# Re: Generalised Procrustes Analysis and Grids?

Thu, 7 May 1998 11:29:13 +0100 (BST)

Hello Devi

On Wed, 6 May 1998, Devi Jankowicz wrote:

> Miguel Sottomayer writes:
>
> >I introduce myself a few months ago saying that I was trying to apply pcp
> >to the study of product (wine) choice perceptual maps and consumer's
> >segmentation.
> >Well, I'm now starting to analyse data from about 60 grids ranging from 3
> >to 25 constructs (ratings 1 to 7).
> >Amonst other things I'm using Generalized Procrustes Analysis (GPA) as a
> >way to reach "aggregated" perceptual maps for pre-defined groups of
> >respondents (on the basis of several background variables) and doing
> >so trying to identify different perceptual segments.
> >
> >Three major problems are allways coming to my mind:
> >
> >1 - How to validate the interpretation of the "aggregated constructs" or
> >factors that are produced by GPA (a list of "real" constructs
> >highly correlated with each factor is given by GPA and is the basis of
> >my interpretation..);
>
> What a fascinating topic for research!
>
> I wish I knew about Generalized Procrustes Analysis: perhaps you might
> give a brief description?

Briefly GPA deals with matrices of ratings having in common the same
elements and the same rating scale (in my case..); e.g. construct 1 for
respondent n is the first construct elicited (in my case too..doesn't
matter really..).
What GPA does is to maximize the agreement amongst respondents on the
evluation of individual constructs. This is done through sucessive stages
of translating, scaling, reflecting and rotating the individual matrices
without changing the relative position of the elements (ratings) within
respondents.
Just to give you a simple example. Imagine that you have a single
respondent, 3 elements and two constructs. If you plot the elements in
a two dimensional system of coordinates (x=constr.1 & y=constr.2) you
would get a triangle (configuration). Then if you add to the same
2dimens. space a second matrix exactly with the same constructs and
ratings but swaping axis (x=constr.2 & y=constr.1) you would get exactly
the same triangle but in a different position in the space. If you rotate
(geometrically) one of the triangles you would be able to justapose
exactly both configurations. The important thing is that the relative
position of the elements within each configuration is kept the same. In
this simple case what you would end up with is an identification of
sinonimous (different construct names for the same thing) as
of GPA in this case would be 0 as both configuration match perfectly after
rotation.

Hopefully, though, it is capable of usefully
> supplementing a content analysis of the constructs themselves? (Or
> perhaps a content analysis forms part of it directly?)

For the interpretation of each factor I've used a list of constructs that
Each construct refers to a
single respondent. As you can have constructs presenting exactly the same
wording but coming from different respondents and with a different
"meaning" you have to be able to trace the construct origin.
So, yes, I'm using content analysis to complement GPA interpretation,
explaining (or
not) why exactly the same (nominal) construct coming from two respondents
correlate to different factors.

> What I have in mind is that a technique which deals only with ratings
> will provide an outcome which, presumably, would state that the following
> two constructs (with the ratings of 7 elements given):
>
> Bland 3 5 7 2 1 5 6 4 Flavoursome
> Cheap 3 5 7 2 1 5 6 4 Expensive
>
> share so much variance that there _must_ be an underlying ("true"?)
> construct to account for the ratings;

Whether the underlying factor is true or whether the correlation is
"spurious" that's the most difficult thing in interpreting the factors! In
average out of ten "explanatory" constructs for each factor I'm getting
1 that I found difficult to explain. Again content analysis is helping in
this respect and also the analysis of correlations between "objective"
characteristics amongst the element set (e.g. only one "Lebanese" wine
which is
"expensive" and presents a "high alcohol content"..)

> yet would do so with exactly the same credibility if the second construct
>
> Mythical 3 5 7 2 1 5 6 4 Historical

During the grid interviews I've asked endessly: "Is this aspect really
influencing your wine choices?". This was a condition to consider the
inclusion of a particular construct in the grid.Sometimes I had to
drop constructs that were elicited but wasn't choice influencing.
Highly correlated constructs were also carefully discussed to check for
"spurious" correlations.
So, your
Mythical/Historical construct wouldn't be included in the grid in the
first place!

> Yet, may I assure you, the third construct is irrelevant to the way in
> which I construe wines! Which makes one wonder about the credibility of
> the "underlying" construct identified in the first instance.
>
> My point being that one seeks an analysis of the meaning, as well as of
> the ratings, to say anything about the similarity of constructs being
> used. Perhaps your question (I've selected only the first of your list,
> since I don't know enough to address the second two) relates to this
> issue?

Yes, I'm using information from the content analysis and from the ratings
for interpretation. My problem is where to draw the border line
between a "spurious" correlation and a "true" correlation and how
sustainable are my decisions?!

> I'm not being a smart-arse here: I really don't know anything about
> Generalized Procrustes Analysis and will freely confess that I need to
> know more before making the above comments more than tentatively. Boorman
> did some work on a problem similar to the one that I've addressed,
> (surely an issue critical in PCP), and I've the glimmerings of an answer
> in the present case, but need to know more about GPA first, along the
> lines of "how does it handle _content_?"
>
> Kind regards,
>
> Devi

Miguel

========================================================
Miguel L. Sottomayor
PhD Student - Dept. Agricultural and Food Economics
University of Reading, 4 Earley Gate