Modeling missing data in knowledge space theory.

Abstract

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.

Cognitive Diagnosis Modeling Incorporating Item-Level Missing Data Mechanism

“…As stated by Heller et al ( 2015 ), CDMs have connections to knowledge space theory (KST), which has been developed by Doignon and Falmagne ( 1985 ) (see also Doignon and Falmagne, 1999 ; Falmagne and Doignon, 2011 ). de Chiusole et al ( 2015 ) and Anselmi et al ( 2016 ) have developed models for the analysis of MCAR, MAR, and MNAR data in the framework of KST. In their work, the MCAR holds if the missing response pattern is independent of the individual's knowledge state (i.e., the collection of all items that an individual is capable of solving in a certain disciplinary domain) and of the observed responses; the MAR holds if the missing response pattern is conditionally independent of the knowledge state given the observed responses; and the MNAR holds if the missing response pattern depends on the knowledge state.…”

Extending the Basic Local Independence Model to Polytomous Data

“…Therefore, the conditional probability P ( R | K ) in the present example is: The , and parameters of the BLIM can be estimated by maximum likelihood (ML) via the expectation–maximization (EM) algorithm (Stefanutti and Robusto 2009 ) or by minimum discrepancy (MD; Heller and Wickelmaier 2013 ). Moreover, methods for obtaining maximum likelihood estimates from data in which some responses are missing are available in the literature (Anselmi et al 2016 ; de Chiusole et al 2015 ), together with procedures for testing the invariance of the and parameters (de Chiusole et al 2013 ). Some extensions of the model have been proposed for the assessment of learning processes, as the gain–loss model (GaLoM; Anselmi et al 2012 , 2017 ; de Chiusole et al 2013 ; Robusto et al 2010 ; Stefanutti et al 2011 ), and a model for the treatment of skills dependence (de Chiusole and Stefanutti 2013 ).…”

Testing the actual equivalence of automatically generated items

“…For deriving the updating equations of the EM for the case of missing-at-random (MAR) data, we draw upon a recent extension of the BLIM to MAR data derived by de Chiusole et al (2015a) and further developed by Anselmi et al (2016). A response pattern where some of the responses are missing is represented by a partition (R, M, W ) of Q.…”

“…Concerning the BLIM, methods for estimating its parameters (Heller & Wickelmaier, 2013;Schrepp, 2005;Stefanutti & Robusto, 2009) and for testing its identifiability (Spoto et al, 2012;Stefanutti et al, 2012) are available, together with methods for testing its assumption about the parameter invariance (de Chiusole et al , 2015b. Furthermore, several extensions of the BLIM have been proposed, which allow for modeling skill dependencies (de Chiusole & Stefanutti, 2013), as well as for treating missing data (de Chiusole et al, 2015a;Anselmi et al, 2016). There is also a model for assessing learning processes (Robusto et al, 2010;Anselmi et al, 2017).…”