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st: PCA vs. Factor Loadings


From   "Michael I. Lichter" <MLichter@Buffalo.EDU>
To   statalist@hsphsun2.harvard.edu
Subject   st: PCA vs. Factor Loadings
Date   Wed, 16 Dec 2009 06:21:17 -0500

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. <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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