2.1 CDM framework

2.1.3 The DINA model

While the methods developed in this article apply to any CDM, we use the DINA model for illustration due to its parsimony and wide use in diagnostic testing applications (Junker & Sijtsma, Reference Junker and Sijtsma2001; de la Torre, Reference de la Torre2009). Under the DINA model, an examinee must master all required attributes to have a high probability of answering an item correctly, making it a conjunctive model. For an examinee i with attribute pattern α responding to item j, the ideal response is

(1) $$\begin{align}{\eta}_{ij}\left(\boldsymbol{\alpha} \right)=\prod \limits_{k=1}^K{\alpha_{ik}}^{q_{jk}},\end{align}$$

where ${\eta}_{ij}\left(\boldsymbol{\alpha} \right)=1$ indicates mastery of all required attributes and ${\eta}_{ij}\left(\boldsymbol{\alpha} \right)=0$ indicates a lack of at least one required attribute (de la Torre, Reference de la Torre2009).

The probability of a correct response of examinee i on item j is given by

(2) $$\begin{align}P\left({X}_{ij}=1|\boldsymbol{\alpha} \right)={\left(1-{s}_j\right)}^{\eta_{ij}\left(\boldsymbol{\alpha} \right)}{g_j}^{1-{\eta}_{ij}\left(\boldsymbol{\alpha} \right)},\end{align}$$

where ${s}_j$ is the slipping parameter (probability of incorrect response despite mastery) and ${g}_j$ is the guessing parameter (probability of correct response despite non-mastery). These item parameters account for the probabilistic nature of the response process, where examinees who have mastered all required attributes may still make mistakes (slips), and those who lack required attributes may still answer correctly through guessing (de la Torre & Douglas, Reference de la Torre and Douglas2004). The DINA model’s simple form makes it particularly useful for understanding the fundamental principles of cognitive diagnosis while still capturing essential features of the response process. Its parsimony in parameter estimation and clear interpretation of results have made it a popular choice in diagnostic testing applications.

2.2 Item selection methods in CD-CAT

Item selection methods in CD-CAT can be broadly categorized into parametric and nonparametric approaches (Chang et al., Reference Chang, Chiu and Tsai2019). While non-parametric methods have emerged recently to address certain limitations of parametric approaches (Chiu & Chang, Reference Chiu and Chang2021), parametric methods remain fundamental to CD-CAT implementation. These parametric methods can be further classified as single-purpose or dual-purpose (Wang et al., Reference Wang, Chang and Douglas2012). Single-purpose methods focus solely on optimizing the measurement of attribute profiles, while dual-purpose methods simultaneously measure both attribute profiles and general ability (Dai et al., Reference Dai, Zhang and Li2016; Kang et al., Reference Kang, Zhang and Chang2017; Wang et al., Reference Wang, Zheng and Chang2014). This article proposes a new algorithm within the framework of parametric single-purpose item selection methods. Therefore, we focus our review on existing methods in this category, which form the foundation for CD-CAT item selection and remain the most widely used in practice.

2.2.1 Basic framework

In CD-CAT, parametric single-purpose item selection methods aim to optimize the measurement of examinees’ attribute mastery profiles. These methods utilize item parameters and probability models within a cognitive diagnostic framework to select items that maximize information about attribute patterns. After J items have been administered to an examinee, let ${\boldsymbol{x}}_{\boldsymbol{J}}=\left({x}_1,\dots, {x}_J\right)$ denote the vector of observed responses, where x_j ∈ {0,1}. Following Bayes’ theorem, the posterior probability of the attribute pattern is

(3) $$\begin{align}\pi \left(\boldsymbol{\alpha} |{\boldsymbol{x}}_{\boldsymbol{J}}\right)\propto {\pi}_0\left(\boldsymbol{\alpha} \right)L\left({\boldsymbol{x}}_{\boldsymbol{J}}|\boldsymbol{\alpha} \right),\end{align}$$

where ${\pi}_0\left(\boldsymbol{\alpha} \right)$ is the prior probability and $L\left({\boldsymbol{x}}_{\boldsymbol{J}}|\boldsymbol{\alpha} \right)={\prod}_{j=1}^JP\left({X}_j={x}_j|\boldsymbol{\alpha} \right)$ is the likelihood function under the specified CDM. Define the item h as a candidate item in the pool of available items, from which the (J+1)-th item is to be selected based on a specified item selection method.

2.2.2 Information-theoretic methods.

Kullback–Leibler-based approaches

The Kullback–Leibler (KL) information (Cover & Thomas, Reference Cover and Thomas1991; Kullback & Leibler, Reference Kullback and Leibler1951) provides a foundation for measuring the distance between probability distributions under different attribute patterns. For item j and two attribute patterns $\boldsymbol{\alpha}, \boldsymbol{\alpha}^{\prime}\in {\left\{0,1\right\}}^K$ , the KL information is defined as

(4) $$\begin{align}D_j\!\left(\boldsymbol{\alpha} \,\|\, \boldsymbol{\alpha}'\right)&=\sum_{x=0}^1P\left({X}_j=x|\boldsymbol{\alpha} \right){log}\;\left[\frac{P\left({X}_j=x|\boldsymbol{\alpha} \right)}{P\left({X}_j=x|{\boldsymbol{\alpha}}^{\prime}\right)}\right],\end{align}$$

where $P\left({X}_j=x|\boldsymbol{\alpha} \right)$ denotes the probability of response x given attribute pattern $\boldsymbol{\alpha}$ under the specified CDM. Building on this framework, the KL index was proposed by Xu et al. (Reference Xu, Chang and Douglas2003) to select the next item maximizing:

(5) $$\begin{align}KL_h\left(\widehat{\boldsymbol{\alpha}}\right)=\sum_{c=1}^{2^K}D_h\!\left(\widehat{\boldsymbol{\alpha}} \,\|\, \boldsymbol{\alpha}_c\right),\end{align}$$

where h represents the candidate item in the pool of available items, $\widehat{\boldsymbol{\alpha}}$ is the current estimate of the examinee’s attribute pattern, and c indexes attribute patterns (c = 1, 2, …, ${2}^K$ ). This index measures the total divergence between the response distributions under the estimated pattern and all other possible patterns. Cheng (Reference Cheng2009) enhanced this approach by incorporating posterior probabilities through the posterior-weighted KL (PWKL) index:

(6) $$\begin{align}PWKL_h\!\left(\widehat{\boldsymbol{\alpha}}\right)&=\sum_{c=1}^{2^K}D_h\!\left(\widehat{\boldsymbol{\alpha}} \,\|\, \boldsymbol{\alpha}_c\right)\pi\!\left(\boldsymbol{\alpha}_c|\boldsymbol{x}_J\right),\end{align}$$

where $\pi \left({\boldsymbol{\alpha}}_c|{\boldsymbol{x}}_J\right)$ is the posterior probability after J items have been answered, and ${\boldsymbol{x}}_J$ is the response vector.

Other KL information-based item selection methods include the modified PWKL (MPWKL) method (Kaplan et al., Reference Kaplan, De La Torre and Barrada2015), and posterior-weighted CDM discrimination index (PWCDI) method (Zheng & Chang, Reference Zheng and Chang2016). This study employs only PWKL for comparison, as MPWKL, while achieving comparable performance to GDI, incurs substantial computational costs (Kaplan et al., Reference Kaplan, De La Torre and Barrada2015). Additionally, PWCDI demonstrates inferior performance to PWKL with small calibration samples (Chang et al., Reference Chang, Chiu and Tsai2019), making PWKL the most suitable KL-based comparator for this investigation.

Shannon entropy-based approaches.

Shannon entropy (Cover & Thomas, Reference Cover and Thomas1991; Shannon, Reference Shannon1948) provides an alternative framework for quantifying uncertainty in the posterior distribution of attribute patterns. In the context of CD-CAT, after J items have been selected, the entropy of a distribution $\pi$ is defined as

(7) $$\begin{align}H\left(\pi \right)=-\sum_{c=1}^{2^K}\pi \left({\boldsymbol{\alpha}}_c|{\boldsymbol{x}}_J\right){log}\;\left[\pi \left({\boldsymbol{\alpha}}_c|{\boldsymbol{x}}_J\right)\right].\end{align}$$

Lower entropy values indicate greater certainty about the true attribute pattern. Building on information theory principles (Cover & Thomas, Reference Cover and Thomas1991), Tatsuoka (Reference Tatsuoka2002) proposed selecting items by minimizing the expected posterior entropy:

(8) $$\begin{align}SH{E}_h=\sum_{x=0}^1H\left(\pi |{X}_h=x,{\boldsymbol{x}}_J\right)P\left({X}_h=x|{\boldsymbol{x}}_J\right),\end{align}$$

where $\pi \mid {X}_h=x,{\boldsymbol{x}}_{\boldsymbol{J}}$ denotes the posterior distribution after observing response x to the candidate item h in the pool of available items, $P\left({X}_h=x|{\boldsymbol{x}}_{\boldsymbol{J}}\right)$ denotes the predicted probability of observing response x conditional on the response vector ${\boldsymbol{x}}_J$ , and

(9) $$\begin{align}P\left({X}_h=x|{\boldsymbol{x}}_J\right)=\sum_{c=1}^{2^K}P\left({X}_h=x|{\boldsymbol{\alpha}}_c\right)\pi \left({\boldsymbol{\alpha}}_c|{\boldsymbol{x}}_J\right).\end{align}$$

Recent methodological advances have extended this framework through the expected mutual information index (Wang, Reference Wang2013) and the Jensen–Shannon divergence (JSD) index (Minchen & de la Torre, Reference Minchen and de la Torre2016; Yigit et al., Reference Yigit, Sorrel and De La Torre2019). Theoretical investigations have established that JSD is mathematically equivalent to mutual information in quantifying the information gain about an examinee’s attribute pattern (Yigit et al., Reference Yigit, Sorrel and De La Torre2019). Furthermore, a linear relationship was found between SHE and JSD (Wang et al., Reference Wang, Song, Wang, Gao and Xiong2020), indicating that these methods yield equivalent item selection decisions in CD-CAT applications. Given these theoretical equivalences, the present study employs the SHE method (Tatsuoka, Reference Tatsuoka2002) as the representative entropy-based approach in our comparative analyses.

2.2.3 GDI approach

Building on the G-DINA framework (de la Torre, Reference de la Torre2011), Kaplan et al. (Reference Kaplan, De La Torre and Barrada2015) introduced the G-DINA discrimination index (GDI). This index offers computational advantages by working with reduced attribute patterns, which are made up by ${K}_h^{\ast }$ attributes required by item h. For example, if a q -vector is defined as (1, 0, 1, 0, 1), ${K}_h^{\ast }=3$ attributes since this item only requires the first, third, and fifth attributes. Consequently, there are eight reduced attribute patterns based on the three required attributes. The GDI for item h is defined as

(10) $$\begin{align}GD{I}_h=\sum_{c=1}^{2^{K_h^{\ast }}}\pi \left({\boldsymbol{\alpha}}_{ch}^{\ast}\right){\left[P\left({X}_h=1|\;{\boldsymbol{\alpha}}_{ch}^{\ast}\right)-{\overline{P}}_h\right]}^2,\end{align}$$

where ${\boldsymbol{\alpha}}_{ch}^{\ast }$ represents the c-th reduced attribute pattern for item h ( $c=1,2,\dots, {2}^{K_h^{\ast }}$ ), $\pi \left({\boldsymbol{\alpha}}_{ch}^{\ast}\right)$ is the posterior probability of the reduced attribute pattern after J items have been selected, and ${\overline{P}}_h=\sum_{c=1}^{2^{K_h^{\ast }}}\pi \left({\boldsymbol{\alpha}}_{ch}^{\ast}\right)P\left({X}_h=1|{\boldsymbol{\alpha}}_{ch}^{\ast}\right)$ is the mean success probability. The GDI measures an item’s ability to differentiate between reduced attribute vectors, emphasizing those with higher success probabilities. The item with the highest GDI in the pool is selected.

2.3 Computational considerations in CD-CAT

The practical implementation of CD-CAT item selection methods faces significant computational challenges, primarily arising from the need to evaluate large numbers of attribute patterns during the item selection process. For KL-based methods, item selection entails computing and summing up the KL information between the current attribute pattern estimate and all ${2}^K$ possible patterns, and this is done for all eligible items in the bank. When using PWKL, additional computational burden comes from calculating posterior probabilities for each pattern combination.

Consider a test measuring K = 5 attributes with an item bank of 300 items as an example. Even in this relatively simple case, 32 possible attribute patterns must be evaluated for each item selection decision. The PWKL method requires computing and summing KL divergence values across all 32 patterns for each item under consideration. This computation must be performed for all eligible items in the bank to select the next item. The Shannon entropy method involves similar computational intensity, requiring the calculation of expected entropy by evaluating posterior distributions for possible responses across all attribute patterns, again repeated for each item in the bank.

The GDI method introduced by Kaplan et al. (Reference Kaplan, De La Torre and Barrada2015) offers some computational advantages by working only with the attributes required for each item. This reduces the pattern space from ${2}^K$ to ${2}^{K_h^{\ast }}$ , where ${K}_h^{\ast}$ is typically much smaller than K, and requires fewer posterior probability calculations. However, even with these improvements, significant computational challenges remain. These computational demands become particularly acute when tests measure many attributes and as the item bank expands.

The practical implications of these computational demands are substantial. They can affect response time between items, overall test administration efficiency, and the system resources required to implement CD-CAT. When multiple examinees are tested simultaneously, as is common in educational settings, these computational requirements become even more demanding. While the GDI method has made progress in reducing computational burden through reduced attribute patterns, there remains a clear need for more efficient approaches that can maintain measurement precision while reducing computation time and scaling effectively with the number of attributes.

3.1 Key ideas

The computational burden of traditional CD-CAT item selection methods grows exponentially with the number of attributes K, as each method evaluates all ${2}^K$ possible attribute patterns for each eligible item in the bank at every item selection decision. However, as testing proceeds, the set of plausible, or most likely, attribute patterns for an examinee typically shrinks based on their response pattern. This observation motivates the key insight of LBPS: by focusing on the most likely attribute patterns for item selection while maintaining full pattern space for estimation, substantial computational savings can be achieved without sacrificing measurement precision.

The likelihood function provides a natural mechanism for identifying these plausible patterns. After each response, patterns with maximum likelihood represent the most probable true states given the observed data. Figure 1 illustrates changes in attribute profiles’ likelihoods using LBPS with KL when K = 5. Early on, multiple profiles may have similar likelihoods, but as the test proceeds, the number of likely profiles shrinks (see Figure 1). By restricting item selection calculations to these patterns while using all patterns for estimation, LBPS balances computational efficiency with measurement accuracy.

Figure 1

An illustration of changes in attribute profiles’ likelihoods using LBPS with KL when K = 5.

Note: An iteration refers to a single cycle of the adaptive testing process: selecting the next item, collecting the examinee’s response, and updating the likelihoods of all attribute profiles and the examinee’s estimated attribute profile based on the accumulated responses.

Figure 1 Long description

Top-left panel, labeled After the First Iteration, shows the y-axis labeled Likelihood from 0 to 1 and the x-axis labeled Attribute Profiles from 0 to 32. Most likelihood values are either near 1 or near 0, with a few profiles at high likelihood. Top-right panel, After the Second Iteration, shows a similar pattern but with more attribute profiles at lower likelihoods and fewer at high likelihood. Bottom-left panel, After the Third Iteration, shows further concentration, with only a few profiles above 0.6 and the rest near zero. Bottom-right panel, After the Fourth Iteration, shows a single profile near 0.6 and all others close to zero. Across panels, the likelihood distribution becomes increasingly peaked, indicating that the adaptive process narrows the estimated attribute profile with each iteration.

3.2 Theoretical framework

Let ${\boldsymbol{x}}_t=\left({x}_1,\dots, {x}_t\right)$ denote the response vector after t items have been administered. For any attribute pattern $\boldsymbol{\alpha} \in A={\left\{0,1\right\}}^K$ , the likelihood under a cognitive diagnostic model is

(11) $$\begin{align}L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_t\right)=\prod \limits_{j=1}^tP\left({X}_j={x}_j|\boldsymbol{\alpha} \right).\end{align}$$

Define the set of attribute patterns with the largest likelihood after t items have been answered as

(12) $$\begin{align}M\left({\boldsymbol{x}}_t\right)=\arg \underset{\boldsymbol{\alpha} \in A}{\max }L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_t\right).\kern0.36em\end{align}$$

For the response pattern ${\boldsymbol{x}}_t$ and a new item j that is answered with a response ${x}_j$ , $L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)=L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_{\boldsymbol{t}}\right)P\left({X}_j={x}_j|\boldsymbol{\alpha} \right)$ .

1. a. Under the DINA model, for item 1 $, L\left(\boldsymbol{\alpha} |{x}_1\right)=P\left({x}_1|\boldsymbol{\alpha} \right)={\left(1-{s}_1\right)}^{\eta_1\left(\boldsymbol{\alpha} \right)}{g_1}^{1-{\eta}_1\left(\boldsymbol{\alpha} \right)}$ , where ${\eta}_1\left(\boldsymbol{\alpha} \right)=\prod_{k=1}^K{\alpha_k}^{q_{1k}}$ .

2. b. For ${x}_1=1$ , $L\left(\boldsymbol{\alpha} |{x}_1=1\right)$ is maximized when ${\eta}_1\left(\boldsymbol{\alpha} \right)=1$ since $\left(1-{s}_1\right)>{g}_1$ . This requires all k attributes specified by item 1 to be mastered. The remaining $K-k$ attributes can be 0 or 1. Therefore, $\lvert M(x_1) \rvert = 2^{K-k}$ .

3. c. For ${x}_1=0$ , $L\left(\boldsymbol{\alpha} |{x}_1=0\right)$ is maximized when ${\eta}_1\left(\boldsymbol{\alpha} \right)=0$ since $\left(1-{g}_1\right)>{s}_1$ . This occurs when any required attribute is 0. So, in this case $\left|M\left({x}_1\right)\right|$ = ${2}^K-{2}^{K-k}$ .

Theorem 2 (Pattern set change). After t items have been answered ( $t\ge 1$ ), response pattern ${\boldsymbol{x}}_t$ and any new response ${x}_j$ from item j, the set of attribute patterns with the largest likelihood is

$$\begin{align*}M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)=\arg \underset{\boldsymbol{\alpha} \in A}{\max }\ L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_t,{x}_j\right)=\arg \underset{\boldsymbol{\alpha} \in A}{\max}\left[L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_t\right)P\left({X}_j={x}_j|\boldsymbol{\alpha} \right)\right].\end{align*}$$

Let ${M}_{1t}=\left\{\boldsymbol{\alpha} \in M\left({\boldsymbol{x}}_t\right):{\eta}_j\left(\boldsymbol{\alpha} \right)=1\right\}$ (i.e., patterns within $M\left({\boldsymbol{x}}_t\right)$ that mastered all attributes required item j), and ${M}_{0t}=\left\{\boldsymbol{\alpha} \in M\left({\boldsymbol{x}}_t\right):{\eta}_j\left(\boldsymbol{\alpha} \right)=0\right\}$ (patterns within $M\left({\boldsymbol{x}}_t\right)$ that miss one or more attributes required by item j). The updated pattern set $M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)$ follows one of the three cases:

Case 1 (Shrinkage): If ${M}_{1t}\ne \varnothing$ , and ${M}_{0t}\ne \varnothing$ (which implies $\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|\ge 2$ ), then $\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)\right|<\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|$ . This occurs because item j separates patterns within $M\left({\boldsymbol{x}}_t\right)$ , with at least one pattern mastering all required attributes, while others miss at least one attribute. If ${x}_j=1,M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)={M}_{1t}$ ; if ${x}_j=0,M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)={M}_{0t}$ . This is the most common case when $\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|$ is large, which tends to be the case at the beginning of the test, especially when K is large.

Case 2 (Stability): If either (a) ${M}_{1t}\ne \varnothing, {M}_{0t}=\varnothing$ and ${x}_j=1$ , or (b) ${M}_{1t}=\varnothing, {M}_{0t}\ne \varnothing$ and ${x}_j=0$ , then $\lvert M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)\rvert=\lvert M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\rvert$ . This occurs when all patterns within $M\left({\boldsymbol{x}}_t\right)$ lead to the same ${\eta}_j$ and the observed response ${x}_j$ matches ${\eta}_j$ .

Case 3: If (1) ${M}_{1t}=\varnothing$ and ${x}_j=1$ , or (2) ${M}_{0t}=\varnothing$ and ${x}_j=0$ :

1. a. Growth occurs when external patterns, i.e., patterns outside of $M\left({\boldsymbol{x}}_t\right)$ , have precisely the threshold likelihood: $L\left({\boldsymbol{\alpha}}^{\prime }|{\boldsymbol{x}}_t\right)={L}^{\ast}\cdot \frac{g_j}{1-{s}_j}$ for ${x}_j=1$ , or $L\left({\boldsymbol{\alpha}}^{\prime }|{\boldsymbol{x}}_t\right)={L}^{\ast}\cdot \frac{s_j}{1-{g}_j}$ for ${x}_j=0$ . Here, ${L}^{\ast }=L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_t\right)$ for any $\boldsymbol{\alpha} \boldsymbol{\in}M\left({\boldsymbol{x}}_t\right)$ . This exact equality is mathematically possible but rare in practice.

2. b. Replacement occurs when there exists at least one external pattern ${\boldsymbol{\alpha}}^{\prime }$ leading to $L\left({\boldsymbol{\alpha}}^{\prime }|{\boldsymbol{x}}_t\right)$ that exceeds the threshold likelihood ratio. This can result in $\lvert M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)\rvert$ being larger, smaller, or equal to $\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|$ depending on the number of qualifying external patterns.

3. c. Stability occurs when no external patterns meet the threshold.

• • For $\boldsymbol{\alpha} \in {M}_{1t}:L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_t,{x}_j\right)={L}^{\ast}\cdot \left(1-{s}_j\right)$ .

• • For $\boldsymbol{\alpha} \in {M}_{0t}:L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_t,{x}_j\right)={L}^{\ast}\cdot {g}_j$ .

• • For ${\boldsymbol{\alpha}}^{\prime}\in {A}_{1t}:L\left({\boldsymbol{\alpha}}^{\prime }|{\boldsymbol{x}}_t,{x}_j\right)={L}^{\ast \prime}\cdot \left(1-{s}_j\right)$ .

• • For ${\boldsymbol{\alpha}}^{\prime}\in {A}_{0t}:L\left({\boldsymbol{\alpha}}^{\prime }|{\boldsymbol{x}}_t,{x}_j\right)={L}^{\ast \prime}\cdot {g}_j$ .

• • For ${\boldsymbol{\alpha}}^{\prime}\in {A}_0,L\left({\boldsymbol{\alpha}}^{\prime }|{\boldsymbol{x}}_t,{x}_j\right)={L}^{\ast \prime}\cdot {g}_j<{L}^{\ast}\cdot {g}_j$ , so those patterns do not enter $M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)$ . Therefore, we can focus on ${\boldsymbol{\alpha}}^{\prime}\in {A}_{1t}$ .

• • For ${\boldsymbol{\alpha}}^{\prime}\in {A}_{1t}$ :

  • ○ if ${L}^{\ast \prime}\cdot \left(1-{s}_j\right)={L}^{\ast}\cdot {g}_j$ , that is, $\frac{L^{\ast \prime }}{L^{\ast }}=\frac{g_j}{1-{s}_j}$ , then pattern ${\boldsymbol{\alpha}}^{\prime }$ ties with patterns in ${M}_{0t}$ and becomes a part of $M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)$ (growth);

  • ○ if ${L}^{\ast \prime}\cdot \left(1-{s}_j\right)>{L}^{\ast}\cdot {g}_j$ , that is, $\frac{L^{\ast \prime }}{L^{\ast }}>\frac{g_j}{1-{s}_j}$ , then pattern ${\boldsymbol{\alpha}}^{\prime }$ achieves a higher likelihood than all patterns in ${M}_{0t}$ , so $M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)=\left\{{\boldsymbol{\alpha}}^{\prime}\in {A}_{1t}:L\left({\boldsymbol{\alpha}}^{\prime }|{\boldsymbol{x}}_t\right)>{L}^{\ast}\cdot \frac{g_j}{1-{s}_j}\right\}$ (replacement), and the size of $M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)$ may increase, decrease, or remain unchanged relative to the size of $M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)$ ;

  • ○ if ${L}^{\ast \prime}\cdot \left(1-{s}_j\right)<{L}^{\ast}\cdot {g}_j$ , that is, $\frac{L^{\ast \prime }}{L^{\ast }}<\frac{g_j}{1-{s}_j}$ , then pattern ${\boldsymbol{\alpha}}^{\prime }$ doesn’t become a member of $M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)$ , so $M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)={M}_{0t}=M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)$ (stability).

A special case: When $\lvert M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\rvert$ = 1

A critical scenario arises when the pattern set contains only a single pattern: $M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)=\left\{{\boldsymbol{\alpha}}^{\ast}\right\}$ . This situation may arise at a late stage of a test. With only one pattern, Case 1 (i.e., mixed mastery) in Theorem 2 is impossible. The single pattern either meets item j’s requirements ( ${M}_{1t}=\left\{{\boldsymbol{\alpha}}^{\ast}\right\},{M}_{0t}=\varnothing$ ) or doesn’t ( ${M}_{1t}=\varnothing, {M}_{0t}=\left\{{\boldsymbol{\alpha}}^{\ast}\right\}$ ). This is a special case of either Case 2 or Case 3 discussed in Theorem 2. According to Theorem 2, $\lvert M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\rvert$ either stays at 1 or possibly expands. There is no chance of further shrinkage in terms of the size of the set.

Conditions driving shrinkage in LBPS. The probability of shrinkage (Case 1) in Theorem 2 depends critically on heterogeneity within $M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)$ —that is, whether some patterns meet all requirements of item j, while others do not. This heterogeneity is likely to occur when K is large and testing is in early stages, due to the combinatorial structure of the pattern space. After t items which have tested m < K attributes, $M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)$ must contain all ${2}^{K-m}$ variants on untested attributes for each viable tested configuration. This ensures a diverse set of attribute patterns that item j can potentially differentiate.

While early stages benefit from guaranteed shrinkage, repeated shrinkage often drives $\lvert M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\rvert$ to a small number of patterns at the later stage of testing. This results in a computational advantage: with LBPS, item selection methods like KL, PWKL, or SHE need to evaluate items against only a handful of patterns remaining in $M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)$ ; while without LBPS, they must evaluate all ${2}^K$ patterns at every stage. Thus, LBPS effectively leverages the structure of high-dimensional attribute spaces to mitigate the computational burden of exhaustive pattern evaluation. Its advantage is pronounced when K is large, offering the greatest benefit precisely when traditional methods become computationally prohibitive.

Theorem 3 (Pattern set size reduction). General reduction: When Case 1 (shrinkage) occurs, the reduction in the pattern set size of $M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)$ is

$$\begin{align*}\vert M\left({\boldsymbol{x}}_{\boldsymbol{t}},x_{j}\right)\vert =\left\{ \begin{array}{c}\vert M_{1t}\vert ,\quad if\; x_{j}=1,\\ \vert M_{0t}\vert ,\quad if\; x_{j}=0. \end{array}\right.\end{align*}$$

The reduction ratio is ${p}_j=\frac{\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}},x_{j}\right)\right|}{\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|}=\left\{ \begin{array}{c} \frac{\left|{M}_{1t}\right|}{\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|},\quad if\;{x}_{j}=1,\\ \frac{\left|{M}_{0t}\right|}{\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|},\quad if\;x_{j}=0.\end{array}\right.$

This ratio depends on the proportion of patterns in $M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)$ that would ideally yield each response. As a special case, when item j measures only untested attributes:

1. 1. $If\;{x}_j=1:\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)\right|=\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|\cdot \frac{1}{2^{k_{new}}}$

2. 2. $If\;{x}_j=0:\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)\right|=\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|\cdot \left(1-\frac{1}{2^{k_{new}}}\right)$

1. a. When ${x}_j=1$ , $L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)$ is maximized when ${\eta}_j\left(\boldsymbol{\alpha} \right)=1$ . This means that each new required attribute must be mastered. Only $\frac{1}{2^{k_{new}}}$ of all patterns in $M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)$ can have 1’s on all the new attributes. Therefore, $\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)\right|=\frac{\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|}{2^{k_{new}}}$ .

2. b. When ${x}_j=0$ , $L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)$ is maximized when ${\eta}_j\left(\boldsymbol{\alpha} \right)=0$ . This means that at least one of the new required attributes is not mastered, and the proportion of such patterns in $M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)$ is $1-\frac{1}{2^{k_{new}}}$ . Therefore, $\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}},{x}_j\right)\right|=\left|M\left({\boldsymbol{x}}_{\boldsymbol{t}}\right)\right|\cdot \left(1-\frac{1}{2^{k_{new}}}\right)$ .

The aforementioned theorems suggest that the efficiency of LBPS is influenced by the q-vectors of the selected items. The sequential selection of items that assess new attributes leads to a reduced pattern space for item selection in LBPS. This insight coincides with Xu et al.’s (Reference Xu, Wang and Shang2016) optimal initial item selection theory for CD-CAT. Specifically, Xu et al. (Reference Xu, Wang and Shang2016) demonstrated that to achieve minimum test length, the first administered item must assess exactly one attribute, followed by items that sequentially introduce single, previously unmeasured attributes. If this condition is not met, identifying all attribute profiles within K items becomes infeasible, resulting in test lengths exceeding K. If the condition is met, following Theorems 2 and 2, LBPS should help shrink the search space by half at each step during the early stage of the test when K is large, thereby achieving substantial computational gains.

Note that the above theorems are built on the DINA model, but they can be extended to other CDMs with ideal response functions. For example, for the DINO model with ${\omega}_j\left(\boldsymbol{\alpha} \right)=1-\prod_{k=1}^K\left(1-{\alpha_k}^{q_{jk}}\right)$ , the theorems hold with ${\omega}_j$ replacing ${\eta}_j$ . For CDMs without ideal response functions (e.g., general CDMs such as LCDM and G-DINA), LBPS can still be implemented, as it operates directly on likelihood values and relies on likelihood updating. Shrinkage of $M\left({\boldsymbol{x}}_t\right)$ may occur when patterns within $M\left({\boldsymbol{x}}_t\right)$ yield different response probabilities, because likelihood updating favors patterns whose predicted probabilities better align with the observed outcome. Such behavior is more likely when $\mid M\left({x}_t\right)\mid$ is large, as characterized under the DINA model. That said, the specific theoretical properties established in this article (e.g., the characterization of shrinkage via ${M}_{1t}$ and ${M}_{0t}$ ) do not directly apply, and further investigation of LBPS performance under these models is needed.

While the above theorems demonstrate how the set of potential likely attribute patterns changes with each item response, practical implementation requires a concrete algorithm. The following section outlines the step-by-step LBPS procedure.

3.3 Algorithm description

The LBPS algorithm maintains two key sets: (1) $M\left({\boldsymbol{x}}_t\right)$ : most likely patterns or patterns that lead to the maximum likelihood, and (2) ${M}^{\ast}\left({\boldsymbol{x}}_t\right)$ : working pattern set used for item selection. Note that the estimation of each examinee’s attribute profile uses the full pattern space (all possible ${2}^K$ attribute profiles); the working pattern set is only used for item selection. In terms of the working set size, at any stage t, $2\le \mid {M}^{\ast}\left({\boldsymbol{x}}_t\right)\mid \le {2}^K$ . This ensures minimally two distinct patterns for item selection decisions. The algorithm proceeds as follows:

Step 1: First item selection

1. a. Use full pattern space A

2. b. Select an item using the traditional method (or randomization)

3. c. Obtain response ${x}_1$

4. d. Calculate $L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_t\right)$ for all α ∈ A

5. e. Define initial $\mathrm{M}({x}_1) $ and ${M}^{\ast}({x}_1)$ : $\mathrm{M}({x}_1)$ is the set of attribute patterns with the largest likelihood after the first item has been answered. ${M}^{\ast}({x}_1)$ is the working pattern set used for item selection defined as follows:

   $$\begin{align*}{M}^{\ast}\left({x}_{1}\right)=\left\{\begin{array}{c} \mathrm{M}\left({x}_{1}\right),\quad \mathrm{if}\;\left|\mathrm{M}\left({x}_{1}\right)\right|\ge 2,\\ \left\{{\boldsymbol{\unicode{x3b1}}}_{(1)}^1,{\boldsymbol{\unicode{x3b1}}}_{(2)}^1\right\},\quad \mathrm{if}\;\left|\mathrm{M}\left({x}_{1}\right)\right|=1, \end{array}\right.\end{align*}$$
   where ${\boldsymbol{\unicode{x3b1}}}_{(1)}^1$ has maximum likelihood and ${\boldsymbol{\unicode{x3b1}}}_{(2)}^1$ has the second-highest likelihood at stage 1 (t = 1).

6. f. Estimate the examinee’s attribute profile (using full pattern space A) based on the response $\mathrm{M}({x}_1)$ using the maximum likelihood estimation (MLE).

Step 2: Subsequent items (t > 1)

For each eligible item in the pool:

1. a. Pattern space update

   • ○ Calculate $L\left(\boldsymbol{\alpha} |{\boldsymbol{x}}_t\right)$ for all α ∈ A

   • ○ Identify $M\left({\boldsymbol{x}}_t\right)=\left\{\boldsymbol{\alpha} :L\left({\boldsymbol{x}}_t\right)={L}^{\ast}\right\}$ , ${L}^{\ast }$ is the current maximum likelihood value among all profiles’ likelihoods

   • ○ Define working set:

     $$\begin{align*}{M}^{\ast}\left({\boldsymbol{x}}_{t}\right)=\left\{ \begin{array}{c}\mathrm{M}\left({\boldsymbol{x}}_t\right),\quad \mathrm{if} \mid \mathrm{M}\left({\boldsymbol{x}}_t\right)\mid \ge 2,\\ \left\{{\boldsymbol{\unicode{x3b1}}}_{(1)}^{t},{\boldsymbol{\unicode{x3b1}}}_{(2)}^{t}\right\},\quad \mathrm{if} \mid \mathrm{M}\left({\boldsymbol{x}}_{t}\right)\mid =1, \end{array}\right.\end{align*}$$
     where ${\boldsymbol{\unicode{x3b1}}}_{(1)}^t$ has maximum likelihood and ${\boldsymbol{\unicode{x3b1}}}_{(2)}^t$ has the second-highest likelihood at stage t.

2. b. Item selection

   • ○ Couple an existing item selection method with ${M}^{\ast}\left({\boldsymbol{x}}_t\right)$ instead of A to select the next item. For example, when KL is used for item selection, the summation in (5) is not over all ${2}^K$ patterns in A, but only over the patterns in ${M}^{\ast}\left({\boldsymbol{x}}_t\right).$

1. a. Response processing

   • - Obtain response ${x}_{t+1}$ and iterate steps a) and b)

Repeat the whole process of Step 2 until the desired number of items has been administered or a prefixed termination criterion has been reached.

Logically, the key efficiency gain of LBPS comes from restricting the item selection computations to within the working set, which shrinks quickly over time, while maintaining estimation accuracy through full pattern space calculations. Practically, the extent to which LBPS helps improve computational efficiency and maintains classification accuracy needs to be evaluated in light of many factors, such as the number of attributes K, the test length, and the underlying CDM model. Therefore, a simulation study was conducted, manipulating these factors to evaluate the practical impact of the LBPS.