3.9 The Generalized DINA model

The generalized DINA model, or G-DINA model, is a general model framework (de la Torre, 2011).

G-DINA model is often called a saturated model because it subsumes many existing CDMs ¹

A revisit of the DINA model

A generalization

The IRF of the G-DINA model can be written by

$$P(Y_j=1|{\alpha }_{lj}^{\ast })={\pi }_{lj}$$

: Although the above IRF of the G-DINA model is very simple, researchers often reparmeterize it in a different, but equivalent, way:

$$P(Y_j=1|{\alpha }_{lj}^{\ast })={\delta }_{j0}+\sum _k^{K_j^{\ast }}{\delta }_{jk}{\alpha }_{lk}+\dots +{\delta }_{j1,2,\dots ,K_j^{\ast }}\prod _k^{K_j^{\ast }}{\alpha }_{lk}$$

Based on this parameterization, which is often referred to as the identity link of the G-DINA model, please calculate the following probabilities of success:

$$\begin{array}{cc}P(Y_j=1|{\alpha }_{lj}^{\ast }=00)& ={\delta }_{j0}\\ P(Y_j=1|{\alpha }_{lj}^{\ast }=10)& =?\\ P(Y_j=1|{\alpha }_{lj}^{\ast }=01)& =?\\ P(Y_j=1|{\alpha }_{lj}^{\ast }=11)& =?\end{array}$$

References

Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61(2), 287–307. https://doi.org/10.1348/000711007X193957

de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76(2), 179–199. https://doi.org/10.1007/s11336-011-9207-7

Henson, R. A., Templin, J. L., & Willse, J. T. (2009). Defining a Family of Cognitive Diagnosis Models Using Log-Linear Models with Latent Variables. Psychometrika, 74(2), 191–210. https://doi.org/10.1007/s11336-008-9089-5

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1. Another two saturated CDMs are loglinear CDM (Henson et al., 2009) and the general diagnostic model (Davier, 2008)↩︎