R help - Aug 2006 - - factanal scores correlated?

Hi,

I wonder why factor scores produced by factanal are correlated, and I'd 
appreciate any hints from people that may help me to get a deeper 
understanding why that's the case. By the way: I'm a psychologist used 
to SPSS, so that question my sound a little silly to your ears.

Here's my minimal example:

***********************************************
      v1 <- c(1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,5,6)
      v2 <- c(1,2,1,1,1,1,2,1,2,1,3,4,3,3,3,4,6,5)
      v3 <- c(3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,5,4,6)
      v4 <- c(3,3,4,3,3,1,1,2,1,1,1,1,2,1,1,5,6,4)
      v5 <- c(1,1,1,1,1,3,3,3,3,3,1,1,1,1,1,6,4,5)
      v6 <- c(1,1,1,2,1,3,3,3,4,3,1,1,1,2,1,6,5,4)
      m1 <- cbind(v1,v2,v3,v4,v5,v6)
      myfac <- factanal(m1, factors=3, scores="regression")#
      cor(myfac$scores)
***********************************************

Tells me
             Factor1     Factor2     Factor3
Factor1 1.000000000 0.001624383 0.002862785
Factor2 0.001624383 1.000000000 0.001956953
Factor3 0.002862785 0.001956953 1.000000000

which means that factor correlations are indeed quite low with regard to 
interpretation issues, but an analysis of a larger dataset yielded 
factor intercorrelations up to .10.

I guess this is an optimization issue because a lower setting of
"lower"
tends to lower factor intercorrelations, but I'm still confused because 
I (misleadingly?) thought that factor scores are (completely) 
independent by definition?

Any hints would be greatly appreciated,

best regards,

Christian


-- 
--------------------------
Dr. Christian Montel
eligo GmbH -- B?ro Berlin
Arndtstr. 34
10965 Berlin
Tel. 030 -- 69 00 11 42
Fax  030 -- 69 00 47 61
christian.montel at eligo.de
http://www.eligo.de/

This is a Heywood case, and you don't have a valid fit:
> myfac
Call:
factanal(x = m1, factors = 3, scores = "regression")

Uniquenesses:
   v1    v2    v3    v4    v5    v6 
0.005 0.101 0.005 0.224 0.084 0.005 

notice no less than 3 very small uniquenesses.


On Fri, 11 Aug 2006, Christian Montel wrote:
> Hi,
> 
> I wonder why factor scores produced by factanal are correlated, and I'd
> appreciate any hints from people that may help me to get a deeper 
> understanding why that's the case. By the way: I'm a psychologist
used
> to SPSS, so that question my sound a little silly to your ears.
> 
> Here's my minimal example:
> 
> ***********************************************
>       v1 <- c(1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,5,6)
>       v2 <- c(1,2,1,1,1,1,2,1,2,1,3,4,3,3,3,4,6,5)
>       v3 <- c(3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,5,4,6)
>       v4 <- c(3,3,4,3,3,1,1,2,1,1,1,1,2,1,1,5,6,4)
>       v5 <- c(1,1,1,1,1,3,3,3,3,3,1,1,1,1,1,6,4,5)
>       v6 <- c(1,1,1,2,1,3,3,3,4,3,1,1,1,2,1,6,5,4)
>       m1 <- cbind(v1,v2,v3,v4,v5,v6)
>       myfac <- factanal(m1, factors=3, scores="regression")#
>       cor(myfac$scores)
> ***********************************************
> 
> Tells me
>              Factor1     Factor2     Factor3
> Factor1 1.000000000 0.001624383 0.002862785
> Factor2 0.001624383 1.000000000 0.001956953
> Factor3 0.002862785 0.001956953 1.000000000
> 
> which means that factor correlations are indeed quite low with regard to 
> interpretation issues, but an analysis of a larger dataset yielded 
> factor intercorrelations up to .10.
> 
> I guess this is an optimization issue because a lower setting of
"lower"
> tends to lower factor intercorrelations, but I'm still confused because
> I (misleadingly?) thought that factor scores are (completely) 
> independent by definition?
> 
> Any hints would be greatly appreciated,
> 
> best regards,
> 
> Christian
> 
> 
> 
-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595