From "Michael I. Lichter" 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-.
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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
--------------------------------------
. 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
```

---------------------------------------
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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```