







Using formal concept analysis to analyse repertory grid data



Peter Caputi and Desley Hennessy 


School of Psychology, University of Wollongong, Wollongong, NSW, Australia







Abstract
Repertory
grids are used widely in Personal Construct Psychology (PCP) research. An array
of accepted techniques for analysing grid data is available to researchers. This
paper revisits the use of formal concept analysis (FCA) in analysing repertory
grids. It particular, this paper focuses on using FCA to explore hierarchical
structures in grid data. A description of the technique is provided, along with
a review of its application in PCP research. The technique is also compared
with other approaches for identifying structures in grid data.
Keywords: formal concept analysis, repertory grid, construct hiercharchies 







Introduction
Kelly’s
(1955) theory of personal constructs provides a unique way of understanding how individuals make sense
of events, people, and situations in the world. Kelly argued that people make
sense of the world via a process of comparison and contrast referred to as
construing. Constructs, and systems of constructs, are used to discriminate
between events, objects, and people, thereby providing meaning to a person’s
world. Kelly’s repertory grid technique is a method for capturing aspects of an
individual’s construct system.
What is a repertory
grid?
Generally,
repertory grids consist of elements (the objects, people, and events in a
person’s world) and constructs (bipolar discriminations used to make sense of
the world). When administering a repertory grid, a researcher may provide constructs,
or the respondent may generate or elicit a set of constructs using certain
elicitation methods. Likewise elements can also be elicited or provided by the
researcher.
The kinds
of elements in a grid and the nature of the constructs provide a context for
the respondent to identify the relationship between elements and constructs. Most
commonly, the respondent will rate elements for each construct within the
construct set, although other strategies for associating elements to constructs
are available. Let E represent a set
of elements and C a set of
constructs. Furthermore, assume that it is possible to quantify the extent to
which an element is related to a construct. A repertory grid can be thought of
as a table of data that represents the relationship between the members, e_{i}, of an element set E and members, c_{j}, of a construct set C. For example, an element e_{i}
can be given a rating r_{ij}
along a construct c_{j}.
The relation r_{ij} can also
be a ranking relation. This definition provides a formal representation of data
collected using the repertory grid technique.
Some
important issues for repertory grid users emerge from formalising a definition
of repertory grid (see Bell,
1990). For instance, the particular procedure for collecting grid data (type of relation r_{ij}) will suggest a particular statistical model for
analysing the data. In other words, whether one uses a rating or ranking scale
to associate elements and constructs will inform which statistical approaches
are appropriate for data analysis. In turn, one might ask what kinds of
representations are appropriate for grid data collected. For instance, rating
data will allow a researcher to examine the extent to which constructs are
associated by correlating the construct ratings across the elements of the
grid.
Presenting
a formal definition of a repertory grid also suggests that the joint representation
of elements and constructs is also possible. Fransella, Bell and Bannister (2004) propose a number of
multivariate approaches for examining the joint representation of elements and
constructs. They report that a representation in twodimensional space can be
obtained using singularvalue decomposition, correspondence analysis and
multidimensional unfolding. A description of these approaches is provided in
Fransella et al. (2004). These analyses, in general, provide spatial
representations based on the degree to which elements and constructs are
similar to one another.
However, ordinal
relations among constructs are also of interest. Kelly’s (1955) Organisation
Corollary explicitly states that ordinal relationships are a defining feature
of construct systems. There are approaches that represent the degree of superordinacysubordinacy
among constructs. For instance, Fransella et al. (2004) identify hierarchical
classes analysis (De Boeck & Rosenberg, 1988) as one method that attempts
such representations. In this paper we revisit the use of formal concept
analysis (Wille, 1982; Ganter & Wille, 1999) as an approach to examining
not only the joint representation of elements and constructs, but also ways of
representing possible ordinal relations present in the data.
Formal Concept Analysis
Formal Concept Analysis (FCA: Wille, 1982) is a
mathematical technique based on lattice theory. Central to FCA is the notion of
a concept. The term concept comes from logic and refers to a category which can
be used to classify objects. Within the FCA framework a concept is defined as
having two parts: (i) the objects that can be categorised using the concept and
(ii) the attributes or properties that are shared by the objects belonging to the
concept (Wormuth & Becker, 2004). In addition, certain objects have certain
attributes; in other words, objects are related to attributes. Taken together,
the set of objects, the set of attributes and the relation defined among the objects
and attributes is known as a formal
context (Wormuth & Becker, 2004). Let’s consider an example based on
one given by Lienhard,
Ducasse, and Arevalo (2005). Consider a group of people
{Sigmund, George, Anna, Fritz, Carl, Fay},
and a set of beverages {beer, orange juice, tea, wine,
coffee}. We can refer to the set of people as the objects and
the set of beverages as the attributes or properties. We can ask the question
which of these beverages people prefer. The preferences can be represented in
tabular form as presented in Table 1. Table 1 is the formal context.
Table 1: A preference table for beverages for six people
 beer  orange juice  tea  wine  coffee  Sigmund   x   x   George   x   x  x  Anna    x  x   Fritz  x   x    Carl  x   x    Fay    x  x  
A concept is
a set of objects having common attributes. In Table 1 we see that ({Sigmund,
George}, {orange juice, wine}) is a concept, so is ({Anna, Fay}, {tea, wine}). Table
2 presents the sets of concepts for the context in Table 1.
Table 2: Set
of concepts for beverage preference example
Concept_{8}  ({all
objects},{ø})  Concept_{7}  ({Sigmund,
George, Anna, Fay},{wine})  Concept_{6}  ({Anna,
Fritz, Carl, Fay},{tea})  Concept_{5}  ({Fritz,
Carl},{beer, tea})  Concept_{4}  ({Fay,
Anna},{tea, wine})  Concept_{3}  ({Sigmund,
George},{orange juice, wine})  Concept_{2}  ({George},{orange
juice, wine, coffee})  Concept_{1}  ({ø},
{all attributes}) 
The set of
concepts in Table 2 forms a structure known as a complete partial order. A
partial order is one way of formally representing or modelling hierarchy in a
dataset. Lienhard, Ducasse, and Arevalo (2005, p.75) define a context C as the triple (O, A, R), where O and A are the sets of objects and
attributes, and R is a binary
relation between O and A. Let X be a subset of O and Y a subset of A, where σ(X) represents
all the attributes common to X, and
τ(Y) represents all the objects
common to Y. A concept is defined as
the pair (X,Y) such that Y = σ(X) and X = τ(Y). Further, a concept (X_{1}, Y_{1}) is a
subconcept of concept (X_{2}, Y_{2})
if X_{1} is contained in X_{2} or equivalently if Y_{2} is contained in Y_{1}_{ }(Lienhard,
Ducasse, & Arevalo, 2005, p.75). Likewise we can define a concept (X_{1}, Y_{1}) as a
superconcept of a concept (X_{2},
Y_{2}) if the inverse properties hold (Lienhard, Ducasse, & Arevalo,
2005, p.75). Given these relations, the set of concepts for a context can be
represented as a concept lattice. A concept lattice for the concepts in Table 2
is presented in Figure 1.
Figure 1: Concept lattice for the concepts in Table 2
Repertory grids and
FCA
The basic ideas
underpinning FCA have similarities to the formal definition of repertory grids that
we presented in Section 2. With FCA, a context is defined by two sets of
entities; a set of formal objects, O
and a set of formal attributes or properties, A. In addition, we define a relation R that relates members of O
to members of A. In this case the
relation R is binary, whereas with
repertory grids the relation is typically multivalued (a rating).
The
similarities, however, are not limited to the model used for generating data. Formal
concepts can be ordered, that is, there is a hierarchical order among concepts.
Kelly’s (1955) Organisation Corollary posits that constructs are also
hierarchically related, that is, there is an ordinal relation among constructs,
so that some constructs are subordinate to other constructs, while others are
superodinate.
Given that
formal concepts in FCA can be naturally ordered, the notion of a
subconceptsuperconcept hierarchy holds. This idea is not dissimilar to the
subordinatesuperordinate relation that defines the association among
constructs. In other words, it is possible to describe the relationships among attributes
in terms of implications. Interestingly, the notion of context, namely, the
triple of objects, attributes and the relation defined among members of these
sets, is an appropriate representation of Kelly’s view that the interpretation
of a grid analysis should be made with its context, namely, the set of elements
that define the grid. It would appear, then, that on some levels the
mathematical framework behind FCA is theoretically consistent with Kelly’s
theory. In other words, FCA is a theoryappropriate method of analysing
repertory grid data (Bell,
1988).
There have
been applications of FCA to repertory grids. Spangenberg and Wolff (1987)
demonstrate how FCA can be applied to a grid elicited from a patient with
anorexia. Repertory grids have been applied in the area of knowledge
acquisition and numerous studies have adopted FCA as a method of analysing data
(see Delugach & Lampkin, 2000; Erdani, Hunger, Werner, & Mertens, 2004;
Richards & Compton, 1997).
To further
illustrate the application of FCA to grid data, we consider the use of
repertory grids to explore reasons why people get tattoos. Consider the grid
from a 57 year old man with approximately ten tattoos. After obtaining his
first tattoo at the age of 19, this man has continued to have tattoos every few
years, his most recent being a week before the grid was filled out.
Table 3: Grid
from man with ten tattoos
(click on heading)
In order to
apply FCA to this grid, we first need to transform multivalued grid to binary
form. In this instance we assign an ‘x’ to a pole if it is rated 1 or 2, and
‘x’ to the contrast pole if it is rated 4 or 5, as per Spangenberg and Wolff
(1987). Using this approach, we transform the 12 (construct) x 12 (element)
grid to a 24 (poles) x 12 element grid, represented in Table 4.
Table 4: Transformed
repertory grid (click on heading)
We used the program CONEXP
1.3 to analysis this grid (Yevtushenko, 2000) The concept lattice for the grid was extremely
complicated, so three self elements (future self, self as I’d like others to
see me, and self as others see me) and one pair of construct poles (sociable
and hermitlike) were removed to bring out the salient concepts. The resulting
concept lattice is given in Figure 2.
Figure 2. Concept lattice for context in Table 4.
Figure 2 shows that this man sees someone he doesn’t like the look of as slimy,
lazy, deadbeats, who are also aloof, dour and miserable. If we take the
hierarchies into consideration, he further considers these people to be noxious
(his contrast pole to patient) and shambles (his contrast to organised). In
addition, it can be seen that he sees a conformist as callous, and himself
before he had tattoos as weak and neutral (his contrast pole to caring). However,
again the hierarchies here show us he was also happy, friendly, kind, honest,
hard working and active. This man’s ideal self is seen as identical to a rebel,
with the attributes of happy, friendly, kind, honest, hard working, active,
patient, confident, and having a sense of humour.
Comparison of FCA and other
repertory grid analysis approaches
One of
the motivations for considering FCA as an alternative approach for analysing
grid data is that it allows for a joint representation of elements and
constructs. As mentioned in Section 2, there are a number of approaches to
spatial representations of elements and constructs. What are the similarities
and differences between approaches such as singular value decomposition (SVD),
correspondence analysis (CA), multidimensional unfolding (MDU) and FCA? Wolff
(1996) conducted a comparison of graphical data analysis methods, including
SVD, FCA, MDU, and CA. Table 3 is a summary of some of the features of these
spatial representation methods based on Wolff (1996).
Table 5: Comparison of FCA and some multivariate methods
for analysing repertory grids.
Method  Data/Model  Representation  Interpretation  FCA  Contexts/lattice
theory  Hasse
diagram  Order/hierarchy  SVD  Eigen
decomposition  spatial
plot  similarity  CA  Eigen
decomposition  spatial
plot  similarity  MDU  distance  spatial
plot  similarity 
Table 5 distinguishes
among the various methods in terms of three features, namely, the analytical
model for dealing with the data, the type of joint representation obtained from
the analysis, and the primary mechanism underpinning the interpretation of the
solution. FCA differs in these three features, but most notably, FCA is able to
capture hierarchy, and not just equivalence among entities.
Other methods
This paper
has focused on methods for analysing repertory grids that allows for joint
representations of elements and constructs. Methods such as SVD are appropriate
and useful methods for determining the extent to which elements and constructs
are similar or related to each other. However, such methods do not adequately
capture the hierarchical relationship that is assumed among constructs, and the
elements associated with those constructs. In this paper we have argued that
FCA is a natural model for representing ordinacy in grid data.
However,
there are other methods that might also be appropriate for capturing the way
constructs are organised. As mentioned in Section 2, Fransella et al. (2004)
noted the use of hierarchical classes analysis to analyse grids. This technique
identifies hierarchies among sets of constructs and elements. While researchers
(for example, Gara et al., 1989) have used this technique, Fransella et al.
(2004) note that there are some drawbacks to its use given that it is based on
Boolean regression and deals only with binary data.
Partial
order scalogram analysis (POSA: Shye, 1985) is another technique that is worthy
of further investigation. POSA is an extension of Guttman’s (1950)
unidimensional scaling technique. It is a way of displaying multivariate data in a
twodimensional space. The rows of a data matrix are plotted so that the
partial order defined on those rows is maximally preserved. Interestingly, Bell (1986) developed a program for obtaining
partial order scalograms for repertory grids. Future research might consider a
systematic comparison of these techniques with FCA.
Concluding remark
The aim of
this paper was to revisit FCA as an alternative approach to analysing repertory
grid data. An important advantage of this technique is that it allows
hierarchical representations to be captured without loss of information. Future
study of the applicability of this technique to repertory grid data is
warranted.
A version
of this paper was presented at the 17^{th} International Personal
Construct Psychology Congress, July 1620, 2007, Brisbane.
 




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ABOUT THE
AUTHORS
Peter Caputi, PhD is an Associate Professor in the School
of Psychology at the University
of Wollongong. He has
published over 30 journal articles in the areas of Personal Construct
Psychology, information systems and measurement issues in psychology, as well
as coauthoring a textbook in research methods. He has reviewed for the International
Journal of Personal Construct Psychology, now the Journal of
Constructivist Psychology and Personal Construct Theory and Practice. He
has also edited conference abstracts for the Australian Journal of Psychology.
Email:
pcaputi@uow.edu.au
Home Page: http://www.uow.edu.au/health/psyc/staff/UOW024988.html
Desley Hennessy is a PhD candidate at the University of Wollongong, Australia. In
addition to her research investigating tattoos as an expression of identity,
she is interested in using her mathematical background in helping students
overcome their hurdles when learning statistics. Desley has a chapter in
Richard Butler’s book on Reflexivity where she examines the application of PCT
to her research as well as herself. Email: desley@uow.edu.au






REFERENCE
Caputi, P. & Hennessy, D. (2008). Using formal concept analysis to analyse repertory grid data. Personal
Construct Theory & Practice, 5, 165173.
(Retrieved from http://www.pcpnet.org/journal/pctp08/caputi08.html)






Received: 18 April 2008 – Accepted: 23 October 2008 –
Published: 23 December 2008





