PCA analysis of rep grids

Tony Downing (a.c.downing@newcastle.ac.uk)
Sat, 1 Mar 1997 12:10:24 +0000

I have a student investigating the construct systems of people with eating
disorders. She has been analysing their grids using the generally
delightful and wonderful Repgrid2 package, from the Centre for
Person-Computer Studies.

The Princom (i.e., first two principal components) plot for one of the
participants puzzled us, because two of the constructs seemed to lie almost
on top of each other, as if they were very similar, yet their verbal labels
suggested almost opposite meanings. Frankly, we wondered whether there
could be a bug in Repgrid2, perhaps reversing the scoring of a construct
without reversing its labels. There wasn't! But to check on this, I used
Minitab to carry out a principal components analysis on the correlations
between the constructs, and replicated the Princom plot semi-manually,
starting from a Minitab Gplot of the loadings on the first two factors from
the PCA. This led to the following findings, which were far from obvious
from the Repgrid2 manual and output. (These are not offered in the spirit
or any kind of flaming!)

1. Repgrid2 evidently rotates factors - good idea, but not mentioned in
the documentation.

2. In the grid in question, the first 2 factors accounted for 61.6% of the
variance, but 4 factors had eigenvalues > 1. The correlation between the 2
puzzling constructs, that still appeared to plot almost identically in the
Principal Components plot, was actually only about 0.34. Evidently the
vectors representing these two constructs were widely separated in the
hyperspace outside the plane of the Princom plot, though in the favoured
2-D plane their "shadows" were almost superimposed .

These two points surely raise the question as to whether, to avoid
misleading interpretations of grids, programs such as Repgrid2 should not
a) routinely offer to show the construct correlation matrix and
b) either show the eigenvalues numerically or do a scree plot, and
c) whether they should offer the option of multi-factor solutions, rather
than always present the principal components analysis as 2-dimensional.

Finally, having plotted the constructs on my PCA chart, I wanted to add in
the elements in the same space generated by the first two principal
components, as Princom does and as been done so instructively by many
authors. Not realising that there would be any problem in doing this, I
got Minitab to give the factor scores for the elements on the first two
components - and I immediately ran into the fact that, while the axes of
coefficients) and therefore run from -1 to +1, the factor scores for the
elements go way outside these ranges. So my question to the PCP forum is:

HOW SHOULD ONE DECIDE ON THE SCALING OF THE ELEMENT SCORES, WHEN PLOTTING
THEM IN THE COMPONENT-SPACE OF THE CONSTRUCTS?

I looked in Fransella & Bannister's 1977 "Manual of Repertory Grid
Technique", and while I found on p. 74 a rather inscrutible reference to
the problem, in a quotation from an unpublished manuscript by Wilson
(1976), I found no solution. It seems to me that it must be arbitrary how
one scales the element scores to plot them in the contruct space, and that
therefore we should all be careful not to read much meaning into proximity
or distance between plotted positions of elements and constructs. But I am
a relative novice in pcp psychology, so quite likely I have missed some key
point and I really would be glad to be enlightened. I do feel, however,
that grid analysis packages should give the user documentation on what
algorithms have been applied, just as general data analysis packages such
as SPSS do. Is it healthy for PCP to pass over technicalities as lightly
as it seems, rather commonly, to do?

===========================================================================
Tony Downing

Dept. of Psychology, University of Newcastle upon Tyne,
Ridley Building, Claremont Place, Newcastle upon Tyne, NE1 7RU, England.
Telephone: +44 (0)191 222 6184, Mobile: +44 (0)468427481
Fax: +44 (0)191 222 5622

email: A.C.Downing@Newcastle.ac.uk

===========================================================================

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%