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 $\pi_j^*$

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

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:

$$P[Y_j=1|\alpha_{lj}^*=(0,0)]=?\\ P[Y_j=1|\alpha_{lj}^*=(1,0)]=?\\ P[Y_j=1|\alpha_{lj}^*=(0,1)]=?\\ P[Y_j=1|\alpha_{lj}^*=(1,1)]= \pi_j^*$$

The item response function of the Reduced RUM is

$$P[Y_j=1|\alpha_{lj}^*]=\pi_j^*\prod_kr_{jk}^{*q_{jk}(1-\alpha_{jk})}$$

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

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