# Re; Grid Analysis

Hemant Desai (hdesai@unlinfo.unl.edu)
Fri, 22 Mar 1996 16:17:05 -0600 (CST)

A few days ago, Rainer Riemman wrote to Tim Anderson on this list:

> Dr. Anderson, in your recent posting you state:
>
> >> 3) the use of principal components, a common practice with gridders,
> >> is absurd (in my not so humble opinion) since the so-called sample size
> >> is necessarily too small for factor analysis.
>
> Although stated frequently, the above statement is simply false. At
> best it reflects a basic misunderstanding of the use of principal
> components analysis for the analysis of grids.
>
> Recently W. Chambers posted a 'mandala grid'. If you are not among the
> happy few people, who can see the basic structure underlying this grid
> without any technical tools, do a PCA and you may be surprised what PCA
> can do for your understanding of relations among elements and constructs
> and between elements and constructs. Kind regards Rainer Riemann
--------------------------------------------------------------------------
Dear Rainer, Tim, Bill and all others interested in grid analyses:

Of late, I have been toying with a somewhat different [_topological_]
approach to analysis of matrices. In particular, I made graphic plots of
the two 8 x 8 grids that Bill Chambers had posted on this list, ref.,
-------
A:=
1 2 3 4 5 6 7 8
2 1 3 4 5 6 7 8
2 3 1 4 5 6 7 8
2 3 4 1 5 6 7 8
2 3 4 5 1 6 7 8
2 3 4 5 6 1 7 8
2 3 4 5 6 7 1 8
2 3 4 5 6 7 8 1
--------
One could call this A, a serial-linear matrix. As Chambers (1985) put it,
it is logical but not integratively complex. [ BTW, when I plotted this
array, I thought it looked like the bleechers at a baseball game or circus;
betchya that must surely be a projective test of some sort! :) ]
-------- Now, let's consider the second grid:
B:=
1 2 3 4 5 6 7 8
2 1 4 3 6 5 8 7
3 4 1 2 7 8 5 6
4 3 2 1 8 7 6 5
5 6 7 8 1 2 3 4
6 5 8 7 2 1 4 3
7 8 5 6 3 4 1 2
8 7 6 5 4 3 2 1
--------
In terms of differences in A and B, note the following: In grid A above,
the Row sums are all the same, 36. However, Column sums vary from 15 to 57.
Matrix two is structurally different from the first because in the latter
BOTH Row and Column sums are identical giving each Row-Column mean has
'equal weightage' in statistical terms. Also, note that no cell-value
is redundant in matrix B, so might be safe to conclude the arrays in B
are not work of random chance but represent an intelligent, more complex
form of order compared to the relations displayed in matrix A.

I realize that some opacity in repertory grid method may exist because a
set of Construct-Element values represent, at the very least, three distinct
perspectives for analysis: 1) Variations in value within rows and columns,
2) correlations between rows and columns, and 3) In indices or row-column
correlation summaries across "subjects". The last mentioned analytic type
appears to be one that Tim Anderson might be referring to. The caveats that
W. V. Chambers posted earlier in the week which refer to reductionism in
grid data by researchers' pet abstractions is, IMO, valid critique of most
mainstream and ironically, some personal construct, psychology literature.

Now consider another grid that Bill also posted with the values of 1,
2.5, 4.5, 6.5, and 8. If you recall this circumplex grid, it goes like this:

1 2.5 4.5 6.5 8 6.5 4.5 2.5
2.5 1 2.5 4.5 6.5 8 6.5 4.5
4.5 2.5 1 2.5 4.5 6.5 8 6.5
6.5 4.5 2.5 1 2.5 4.5 6.5 8
8 6.5 4.5 2.5 1 2.5 4.5 6.5
6.5 8 6.5 4.5 2.5 1 2.5 4.5
4.5 6.5 8 6.5 4.5 2.5 1 2.5
2.5 4.5 6.5 8 6.5 4.5 3.5 1

Plot the values from these three grids in three separate spreadsheet
pages/files and as 3d plots as possible. In observing the patterns made in
grid three, compared with the first two, I predict that you might get my
drift about such dimensional plots/analysis of construct-concept "structure".

Grid three above is different also because five values (not eight as in A
and B above) are used to fill a mtx of 8 rows and 8 columns. In general,
I think the issue of sample size is relevant mainly if someone tries to
extrapolate from one person's worldview, (of course a worthy goal and of
both demographic & semantic interest), but not so much in terms of the
uniqueness of each person's conceptual schemata of interpersonal traits or
their cultural-transpersonal framework of values, hopes, and fears.

I have a hunch that visual analysis of the "spatial" relations between
values in a grid using topological mapping (as an adjunct with euclidean
geometry and correlations and cosines) could somehow be fruitfully used
to extend further such measures, because at the heart of Kelly's theory
of construct systems is the notion that humans make cognitive -- in the
holistic sense -- appraisals of things because they can distinguish how
TWO perceptive-verbal labels are qualitatively different from a THIRD
such perceptive-verbal label in some individually meaningful way.

Kelly's idea created the repertory grid of self-significant others and
the associated method of analysis, termed "nonparametric" or idiographic
technique (a guide for clinical assessment NOT psychometric measurement).
Obviously, after Kelly, the concept of grids went many different, not
altogether unconnected, ways. The two most popular are cluster analysis
which groups together rows or columns with similar raw values, while
primary component analysis, is quite often used to find the first few
(three is fine) factors that account for most variance unrelated to other
components, and then use these factors as vectors (made orthogonal or
at right angles to each other through rotation if necessary) as the X, Y,
and Z, reference axes for plotting semantic constructs, people-as-elements
or perhaps abstractual-ideal selves.

I found, by plotting such numeric tables as 3-dimensional Histograms, one
can graphically "see the shape" of the matrix of values, or of correlation
coefficients as need be. I also realize this approach is derived from
the metaphor of CONSTRUCTION very literally. From such a perspective,
People are ARCHITECTS as much as they are Scientists (or Politicians!).
The questions that must then be asked, in due time: Architects of what?
Of their own destinies/identities perhaps?

Hopefully, this will be possible in future research consonant with the idea
that we create and react from our mental structures when we compare people,
places and things, and speak, listen, read, write, critique, or argue!

The utility of a visual-map approach is that you no longer_have to have_
comparative indices or large sample sizes or even larger grid sizes. I
have also plotted these grids while limiting variance by the number of
rows and columns. It is interesting to note that as the size of the
grid decreases there are fewer possible combinations that are unique.
[ A fact I tacitly knew in preuniversity as, say, eight factorial:
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320 different combinations
of which only n!/n-1! or 540 are unique permutations ]

Perhaps someone else here who has tried some non-Euclidean approach to
matrix analysis and can inform me on if and how this is relevant for further
explorations. BTW, I got excellent graphical results, you might too, depending
on the package you use -- a software program called MAPLE V rel. 3 on a
Windows 3.x ibmPC worked well for me. MAPLE syntax is quite easy, such as:

Z:=matrix(8,8,[x,y,z,a,b,c,..],[x2,y2,z2,a2,b2,c2,..],[etc...]]);

Note that the matrix Z is defined only after the following text is entered
in the Maple kernel or command prompt: with(linalg);
[ and then: with(plots);
and finally: matrixplot(Z);
]