Hello. Why are component loadings from -pca- so much smaller than factor
loadings from -factor-? Is there something about the procedure used by
Stata that makes them systematically smaller? I get the sense (which may
be mistaken; I don't have any evidence in my hand) that in other
packages -pca- and -factor- loadings are more similar.
For example, in the example below the variable -trunk- has a component
loading of 0.5068 and a factor loading of .8807, which is a fairly large
difference. Aside from the difference in the loading sizes, the
solutions look comparable.
My question is prompted by a more fundamental question, which is how
large should a loading be before it is considered significant (in the
sense of "worthy of notice")? Texts that give advice on interpretation
seem to assume that -pca- and -factor- results are on the same scale,
and I am a bit flustered about what to do with the low-ish loadings I'm
getting from -pca-.
Thanks.
Michael
Example:
. sysuse auto
. pca trunk weight length headroom, mineigen(1)
Principal components/correlation Number of obs
= 74
Number of comp.
= 1
Trace
= 4
Rotation: (unrotated = principal) Rho =
0.7551
--------------------------------------------------------------------------
Component | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Comp1 | 3.02027 2.36822 0.7551 0.7551
Comp2 | .652053 .37494 0.1630 0.9181
Comp3 | .277113 .226551 0.0693 0.9874
Comp4 | .0505616 . 0.0126 1.0000
--------------------------------------------------------------------------
Principal components (eigenvectors)
--------------------------------------
Variable | Comp1 | Unexplained
-------------+----------+-------------
trunk | 0.5068 | .2243
weight | 0.5221 | .1768
length | 0.5361 | .1319
headroom | 0.4280 | .4467
--------------------------------------
. factor trunk weight length headroom, pcf
(obs=74)
Factor analysis/correlation Number of obs
= 74
Method: principal-component factors Retained factors
= 1
Rotation: (unrotated) Number of params
= 4
--------------------------------------------------------------------------
Factor | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Factor1 | 3.02027 2.36822 0.7551 0.7551
Factor2 | 0.65205 0.37494 0.1630 0.9181
Factor3 | 0.27711 0.22655 0.0693 0.9874
Factor4 | 0.05056 . 0.0126 1.0000
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(6) = 257.89 Prob>chi2 =
0.0000
Factor loadings (pattern matrix) and unique variances
---------------------------------------
Variable | Factor1 | Uniqueness
-------------+----------+--------------
trunk | 0.8807 | 0.2243
weight | 0.9073 | 0.1768
length | 0.9317 | 0.1319
headroom | 0.7438 | 0.4467
---------------------------------------
--
Michael I. Lichter, Ph.D. <[email protected]>
Research Assistant Professor & NRSA Fellow
UB Department of Family Medicine / Primary Care Research Institute
UB Clinical Center, 462 Grider Street, Buffalo, NY 14215
Office: CC 126 / Phone: 716-898-4751 / FAX: 716-898-3536
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