## Example Spaces

As example data, knowledge spaces provided by the R package pks (Heller & Wickelmaier, 2013;
Wickelmaier et al., 2016) are used. Concretely, the following spaces are used:

- Density 97
- Taagepera et al. (1997) applied knowledge space theory to specific science problems. The
density test was administered to 2060 students, a sub structure of five items is included here.
- Matter 97
- Taagepera et al. (1997) applied knowledge space theory to specific science problems. The conservation
of matter test was administered to 1620 students, a sub structure of five items is included here.
- Doignon & Falmagne
- Fictitious data set from Doignon and Falmagne (1999, chap. 7).

Please note that for "Matter 97" and "Density 97" the structures are not (quasi-ordinal) knowledge spaces. The VC
and γ coefficients work on the underlying surmise relation, i.e. effectively on the closure of the
knowledge structure under union and intersection.

#### References

Doignon, J.-P., & Falmagne, J.-C. (1999).

*Knowledge spaces.* Berlin: Springer.

Heller, J. & Wickelmaier, F. (2013). Minimum discrepancy estimation in probabilistic knowledge structures.
*Electronic Notes in Discrete Mathematics, 42,* 49-56.

Schrepp, M., Held, T., & Albert, D. (1999). Component-based construction of surmise relations for chess problems.
In D. Albert & J. Lukas (Eds.), Knowledge spaces: Theories, empirical research, and applications (pp. 41--66).
Mahwah, NJ: Erlbaum.

Taagepera, M., Potter, F., Miller, G.E., & Lakshminarayan, K. (1997). Mapping students' thinking patterns by
the use of knowledge space theory. *International Journal of Science Education, 19,* 283--302.

Wickelmaier, F., Heller, J., & Anselmi, P. (2016). *pks: Probabilistic Knowledge Structures.* R package
version 0.4-0. https://CRAN.R-project.org/package=kst