[R] Princomp(), prcomp() and loadings()

[F.Tusell]

In comparing the results of princomp and prcomp I find:

   1. The reported standard deviations are similar but about 1% from
      each other, which seems well above round-off error.
   2. princomp returns what I understand are variances and cumulative
      variances accounted for by each principal component which are
      all equal. "SS loadings" is always 1. 
   3. Same happens after the loadings are varimax-rotated, which in 
      general should alter the proportions of variance accounted by 
      each component.

It looks as if the loadings() function were expecting the eigenvectors
to be normalized to the corresponding eigenvalue.

Transcript and version information follow signature. Thank you for any
clues.

ft.
-- 
Fernando TUSELL                                e-mail:
Departamento de Econometría y Estadística           etptupaf at bs.ehu.es 
Facultad de CC.EE. y Empresariales             Tel:   (+34)94.601.3733
Avenida Lendakari Aguirre, 83                  Fax:   (+34)94.601.3754
E-48015 BILBAO  (Spain)                        Secr:  (+34)94.601.3740
----------------------------------------------------------------------




> pca.1 <- prcomp(USArrests)
> pca.1
Standard deviations:
[1] 83.732400 14.212402  6.489426  2.482790

Rotation:
                PC1         PC2         PC3         PC4
Murder   0.04170432 -0.04482166  0.07989066 -0.99492173
Assault  0.99522128 -0.05876003 -0.06756974  0.03893830
UrbanPop 0.04633575  0.97685748 -0.20054629 -0.05816914
Rape     0.07515550  0.20071807  0.97408059  0.07232502
> pca.2 <- princomp(USArrests)
> pca.2
Call:
princomp(x = USArrests)

Standard deviations:
   Comp.1    Comp.2    Comp.3    Comp.4 
82.890847 14.069560  6.424204  2.457837 

 4  variables and  50 observations.
> summary(pca.2)
Importance of components:
                           Comp.1      Comp.2      Comp.3       Comp.4
Standard deviation     82.8908472 14.06956001 6.424204055 2.4578367034
Proportion of Variance  0.9655342  0.02781734 0.005799535 0.0008489079
Cumulative Proportion   0.9655342  0.99335156 0.999151092 1.0000000000
> loadings(pca.2)

Loadings:
         Comp.1 Comp.2 Comp.3 Comp.4
Murder                         0.995
Assault  -0.995                     
UrbanPop        -0.977 -0.201       
Rape            -0.201  0.974       

               Comp.1 Comp.2 Comp.3 Comp.4
SS loadings      1.00   1.00   1.00   1.00
Proportion Var   0.25   0.25   0.25   0.25
Cumulative Var   0.25   0.50   0.75   1.00

> varimax(pca.2$loadings[,1:3])
$loadings

Loadings:
         Comp.1 Comp.2 Comp.3
Murder                       
Assault  -0.998              
UrbanPop        -0.997       
Rape                    0.995

               Comp.1 Comp.2 Comp.3
SS loadings      1.00   1.00   1.00
Proportion Var   0.25   0.25   0.25
Cumulative Var   0.25   0.50   0.75

$rotmat
            [,1]       [,2]       [,3]
[1,]  0.99211386 0.03604908 -0.1200439
[2,] -0.05442524 0.98664663 -0.1535132
[3,]  0.11290692 0.15883603  0.9808278

> R.Version()
$platform
[1] "i386-pc-linux-gnu"

$arch
[1] "i386"

$os
[1] "linux-gnu"

$system
[1] "i386, linux-gnu"

$status
[1] ""

$major
[1] "2"

$minor
[1] "0.0"

$year
[1] "2004"

$month
[1] "10"

$day
[1] "04"

$language
[1] "R"