Mathematical Social Sciences 38 1999 361–375
On the empirical construction of implications between bi-valued test items
Martin Schrepp
Schwetzinger straße 86, 68766 Hockenheim, Germany
Abstract
We describe a data-analytic method which allows one to derive implications between test items from observed response patterns to these items. This method is a further development of item tree
analysis J.F.J.v. Leeuwe, Nederlands Tijdschrift voor de Psychologie, 29 1974 475–484. A simulation study shows that the improvements have a positive effect on the ability of the method
to reconstruct the valid implications from data. The improved method is able to reconstruct the underlying structure on the items with high accuracy. As a concrete example the implications in a
set of letter series completion problems are analysed.
1999 Elsevier Science B.V. All rights
reserved.
Keywords : Item tree analysis; Test items; Improved method
1. Introduction
We deal with the question of how a structure on an item set can be derived from observed response patterns to these items. Assume that we have a binary item set I, i.e. a
set of n items to which every subject may respond positive 1 or negative 0. We try to derive the logical implications between the items from the observed data.
Assume, for example, that I is a problem set. In this case a subject can either solve 1 a problem or fail 0 in solving the problem. In this example an implication j
→ i can be
interpreted as ‘every subject who is able to solve j is also able to solve i’. As a second example assume that I is a set of statements from a questionnaire to
which subjects can either agree 1 or disagree 0. In this case an implication j →
i can be interpreted as ‘every subject who agrees to statement j also agrees to statement i’.
Because of our interpretation of →
as a logical implication, the set of implications on
Tel.: 149-6205-189-696. E-mail address
: martin.schreppsap-ag.de M. Schrepp 0165-4896 99 – see front matter
1999 Elsevier Science B.V. All rights reserved.
P I I : S 0 1 6 5 - 4 8 9 6 9 9 0 0 0 2 5 - 6
362 M
. Schrepp Mathematical Social Sciences 38 1999 361 –375
I should form a reflexive and transitive relation on I. We call such a relation in accordance with Doignon and Falmagne 1985 a surmise relation on I.
There are a number of data-analytic techniques which try to derive such logical implications between items from binary datasets. See, for example, Leeuwe 1974,
Flament 1976, Buggenhaut and Degreef 1987, Duquenne 1987, and Theuns 1994. There are some obvious connections between these data-analytic techniques and the
theory of knowledge spaces Doignon and Falmagne, 1985. In this approach a knowledge domain is represented as a finite set I of questions. The subset of questions
from I that a subject is capable of solving is called the knowledge state of that subject. In general, not every subset of I will be a possible knowledge state of a subject. Thus,
the set of possible knowledge states, which is called the knowledge structure, will be a proper subset of the power set of I. This can be used for an efficient adaptive diagnosis
of students knowledge see Falmagne and Doignon, 1988.
A data-analytic technique which derives the set of valid logical implications from observed response patterns determines a knowledge structure on the item set. This
knowledge structure consists of all subsets A of I which are consistent with the derived implications, i.e. which fulfil the condition j
→ i
∧ j [ A
⇒ i [ A for all i, j [ I for
details see Doignon and Falmagne, 1985.
2. Problems in reconstructing implications from data
