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Re: st: RE: PCA vs. Factor Loadings


From   "Michael I. Lichter" <MLichter@Buffalo.EDU>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: RE: PCA vs. Factor Loadings
Date   Wed, 16 Dec 2009 08:16:55 -0500

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 n.j.cox@durham.ac.uk
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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--
Michael I. Lichter, Ph.D. <mlichter@buffalo.edu>
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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