3.4 Condensation Rules

CDMs are models at item response level

The item response function, or the conditional probability of success on attribute profile, can be expressed as

$$P(Y_{ij}=1|\alpha_c)= P(Y_{ij}=1|\alpha_{lj}^*)=f( \alpha_{lj}^*,\delta_j )$$

To define the model, we need to specify the functional form $f(\cdot)$

In other words, we need to have some assumptions about how students use attributes to solve item $j$ (i.e., condensation rule)

• Models are conjunctive if all the required attributes are necessary for successful completion of the item

• Models are compensatory if the absence of one attribute can be made up for by the presence of other attributes

• For most part, these two schemes of classifying CDMs are used interchangeably, specifically,

• conjunctive ≈ non-compensatory

• disjunctive ≈ compensatory

• Depending on how the terms are defined, the two classification schemes may not be identical