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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.

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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.

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.