Quality & Quantity 35: 77–89, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
On the Structure of Measurements in Facet Theory
GODFRIED VAN DEN WITTENBOER
University of Amsterdam, Department of Education, P.O. Box 94208, 1090 GE Amsterdam,
Abstract. The mapping sentence in facet theory suggests constructing items according to a factorial
design of facets. Analyzing the structure of the responses by multidimensional scaling, theorists
usually expect the solution to reﬂect the constituting facets, so that it can be partitioned by levels of
the facets. In the paper we elucidate that each dimension relates to the response function of a speciﬁc
subgroup, rather than to one or more separate facets. Partitioning by levels of a facet may lead to
neglecting the real structure of the response function to be analyzed.
Key words: facet theory, mapping sentence, multidimensional scaling, error, response function,
principal components, homogeneous groups, inhomogeneous groups, dimensionality
Facet theory is a methodological approach that links a systematic way of designing
questions or items of questionnaires with speciﬁc forms of data analysis by corres-
pondence hypotheses (Canter, 1985; Borg and Shye, 1995; Hox et al., 1995). The
most characteristic part of the approach is the mapping sentence, which must be
seen as a ‘mapping’ in the mathematical sense of the word. The Cartesian product
of a set of persons (P ) and a set of items or questions (Q), is mapped into a range
of responses R, M: PxQ → R, in that each ‘person x stimulus’ combination leads
to exactly one response out of the range of possible responses R. In other words,
each person gives exactly one response to each question or item, and that response
must be the same for identical questions.
The set of items Q is usually a Cartesian product itself consisting of combina-
tions of observable characteristics (called facets). These facets could be anything.
In this paper we will use boxes – to use an analogy of Thurstone (1947) – leading
to a mapping sentence with facets length, width and height:
People estimate the size of boxes with a speciﬁc