# Re: PCA analysis of rep grids

Dr Stephen K Tagg (s.k.tagg@strath.ac.uk)
Wed, 5 Mar 1997 10:11:58 -0000

I'm investigating using PRINCALS for rep-grids. This allows appropriate =
level of representation for different levels of measurement.=20
see a conference paper at
http://s-tagg.market.strath.ac.uk/princals/princals.htm

----------
> From: Richard Bell <rcbell@rubens.its.unimelb.edu.au>
> To: pcp@mailbase.ac.uk
> Subject: Re: PCA analysis of rep grids
> Date: 04 March 1997 23:12
>=20
> Tony Downing
> Dept. of Psychology, University of Newcastle upon Tyne,
> Ridley Building, Claremont Place, Newcastle upon Tyne, NE1 7RU, =
England.
>=20
> Dear Tony,
>=20
> 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.
>=20
> 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.=20
>=20
> 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.
>=20
> 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.
>=20
> 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.
>=20
> 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
>=20
> 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.
>=20
> 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?)
>=20
> Regards,
>=20
> 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
>

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