Basic Local Independence Model Research Articles

Assessing Parameter Invariance in the BLIM: Bipartition Models

In knowledge space theory, the knowledge state of a student is the set of all problems he is capable of solving in a specific knowledge domain and a knowledge structure is the collection of knowledge states. The basic local independence model (BLIM) is a probabilistic model for knowledge structures. The BLIM assumes a probability distribution on the knowledge states and a lucky guess and a careless error probability for each problem. A key assumption of the BLIM is that the lucky guess and careless error probabilities do not depend on knowledge states (invariance assumption). This article proposes a method for testing the violations of this specific assumption. The proposed method was assessed in a simulation study and in an empirical application. The results show that (1) the invariance assumption might be violated by the empirical data even when the model's fit is very good, and (2) the proposed method may prove to be a promising tool to detect invariance violations of the BLIM.

On the unidentifiability of a certain class of skill multi map based probabilistic knowledge structures

In the process of fitting a probabilistic knowledge structure to data, standard goodness-of-fit statistics only partially describe the correctness of the fitted model. Irrespectively of how good the fit is, a too-high value of the error rates (careless error and lucky guess probabilities) might be a symptom of a misspecification of the model. In this situation, it could be critical to interpret those values as error rates. A more reasonable solution would be to hypothesize that some modifications have to be introduced in the model. In this paper, we show that in specific cases, these modifications yield basic local independence model parameterizations that are not identifiable. The applicative consequences of the theoretical results are displayed by means of an example carried out on a set of clinical data collected through the Maudsley Obsessional-Compulsive Questionnaire.

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.

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.

Recovering a Probabilistic Knowledge Structure by Constraining its Parameter Space

In the Basic Local Independence Model (BLIM) of Doignon and Falmagne (Knowledge Spaces, Springer, Berlin, 1999), the probabilistic relationship between the latent knowledge states and the observable response patterns is established by the introduction of a pair of parameters for each of the problems: a lucky guess probability and a careless error probability. In estimating the parameters of the BLIM with an empirical data set, it is desirable that such probabilities remain reasonably small. A special case of the BLIM is proposed where the parameter space of such probabilities is constrained. A simulation study shows that the constrained BLIM is more effective than the unconstrained one, in recovering a probabilistic knowledge structure.

Nonparametric item response theory axioms and properties under nonlinearity and their exemplification with knowledge space theory

This paper investigates the dichotomous Mokken nonparametric item response theory (IRT) axioms and properties under incomparabilities among latent trait values and items. Generalized equivalents of the unidimensional nonparametric IRT axioms and properties are formulated for nonlinear (quasi-ordered) person and indicator spaces. It is shown that monotone likelihood ratio (MLR) for the total score variable and nonlinear latent trait implies stochastic ordering (SO) of the total score variable, but may fail to imply SO of the nonlinear latent trait. The reason for this and conditions under which the implication holds are specified, based on a new, simpler proof of the fact that in the unidimensional case MLR implies SO. The approach is applied in knowledge space theory (KST), a combinatorial test theory. This leads to a (tentative) Mokken-type nonparametric axiomatization in the currently parametric theory of knowledge spaces. The nonparametric axiomatization is compared with the assumptions of the parametric basic local independence model which is fundamental in KST. It is concluded that this paper may provide a first step toward a basis for a possible fusion of the two split directions of psychological test theories IRT and KST.

Estimation of careless error and lucky guess probabilities for dichotomous test items: A psychometric application of a biometric latent class model with random effects

Medical research has extensively dealt with the estimation of the accuracy (sensitivity and specificity) of a diagnostic test for screening individuals. In this paper we apply the biometric latent class model with random effects by Qu, Tan, and Kutner [(1996). Random effects models in latent class analysis for evaluating accuracy of diagnostic tests. Biometrics, 52, 797–810] to estimate the response error (careless error and lucky guess) probabilities for dichotomous test items in the psychometric theory of knowledge spaces. The approach is illustrated with simulated data. In particular, we extend this approach to give a generalization of the basic local independence model in knowledge space theory. This allows for local dependence among the indicators given the knowledge state of an examinee and/or for the incorporation of covariates.