Journal of Mathematical Psychology 57 (2013) 249–254 Contents lists available at ScienceDirect Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp Considerations about the identification of forward- and backward-graded knowledge structures Andrea Spoto a,∗ , Luca Stefanutti b , Giulio Vidotto a a Department of General Psychology, University of Padua, Italy b FISPPA, University of Padua, Italy highlights • The BLIM is unidentifiable for FG and BG knowledge structures. • A relevant number of knowledge structures is FG and/or BG. • The introduction of equally informative items solves unidentifiability. • A structure where a tradeoff dimension involves one error parameter is FG or BG. • A relationship exists between unidentifiability and forward- and backward-gradedness. article info Article history: Received 19 December 2012 Received in revised form 10 September 2013 Available online 7 October 2013 Keywords: Identifiability Forward-graded knowledge structures Backward-graded knowledge structures Probabilistic knowledge structures Learning spaces Equally informative items abstract The application of the basic local independence model (BLIM) to a knowledge structure (Q , K) that satisfies a particular kind of gradation (namely forward- or backward-gradedness) leads the model to be not identifiable. In the present article, we show that many important types of knowledge structures happen to be either forward- or backward-graded. This means that the application of the BLIM to these structures leads to unidentifiable models. No universal remedy for recovering identifiability is presently known. However, we propose a construction that consists in introducing an equally informative item for each item in Q . We conjecture that the BLIM based on the resulting knowledge structure is always identifiable. This conjecture is proven to be true for knowledge structures on small sets of items. © 2013 Elsevier Inc. All rights reserved. 1. Introduction In knowledge space theory (KST; Doignon & Falmagne, 1985, 1999; Falmagne & Doignon, 2011) the knowledge state of a student is the collection of all problems that she/he masters in a given field of knowledge, and a knowledge structure is the collection of all possible knowledge states in a given population of students. The basic local independence model (BLIM; Doignon & Falmagne, 1999; Falmagne & Doignon, 1988) is a probabilistic model applied, in KST, for the stochastic assessment of knowledge, and for the empirical validation of knowledge structures. This model has lately received some attention (de Chiu- sole, Stefanutti, Anselmi, & Robusto, in press; Heller, under re- vision; Schrepp, 2005; Stefanutti & Robusto, 2009), particularly ∗ Correspondence to: Department of General Psychology, Via Venezia 8, 35131 Padua, Italy. E-mail addresses: andrea.spoto@unipd.it (A. Spoto), luca.stefanutti@unipd.it (L. Stefanutti), giulio.vidotto@unipd.it (G. Vidotto). concerning its identifiability. At the moment, there is still in- complete knowledge about this fundamental issue of the BLIM. Notwithstanding, a few results have been recently obtained in this direction. For instance, Spoto, Stefanutti, and Vidotto (2012) have shown that the BLIM is not identifiable for two broad classes of knowledge structures named, respectively, forward-graded and backward-graded. Moreover, Stefanutti, Heller, Anselmi, and Ro- busto (2012) developed a procedure for testing the local identifi- ability of the BLIM with arbitrary knowledge structures on sets of items having a moderate size. Furthermore, some connections be- tween the BLIM and latent class models, including identifiability issues, have been pointed out by Schrepp (2005) and Ünlü (2011). While it is known that the identifiability of the BLIM strictly de- pends on the specific knowledge structure to which it is applied, it is still not clear how to separate all knowledge structures, on a finite set of items, that make the BLIM unidentifiable, from the re- maining ones. In other words it is not known, in general, how to read the identifiability of the BLIM, directly from the combinato- rial properties of the knowledge structure to which it is applied. 0022-2496/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmp.2013.09.002