Algorithms for the adaptive assessment of procedural knowledge and skills.

Procedural knowledge space theory (PKST) was recently proposed by Stefanutti (British Journal of Mathematical and Statistical Psychology, 72(2) 185-218, 2019) for the assessment of human problem-solving skills. In PKST, the problem space formally represents how a family of problems can be solved and the knowledge space represents the skills required for solving those problems. The Markov solution process model (MSPM) by Stefanutti et al. (Journal of Mathematical Psychology, 103, 102552, 2021) provides a probabilistic framework for modeling the solution process of a task, via PKST. In this article, three adaptive procedures for the assessment of problem-solving skills are proposed that are based on the MSPM. Beside execution correctness, they also consider the sequence of moves observed in the solution of a problem with the aim of increasing efficiency and accuracy of assessments. The three procedures differ from one another in the assumption underlying the solution process, named pre-planning, interim-planning, and mixed-planning. In two simulation studies, the three adaptive procedures have been compared to one another and to the continuous Markov procedure (CMP) by Doignon and Falmagne (1988a). The last one accounts for dichotomous correct/wrong answers only. Results show that all the MSP-based adaptive procedures outperform the CMP in both accuracy and efficiency. These results have been obtained in the framework of the Tower of London test but the procedures can also be applied to all psychological and neuropsychological tests that have a problem space. Thus, the adaptive procedures presented in this paper pave the way to the adaptive assessment in the area of neuropsychological tests.

Extracting partially ordered clusters from ordinal polytomous data.

In practical applications of knowledge space theory, knowledge states can be conceived as partially ordered clusters of individuals. Existing extensions of the theory to polytomous data lack methods for building "polytomous" structures. To this aim, an adaptation of the k-median clustering algorithm is proposed. It is an extension of k-modes to ordinal data in which the Hamming distance is replaced by the Manhattan distance, and the central tendency measure is the median, rather than the mode. The algorithm is tested in a series of simulation studies and in an application to empirical data. Results show that there are theoretical and practical reasons for preferring the k-median to the k-modes algorithm, whenever the responses to the items are measured on an ordinal scale. This is because the Manhattan distance is sensitive to the order on the levels, while the Hamming distance is not. Overall, k-median seems to be a promising data-driven procedure for building polytomous structures.

Does discrimination beat association in the IAT? The discrimination-association model reconceived.

The discrimination-association model (DAM; Stefanutti et al. 2013) disentangles two components underlying the responses to the implicit association test (IAT), which pertain to stimuli discrimination (the strength of the association of the stimuli with their own category) and automatic association (the strength of the association between targets and attributes). The assumption of the DAM that these two components sum into a single process generates critical drawbacks. The present work provides a new formulation of the model, called DAM-4C, in which stimuli discrimination and automatic association are separate, independent, and competing processes. Results of theoretical and simulation studies suggest that the DAM-4C outperforms the DAM. The IAT effect is found to vary with the association rates of the DAM-4C and not with those of the DAM. The parameters of the DAM-4C fitted on data from a Coca-Pepsi IAT are found to account for variance in brand attractiveness, taste preference, and cola choice that is not accounted for by the D score and the diffusion model. In addition, the association rates estimated on data from a Black-White IAT are in line with expectations.

Testing the actual equivalence of automatically generated items.

If the automatic item generation is used for generating test items, the question of how the equivalence among different instances may be tested is fundamental to assure an accurate assessment. In the present research, the question was dealt by using the knowledge space theory framework. Two different ways of considering the equivalence among instances are proposed: The former is at a deterministic level and it requires that all the instances of an item template must belong to exactly the same knowledge states; the latter adds a probabilistic level to the deterministic one. The former type of equivalence can be modeled by using the BLIM with a knowledge structure assuming equally informative instances; the latter can be modeled by a constrained BLIM. This model assumes equality constraints among the error parameters of the equivalent instances. An approach is proposed for testing the equivalence among instances, which is based on a series of model comparisons. A simulation study and an empirical application show the viability of the approach.

A class of k-modes algorithms for extracting knowledge structures from data.

One of the most crucial issues in knowledge space theory is the construction of the so-called knowledge structures. In the present paper, a new data-driven procedure for large data sets is described, which overcomes some of the drawbacks of the already existing methods. The procedure, called k-states, is an incremental extension of the k-modes algorithm, which generates a sequence of locally optimal knowledge structures of increasing size, among which a "best" model is selected. The performance of k-states is compared to other two procedures in both a simulation study and an empirical application. In the former, k-states displays a better accuracy in reconstructing knowledge structures; in the latter, the structure extracted by k-states obtained a better fit.

An iterative procedure for extracting skill maps from data.

The methodologies for the construction of a knowledge structure mainly refer to the query to experts, the skill maps, and the data-driven approaches. This last method is of growing interest in recent literature. In this paper, an iterative procedure for building a skill map from a set of data is introduced. This procedure is based on the minimization of the distance between the knowledge structure delineated by a given skill map and the data. The accuracy of the proposed method is tested through a number of simulation studies where the amount of noise in the data is manipulated as well as the kind of structure to be reconstructed. Results show that the procedure is accurate and that its performance tends to be sufficiently stable even with high error rates. The procedure is compared to two already-existing methodologies to derive knowledge structures from a set of data. The use of the corrected Akaike Information Criterion (AICc) as a stopping criterion of the iterative reconstruction procedure is tested against the app criterion introduced by Schrepp. Moreover, two empirical applications on clinical data are reported, and their results show the applicability of the procedure.

PsycAssist: A Web-Based Artificial Intelligence System Designed for Adaptive Neuropsychological Assessment and Training.

Assessing executive functions in individuals with disorders or clinical conditions can be challenging, as they may lack the abilities needed for conventional test formats. The use of more personalized test versions, such as adaptive assessments, might be helpful in evaluating individuals with specific needs. This paper introduces PsycAssist, a web-based artificial intelligence system designed for neuropsychological adaptive assessment and training. PsycAssist is a highly flexible and scalable system based on procedural knowledge space theory and may be used potentially with many types of tests. We present the architecture and adaptive assessment engine of PsycAssist and the two currently available tests: Adap-ToL, an adaptive version of the Tower of London-like test to assess planning skills, and MatriKS, a Raven-like test to evaluate fluid intelligence. Finally, we describe the results of an investigation of the usability of Adap-ToL and MatriKS: the evaluators perceived these tools as appropriate and well-suited for their intended purposes, and the test-takers perceived the assessment as a positive experience. To sum up, PsycAssist represents an innovative and promising tool to tailor evaluation and training to the specific characteristics of the individual, useful for clinical practice.

A discrimination-association model for decomposing component processes of the implicit association test.

A formal model is proposed that decomposes the implicit association test (IAT) effect into three process components: stimuli discrimination, automatic association, and termination criterion. Both response accuracy and reaction time are considered. Four independent and parallel Poisson processes, one for each of the four label categories of the IAT, are assumed. The model parameters are the rate at which information accrues on the counter of each process and the amount of information that is needed before a response is given. The aim of this study is to present the model and an illustrative application in which the process components of a Coca-Pepsi IAT are decomposed.

Empirical indistinguishability: From the knowledge structure to the skills.

Recent literature has pointed out that the basic local independence model (BLIM) when applied to some specific instances of knowledge structures presents identifiability issues. Furthermore, it has been shown that for such instances the model presents a stronger form of unidentifiability named empirical indistinguishability, which leads to the fact that the existence of certain knowledge states in such structures cannot be empirically tested. In this article the notion of indistinguishability is extended to skill maps and, more generally, to the competence-based knowledge space theory. Theoretical results are provided showing that skill maps can be empirically indistinguishable from one another. The most relevant consequence of this is that for some skills there is no empirical evidence to establish their existence. This result is strictly related to the type of probabilistic model investigated, which is essentially the BLIM. Alternative models may exist or can be developed in knowledge space theory for which this indistinguishability problem disappears.

Assessing the local identifiability of probabilistic knowledge structures.

Given a collection Q of problems, in knowledge space theory Doignon & Falmagne, (International Journal of Man-Machine Studies 23:175-196, 1985) the knowledge state of a student is the collection K ⊆ Q of all problems that this student is capable of solving. A knowledge structure is a pair (Q, ), where is a collection of knowledge states that contains at least the empty set and Q. A probabilistic knowledge structure (PKS) is a knowledge structure (Q, , π), where π is a probability distribution on the knowledge states. The PKS that has received the most attention is the basic local independence model BLIM; Falmagne & Doignon, (British Journal of Mathematical and Statistical Psychology 41:1-23, 1988a, Journal of Mathematical Psychology 32:232-258, 1988b). To the best of our knowledge, systematic investigations in the literature concerning the identifiability of the BLIM are totally missing. Based on the theoretical work of Bamber and van Santen (Journal of Mathematical Psychology 29:443-473, 1985), the present article is aimed to present a method and a corresponding computerized procedure for assessing the local identifiability of the BLIM, which is applicable to any finite knowledge structure of moderate size.

Assessing learning processes with the gain-loss model.

Within the framework of knowledge space theory, a probabilistic skill multimap model for assessing learning processes is proposed. The learning process of a student is modeled as a function of the interaction between his competence state and the effect of a learning object on specific skills. Model parameters are initial probabilities of the skills, effects of learning objects on gaining and losing the skills, careless error, and lucky guess probabilities of the problems. A simulation study assessed model identifiability and goodness-of-recovery under several conditions. Practical implications of using the model are discussed, and the MATLAB code for simulating, estimating and testing it is available in the Psychonomic Society supplemental archive.

On the empirical indistinguishability of knowledge structures.

In recent years a number of articles have focused on the identifiability of the basic local independence model. The identifiability issue usually concerns two model parameter sets predicting an identical probability distribution on the response patterns. Both parameter sets are applied to the same knowledge structure. However, nothing is known about cases where different knowledge structures predict the same probability distribution. This situation is referred to as 'empirical indistinguishability' between two structures and is the main subject of the present paper. Empirical indistinguishability is a stronger form of unidentifiability, which involves not only the parameters, but also the structural and combinatorial properties of the model. In particular, as far as knowledge structures are concerned, a consequence of empirical indistinguishability is that the existence of certain knowledge states cannot be empirically established. Most importantly, it is shown that model identifiability cannot guarantee that a certain knowledge structure is empirically distinguishable from others. The theoretical findings are exemplified in a number of different empirical scenarios.

Knowledge space theory, formal concept analysis, and computerized psychological assessment.

In the present study, the use of knowledge space theory (KST), jointly with formal concept analysis (FCA), is proposed for developing a formal representation of the relations between the items of a questionnaire and a set of psychodiagnostic criteria. This formal representation can be used to develop an efficient adaptive tool for psychological assessment. Rusch and Wille (1996) have shown some interesting connections between KST and FCA; these connections are applied in the construction of knowledge structures, starting from a formal context representing the relations between items and criteria. The proposed general methodology was applied, as an example, to the Maudsley Obsessional-Compulsive Questionnaire. We used a data set provided by a sample of patients with a diagnosis of obsessive-compulsive disorder to validate the obtained structures. The parameters of the basic local independence model (BLIM) were estimated for the obtained knowledge structures. The fit of each model was tested by parametric bootstrap because of the sparseness of the derived data matrix. The results are discussed in light of both psychological and methodological relapses. In particular, we propose a reinterpretation of the BLIM parameters that seems suitable for testing reliability and construct validity; furthermore, it is pointed out how the obtained structures could represent the starting point for the development of a computerized assessment tool.

Extending the Basic Local Independence Model to Polytomous Data.

A probabilistic framework for the polytomous extension of knowledge space theory (KST) is proposed. It consists in a probabilistic model, called polytomous local independence model, that is developed as a generalization of the basic local independence model. The algorithms for computing "maximum likelihood" (ML) and "minimum discrepancy" (MD) estimates of the model parameters have been derived and tested in a simulation study. Results show that the algorithms differ in their capability of recovering the true parameter values. The ML algorithm correctly recovers the true values, regardless of the manipulated variables. This is not totally true for the MD algorithm. Finally, the model has been applied to a real polytomous data set collected in the area of psychological assessment. Results show that it can be successfully applied in practice, paving the way to a number of applications of KST outside the area of knowledge and learning assessment.

On the assessment of procedural knowledge: From problem spaces to knowledge spaces.

By generalizing and completing the work initiated by Stefanutti and Albert (2003, Journal of Universal Computer Science, 9, 1455), this article provides the mathematical foundations of a theoretical approach whose primary goal is to construct a bridge between problem solving, as initially conceived by Newell and Simon (1972, Human problem solving. Englewood Cliffs, NJ: Prentice-Hall.), and knowledge assessment (Doignon and Falmagne, 1985, International Journal of Man-Machine Studies, 23, 175; Doignon and Falmagne, 1999, Knowledge spaces. Berlin, Germany: Springer-Verlag.; Falmagne et al., 2013, Knowledge spaces: Applications in education. New York, NY: Springer-Verlag; Falmagne and Doignon, 2011, Learning spaces: Interdisciplinary applied mathematics. Berlin, Germany: Springer-Verlag.). It is shown that the collection of all possible knowledge states for a given problem space is a learning space. An algorithm for deriving a learning space from a problem space is illustrated. As an example, the algorithm is used to derive the learning space of a neuropsychological test whose problem space is well known: the Tower of London (TOL; Shallice, 1982, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 298, 199). The derived learning space could then be used for adaptively assessing individual planning skills with the TOL.

The assessment of knowledge and learning in competence spaces: The gain-loss model for dependent skills.

The gain-loss model (GaLoM) is a formal model for assessing knowledge and learning. In its original formulation, the GaLoM assumes independence among the skills. Such an assumption is not reasonable in several domains, in which some preliminary knowledge is the foundation for other knowledge. This paper presents an extension of the GaLoM to the case in which the skills are not independent, and the dependence relation among them is described by a well-graded competence space. The probability of mastering skill s at the pretest is conditional on the presence of all skills on which s depends. The probabilities of gaining or losing skill s when moving from pretest to posttest are conditional on the mastery of s at the pretest, and on the presence at the posttest of all skills on which s depends. Two formulations of the model are presented, in which the learning path is allowed to change from pretest to posttest or not. A simulation study shows that models based on the true competence space obtain a better fit than models based on false competence spaces, and are also characterized by a higher assessment accuracy. An empirical application shows that models based on pedagogically sound assumptions about the dependencies among the skills obtain a better fit than models assuming independence among the skills.

An Upgrading Procedure for Adaptive Assessment of Knowledge.

In knowledge space theory, existing adaptive assessment procedures can only be applied when suitable estimates of their parameters are available. In this paper, an iterative procedure is proposed, which upgrades its parameters with the increasing number of assessments. The first assessments are run using parameter values that favor accuracy over efficiency. Subsequent assessments are run using new parameter values estimated on the incomplete response patterns from previous assessments. Parameter estimation is carried out through a new probabilistic model for missing-at-random data. Two simulation studies show that, with the increasing number of assessments, the performance of the proposed procedure approaches that of gold standards.

On the Link between Cognitive Diagnostic Models and Knowledge Space Theory.

The present work explores the connections between cognitive diagnostic models (CDM) and knowledge space theory (KST) and shows that these two quite distinct approaches overlap. It is proved that in fact the Multiple Strategy DINA (Deterministic Input Noisy AND-gate) model and the CBLIM, a competence-based extension of the basic local independence model (BLIM), are equivalent. To demonstrate the benefits that arise from integrating the two theoretical perspectives, it is shown that a fairly complete picture on the identifiability of these models emerges by combining results from both camps. The impact of the results is illustrated by an empirical example, and topics for further research are pointed out.

Modeling missing data in knowledge space theory.

Missing data are a well known issue in statistical inference, because some responses may be missing, even when data are collected carefully. The problem that arises in these cases is how to deal with missing data. In this article, the missingness is analyzed in knowledge space theory, and in particular when the basic local independence model (BLIM) is applied to the data. Two extensions of the BLIM to missing data are proposed: The former, called ignorable missing BLIM (IMBLIM), assumes that missing data are missing completely at random; the latter, called missing BLIM (MissBLIM), introduces specific dependencies of the missing data on the knowledge states, thus assuming that the missing data are missing not at random. The IMBLIM and the MissBLIM modeled the missingness in a satisfactory way, in both a simulation study and an empirical application, depending on the process that generates the missingness: If the missing data-generating process is of type missing completely at random, then either IMBLIM or MissBLIM provide adequate fit to the data. However, if the pattern of missingness is functionally dependent upon unobservable features of the data (e.g., missing answers are more likely to be wrong), then only a correctly specified model of the missingness distribution provides an adequate fit to the data.

Naïve Tests of Basic Local Independence Model's Invariance.

The basic local independence model (BLIM) is a probabilistic model for knowledge structures, characterized by the property that lucky guess and careless error parameters of the items are independent of the knowledge states of the subjects. When fitting the BLIM to empirical data, a good fit can be obtained even when the invariance assumption is violated. Therefore, statistical tests are needed for detecting violations of this specific assumption. This work provides an extension to theoretical results obtained by de Chiusole, Stefanutti, Anselmi, and Robusto (2013), showing that statistical tests based on the partitioning of the empirical data set into two (or more) groups are not adequate for testing the BLIM's invariance assumption. A simulation study confirms the theoretical results.