Nonparametric CD-CAT for multiple-choice items: Item selection method and Q-optimality.

Computerized adaptive testing for cognitive diagnosis (CD-CAT) achieves remarkable estimation efficiency and accuracy by adaptively selecting and then administering items tailored to each examinee. The process of item selection stands as a pivotal component of a CD-CAT algorithm, with various methods having been developed for binary responses. However, multiple-choice (MC) items, an important item type that allows for the extraction of richer diagnostic information from incorrect answers, have been underemphasized. Currently, the Jensen-Shannon divergence (JSD) index introduced by Yigit et al. (Applied Psychological Measurement, 2019, 43, 388) is the only item selection method exclusively designed for MC items. However, the JSD index requires a large sample to calibrate item parameters, which may be infeasible when there is only a small or no calibration sample. To bridge this gap, the study first proposes a nonparametric item selection method for MC items (MC-NPS) by implementing novel discrimination power that measures an item's ability to effectively distinguish among different attribute profiles. A Q-optimal procedure for MC items is also developed to improve the classification during the initial phase of a CD-CAT algorithm. The effectiveness and efficiency of the two proposed algorithms were confirmed by simulation studies.

Commentary on "Extending the Basic Local Independence Model to Polytomous Data" by Stefanutti, de Chiusole, Anselmi, and Spoto.

The Polytomous Local Independence Model (PoLIM) by Stefanutti, de Chiusole, Anselmi, and Spoto, is an extension of the Basic Local Independence Model (BLIM) to accommodate polytomous items. BLIM, a model for analyzing responses to binary items, is based on Knowledge Space Theory, a framework developed by cognitive scientists and mathematical psychologists for modeling human knowledge acquisition and representation. The purpose of this commentary is to show that PoLIM is simply a paraphrase of a DINA model in cognitive diagnosis for polytomous items. Specifically, BLIM is shown to be equivalent to the DINA model when the BLIM-items are conceived as binary single-attribute items, each with a distinct attribute; thus, PoLIM is equivalent to the DINA for polytomous single-attribute items, each with a distinct attribute.

Consistency Theory for the General Nonparametric Classification Method.

Parametric likelihood estimation is the prevailing method for fitting cognitive diagnosis models-also called diagnostic classification models (DCMs). Nonparametric concepts and methods that do not rely on a parametric statistical model have been proposed for cognitive diagnosis. These methods are particularly useful when sample sizes are small. The general nonparametric classification (GNPC) method for assigning examinees to proficiency classes can accommodate assessment data conforming to any diagnostic classification model that describes the probability of a correct item response as an increasing function of the number of required attributes mastered by an examinee (known as the "monotonicity assumption"). Hence, the GNPC method can be used with any model that can be represented as a general DCM. However, the statistical properties of the estimator of examinees' proficiency class are currently unknown. In this article, the consistency theory of the GNPC proficiency-class estimator is developed and its statistical consistency is proven.

A Procedure for Assessing the Completeness of the Q-Matrices of Cognitively Diagnostic Tests.

The Q-matrix of a cognitively diagnostic test is said to be complete if it allows for the identification of all possible proficiency classes among examinees. Completeness of the Q-matrix is therefore a key requirement for any cognitively diagnostic test. However, completeness of the Q-matrix is often difficult to establish, especially, for tests with a large number of items involving multiple attributes. As an additional complication, completeness is not an intrinsic property of the Q-matrix, but can only be assessed in reference to a specific cognitive diagnosis model (CDM) supposed to underly the data-that is, the Q-matrix of a given test can be complete for one model but incomplete for another. In this article, a method is presented for assessing whether a given Q-matrix is complete for a given CDM. The proposed procedure relies on the theoretical framework of general CDMs and is therefore legitimate for CDMs that can be reparameterized as a general CDM.

Joint Maximum Likelihood Estimation for Diagnostic Classification Models.

Joint maximum likelihood estimation (JMLE) is developed for diagnostic classification models (DCMs). JMLE has been barely used in Psychometrics because JMLE parameter estimators typically lack statistical consistency. The JMLE procedure presented here resolves the consistency issue by incorporating an external, statistically consistent estimator of examinees' proficiency class membership into the joint likelihood function, which subsequently allows for the construction of item parameter estimators that also have the consistency property. Consistency of the JMLE parameter estimators is established within the framework of general DCMs: The JMLE parameter estimators are derived for the Loglinear Cognitive Diagnosis Model (LCDM). Two consistency theorems are proven for the LCDM. Using the framework of general DCMs makes the results and proofs also applicable to DCMs that can be expressed as submodels of the LCDM. Simulation studies are reported for evaluating the performance of JMLE when used with tests of varying length and different numbers of attributes. As a practical application, JMLE is also used with "real world" educational data collected with a language proficiency test.

Consistency of Cluster Analysis for Cognitive Diagnosis: The Reduced Reparameterized Unified Model and the General Diagnostic Model.

The asymptotic classification theory of cognitive diagnosis (ACTCD) provided the theoretical foundation for using clustering methods that do not rely on a parametric statistical model for assigning examinees to proficiency classes. Like general diagnostic classification models, clustering methods can be useful in situations where the true diagnostic classification model (DCM) underlying the data is unknown and possibly misspecified, or the items of a test conform to a mix of multiple DCMs. Clustering methods can also be an option when fitting advanced and complex DCMs encounters computational difficulties. These can range from the use of excessive CPU times to plain computational infeasibility. However, the propositions of the ACTCD have only been proven for the Deterministic Input Noisy Output "AND" gate (DINA) model and the Deterministic Input Noisy Output "OR" gate (DINO) model. For other DCMs, there does not exist a theoretical justification to use clustering for assigning examinees to proficiency classes. But if clustering is to be used legitimately, then the ACTCD must cover a larger number of DCMs than just the DINA model and the DINO model. Thus, the purpose of this article is to prove the theoretical propositions of the ACTCD for two other important DCMs, the Reduced Reparameterized Unified Model and the General Diagnostic Model.

An evaluation of exact methods for the multiple subset maximum cardinality selection problem.

The maximum cardinality subset selection problem requires finding the largest possible subset from a set of objects, such that one or more conditions are satisfied. An important extension of this problem is to extract multiple subsets, where the addition of one more object to a larger subset would always be preferred to increases in the size of one or more smaller subsets. We refer to this as the multiple subset maximum cardinality selection problem (MSMCSP). A recently published branch-and-bound algorithm solves the MSMCSP as a partitioning problem. Unfortunately, the computational requirement associated with the algorithm is often enormous, thus rendering the method infeasible from a practical standpoint. In this paper, we present an alternative approach that successively solves a series of binary integer linear programs to obtain a globally optimal solution to the MSMCSP. Computational comparisons of the methods using published similarity data for 45 food items reveal that the proposed sequential method is computationally far more efficient than the branch-and-bound approach.

The Reduced RUM as a Logit Model: Parameterization and Constraints.

Cognitive diagnosis models (CDMs) for educational assessment are constrained latent class models. Examinees are assigned to classes of intellectual proficiency defined in terms of cognitive skills called attributes, which an examinee may or may not have mastered. The Reduced Reparameterized Unified Model (Reduced RUM) has received considerable attention among psychometricians. Markov Chain Monte Carlo (MCMC) or Expectation Maximization (EM) are typically used for estimating the Reduced RUM. Commercial implementations of the EM algorithm are available in the latent class analysis (LCA) routines of Latent GOLD and Mplus, for example. Fitting the Reduced RUM with an LCA routine requires that it be reparameterized as a logit model, with constraints imposed on the parameters. For models involving two attributes, these have been worked out. However, for models involving more than two attributes, the parameterization and the constraints are nontrivial and currently unknown. In this article, the general parameterization of the Reduced RUM as a logit model involving any number of attributes and the associated parameter constraints are derived. As a practical illustration, the LCA routine in Mplus is used for fitting the Reduced RUM to two synthetic data sets and to a real-world data set; for comparison, the results obtained by using the MCMC implementation in OpenBUGS are also provided.

An Exact Method for Partitioning Dichotomous Items Within the Framework of the Monotone Homogeneity Model.

The monotone homogeneity model (MHM-also known as the unidimensional monotone latent variable model) is a nonparametric IRT formulation that provides the underpinning for partitioning a collection of dichotomous items to form scales. Ellis (Psychometrika 79:303-316, 2014, doi: 10.1007/s11336-013-9341-5 ) has recently derived inequalities that are implied by the MHM, yet require only the bivariate (inter-item) correlations. In this paper, we incorporate these inequalities within a mathematical programming formulation for partitioning a set of dichotomous scale items. The objective criterion of the partitioning model is to produce clusters of maximum cardinality. The formulation is a binary integer linear program that can be solved exactly using commercial mathematical programming software. However, we have also developed a standalone branch-and-bound algorithm that produces globally optimal solutions. Simulation results and a numerical example are provided to demonstrate the proposed method.

A general proof of consistency of heuristic classification for cognitive diagnosis models.

The Asymptotic Classification Theory of Cognitive Diagnosis (Chiu et al., 2009, Psychometrika, 74, 633-665) determined the conditions that cognitive diagnosis models must satisfy so that the correct assignment of examinees to proficiency classes is guaranteed when non-parametric classification methods are used. These conditions have only been proven for the Deterministic Input Noisy Output AND gate model. For other cognitive diagnosis models, no theoretical legitimization exists for using non-parametric classification techniques for assigning examinees to proficiency classes. The specific statistical properties of different cognitive diagnosis models require tailored proofs of the conditions of the Asymptotic Classification Theory of Cognitive Diagnosis for each individual model ­ a tedious undertaking in light of the numerous models presented in the literature. In this paper a different way is presented to address this task. The unified mathematical framework of general cognitive diagnosis models is used as a theoretical basis for a general proof that under mild regularity conditions any cognitive diagnosis model is covered by the Asymptotic Classification Theory of Cognitive Diagnosis.

Consistency of Cluster Analysis for Cognitive Diagnosis: The DINO Model and the DINA Model Revisited.

The Asymptotic Classification Theory of Cognitive Diagnosis (ACTCD) developed by Chiu, Douglas, and Li proved that for educational test data conforming to the Deterministic Input Noisy Output "AND" gate (DINA) model, the probability that hierarchical agglomerative cluster analysis (HACA) assigns examinees to their true proficiency classes approaches 1 as the number of test items increases. This article proves that the ACTCD also covers test data conforming to the Deterministic Input Noisy Output "OR" gate (DINO) model. It also demonstrates that an extension to the statistical framework of the ACTCD, originally developed for test data conforming to the Reduced Reparameterized Unified Model or the General Diagnostic Model (a) is valid also for both the DINA model and the DINO model and (b) substantially increases the accuracy of HACA in classifying examinees when the test data conform to either of these two models.

Heuristic cognitive diagnosis when the Q-matrix is unknown.

Cognitive diagnosis models of educational test performance rely on a binary Q-matrix that specifies the associations between individual test items and the cognitive attributes (skills) required to answer those items correctly. Current methods for fitting cognitive diagnosis models to educational test data and assigning examinees to proficiency classes are based on parametric estimation methods such as expectation maximization (EM) and Markov chain Monte Carlo (MCMC) that frequently encounter difficulties in practical applications. In response to these difficulties, non-parametric classification techniques (cluster analysis) have been proposed as heuristic alternatives to parametric procedures. These non-parametric classification techniques first aggregate each examinee's test item scores into a profile of attribute sum scores, which then serve as the basis for clustering examinees into proficiency classes. Like the parametric procedures, the non-parametric classification techniques require that the Q-matrix underlying a given test be known. Unfortunately, in practice, the Q-matrix for most tests is not known and must be estimated to specify the associations between items and attributes, risking a misspecified Q-matrix that may then result in the incorrect classification of examinees. This paper demonstrates that clustering examinees into proficiency classes based on their item scores rather than on their attribute sum-score profiles does not require knowledge of the Q-matrix, and results in a more accurate classification of examinees.

The p-median model as a tool for clustering psychological data.

The p-median clustering model represents a combinatorial approach to partition data sets into disjoint, nonhierarchical groups. Object classes are constructed around exemplars, that is, manifest objects in the data set, with the remaining instances assigned to their closest cluster centers. Effective, state-of-the-art implementations of p-median clustering are virtually unavailable in the popular social and behavioral science statistical software packages. We present p-median clustering, including a detailed description of its mechanics and a discussion of available software programs and their capabilities. Application to a complex structured data set on the perception of food items illustrates p-median clustering.

Comment on "Clustering by passing messages between data points".

Frey and Dueck (Reports, 16 February 2007, p. 972) described an algorithm termed "affinity propagation" (AP) as a promising alternative to traditional data clustering procedures. We demonstrate that a well-established heuristic for the p-median problem often obtains clustering solutions with lower error than AP and produces these solutions in comparable computation time.