3.7 Reduced RUM

The Reduced RUM is a reduction of the Reparameterized Unified Model (Hartz, n.d.)

The Reduced RUM allows each required attribute to contribute differentially to the probability of success

The probability of a correct response to item $j$ for examinees who have mastered all the required attributes for the item is given by ${\pi }_j^{\ast }$

The penalty for not mastering attribute $k$ is $r_{jk}^{\ast }$

An example

If an item measures attributes 1 and 2, there are four latent groups with reduced attribute profiles (00), (10), (01) and (11). Find the following success probabilities:

$$P[Y_j=1|{\alpha }_{lj}^{\ast }=(0,0\left)\right]=?P[Y_j=1|{\alpha }_{lj}^{\ast }=(1,0\left)\right]=?P[Y_j=1|{\alpha }_{lj}^{\ast }=(0,1\left)\right]=?P[Y_j=1|{\alpha }_{lj}^{\ast }=(1,1\left)\right]={\pi }_j^{\ast }$$

The item response function of the Reduced RUM is

$$P[Y_j=1|{\alpha }_{lj}^{\ast }]={\pi }_j^{\ast }\prod _k{r}_{jk}^{\ast q_{jk}(1-{\alpha }_{jk})}$$

The plot below shows the RRUM with ${\pi }_j^{\ast }=.9$,$r_{j1}^{\ast }=.5$, and $r_{j2}^{\ast }=.8$

References

Hartz, S. M. (n.d.). A bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality [PhD thesis]. https://www.proquest.com/docview/305590285/abstract/20500E9536DD4FB4PQ/1