Ken Reed schrieb:
>
> It's not really possible to explain this in lay person's terms. The
> difference between principal factor analysis and common factor analysis is
> roughly that PCA uses raw scores, whereas factor analysis uses scores
> predicted from the other variables and does not include the residuals.
> That's as close to lay terms as I can get.
If you have a correlation-matrix of
itm itm
___1_____2__
item 1 | 1 0.8 | = R
item 2 | 0.8 1 |
then you can factor it into a loadingsmatrix L, so that L*Lī = R.
There is a infinite number of options for L; especially there can
be found infinitely many just in different rotational positions
(pairs of columns rotated). In the list below there are 5 examples
from the extreme solutions (the extremes and 3 rotational positions
betweeen)
============================================================================
PCA CFA
factor factor factor factor factor
1 2 1 2 3
----------------------------------------------------------
1)
Item 1 1.000 0 1.000 0 0
Item 2 0.800 0.600 0.800 0 0.600
2)
Item 1 0.987 -0.160 0.949 0.316 0
Item 2 0.886 0.464 0.843 0 0.537
3)------------------------------------------------------------------+
Item 1 0.949 -0.316 0.894 0.447 0 |
Item 2 0.949 0.316 0.894 0 0.447 |
--------------------------------------------------------------------+
4)
Item 1 0.886 0.464 0.843 0.537 0
Item 2 0.987 -0.160 0.949 0 0.316
5)
Item 1 0.800 0.600 0.800 0.600 0
Item 2 1.000 0 1.000 0 0
===========================================================================
PCA:
Left list shows, how a components-analyst would attack the problem:
two factors; *principal* components analysis starts from the configu-
ration 3, where the sum of the squares of the entries is at a maximum.
The reduced number of factors, which a PCA-analyst will select for
further work, will be determined by criteria like "use all factors with
sum of loadingssquares>1" (equivalent eigenvalues>1) or "apply scree test"
or something like that.
CFA:
Right list shows, how a *common* factor analyst would attack the problem:
a common factor, plus an itemspecific factor for each item (like measuring
error etc). There are again infinite options, how to position the factors.
One could assume, that again example 3 was taken as default; but this is
not the case, as the algorithms, how to identify itemspecific variance
are iterative and not restricted to a special outcome (only the startposi-
tion is given). The only restriction is, that *for each item* there must
be an itemspecific factor. (In a two-item-case however the CFA-iteration
converges to position 3).
The itemspecific factors are not used in further analysis, only the common
one(s). Hence the name.
--------
>
> I have never heard a simple explanation of maximum likelihood estimation,
> but -- MLE compares the observed covariance matrix with a covariance
> matrix predicted by probability theory and uses that information to estimate
> factor loadings etc that would 'fit' a normal (multivariate) distribution.
It estimates the population-covariance matrix in that way, that your
empirical matrix is the most-likely-one, when randomly selected samples
would be taken.
Gottfried Helms
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