# Re: PCA analysis of rep grids

Brian Gaines (gaines@cpsc.ucalgary.ca)
Sat, 1 Mar 1997 09:34:18 -0700

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

RepGrid does a principal components analysis, not a factor analysis, and
performs no rotations of the components.

>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 .

There is not one PrinCom plot but an indefinite number that can be chosen, e.g.
PC1 against PC2, PC1 against PC3, etc. PrinCom provides the % variance
accounted
for by each component so that one can see which plots may be worth examining.

>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

PrinCom does this.

>b) either show the eigenvalues numerically or do a scree plot, and

PrinCom shows the eigenvalues numerically.

>c) whether they should offer the option of multi-factor solutions, rather
>than always present the principal components analysis as 2-dimensional.

There are many approaches to factor analysis and none has become widely
adopted in PCP. One reason is that typical grids are small datasets,
with 7-20 elements, whereas a typical factor analytic study will involve
100's of elements and more complex statistics may (may!) be justified.

The appropriate stats for small grids are descriptive ones that present
the grid data in ways that give clients new insights -- FOCUS and PrinCom
give different views of the raw grid.

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

It does not matter. The constructs are axes and the elements are points in
space. Scaling makes no difference to the direction of the axes.

I think all these issues arise through thinking of principal components
analysis as a statistical technique involving a lot of mathematical
mumbo jumbo. It is far better to think of it in terms of Kelly's
psychological geometry: n constructs are axes in n-dimensional space;
elements are points plotted in that space. Principal component analysis
moves the eye of the observer around in that space so as to achieve
the maximum separation of points. 2-D plots are cross sections of the
space.

Imagine yourself doing this with objects in a room and you can see how
the various phenomena described in the original mail arise. This analysis
also makes it clear that principal components analysis is a purely
descriptive technique -- a changing of viewpoints with no deep statistical
analysis involved. That is its attraction as a way of re-presenting
grid data to clients.

b.

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