|
Asymmetric Triangulation
Scaling |
|
This site provides an introduction
to asymmetric
triangulation scaling (ATRISCAL), which is a kind of multidimensional scaling
(MDS) developed to visualize inter-item dependency structure in test data. Quick
Introduction
PPT |
|
The
method that is currently most widely used for analyzing test data is item
response theory (IRT), which is a well-polished and highly complete theory.
However, all statistical models involve the extraction of information from
data. Consequently, a characteristic of these models is that information that
conforms to their internal mechanisms is abstracted through a process of
generalization, while all other information is discarded. As a result, IRT by
itself results in a one-sided extraction of information from test data. For
example, if we use only an astronomical telescope to study the moon, we would
simply be repeating the same detailed observations of the lunar surface. |
|
We developed ATRISCAL as a method
for studying the dependency relationships between items, which is one of the
things that IRT is less good at (not designed for). This
method is still in the early stages of development and has not yet been fully
discussed in academic meetings. We set up this website as a way of attracting
comments from as many people as possible. |
|
ATRISCAL can be applied to the
analysis of conditional correct response rate matrix between items. An
element ij of such a matrix shows the
correct response rate of item j
under that item i
is correctly answered, as expressed by the formula P(j|i) = P(i,j)/P(i),
where P(i,j) is the joint correct response rate of
items i
and j, and P(i) is the correct response rate of item i. When P(j|i) is
large, it is highly likely that the knowledge needed to answer item i correctly is
also a prerequisite for answering item j
correctly. In general, P(j|i) and P(i|j) are not equal, so this matrix is asymmetric.
Also, the diagonal elements of this matrix are all 1.0 since P(j|j) = P(j,j)/P(j) = P(j)/P(j) = 1.0. |