Imagine a surmise relationship between to items x and y(ySx), where item y is prerquisite for item x.
The admissible solution patterns for the items are that both items
are solved correctly (1,1), neither of the items is solved
correctly (0,0), or only item y ist solved correctly (0,1).
In other words, x can only be solved in combination with
a correct solution to y, while y can also be solved
by itself. Therefore, we expect that the solution frequency for
item y is equal or higher than the solution frequency for
item x. In the Hasse diagram the relationship ySx
is presented by a line going down to the hypothesized prerequisite
(see Figure 4).
Figure 4: Surmise relationship between two items x
and y (ySx)
Symmetric distances
Symmetric distances between a knowledge structure and a binary
data matrix denote the averaged minimal distance or number of deviations
between each person's response pattern and the closest hypothesized
knowledge state. Formally, the symmetric distance d between two
sets A and B is defined as (4)
Distance agreement coeffcient
For a comparison of di erent models, the symmetric distances have
to be related to the number of states in each model. This means,
that with regard to the varying sizes of the hypothetical knowledge
structures, a comparison of the structures' fit is only possible
in consideration of the number of knowledge states within the powersets
()
of the respective structures. Hence, as further validation method
to test the fit between a knowledge structure and a set of data
in consideration of the structures' sizes we can calculate the distance
agreement coefficient (DA). (5)
A lower value of the distance agreement coeffcient indicates a better
fit of a knowledge structure to a given set of data.
References
Albert, D. (1995). Surmise relations between
tests. Talk at the 28th Annual Meeting of the Society for Mathematical
Psychology, University of California, Irvine, August.
Albert, D., Brandt, S., Hockemeyer, C., & Schappacher,
W. (in preparation). Properties of surmise relations between
tests. Manuscript in preparation.
Brandt, S., Albert, D., & Hockemeyer, C. (1999).
Surmise relations between tests - preliminary results of the mathematical
modelling. Electronic Notes in Discrete Mathematics, 2.
Brandt, S., Albert, D., & Hockemeyer, C. (2000).
Surmise relations between tests - mathematical considerations. Submitted
to Discrete Applied Mathematics.
(4)
)
of the respective structures. Hence, as further validation method
to test the fit between a knowledge structure and a set of data
in consideration of the structures' sizes we can calculate the distance
agreement coefficient (DA). 
(5)
