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st: RE: PCA vs. Factor Loadings
I think the short answer is that you are not comparing like with like.
Loadings can be presented in various ways. See for example the help for
-pca postestimation- and then experiment with the different
normalisations on offer for -estat loadings-.
The default presentation of PCA loadings is not what you want, but a
different normalisation shows that PCA and factor analysis coincide in
the limit:
. pca headroom trunk weight length displacement
Principal components/correlation Number of obs =
74
Number of comp. =
5
Trace =
5
Rotation: (unrotated = principal) Rho =
1.0000
------------------------------------------------------------------------
--
Component | Eigenvalue Difference Proportion
Cumulative
-------------+----------------------------------------------------------
--
Comp1 | 3.76201 3.026 0.7524
0.7524
Comp2 | .736006 .427915 0.1472
0.8996
Comp3 | .308091 .155465 0.0616
0.9612
Comp4 | .152627 .111357 0.0305
0.9917
Comp5 | .0412693 . 0.0083
1.0000
------------------------------------------------------------------------
--
Principal components (eigenvectors)
------------------------------------------------------------------------
------
Variable | Comp1 Comp2 Comp3 Comp4 Comp5 |
Unexplained
-------------+--------------------------------------------------+-------
------
headroom | 0.3587 0.7640 0.5224 -0.1209 0.0130 |
0
trunk | 0.4334 0.3665 -0.7676 0.2914 0.0612 |
0
weight | 0.4842 -0.3329 0.0737 -0.2669 0.7603 |
0
length | 0.4863 -0.2372 -0.1050 -0.5745 -0.6051 |
0
displacement | 0.4610 -0.3390 0.3484 0.7065 -0.2279 |
0
------------------------------------------------------------------------
------
. estat loadings, cnorm(eigen)
Principal component loadings (unrotated)
component normalization: sum of squares(column) = eigenvalue
----------------------------------------------------------------
| Comp1 Comp2 Comp3 Comp4 Comp5
-------------+--------------------------------------------------
headroom | .6958 .6554 .29 -.04724 .002635
trunk | .8405 .3144 -.4261 .1138 .01243
weight | .9392 -.2856 .04092 -.1043 .1545
length | .9432 -.2035 -.05829 -.2245 -.1229
displacement | .8942 -.2909 .1934 .276 -.04629
----------------------------------------------------------------
. factor headroom trunk weight length displacement, pcf
(obs=74)
Factor analysis/correlation Number of obs =
74
Method: principal-component factors Retained factors =
1
Rotation: (unrotated) Number of params =
5
------------------------------------------------------------------------
--
Factor | Eigenvalue Difference Proportion
Cumulative
-------------+----------------------------------------------------------
--
Factor1 | 3.76201 3.02600 0.7524
0.7524
Factor2 | 0.73601 0.42791 0.1472
0.8996
Factor3 | 0.30809 0.15546 0.0616
0.9612
Factor4 | 0.15263 0.11136 0.0305
0.9917
Factor5 | 0.04127 . 0.0083
1.0000
------------------------------------------------------------------------
--
LR test: independent vs. saturated: chi2(10) = 373.68 Prob>chi2 =
0.0000
Factor loadings (pattern matrix) and unique variances
---------------------------------------
Variable | Factor1 | Uniqueness
-------------+----------+--------------
headroom | 0.6958 | 0.5159
trunk | 0.8405 | 0.2935
weight | 0.9392 | 0.1180
length | 0.9432 | 0.1103
displacement | 0.8942 | 0.2003
---------------------------------------
On the broader question, the question has some similarity with the
question of how big should a correlation be before one should pay
attention. I doubt there's an answer independent of discipline and
problem.
Nick
[email protected]
Michael I. Lichter
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-.
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
---------------------------------------
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