If you're using Stata 10, please do make that clear in future postings.
I often calculate correlations directly between PCs (in my case always
vanilla flavour) and original variables. That way I know exactly what I
am getting without having to decode eigenspeak. -correlate- will do
that, naturally, but I sometimes use -cpcorr- from SSC, which shows me
just the submatrix I care about. Presumably that could also be used for
looking at results of other modes of PCA or factor analysis.
. pca trunk displacement weight length
Principal components/correlation Number of obs =
74
Number of comp. =
4
Trace =
4
Rotation: (unrotated = principal) Rho =
1.0000
------------------------------------------------------------------------
--
Component | Eigenvalue Difference Proportion
Cumulative
-------------+----------------------------------------------------------
--
Comp1 | 3.35594 2.91098 0.8390
0.8390
Comp2 | .444963 .287221 0.1112
0.9502
Comp3 | .157742 .11639 0.0394
0.9897
Comp4 | .0413522 . 0.0103
1.0000
------------------------------------------------------------------------
--
Principal components (eigenvectors)
--------------------------------------------------------------------
Variable | Comp1 Comp2 Comp3 Comp4 | Unexplained
-------------+----------------------------------------+-------------
trunk | 0.4418 0.8721 0.1981 -0.0700 | 0
displacement | 0.5007 -0.4020 0.7328 0.2252 | 0
weight | 0.5272 -0.2661 -0.2728 -0.7595 | 0
length | 0.5255 -0.0833 -0.5910 0.6063 | 0
--------------------------------------------------------------------
. predict pc1-pc4
(score assumed)
Scoring coefficients
sum of squares(column-loading) = 1
------------------------------------------------------
Variable | Comp1 Comp2 Comp3 Comp4
-------------+----------------------------------------
trunk | 0.4418 0.8721 0.1981 -0.0700
displacement | 0.5007 -0.4020 0.7328 0.2252
weight | 0.5272 -0.2661 -0.2728 -0.7595
length | 0.5255 -0.0833 -0.5910 0.6063
------------------------------------------------------
. cpcorr trunk displacement weight length \ pc1-pc4
(obs=74)
pc1 pc2 pc3 pc4
trunk 0.8094 0.5818 0.0787 -0.0142
displacement 0.9172 -0.2682 0.2910 0.0458
weight 0.9659 -0.1775 -0.1084 -0.1544
length 0.9626 -0.0556 -0.2347 0.1233
Nick
[email protected]
Michael I. Lichter
Thanks, Nick! This would give me exactly what I was looking for ...
Except that -estat loadings- ignores rotations, as far as I can tell, in
Stata 10. I can do the normalization "by hand" in Excel, though, so
that's better than nothing.
Michael
Nick Cox wrote:
> 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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