rescaling PCA scores

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vatik51

Posted: Mon Apr 21, 2008 4:18 am

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On a set of 4 psychometric 5-point likert scales I performed PCA and
I saved the new PC scores. Then I rescaled the PC scores in a range
where min=0 and max=100. My goal was to perform a regression using
the rescaled PC's as independent variables in a binary logistic
regression so that betas would have meaningful interpretation.

I have some doubts about the result as I couldn't find any relative
article.

A brief description of what I did is presented below.
Any suggestion is greatly appreciated.

-The Component Score Coefficient Matrix is:

PC1 PC2
x1 0.80 -0.41
x2 0.55 -0.15
x3 -0.11 0.53
x4 -0.43 0.84

-The maximum and minimum values of the standardized original scales
are:

min max
zx1 -1.02 1.76
zx2 -0.97 1.86
zx3 -1.28 1.33
zx4 -1.26 1.30

-the maximum PC-values are calculated by multiplying positive factor
score coefficients with the maximum z-scores and negative coefficients
with the minimum z-scores
-the minimum PC-value are calculated by multiplying positive factor
score coefficients with the minimum z-scores and negative coefficients
with the maximum z-scores

-so, the maximum and minimum for PC1 are calculated as:

max(PC1)=0.80*1.76+0.55*1.86-0.11*(-1.2 Cool

-0.43*(-1.26)=3.12
min(PC1)=0.80*(-1.02)+0.55*(-0.97)-0.11*1.33-0.43*1.30.=-2.06.

-and finally, the rescaled PC1 [0,100] is calculated as:

newPC1=0+100*((PC1-min(PC1)) / (max(PC1)-min(PC1)).

Richard Ulrich

Posted: Tue Apr 22, 2008 8:42 pm

Guest

On Mon, 21 Apr 2008 07:18:52 -0700 (PDT), vatik51
<vatik51@hotmail.com> wrote:

Quote:

Performing PCA is not a way to head directly to
"meaningful interpretation" unless the factors turn
out to be unambiguous. The example below is
*worse* for interpretation that any that I can recall
every doing.

Scaling from Min to Max is an odd choice. PC scores
either have equal variance as they are produced, or they
reflect the variance of the underlying eigen vectors -- and,
as such, either version of PC scores is more readily interpretable,
in terms of variances, than your Min-Max thing.

Quote:

I have some doubts about the result as I couldn't find any relative
article.

A brief description of what I did is presented below.
Any suggestion is greatly appreciated.

-The Component Score Coefficient Matrix is:

PC1 PC2
x1 0.80 -0.41
x2 0.55 -0.15
x3 -0.11 0.53
x4 -0.43 0.84

I've never obtained PC scores that seemed so nearly
identical. Let's see: PC1 is characterized by X1 and
X4 with largest weights, in the opposite direction,
whereas PC2 has exactly the same.

If this was supposed to be an unrotated structure, then
I would expect "scales" on a subject to provide an overall
PC1 with positive loadings on everything.

Whatever the source -- You can't say anything clear about
"two factors" separately, when they look so much alike.

Quote:

-The maximum and minimum values of the standardized original scales
are:

min max
zx1 -1.02 1.76
zx2 -0.97 1.86
zx3 -1.28 1.33
zx4 -1.26 1.30

-the maximum PC-values are calculated by multiplying positive factor
score coefficients with the maximum z-scores and negative coefficients
with the minimum z-scores
-the minimum PC-value are calculated by multiplying positive factor
score coefficients with the minimum z-scores and negative coefficients
with the maximum z-scores

-so, the maximum and minimum for PC1 are calculated as:

max(PC1)=0.80*1.76+0.55*1.86-0.11*(-1.2 Cool

-0.43*(-1.26)=3.12
min(PC1)=0.80*(-1.02)+0.55*(-0.97)-0.11*1.33-0.43*1.30.=-2.06.

-and finally, the rescaled PC1 [0,100] is calculated as:

newPC1=0+100*((PC1-min(PC1)) / (max(PC1)-min(PC1)).

Assuming that your "coefficients" are the right set of
scoring coefficients, I see that you are computing the
maximum and minimum scores that are possible on the
factors. That's a bit clever. It will make more sense if you
do characterize your sample well, in terms if how limited
the *actual* range is. And, again, those factors look
unbelievably similar.

--
Rich Ulrich

http://www.pitt.edu/~wpilib/index.html

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