







WHEN IS MY GRID COGNITIVELY
COMPLEX AND WHEN IS IT SIMPLE?
SOME APPROACHES TO DECIDING



Richard C. Bell



Department of Psychology, University of
Melbourne, Australia 






Indices which
purport to measure cognitive
complexitysimplicity in repertory grid data have proliferated since
Bieri
introduced the concept in 1955, and a number of studies have shown how
two
correlation based measures, Bannister’s intensity index (the average
correlation) and the amount of variance accounted for by the first
principal
component of these correlations [PVAFF], are related. The usefulness of
these
indices in decision making contexts is limited by the fact that users,
unless
they have employed a standard grid on many occasions, have no idea what
constitutes a normal index value and what is abnormal. Here we use
comparisons
with indices derived from random data to identify abnormal indices and
in doing
so show limitations with both intensity and PVAFF, and show how an
elaboration
of the latter, determination of the number of components, provides a
more
informative basis for classifying construct systems as fragmented,
monolithic,
or complex.
Keywords: cognitive complexity, intensity,
PVAFF, random grids, number of factors








INTRODUCTION
Indices derived
from the repertory grid are
sometimes used as indicators of a characteristic of a person’s mental
functioning. The oldest and most widely known of these is the notion of
cognitive
complexitysimplicity (Bieri, 1955) which can be assessed in a number
of ways
in a repertory grid, including some measures based on correlations; the
average
correlation, the number of factors, or the size of the first
eigenvalue. There
are a number of others. From a decisionmaking perspective, such
indices
require a threshold against which they may be judged.
In psychological
testing decisions are
usually based on the performance of a comparable reference group
through norms.
Norms however require a standard data format and usage of the repertory
grid
often occurs in far from standard situations. Another approach is to
use
statistical significance testing. There have been a few evaluations of
indices
in grids in this way. Several authors have tested correlations between
constructs for a significant difference from zero. For example,
MahkloufNorris,
Jones and Norris (1970) proposed a system termed ‘articulation’ (which
can be
interpreted in terms of cognitive simplicity – complexity) and
identified
construct groupings by identifying correlations significantly greater
than zero
at the .05 level. There are several problems associated with this
approach. A
number of significance tests are conducted at the same time, the
correlations
are not independent of each other, and the testing is not simple for
the sample
sizes (i.e., the numbers of elements) involved. Since the number of
elements
does not approach the sample size required for normal distribution
testing,
ttests involving degrees of freedom are involved. Figure 1 below shows
the
level of significance for a correlation to be different from zero at
the .05
level.
Figure 1. Level of Construct Correlation
significantly different from 0.0 at p < .05 by Number of Elements.
However such a test is not available for
average correlations such as are involved in Bannister’s intensity or
for size
of the first component, since the distributional properties of these
indices
have not been studied. Another strategy must be adopted there.
One strategy adopted elsewhere in
psychometrics
is the evaluation of indices by comparison with distributions of such
indices
from random data. Horn (1965) proposed using such an approach to the
determination
of the number of components in factor analysis, and Spence and Ogilvie
(1973)
and others advocated its use in evaluating the stress index associated
with
multidimensional scaling. In the grid field Patrick Slater (1977) was
first to
examine the statistics from grids comprised of random data, with the
examination
of interelement distances since such distances are determined by the
scale
used in making the grid ratings. This work has been extended by
Hartmann (1992)
and Schoeneich and Klapp (1998) but the principle does not appear to
have been
extended to a more general consideration of indices in grids.
How does this
approach compare with the
more traditional significance testing above? To examine this, random
ratings
(on a 1 to 7 scale) were generated for pairs of variables (representing
constructs) where the numbers of cases (representing elements) was
varied from
6 to 30 in steps of two. For each sample size of elements, 100
replications
were carried out. Figure 2 shows the median value of the correlations
from
random data, and the 95th percentile of the distributions. The
significance
level from figure 1 is superimposed on this graph for comparison.
Figure 2. Median, 95th percentile values for
Construct Correlations from random data by No. of Elements.
It can be seen that the 95th percentile
values correspond in general to the value of correlations significantly
different from zero at a .05 significance level. It would appear thus
that we
may use the distribution of indices derived from random data as a basis
for
evaluating indices derived from actual grid data.
THE DATA
Here we
considered grids of varying kinds
from a number of studies involving different kinds of participants and
different kinds of grids. In total there were 834 grids. For each grid
100
replications were constructed by randomly permuting the grid data. The
distribution
of overall grid ratings thus remained the same, but the structure of
the grid
in the replications was randomized.
57% of these grids had
average correlations
(intensity) greater than that of the value associated with 95th
percentile of
the distribution of correlations from random data. If we look at
variance
associated with the first factor, then 88% of the 834 grids have first
factor
variances greater than that associated with the 95th percentile of the
distribution
of first factor variance of random data.
It is wellknown
that intensity and first
factor variance correlate strongly. For these grids the correlation was
0.98.
It is somewhat disconcerting therefore that the two indices do not
agree with
respect to their classification against random data. Table 1 shows the
crosstabulation of classifications.
Table 1. Crosstabulation of Av correlation
(intensity) permuted data classification by PVAFF permuted data
classification

PVAFF
permuted data classification 

Between 5th and 95th percentiles 
Above 95th
percentile

Total 
Av
RMS correlation permuted data classification 
Between 5th and 95th percentiles 
7.4% 
36.1% 
43.5% 
Above 95th
percentile 
4.2% 
52.3% 
56.5% 
Total 
11.6% 
88.4% 
100.0% 
% of Total
The two agree only in
60% of grids and disagree
in the other 40%. Bell (2003) and Fransella, Bell, and Bannister (2004,
p.118)
showed with hypothetical examples, why intensity as an average
correlation may
not be a good index of complexity or simplicity, since the mean of
heterogeneous
correlations can be the same as the mean of homogeneous correlations.
These present
results add to those concerns. There is a different concern with the
first
factor variance measure. Nearly 12% of grids do not differ from the
first factor
variance of random data. These grid structures could be described as
having no
structure or a fragmented structure. But what of the other 88% of grids
with
first factor variances greater than that associated with the 95th
percentile of
the distribution of first factor variance of random data? Are these
monolithically
structured or do they represent cognitively complex systems of
constructs? We
cannot tell.
One solution is to follow the factor
analytic strategy of Horn (1969) mentioned above and test the variance
associated with each component in turn against its corresponding random
distribution.
This method of determining the number of factors (known as ‘parallel
analysis’)
has become fairly widespread in its use as O’Connor (2002) has provided
syntax
for carrying this out in standard statistical packages. The number of
factors,
while an obvious candidate for assessing the cognitive complexity
evident in a
grid, has rarely been examined as such. Kuusinen and Nystedt (1975)
compared it
with other traditional measures, such as average correlation and
variance explained
by the first factor, and found it unrelated. One drawback to their
approach was
that they determined the number of factors by the
‘eigenvaluesgreaterthanone’
rule, a rule notable for its failure to correctly determine the number
of
factors (Wood, Tartaryn and Gorsuch, 1996; Zwick and Velicer, 1982,
1986). However,
Bell (2003) has shown an example of how the number of factors can be a
better
indicator of construct structure than the traditional indices of
average correlation
and variance explained by the first factor.
If
we apply this procedure to these grids we obtain the distribution shown
in
Table 2.
Table 2. No. of Components (Parallel
Analysis)
No. of Components

Frequency 
Percent 
0
1
2
3

97
587
149
1

11.6
70.4
17.9
.1

Total 
834 
100.0

The information available is somewhat
greater. Now we can conclude that nearly 12% of grids have a fragmented
structure (i.e., no factors – an outcome not realisable with either the
average
correlation or PVAFF approaches), over 70% can be accounted for by one
factor
only and could be termed monolithic in structure, while 18% are more
cognitively complex. The extent of monolithic construing is somewhat
disappointing
given the variety of grids involved. The number of components is in
part a function
of the number of constructs. Here the number of constructs ranged
between 8 and
15 with an average of 11, with 40% of grids having 12 or more
constructs. Since
the maximum possible number of fully independent factors can be
approximated by
onethird the number of variables we might have expected a greater
incidence of
3 and 4 factor solutions.
One possibility might be that the
parallel
test is conservative and tends to choose a lower number of factors (see
Turner,
1998). Another possibility is the number of random replications being
too few
(as shown by the discrepancies between the 95th percentile of random
data and
the theoretical .05 line in Figure 2). Another test commonly used to
determine
the number of factors is the "minimum average partial correlation"
approach
devised by Velicer (1976). This test has the advantage of not requiring
random
data for its computation[1],
and has been shown to
be able to identify the correct number of factors in simulation studies
(Zwick
and Velicer, 1982, 1986).
After each component is extracted, its
effect (and those extracted before it) is "partialled out" of the
correlation
matrix of original variables (through computing partial correlations)
and the
average of the resulting partial correlations is calculated. Velicer
reasoned
that after more and more factors which accounted for the original
correlations
were partialled out, the resulting partial correlations would approach
zero. However
when further components were partialled out, components which reflected
unique
or "noise" components, rather than "common" factors, the average
partial
correlation would begin to rise. Velicer’s rule was that the number of
factors
corresponding to the minimum average partial correlation would be
closest to
the correct number.
Table 3 shows the
number of components per
grid by this rule. While the number of fragmented grids remains
similar, there
are fewer monolithically structured grids (less than 40% instead 77%).
Table 3. No. of Components (Min Av. Partial
Correlation)
No. of Components

Frequency 
Percent 
0
1
2
3
4
5

99
324
290
95
24
2

11.9
38.8
34.8
11.4
2.9
.2

Total 
834 
100.0

Unfortunately the
results from this test do
not correspond closely to those of the parallel approach as Table 4
following
shows. Neither are they independent however; with a chisquare test of
association being significant (chisquare = 324, df=15, p <.001),
numbers of
factors correlating 0.36, and percentage of agreement being about 54%.
Table 4. Crosstabulation of No. of
Components (Min Av. Partial Correlation) by No. of Components (Parallel
Analysis)

No. of Components (Parallel Analysis)


0

1

2

3

Total 
No.
of Components
(Min Av. Partial Correlation Criterion) 
0
1
2
3
4
5

7.4%
1.9%
1.7%
.6% 
2.4%
36.0%
22.3%
7.1%
2.4%
.2%

2.0%
1.0%
10.7%
3.7%
.5%

.1%

11.9%
38.8%
34.8%
11.4%
2.9%
.2%


Total 
11.6% 
70.4% 
17.9% 
.1% 
100.0% 
Further
comparisons of these approaches require
external criterion information as a basis for decision making although
clearly
both approaches provide a basis for deciding about the simplicity or
complexity
of construct structure which is sufficiently differentiated to allow
differential decisions to be made.
SUMMARY AND
CONCLUSIONS
Making decisions
about the complexity or simplicity
of construct structure has not featured strongly in grid research. One
reason
for this has been the lack of criteria for making such decisions. At
present
norms are not really feasible for grids since they demand a standard
form, and
while statistical testing is not generally available, comparisons with
distributions
of indices derived from random data provide a promising alternative.
This study
showed that intensity was an index that provided little useful
differentiation
in terms of comparison with random baselines. PVAFF was somewhat more
informative in identifying grids with a fragmented structure, but could
not
distinguish between monolithic and complex structures. Determining the
number
of components, either by comparisons with random data, or through the
use of
another statistic, the minimum average partial correlation, provides
information
that enabled grids to be classified as monolithic, complex, or
fragmented in
structure. While the minimum average partial correlation approach
appeared to
provide finer discrimination for these grids, the generality of the
random comparison
approach suggested that it could be useful in evaluating many other
grid indices.
[1] This
measure is included in the
current version of GRIDSTAT (Bell, 2004) 





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ABOUT THE
AUTHOR
Richard
Bell, Ph.D., is an Associate Professor of Psychology at the
University of Melbourne, Australia. He is interested in practical
problems of measurement in clinical, organizational and educational
settings. He has written extensively on the analysis of repertory grid
data and has authored widely used software for the analysis of such
data. Email: rcb@unimelb.edu.au.






REFERENCE
Bell, R. C.
(2004). When is my grid cognitively complex and when is it simple? Some
approaches to deciding. Personal
Construct Theory & Practice,
1, 3136
(Retrieved from http://www.pcpnet.org/journal/pctp04/bell04.html)






Received: 31 Dec 2003  Accepted: 12 Jan 2004 
Published: 31 Jan 2004 




