In cognitive diagnosis models, the condensation rule reflects how latent attributes influence individuals’ observed item responses; in practice, multiple condensation rules may be involved in an item simultaneously

Zhan, Peida. 2020. “Deterministic-inputs, Noisy Mixed Modeling for Identifying Coexisting Condensation Rules.” PsyArXiv. February 12. doi:10.31234/osf.io/qwx2m

Abstract: In cognitive diagnosis models (CDMs), the condensation rule reflects how latent attributes influence individuals’ observed item responses. In practice, multiple condensation rules may be involved in an item simultaneously, which indicates that the contribution of required attributes to the correct item response probability follows multiple condensation rules with different proportions. To consider the coexisting condensation rules while keeping the interpretability of model parameters, this study proposed the deterministic-inputs, noisy mixed (DINMix) model. Two simulation studies were conducted to evaluate the psychometric properties of the proposed model. The results indicate that the model parameters for the DINMix model can be well recovered, and the DINMix model can accurately identify coexisting condensation rules. An empirical example was also analyzed to illustrate the applicability and advantages of the proposed model.

6. Summary and Discussion
The condensation rule describes the logical relationship between the required attributes and the item response. When an item contains coexisting condensation rules, it means that the contribution of required attributes to the correct item response probability follows multiple condensation rules with different proportions. Coexisting condensation rules reflect the complexity of cognitive processes in problem-solving. To take into account coexisting condensation rules while keeping the interpretability of model parameters, this study proposed the DINMix model. Two simulation studies were conducted to evaluate the psychometric properties of the proposed model. The simulation results indicate that (a) the model parameters for the DINMix model can be well recovered, especially in the conditions with a larger sample, longer test length, and higher item quality; (b) the DINMix model can adaptively and accurately identify coexisting condensation rules, either existing simultaneously in an item or existing separately in multiple items. An empirical example was also analyzed to illustrate the applicability and advantages of the proposed model.

As aforementioned, the DINMix model can be viewed as a constraint model from the GDINA model after some parameter transformations. Thus, the number of item parameters of the DINMix model is larger than that of the reduced models but smaller than that of the general models. For example, in the simulated condition in simulation Study 2, there were 60, 60, 60, 100, and 140 items parameters for the DINA, DINO, DINR, DINMix, and GDINA models, respectively. To explore the differences between the performance of the DINMix and GDINA models, we also used the GDINA model to conduct a simple analysis of the data in simulation Study 2, based on the GDINA package (Ma & de la Torre, 2020) in R software. The results (see Tables S4 and S5 in online supplements) indicate that the performance of the DINMix and GDINA models was almost identical in the recovery of attributes and item parameters. Specifically, the DINMix and GDINA models have almost the same diagnostic capabilities, but the former is more concise and easier to be interpreted.

The work represented in this article is an initial attempt to simultaneously consider multiple condensation rules in a single CDM. Despite promising results, some limitations still exist. First, the utilized model framework (see Equations 1 and 5) models the aberrant responses at the item level. However, in practice, such aberrant responses may occur at the attribute rather than item level, such as the noisy inputs, deterministic, ‘and’ gate model (Junker & Sijtsma, 2001). Ways to incorporate attribute-level aberrant responses into the proposed model are worthy of further research, as Equation 11 in de la Torre (2011) seems to give us a reference. Second, within-item characteristic dependency (Zhan, Jiao, Liao et al., 2019), which means that the dependency exists between the guessing and slip parameters within an item, was not considered in the proposed model. It can be incorporated into the proposed model to increase the estimation accuracy of the item parameters in a future study. Third, only the dichotomous scoring item and dichotomous attribute were modeled in the proposed model. It would be meaningful and practical to extend the current model to consider polytomous scoring items (e.g., Ma & de la Torre, 2016) and polytomous attributes (e.g., Zhan et al., 2020). Fourth, in recent years, some studies have focused on the Q-matrix validation or estimation (Chen et al., 2018; de la Torre & Chiu, 2016) and the multiple strategies for problem-solving (Ma & Guo, 2019), which are not covered in current study. Fifth, notably, the generalizability of the findings of this study is dependent upon the limitations of the design of the simulation studies, such as a fixed number of attributes and assuming the Q-matrix is correct. To further generalize these findings, a wider range of simulated conditions should be considered in future studies.