Surmise Relations between Tests
So far, we have referred to single tests. However, in common psychological
assessment procedures we often deal with a set 
of
different tests that are usually related. On the background of Doignon
and Falmagne's framework, Albert (1995,Albert, Brandt, Hockemeyer,
and Schappacher,in preparation; Brandt, Albert, and Hockemeyer,1999;
2000) extended the concept of the nonsymmetric surmise relation between
items (i. e. within tests) to surmise relations between
tests. The interpretation of a surmise or prerequisite relation
between tests, i. e. 
,
is that two tests 
are in surmise relation from A to B, if one can surmise
from the correct solution of a given set of items in test A
the correct solution of a particular non-empty subset of items in
test B (see Figure 1).
Figure 1: Two tests A and B are in surmise
relation from A to B (
)
is defined by
is the set of all knowledge states containing item a. For a
set of tests 
,
a surmise relation between tests has the property of reflexivity but
not necessarily transitivity, i. e. in general, it is not a quasi
order. However, there are special cases for which transitivity holds,
namely left- and right-covering surmise relations.
,
see Figure 2) is that for each item 
there exists a nonempty subset of prerequisites in test B,
i. e. a person who doesn't solve any item in B will not be
able to solve any item in A, either. There is no need to test
this person on test A. Formally, we say that two tests 
,
there exists at least one item 
for which b is a prerequisite 
,
i. e. failing to solve any item in test B implies a failure
on a subset of items in test A (see Figure 3). In other words,
a person who solves all items in test A is also able to solve
all items in test B. Hence, there is no further need to test
the person on test B. Formally, we say that two tests 
)
)
is
defined as the combination of item subsets per test 
person i is capable of mastering. The collection of all test
knowledge states 
is called the test knowledge structure, which is defined as the
pair 
,
with 
.
If a test knowledge structure is closed under union 
it is called test knowledge space, if it is also closed under
intersection 
we speak of a quasi ordinal test kowledge space.