Inductive Item Tree Analysis (IITA)

About the Theory

Schrepp (1999, 2003) developed IITA (Inductive itemm Tree Analysis) as a means to derive a surmise relation from dichotomous response patterns. Sargin and Ünlü (2009; Ünlü & Sargin, 2010) implemented two advanced versions of that procedure.

The three inductive item tree analysis algorithms are exploratory methods for extracting quasi orders (surmise relations) from data. In each algorithm, competing binary relations are generated (in the same way for all three versions), and a fit measure (differing from version to version) is computed for every relation of the selection set in order to find the quasi order that fits the data best. In all three algorithms, the idea is to estimate the numbers of counterexamples for each quasi order, and to find, over all competing quasi orders, the minimum value for the discrepancy between the observed and expected numbers of counterexamples. The three data analysis methods differ in their choices of estimates for the expected numbers of counterexamples. (For an item pair (i,j), the number of subjects solving item j but failing to solve item i, is the corresponding number of counterexamples. Their response patterns contradict the interpretation of (i,j) as `mastering item j implies mastering item i.')


  • Sargin, A. and Ünlü, A. (2009) Inductive item tree analysis: Corrections, improvements, and comparisons. Mathematical Social Sciences, 58, 376--392.
  • Schrepp, M. (1999) On the empirical construction of implications between bi-valued test items. Mathematical Social Sciences, 38, 361--375.
  • Schrepp, M. (2003) A method for the analysis of hierarchical dependencies between items of a questionnaire. Methods of Psychological Research, 19, 43--79.
  • Ünlü, A. and Sargin, A. (2010) DAKS: An R package for data analysis methods in knowledge space theory. Journal of Statistical Software, 37(2), 1--31.

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:
The accompanying binary dataset is part of the empirical 2003 Programme for International Student Assessment (PISA) data. It contains the item responses by 340 German students on a 5-item dichotomously scored mathematical literacy test. The dataset was obtained after dichotomizing the original multiple-choice or open format test data. Wording of the test items used in the assessment is not known (not available publicly).
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.
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).
Strict Linear Order
Fictitious data set describing a linear order 1 ≾ 2 ≾ 3 ≾ 4 ≾ 5 almost without noise.


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.

OECD Programme for International Student Assessment (PISA)

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.

Do not forget to press the "Done" button when you have finished!

Please note that each item must be solved at least once within a non-trivial pattern, i.e. within a pattern unequal to the full item set Q.

About this App

This App was created within the TquanT project.
TquanT was co-funded by the Erasmus+ Programme of the European Commission. csm_logo-erasmus-plus_327d13b53f.png

© 2018 Christoph Anzengruber & Cord Hockemeyer, University of Graz, Austria