R help - Apr 2013 - prcomp() and varimax()

Hello,
I am attempting to do a principal components analysis on 15 survey 
items. I want to use a varimax rotation on the retained components, but 
I am dubious of the output I am getting, and so I suspect I am doing 
something wrong. I proceed in the following steps:

  1) use prcomp() to inspect all 15 components, and decide which to retain
  2) run prcomp() again, using the "tol" parameter to omit unwanted 
components
  3) pass the output of step 2 to varimax()

My concern is with the reported proportions of variance for the 3 
components after varimax rotation. It looks like each of my 3 components 
explains 1/15 of the total variance, summing to a cumulative proportion 
of 20% of variance explained. But those 3 components I retained should 
now be the only components in the analysis, so they should be able to 
account for 100% of the explained variance.

I am able to get reliable seeming results using principal() from the 
"psych" package, in which the total amount of variance explained by my
retained components does not differ before or after rotation. But 
principal() uses varimax(), so I suspect I am either doing something 
wrong or misinterpreting the output when using the base package functions.

Am I doing something wrong when attempting to retain only 3 components?
Am I using varimax() incorrectly?
Am I misinterpreting the returned values from varimax()?

Thanks for any help,
Mike



Here is a link to the data file I am using: 
https://www.dropbox.com/s/scypebzy0nnhlwk/pca_sampledata.txt

### step 1 ###
 > d1 = read.table("pca_sampledata.txt", T)
 > m1 = with(d1, ~ g.enjoy + g.look + g.cost + g.fit + g.health + 
g.resale + b.withstand + b.satisfy + b.vegetated + b.everyone + b.harmed 
+ b.eco + b.ingenuity + b.security + b.proud)
 > pca1 = prcomp(m1)
 > summary(pca1) #output truncated for this posting
Importance of components:
                           PC1    PC2    PC3     PC4     PC5 ...    PC15
Standard deviation     1.5531 1.3064 1.1695 0.93512 0.92167 ... 0.35500
Proportion of Variance 0.2199 0.1556 0.1247 0.07972 0.07744 ... 0.01149
Cumulative Proportion  0.2199 0.3755 0.5002 0.57988 0.65732 ... 1.00000


### step 2 ###
 > pca2 = prcomp(m1, tol=.75)
 > summary(pca2) #full output shown
Importance of components:
                           PC1    PC2    PC3
Standard deviation     1.5531 1.3064 1.1695
Proportion of Variance 0.4397 0.3111 0.2493
Cumulative Proportion  0.4397 0.7507 1.0000


### step 3 ###
 > pca3 = varimax(pca2$rotation)
 > pca3
 > ...
 >                  PC1   PC2   PC3
 > SS loadings    1.000 1.000 1.000
 > Proportion Var 0.067 0.067 0.067
 > Cumulative Var 0.067 0.133 0.200

> 
> My concern is with the reported proportions of variance for the 3 
> components after varimax rotation. It looks like each of my 3 components 
> explains 1/15 of the total variance, summing to a cumulative proportion 
> of 20% of variance explained. But those 3 components I retained should 
> now be the only components in the analysis, so they should be able to 
> account for 100% of the explained variance.
Am I misreading what you just wrote? One percentage (20%) is a proportion of the
total variance in the data; the other is the proportion of the variance
explained by the model. These are different things; they should not usually be
the same.

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