta) \end{bmatrix}= \begin{bmatrix} (1-\eta_1)(1-\eta_{2}) & \beta_{1}(1-\eta_{2}) & \beta_{1}\beta_{2}\\ (1-\eta_1)\eta_{2} &\beta_1\eta_{2}& \beta_1(1-\beta_{2})\\ \eta_{1}(1-\eta_{2}) & (1-\beta_1)(1-\eta_{2}) & (1-\beta_1)\beta_{2} \\ \eta_1\eta_{2} & (1-\beta_1)\eta_{2} & (1-\beta_1)(1-\beta_{2}) \end{bmatrix}\begin{bmatrix} \pi(\emptyset|\theta)\\ \pi(\{q_1\}|\theta)\\ \pi(\{q_1,q_{2}\}|\theta) \end{bmatrix}, \end{align} $$

where the SRFs $\pi (K|\theta )$ capture the p-process $(\mathcal {C}, \mathcal {K}_A)$ . Once the SRFs have been specified, integration of Eq. (37) yields the proportions $p_{x_1x_{2}}$ in the contingency table. The SRFs can be given either a sequential form as in the $\Theta $ -SLM (21) or a divide-by-total form as in the LKS (24). As the former is akin to RD (26), while the latter is akin to OD in Eq. (27), we reparameterize the SRF $\pi (K|\theta )$ by, respectively, using a logit link function and a softmax function for sake of comparability, thus obtaining the SRFs:

$$ \begin{align*}\nonumber \Theta\text{-SLM}: \begin{cases} \pi(X_1 = 0, X_{2} = 0 |\theta) = 1 - \frac{e^{\theta-b_{1}}}{1+e^{\theta-b_{1}}}\\ \pi(X_1 = 1, X_{2} = 0 |\theta) = \frac{e^{\theta-b_{1}}}{1+e^{\theta-b_{1}}}(1 - \frac{e^{\theta-b_{2}}}{1+e^{\theta-b_{2}}})\\ \pi(X_1 = 1, X_{2} = 1 |\theta) = \frac{e^{\theta-b_{1}}}{1+e^{\theta-b_{1}}}\frac{e^{\theta-b_{2}}}{1+e^{\theta-b_{2}}} \end{cases}\quad \!\!\!\!\!\!\text{LKS}: \begin{cases} \pi(X_1 = 0, X_{2} = 0 |\theta) = \frac{1}{1+e^{(\theta-b_{1})}+e^{(\theta-b_{1})+(\theta-b_{2})}}\\ \pi(X_1 = 1, X_{2} = 0 |\theta) = \frac{e^{(\theta-b_{1})}}{1+e^{(\theta-b_{1})}+e^{(\theta-b_{1})+(\theta-b_{2})}}\\ \pi(X_1 = 1, X_{2} = 1 |\theta) = \frac{e^{(\theta-b_{1})+(\theta-b_{2})}}{1+e^{(\theta-b_{1})}+e^{(\theta-b_{1})+(\theta-b_{2})}}, \end{cases} \end{align*} $$

so that we can compare the contingency table associated with the $\Theta $ -SLM (LKS) with those of LI and RD (OD). More in detail, the following comparisons for Table A report:

1. 1. Table LI-3 from Section 2 for reference and an LD table based on a traditional approach (i.e., Tables RD-1 and OD-1) using the same IRT parameter values as the LI table ( $a_1=a_2=1$ , $b_1=0$ , and $b_2 = 4$ );

2. 2. an incomplete (no g-process) LD table based on the corresponding KST-IRT model (i.e., Tables $\Theta $ -SLM-1 and LKS-1) that requires less extreme values of the item parameters in the traditional LD;

3. 3. an LD table (i.e., Tables $\Theta $ -SLM-2 and LKS-2) based on the same KST-IRT model but assuming a g-process to stochastically generate the same complete LD table as in the traditional method;

4. 4. two LD tables (i.e., Tables $\Theta $ -SLM-3-4 and LKS-3-4) showing the behavior of the KST-IRT model for a different choice of parameters.

Comparison of RD and $\Theta $ -SLM yields the following contingency tables:

$$ \begin{align*}\nonumber \underset{(b_1=0, b_2=4)}{\text{Table LI-3:}} \quad \left( \begin{matrix} 0.491 & 0.009 \\ 0.481 & 0.019 \end{matrix} \right) \quad \underset{(b_1=0, b_2=4, d=1)}{\text{Table RD-1:}} \quad \left( \begin{matrix} 0.497 & 0.003 \\ 0.453 & 0.047 \end{matrix} \right) \quad\underset{\substack{(\eta_1=0, \eta_2=0,\\ \beta_1=0, \beta_2=0,\\ b_1=0, b_2=3)}}{\text{Table } \Theta\text{-SLM-1:}} \quad \left( \begin{matrix} 0.500 & 0.000 \\ 0.453 & 0.047 \end{matrix} \right) \end{align*} $$

$$ \begin{align*}\nonumber \underset{\substack{(\eta_1 = 0, \eta_2 =0 .006,\\ \beta_1 = 0, \beta_2 =0 .06,\\ b_1=0, b_2=3)}}{\text{Table } \Theta\text{-SLM-2:} } \quad\!\!\!\! \left( \begin{matrix} 0.497 & 0.003 \\ 0.453 & 0.047 \end{matrix} \right) \quad\!\!\!\! \underset{\substack{(\eta_1=0.05, \eta_2=0.01,\\ \beta_1=0.01, \beta_2=0.05,\\ b_1=0, b_2=3)}}{\text{Table } \Theta\text{-SLM-3:}} \quad \!\!\!\!\left( \begin{matrix} 0.475 & 0.005 \\ 0.471 & 0.049 \end{matrix} \right) \quad \!\!\!\!\underset{\substack{(\eta_1=0.15, \eta_2=0.1,\\ \beta_1=0.1, \beta_2=0.15,\\ b_1=0, b_2=3)}}{\text{Table } \Theta\text{-SLM-4:} } \quad\!\!\!\! \left( \begin{matrix} 0.424 & 0.051 \\ 0.441 & 0.084 \end{matrix} \right) \end{align*} $$

In order to obtain the same proportions of Table RD-1 with the $\Theta $ -SLM (Tables $\Theta $ -SLM-1 and -2), a value of $b_2 = 3$ is sufficient, which is less extreme than $b_2 = 4$ (notice that a correction $d=1$ under RD is required). Moreover, no inflation of the a-parameters is required in Table $\Theta $ -SLM-1 to have $p_{01}=0$ since the response pattern is missing by design. A g-process assuming that about $6\%$ of individuals might make a mistake in the second item ( $\beta _2 = 0.06$ ) while about the 0. $06\%$ of individuals might guess it ( $\eta _2 = 0.006$ ), as in Table $\Theta $ -SLM-2, recovers the proportion $p_{01}=0.003$ while keeping $b_2$ smaller and without inflating the a-parameters. Tables $\Theta $ -SLM-3 and -4 provide further examples for different values of the parameters.

If we consider now a divide-by-total approach, we can compare the contingency table associated with an LKS to those of LI and OD. Table LI-3 is reported for comparison. For OD, we set the interaction parameter to $b_{12}=-d$ for comparison with RD. Due to the way OD is defined in Eq. (28), the interaction parameter $b_{12}$ increases (the sign is negative), the difficulty of the first item to $b_1-b_{12}=b_1+d=1$ . As it can be seen from the contingency tables below, to obtain similar proportions of Table OD-1 with the LKS in the absence of a g-process (Table LKS-A1), it is sufficient to assume $b_1 = 1$ (as in the OD case) and $b_2 = 2$ , which is less extreme than $b_2 = 4$ . Once again, no inflated a-parameters are needed to obtain $p_{01}=0$ . If we add a g-process as in Table LKS-A2, the small proportions $p_{01} = 0.013$ can be achieved while keeping the parameter $b_2$ smaller than the LI or OD cases. Very close values to Table OD-1 are obtained in Table LKS-A2 by assuming that small percentages of individuals can guess or make mistakes on the items. Tables LKS-A3 and LKS-A4 provide further examples using the same values of slipping and guessing parameters used in Tables $\Theta $ -SLM-3 and $\Theta $ -SLM-4.

$$ \begin{align*}\nonumber \underset{(b_1=0, b_2=4)}{\text{Table LI-3:}} \quad \left( \begin{matrix} 0.491 & 0.009 \\ 0.481 & 0.019 \end{matrix} \right) \quad \underset{(b_1=0, b_2=4, b_{12}=-1)}{\text{Table OD-1:}} \quad \left( \begin{matrix} 0.653 & 0.013 \\ 0.257 & 0.078 \end{matrix} \right) \quad \underset{\substack{(\eta_1=0,\eta_2=0,\\ \beta_1=0,\beta_2=0,\\ b_1=1, b_2 = 2)}}{\text{Table LKS-A1:} } \quad \left( \begin{matrix} 0.660 & 0.000 \\ 0.261 & 0.079 \end{matrix} \right) \end{align*} $$

$$ \begin{align*}\nonumber \underset{\substack{(\eta_1=0.006, \eta_2=0.016,\\ \beta_1=0.025, \beta_2=0.03,\\ b_1=1, b_2=2)}}{\text{Table LKS-A2:} } \quad\!\! \left( \begin{matrix} 0.652 & 0.013 \\ 0.257 & 0.079 \end{matrix} \right) \quad \!\!\underset{\substack{(\eta_1=0.05, \eta_2=0.01,\\ \beta_1=0.01, \beta_2=0.5,\\ b_1=1, b_2=2)}}{\text{Table LKS-A3:}} \quad \!\!\left( \begin{matrix} 0.623 & 0.007 \\ 0.292 & 0.077 \end{matrix} \right) \quad \!\!\underset{\substack{(\eta_1=0.15, \eta_2=0.1,\\ \beta_1=0.1, \beta_2=0.15,\\ b_1=1, b_2=2)}}{\text{Table LKS-A4:} } \quad \left( \begin{matrix} 0.529 & 0.065 \\ 0.311 & 0.094 \end{matrix} \right) \end{align*} $$

Table B: Application of the $\Theta $ -BLIM (19) to Table B, yields instead

(38) $$ \begin{align} \begin{bmatrix} P(X_1 = 0, X_{2} = 0|\theta)\\ P(X_1 = 0, X_{2} = 1|\theta)\\ P(X_1 = 1, X_{2} = 0|\theta)\\ P(X_1 = 1, X_{2} = 1|\theta) \end{bmatrix}= \begin{bmatrix} (1-\eta_1)(1-\eta_{2}) & \beta_{1}\beta_{2}\\ (1-\eta_1)\eta_{2} & \beta_1(1-\beta_{2})\\ \eta_{1}(1-\eta_{2}) & (1-\beta_1)\beta_{2} \\ \eta_1\eta_{2} & (1-\beta_1)(1-\beta_{2}) \end{bmatrix}\begin{bmatrix} \pi(\emptyset|\theta)\\ \pi(\{q_1,q_{2}\}|\theta) \end{bmatrix}, \end{align} $$

where the SRFs $\pi (K|\theta )$ capture the p-process $(\mathcal {C}, \mathcal {K}_B)$ and, by applying the LKS (the $\Theta $ -SLM cannot be applied since $\mathcal {K}_B$ is not a learning space), can be written as

(39) $$ \begin{align} \begin{cases} \pi(X_1 = 0, X_{2} = 0 |\theta) = \frac{1}{1+e^{(\theta-b_{1})+(\theta-b_{2})}} = \frac{1}{1+e^{2(\theta-\overline{b}_{12})}}\\ \pi(X_1 = 1, X_{2} = 1 |\theta) = \frac{e^{(\theta-b_{1})+(\theta-b_{2})}}{1+e^{(\theta-b_{1})+(\theta-b_{2})}} = \frac{e^{2(\theta-\overline{b}_{12})}}{1+e^{2(\theta-\overline{b}_{12})}}, \end{cases} \end{align} $$

in which $\overline {b}_{12} = \frac {b_1+b_{2}}{2}$ is the average value of the two difficulties. Such a dependence of the SRF on the average value implies that different pairs of items with different values of difficulties will return the same proportions as long as their average is the same. In the absence of a g-process, items with null average will return $p_{00}=p_{11}=0.5$ , items with positive average will return $p_{00}>p_{11}$ , and items with negative average will return $p_{00} < p_{11}$ , while always keeping $p_{10}=p_{01}=0$ . For instance, the following tables

$$ \begin{align*}\nonumber \underset{\substack{(\eta_1=0,\eta_2=0,\\ \beta_1=0,\beta=0,\\ a_1=1,a_2=1,\\ b_1=-2,b_2=0)}}{\text{Table LKS-B1:}} \quad \left( \begin{matrix} 0.225 & 0.000 \\ 0.000 & 0.775 \end{matrix} \right) \quad \underset{\substack{(\eta_1=0,\eta_2=0,\\ \beta_1=0,\beta=0,\\ a_1=1,a_2=1,\\ b_1=0,b_2=0)}}{\text{Table LKS-B2:}} \quad \left( \begin{matrix} 0.500 & 0.000 \\ 0.000 & 0.500 \end{matrix} \right) \quad \underset{\substack{(\eta_1=0,\eta_2=0,\\ \beta_1=0,\beta=0,\\ a_1=1,a_2=1,\\ b_1=0,b_2=2)}}{\text{Table LKS-B3:}} \quad \left( \begin{matrix} 0.775 & 0.000 \\ 0.000 & 0.225 \end{matrix} \right) \end{align*} $$

would be returned by any pair of items with the same average: a pair like $b_1= -3, b_2=1$ would return Table LKS-B1, a pair like $b_1= -1, b_2=1$ would return Table LKS-B2, while a pair like $b_1= -1, b_2=3$ would return Table LKS-B3. As an example of equally informative items with different difficulties, consider two items like “ $21+32=?$ ” and “ $123+342=?$ .” It is not unreasonable that these items might be mostly failed together or solved together, in spite of the fact that the latter is slightly more difficult than the former.Footnote ³ Therefore, contrary to the LI and CD cases, in an LKS, items do not need to have the same difficulty to obtain Table B. If, nonetheless, we assume for the sake of simplicity $b_1=b_2=0,$ then comparison of the contingency tables in the cases of LI, CD, and LKS in the presence of a g-process shows that CD and LKS tables can be matched by setting slipping and guessing parameters, but the LKS does not require inflated discrimination parameters to approximate Table B:

$$ \begin{align*}\nonumber \underset{\substack{(a_1=1,a_2=1,\\ b_1=0,b_2=0)}}{\text{Table LI-1:}} \quad \left( \begin{matrix} 0.293 & 0.207 \\ 0.207 & 0.293 \end{matrix} \right) \quad \underset{\substack{(a_1=1,a_2=1,\\ b_1=0,b_2=0, b_{12}=-1)}}{\text{Table CD-B1:}} \quad \left( \begin{matrix} 0.239 & 0.145 \\ 0.145 & 0.470 \end{matrix} \right) \quad \underset{\substack{(\eta_1=0.313,\eta_2=0.313,\\ \beta_1=0.082,\beta=0.082,\\ a_1=1, a_2=1,\\ b_1=0,b_2=0)}}{\text{Table LKS-B4:}} \quad \left( \begin{matrix} 0.239 & 0.145 \\ 0.145 & 0.470 \end{matrix} \right) \end{align*} $$

$$ \begin{align*}\nonumber \underset{\substack{(a_1=10,a_2=10,\\ b_1=0,b_2=0)}}{\text{Table LI-4:}} \quad \left( \begin{matrix} 0.461 & 0.039 \\ 0.039 & 0.461 \end{matrix} \right) \quad \underset{\substack{(a_1=10,a_2=10,\\ b_1=0,b_2=0, b_{12}=-1)}}{\text{Table CD-B2:}} \quad \left( \begin{matrix} 0.453 & 0.027 \\ 0.027 & 0.492 \end{matrix} \right) \quad \underset{\substack{(\eta_1=0.048,\eta_2=0.048,\\ \beta_1=0.009,\beta=0.009, \\a_1=1,a_2=1,\\ b_1=0,b_2=0)}}{\text{Table LKS-B5:}} \quad \left( \begin{matrix} 0.453 & 0.027 \\ 0.027 & 0.492 \end{matrix} \right) \end{align*} $$

Parameter interpretation: Interpretation of the item parameters can be given by checking the reproducibility of the marginal IRFs. Interestingly, parameters in the previously discussed KST-IRT approaches to LD can be given interpretations similar to those of polytomous IRT models since the resulting non-reproducible marginal IRFs are formally equivalent to left side-added models for polytomous items. A little algebra shows that for Table A, the marginal IRFs as given by system (37) are

(40) $$ \begin{align} P(X_1=1|\theta) & = \begin{cases} \eta_1+(1-\beta_1-\eta_1)\pi(K_1=1|\theta) &\quad\text{for the } \Theta\text{-SLM}\\ \eta_1+(1-\beta_1-\eta_1)\frac{e^{\theta-b_1}(1+e^{\theta-b_{2}})}{1+e^{\theta-b_1}+e^{2\theta-b_1-b_{2}}} & \quad\text{for the LKS} \end{cases}\end{align} $$

(41) $$ \begin{align} P(X_{2}=1|\theta) & = \begin{cases} \eta_{2}+(1-\beta_{2}-\eta_{2})\pi(K_1=1|\theta)\pi(K_{2}=1|\theta)&\quad\text{for the } \Theta\text{-SLM}\\ \eta_{2}+(1-\beta_{2}-\eta_{2})\frac{e^{2\theta-b_1-b_{2}}}{1+e^{\theta-b_1}+e^{2\theta-b_1-b_{2}}} & \quad\text{for the LKS,} \end{cases} \end{align} $$

while for Table B, the marginal IRFs as given by system (38) are

(42) $$ \begin{align}\kern-3pt P(X_1=1|\theta) & = \eta_1+(1-\beta_1-\eta_1)\frac{e^{2(\theta-\overline{b}_{12})}}{1+e^{2(\theta-\overline{b}_{ii'})}} \end{align} $$

(43) $$ \begin{align} P(X_{2}=1|\theta) & = \eta_{2}+(1-\beta_{2}-\eta_{2})\frac{e^{2(\theta-\overline{b}_{12})}}{1+e^{2(\theta-\overline{b}_{ii'})}}. \end{align} $$

A graphical representation of these IRFs is available in the Supplementary Material. As the $\Theta $ -SLM case for Table A is a sequential model, the reproducible marginal $P(X_1=1|\theta )$ coincides with a 4PL (2) if $\pi (K_1=1|\theta )$ is a 2PL, while the IRF $\pi (K_{2}=1|\theta )$ differs from the LI case since $q_{2}$ can be mastered only if $q_1$ is mastered. The difficulty $b_{2}$ for a Rasch-like $\pi (K_{2}=1|\theta )$ is the location at which half of the individuals that correctly answered $q_1$ are also able to correctly answer $q_{2}$ . The LKS approach for Table A is instead a divide-by-total model, hence the difficulty parameter is the location at which one has the same probability to belong to a lower or higher category (i.e., state). Finally, an interpretation of the IRFs for Table B can be given by noticing that equally informative items behave like a “virtual” item with difficulty given by the average $\overline {b}_{12}$ of the item difficulties and doubled discrimination parameter. In principle, the error parameters $\eta , \beta $ can differ for the two items and, contrary to the upper Fréchet–Hoeffding bound for copulas (32) and mixtures of copulas (34), the difficulties can differ as previously discussed.

Identifiability: If only a $2\times 2$ contingency table were considered, there would be an identifiability issue since there are four parameters per item but only three independent conditions. One needs at least $n=4$ items to have $16$ item parameters and $18$ independent conditions, that is, the condition $3\frac {n(n-1)}{2}\geq 4n$ must be satisfied in a pairwise approach. If one were instead to consider a knowledge structure encompassing all items, then one would require at least $n=5$ items since there would be $2^n-1$ independent response patterns. In other words, the condition $2^n-1\geq 4n$ must be satisfied. It is also worth mentioning that these simple calculations do not however account for more complex identifiability issues that might arise when specific models are applied to specific structures. Although the identifiability issue in KST has been extensively explored (see, e.g., Heller, Reference Heller2017; Heller et al., Reference Heller, Stefanutti, Spoto, Heller and Stefanutti2024; Spoto et al., Reference Spoto, Stefanutti and Vidotto2013; Stefanutti & Spoto, Reference Stefanutti and Spoto2020), the identifiability of KST-IRT models is a currently open problem since it is more general than the identifiability issue in KST. Nonetheless, some promising results were given, for instance, by Noventa et al. (Reference Noventa, Heller and Kelava2024) that explored the identifiability issue of KST-IRT models under the power set conditions to provide a full account of the identifiability problem of 3- and 4-parameter IRT models.

LD items and testlets: As an example of a potential application of KST-IRT modeling, we briefly consider testlets. Indeed, grouping of dependent items has been advocated under the similar notions of superitems (Cureton, Reference Cureton1965), subtests (Andrich, Reference Andrich and Embretson1985; Wilson, Reference Wilson1988), testlets (Wainer & Kiely, Reference Wainer and Kiely1987), and bundles (Rosenbaum, Reference Rosenbaum1988). Polytomous IRT models are then used to capture the aggregated behavior (see, e.g., Andrich, Reference Andrich and Embretson1985; Thissen et al., Reference Thissen, Steinberg and Mooney1989; Wainer & Kiely, Reference Wainer and Kiely1987; Wilson, Reference Wilson1988). A typical approach is to apply a PCM to the sum-score of the dependent items. Huynh (Reference Huynh1994, Reference Huynh1996) discussed the necessary and sufficient conditions under which a PCM captures the distribution of the sum-score of a group of locally independent items and concluded that while “close enough” step difficulties in a PCM are an indication of LD, “distanced enough” step difficulties were an indication of LI. However, it can be shown that if one extends the results of Huynh (Reference Huynh1994, Reference Huynh1996) to probabilistic and deterministic LD the situation becomes more nuanced: while “close enough” step difficulties in a PCM are still an indication of LD, “distanced enough” step difficulties are not anymore sufficient to discriminate between LI from probabilistic or deterministic LD. A detailed discussion and proof of this result are given in Appendix B. Its implications for the so-called “disordered threshold controversy” are instead discussed in Appendix C. For the purposes of the present section, it is here argued that, as a consequence of the fact that the step difficulties obtained from fitting a PCM cannot discriminate between LI and probabilistic or deterministic LD, since these latter can be modeled by means of general models like the $\Theta $ -BLIM (19) one does not require to model the conditional probability of the sum-score via polytomous models to capture locally dependent items but can directly apply KST-IRT models as discussed in the present section so that the dependence between the items is captured by the knowledge structure. To illustrate the direct item-based approach to testlets offered by KST-IRT models, we here briefly apply such an idea to the modeling of both linear and hierarchical testlets. Let us consider the three testlets represented in Figure 2 and borrowed from Wainer and Kiely (Reference Wainer and Kiely1987), that is, a linear testlet with three items, a fully hierarchical testlet with seven items leading to pattern of responses that are unambiguously associated with the sum-scores, and a partially hierarchical testlet of six items leading to pattern of responses that are not unambiguously associated with the sum-scores. The underlined responses to the items are those identified by the outcomes of the testlet. The other responses are surmised given the outcome. The diagrams in Figure 2 are not structures as in Figure 1 but represent the order in which the items are administered, by (solving) failing an item, one progresses (upward) downward in the tree.

Figure 2

Examples of testlets from Wainer and Kiely (Reference Wainer and Kiely1987), the underlined responses to the items for both fully and partially hierarchical testlets are those identified by the outcomes of the testlet. The other responses are surmised given the outcome of the testlet.

Figure 2 Long description

The diagram consists of three panels.

1. Linear testlet. A horizontal line connects three nodes labeled q sub 1, q sub 2, and q sub 3 from left to right.

2. Fully hierarchical testlet. A branching tree structure starting from a root node q sub 4 on the left.

- q sub 4 branches into q sub 6 (top) and q sub 2 (bottom).

- q sub 6 branches into q sub 7 (top) and q sub 5 (bottom).

- q sub 2 branches into q sub 3 (top) and q sub 1 (bottom).

- Each of these four nodes (q sub 7, q sub 5, q sub 3, q sub 1) branches into two terminal binary strings.

- From top to bottom, the strings are: 1111111 (last four underlined), 1111110 (last two underlined), 1111100 (last two underlined), 1111000 (last two underlined), 1110000 (last two underlined), 1100000 (last two underlined), 1000000 (last two underlined), and 0000000 (first, third, and fifth zeros underlined).

3. Partially hierarchical testlet. A branching structure starting from root node q sub 4.

- q sub 4 branches into q sub 5 (top) and q sub 2 (bottom).

- q sub 5 branches into q sub 6 (top) and q sub 3 (bottom).

- q sub 2 branches into q sub 3 (top) and q sub 1 (bottom).

- Note that q sub 3 receives inputs from both q sub 5 and q sub 2.

- Terminal strings from top to bottom: 111111 (q sub 6 top), 111110 (q sub 6 bottom, last digit underlined), 111100 (q sub 3 top, last two underlined), 111000 (q sub 3 middle-top, last two underlined), 110100 (q sub 3 middle-bottom, last two underlined), 110000 (q sub 3 bottom, last two underlined), 100000 (q sub 1 top, last two underlined), and 000000 (q sub 1 bottom, first, third, and fifth zeros underlined).

The linear testlet has domain $Q_L=\{q_1,q_2,q_3\}$ and, unless one wants to impose deterministic LD between the items, can be modeled as a power set $\mathcal {K}_L=2^{Q_L}$ . One can then either consider only a p-process with SRF $P(X|\theta )=\pi (K|\theta )$ and apply LI to describe items with, for example, a 2PL model, or consider both a p-process and a g-process and apply LI to describe items with, for example, a 4PL model. Probabilistic LD can also be imposed in any of the form discussed above via assumptions on the SRF.

The fully hierarchical testlet model has domain $Q_{F}=\{q_1, q_2, q_3, q_4, q_5, q_6, q_7\}$ and is modeled by means of a chain $\mathcal {K}_{F}=\{\emptyset , \{q_1\}, \{q_1, q_2\}, \ldots , Q_{F}\},$ each state of which is associated with an outcome of the testlet. The outcomes are obtained by administering an item from the middle of the chain, and then progressing either left or right by pre-coded jumps according to failure or success. Items that are prerequisite to a solved (failed) item are surmised to be solved (failed). Since a g-process is not needed, the probability of each sum-score equals the probability of a knowledge state in the chain, that is, $P(r|\theta ) = \pi (\{q_i, \ldots , q_r\}|\theta )$ .

The partially hierarchical testlet model has instead domain $Q_{P}=\{q_1, q_2, q_3, q_4, q_5, q_6\}$ and is also modeled by means of a chain $\mathcal {K}_{P}=\{\emptyset , \{q_1\}, \{q_1, q_2\}, \ldots , Q_{P}\}$ . Items that are prerequisite to a solved (failed) item are surmised to be solved (failed) unless explicitly failed (solved). Indeed, contrary to the hierarchical case, some items can occur in multiple administration paths, thus yielding additional patterns of responses that belong to $2^Q\setminus \mathcal {K}_P$ and require a g-process to be modeled. Different assumptions on the conditional probabilities of the $\Theta $ -BLIM (19) thus offer different modeling of the testlet. As many forms of the SRF $\pi (K|\theta )$ and of the g-process matrix are possible, this allows for a wide range of possibilities to model both fully and partially hierarchical testlets. For instance, given a partially hierarchical testlet as in Figure 2, a hypothesis close to the one discussed by Wainer & Kiely (Reference Wainer and Kiely1987, Table 2) is that the pattern $110100$ might be generated by guessing item $q_4$ from the state $\{q_1,q_2\}$ or by a mistake on item $q_3$ from the state $\{q_1,\ldots , q_4\}$ , that is, by a g-process of the form given in Eq. (44) below, in which clearly mistakes or guesses are assumed to occur at most in one item. A more complex hypothesis could be that the pattern $110100$ might emerge from mistakes/guesses on both items $q_3$ and $q_4$ , thus yielding a g-process of the form given in Eq. (45) below, in which more states can generate the pattern $110100$ :

(44) $$ \begin{align} \begin{bmatrix} P(\underline{00}0\underline{0}00|\theta)\\ P(\underline{10}0\underline{0}00|\theta)\\ P(1\underline{100}00|\theta)\\ P(11\underline{010}0|\theta)\\ P(1\underline{110}00|\theta)\\ P(11\underline{110}0|\theta)\\ P(111\underline{11}0|\theta)\\ P(111\underline{111}|\theta) \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 -\eta_4 & 0 & 0 & 0 & 0\\ 0 & 0 & \eta_4 & 0 & \beta_3 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1-\beta_3 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} \pi(\emptyset|\theta)\\ \pi(\{q_1\}|\theta)\\ \pi(\{q_1,q_{2}\}|\theta)\\ \pi(\{q_1, q_2, q_{3}\}|\theta)\\ \pi(\{q_1,\ldots, q_{4}\}|\theta)\\ \pi(\{q_1,\ldots, q_{5}\}|\theta)\\ \pi(\{q_1,\ldots, q_{6}\}|\theta) \end{bmatrix} \end{align} $$

(45) $$ \begin{align} \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & (1-\eta_3)(1-\eta_4) & \beta_3(1-\eta_4) & \beta_3\beta_4 & 0 & 0\\ 0 & 0 & (1-\eta_3)\eta_4 & \beta_3\eta_4 & \beta_3(1-\beta_4) & 0 & 0\\ 0 & 0 & \eta_3(1-\eta_4) & (1-\beta_3)(1-\eta_4) & (1-\beta_3)(1-\beta_4) & 0 & 0\\ 0 & 0 & \eta_3\eta_4 & (1-\beta_3)\eta_4 & (1-\beta_3)\beta_4 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}. \end{align} $$

Moreover, the testlets here considered are based on a chain, which is the structure capturing a polytomous item for which the application of a PCM is reasonable. However, the possibility of using an arbitrary structure allows for a broader range of KST-IRT models and dependencies.

3.2.3 New and/or currently unused KST-IRT modeling of LD

As it can be seen from Tables 1 and 2, the deterministic approaches to LD discussed so far do not cover all possible approaches in the given taxonomy. Indeed, by suitably selecting processes and imposing constraints on their conditional probabilities, one can generate new families of models that, up to our knowledge, have yet to be applied in the literature and some existing families of model that have never been applied. As exploration of these (to the best of our knowledge) new models goes beyond the scope of the present work, we only provide three brief remarks on possible extensions:

1. 1. ULD/trait dependence can be modeled independently of whether a probabilistic or a deterministic LD approach is followed. As the modeling mechanisms (structure vs. SRF) are complementary but distinct, multidimensional extensions of the SRF can be carried out on top of any other assumption.

2. 2. For the same rationale of ULD, application of the constraints to the SRF that are used in probabilistic LD can also be applied in the deterministic LD case as they pertain to the SRF, and not to the structure. This potentially generates a quite large family of models that have not been explored in the literature so far. For instance, one might still introduce additional parameters as in the RD, CD, or OD approaches, while at the same time implementing an underlying structure.

3. 3. KST-IRT models were systematized as deterministic LD in the presence of a competence-independent g-process. However, as in the last row of Table 2, one can consider a competence-dependent g-process. In such a case, one can replace the conditional error parameters $\eta $ and $\beta $ in all previous KST-IRT models with functions like $\eta _i(\theta )=(1+\exp {(\tilde {\eta }_i-\theta )})^{-1}$ and $\beta _i(\theta )=(1+\exp {(\theta - \tilde {\beta }_i)})^{-1}$ for some $\tilde {\eta }_i, \tilde {\beta }_i\in \mathbb {R}$ , exactly as discussed for the 1PLAG model in Section 2.1.