Re: PCA analysis of rep grids

Richard Bell (
Wed, 05 Mar 1997 10:12:31 +1100

Tony Downing
Dept. of Psychology, University of Newcastle upon Tyne,
Ridley Building, Claremont Place, Newcastle upon Tyne, NE1 7RU, England.

Dear Tony,

As one of the writers of grid analysis software (the hopefully soon defunct
G-pack), I was interested in your exchange for several reasons. The issue of
principal components is the first one. Principal components of what? We are
classically taught that what one factor analyzes is a correlation amtrix,
but this is not necessarily true - we can find the 'principal components' of
pretty well any sort of matrix. In the classical approach we find the
principal components of one set of correlations (say the constructs) and
then post-hoc construct the component score for the other set (say the
elements). We could perhaps even do it the other way around. As you observe
the numbers are scaled differently, though it ought to be possible to
rescale them into some common form. But this is not the only way to go.
Other approaches are based on what is known as the
singular-value-decomposition (or Eckart-Young decomposition) approach. Here
the grid is factored directly to produce two sets of factor loadings, one
for elements and one for constructs. Three programmes (that I know of) do
this: Patrick Slater's INGRID, my (soon to be replaced) G-pack, and the
Feixas and Cornejo GRIDCOR. Each of these pre-processes the grid in some
way; INGRID re-scales by construct, and G-pack's default is to remove
construct and element means. GRIDCOR employs a correspondence analysis
approach so I would expect that its rescaling is governed by sums of ratings
for both elements and constructs in some way.

Your other issue of rotation is less technical. As you observed
> On the one grid which I analysed using the PCA
>command in Minitab and for which I then plotted the projections of the
>construct axes manually, the pattern of axes that I got was the same as we
>had got from the RepGrid2 PrinCom output for the same grid - except that
>the whole plot definitely was swung round a bit in relation to PrinCom's
>orthogonal axes for Components 1 and 2.

If you do orthogonal rotation (such as varimax) then you will observe
precisely the phenomena you did: since rotation does not change the
configuration of the constructs (or elements), merely the angles of the
axes. So if you are interested in which construct (or element) is close to
which other ones, then (a) you need to plot the loadings and (b) it doesn't
matter whether you rotate them or not.

This works fine for two components. For more than two you have to do
multiple plots - and it becomes very hard to co-ordinate the information
across plots (cluster analysis of the constructs across loadings is a much
safer way to go) - even for a three-D plot it is difficult to see what is where.

I have noticed when people try to make sense of these multiple
two-dimensional plots they ofteen start to refer to the axes with meaning -
i.e. they start to interpret the factors.

If you are going to interpret the factors (i.e., see bases of commonality
between constructs [or elements]) then you are right - it is much better to
rotate the factor loadings.

If you have more than two or three factors you are almost always committed
to this (a one factor solution cannot be rotated). However it takes at least
three variables to define a factor (analogous to the 3 elements defining a
construct). If you think about the size of most grids, you will see that it
is unlikely that you will be in a position to extract substantial numbers of
factors - and hence the need for this does not arise very often. Finn
Tschudi's FLEXIGRID program does have some rotation built in.

The real problem is that there is no 'true' representation of a grid. What
your queries raise is the extent of our lack of knowledge about how the
various methods of analysis impose their own artefactual structure on the
solution (ph.d. anyone?)


Richard Bell
Richard C Bell
Department of Psychology
University of Melbourne
Parkville Vic 3052 Australia
Phone: +61 (0)3 9344 6364
Fax: +61 (0)3 9347 6618