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The Probabilistic-Inputs, Noisy "And" Gate Model
Author(s): 
Pages: 1230-1238
Year: Issue:  5
Journal: Psychological Science

Keyword:  cognitive diagnosisattribute mastery probabilityPINAhigher-order latent traitsQ matrixitem response theoryDINA;
Abstract: Cognitive diagnosis(CD), which is also referred to as skill assessment or skill profiling, utilizes latent class models to provide finegrained information about students’ strengths and weaknesses in the learning process. One major advantage of CD is the capacity to provide additional information about the instructional needs of students. In the past decades, extensive research has been conducted in the area of CD and many statistical models based on a probable approach have been proposed. Examples of cognitive diagnostic models(CDMs) include the deterministic-inputs, noisy "and" gate(DINA) model(Haertel, 1989; Junker & Sijtsma, 2001), the deterministic-inputs, noisy "or" gate(DINO) model(Templin & Henson, 2006), C-RUM(Hartz, 2002), the higher-order DINA model(de la Torre & Douglas, 2004), the log-linear cognitive diagnosis model(Henson, Templin, & Willse, 2009), and the generalized DINA model(de la Torre, 2011).Currently, the outcome of CDMs is a profile with a binary element for each examination to indicate the mastery/non-mastery status of every attribute/skill, i.e. the attribute mastery status(AMS). But this coarse classification or diagnosis cannot distinguish the subtle, individual differences between different students. Those students that are assigned with the same category(e.g., mastery) may not possess the same degree of certainty(i.e., some students may be more likely to master than other examinees). As a consequence, the AMS may not conducive to being used by teachers to make decisions regarding the optimal intervention that should be put into place for the students. In existing CDMs, the attributes are viewed as specific knowledge required for examination performance. de la Torre and Douglas(2004) treated these attributes as arising from a higher-order latent trait to reduce computational burden.In order to obtain a nuanced profile of the student with respect to the student’s characteristics, this study proposed the probabilistic-inputs, noise "and" gate(PINA) model based on the attribute mastery probability(AMP; i.e., the probability of attribute mastery that can be calculated by the higher-order latent trait model(de la Torre & Douglas, 2004)), which means that the AMP was directly used in modeling rather than the AMS. The multicomponent latent traits model(Embretson, 1980, 1984) has been taken as a template from the PINA model. Besides the same advantages(e.g., reduce computational burden), there are two major differences between the PINA model and the higher-order DINA model. Firstly, outcome of the higher-order DINA model is a profile with binary attribute mastery status, whereas outcome of the PINA model is a profile with attribute mastery probability. Secondly, the PINA model directly nests the higher-order latent trait model within the DINA model, but the higher-order latent trait model in the higher-order DINA model is separated from the DINA model. In this sense, the PINC model can be seen as a variation of the higher-order DINA model.Simulation studies were conducted to evaluate the parameter recovery of the new model and the results revealed that the parameters can be recovered well with the freeware Win BUGS(Spiegelhalter, Thomas, & Best, 2003), which implemented MCMC algorithms.An empirical example of the Examination for the Certificate of Proficiency in English was provided to demonstrate applications of the new model. We analyzed a dataset of 2,922 examinees on 28 multiple-choice items, which required mastery of three attributes(morphosyntactic rules, cohesive rules, and lexical rules). The Q matrix can be found in Templin and Hoffman(2013). The AIC and BIC values were 80964.07 and 81340.81, respectively, for the DINA model; 84557.31 and 84928.07, respectively, for the higher-order DINA model; 80984.60 and 81373.15, respectively, for the PINA model. The higher-order DINA model had the worst model-data fit. The DINA model seems to have a better fit, because the number of attributes is just 3, which means that the PINA model can not show it’s advantages(i.e., the number of model parameters). Under the PINA model, the estimates were-0.058,-0.792, and-1.646 for the three attribute difficulty parameters, respectively. That is, the difficulty of morphosyntactic rules, cohesive rules, and lexical rules decreased progressively, meaning that a linear hierarchical structure might exist among these three attributes(Templin & Bradshaw, 2014).
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