Principal component analysis with the scale of original data

[Elien Hendrickx]

Hi,

I conducted a survey in which I asked 9 questions, each ranging from a scale from 1 to 7.
I then conducted a PCA in Stata in order to reduce these 9 questions to 3 components:

pca Vraag4contr Vraag5contr Vraag6contr Vraag10comp Vraag11comp Vraag12comp Vraag4DTT Vraag5DTT Vraag6DTT

Principal components/correlation Number of obs = 294
Number of comp. = 9
Trace = 9
Rotation: (unrotated = principal) Rho = 1.0000

--------------------------------------------------------------------------
Component | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Comp1 | 3.03713 .969488 0.3375 0.3375
Comp2 | 2.06764 .958044 0.2297 0.5672
Comp3 | 1.1096 .473579 0.1233 0.6905
Comp4 | .636017 .0811268 0.0707 0.7612
Comp5 | .55489 .0484921 0.0617 0.8228
Comp6 | .506398 .127643 0.0563 0.8791
Comp7 | .378756 .00147694 0.0421 0.9212
Comp8 | .377279 .0449838 0.0419 0.9631
Comp9 | .332295 . 0.0369 1.0000
--------------------------------------------------------------------------

Principal components (eigenvectors)

----------------------------------------------------------------------------------------------------------------------
Variable | Comp1 Comp2 Comp3 Comp4 Comp5 Comp6 Comp7 Comp8 Comp9 | Unexplained
-------------+------------------------------------------------------------------------------------------+-------------
Vraag4contr | 0.3808 -0.0640 -0.5154 -0.3803 0.1380 0.1632 0.2388 0.3757 0.4436 | 0
Vraag5contr | 0.4395 -0.0282 -0.3784 -0.1669 0.0883 0.1366 -0.1332 -0.6362 -0.4314 | 0
Vraag6contr | 0.4135 -0.0745 -0.1991 0.5100 0.0114 -0.6957 -0.1486 0.0940 0.0930 | 0
Vraag10comp | 0.3706 0.0149 0.5657 -0.2592 -0.0387 -0.2465 0.4714 -0.3485 0.2632 | 0
Vraag11comp | 0.4113 -0.0417 0.4413 -0.2787 0.2342 0.0481 -0.4903 0.4285 -0.2747 | 0
Vraag12comp | 0.4136 -0.0570 0.1517 0.5575 -0.3276 0.5961 0.1430 0.0916 0.0189 | 0
Vraag4DTT | 0.0729 0.5911 -0.0083 -0.1059 -0.3631 0.0445 -0.5153 -0.1808 0.4494 | 0
Vraag5DTT | 0.0871 0.5857 -0.1057 -0.1110 -0.3246 -0.1729 0.3728 0.3090 -0.5060 | 0
Vraag6DTT | 0.0161 0.5402 0.0498 0.2972 0.7564 0.1427 0.1247 -0.0607 0.0733 | 0
----------------------------------------------------------------------------------------------------------------------

. pca Vraag4contr Vraag5contr Vraag6contr Vraag10comp Vraag11comp Vraag12comp Vraag4DTT Vraag5DTT Vraag6DTT, mineigen(1)

Principal components/correlation Number of obs = 294
Number of comp. = 3
Trace = 9
Rotation: (unrotated = principal) Rho = 0.6905

--------------------------------------------------------------------------
Component | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Comp1 | 3.03713 .969488 0.3375 0.3375
Comp2 | 2.06764 .958044 0.2297 0.5672
Comp3 | 1.1096 .473579 0.1233 0.6905
Comp4 | .636017 .0811268 0.0707 0.7612
Comp5 | .55489 .0484921 0.0617 0.8228
Comp6 | .506398 .127643 0.0563 0.8791
Comp7 | .378756 .00147694 0.0421 0.9212
Comp8 | .377279 .0449838 0.0419 0.9631
Comp9 | .332295 . 0.0369 1.0000
--------------------------------------------------------------------------

Principal components (eigenvectors)

----------------------------------------------------------
Variable | Comp1 Comp2 Comp3 | Unexplained
-------------+------------------------------+-------------
Vraag4contr | 0.3808 -0.0640 -0.5154 | .2563
Vraag5contr | 0.4395 -0.0282 -0.3784 | .2527
Vraag6contr | 0.4135 -0.0745 -0.1991 | .4252
Vraag10comp | 0.3706 0.0149 0.5657 | .2273
Vraag11comp | 0.4113 -0.0417 0.4413 | .2664
Vraag12comp | 0.4136 -0.0570 0.1517 | .4482
Vraag4DTT | 0.0729 0.5911 -0.0083 | .2613
Vraag5DTT | 0.0871 0.5857 -0.1057 | .2552
Vraag6DTT | 0.0161 0.5402 0.0498 | .393
----------------------------------------------------------

. pca Vraag4contr Vraag5contr Vraag6contr Vraag10comp Vraag11comp Vraag12comp Vraag4DTT Vraag5DTT Vraag6DTT, mineigen(1) blanks(0.3)

Principal components/correlation Number of obs = 294
Number of comp. = 3
Trace = 9
Rotation: (unrotated = principal) Rho = 0.6905

--------------------------------------------------------------------------
Component | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Comp1 | 3.03713 .969488 0.3375 0.3375
Comp2 | 2.06764 .958044 0.2297 0.5672
Comp3 | 1.1096 .473579 0.1233 0.6905
Comp4 | .636017 .0811268 0.0707 0.7612
Comp5 | .55489 .0484921 0.0617 0.8228
Comp6 | .506398 .127643 0.0563 0.8791
Comp7 | .378756 .00147694 0.0421 0.9212
Comp8 | .377279 .0449838 0.0419 0.9631
Comp9 | .332295 . 0.0369 1.0000
--------------------------------------------------------------------------

Principal components (eigenvectors) (blanks are abs(loading)<.3)

----------------------------------------------------------
Variable | Comp1 Comp2 Comp3 | Unexplained
-------------+------------------------------+-------------
Vraag4contr | 0.3808 -0.5154 | .2563
Vraag5contr | 0.4395 -0.3784 | .2527
Vraag6contr | 0.4135 | .4252
Vraag10comp | 0.3706 0.5657 | .2273
Vraag11comp | 0.4113 0.4413 | .2664
Vraag12comp | 0.4136 | .4482
Vraag4DTT | 0.5911 | .2613
Vraag5DTT | 0.5857 | .2552
Vraag6DTT | 0.5402 | .393
----------------------------------------------------------

. rotate, promax(3) oblique

Principal components/correlation Number of obs = 294
Number of comp. = 3
Trace = 9
Rotation: oblique promax (Kaiser off) Rho = 0.6905

--------------------------------------------------------------------------
Component | Variance Proportion Rotated comp. are correlated
-------------+------------------------------------------------------------
Comp1 | 2.09808 0.2331
Comp2 | 2.0778 0.2309
Comp3 | 1.90081 0.2112
--------------------------------------------------------------------------

Rotated components

----------------------------------------------------------
Variable | Comp1 Comp2 Comp3 | Unexplained
-------------+------------------------------+-------------
Vraag4contr | 0.6396 0.0031 -0.1564 | .2563
Vraag5contr | 0.5813 0.0383 -0.0135 | .2527
Vraag6contr | 0.4407 -0.0193 0.1103 | .4252
Vraag10comp | -0.1375 0.0272 0.6749 | .2273
Vraag11comp | -0.0156 -0.0184 0.6055 | .2664
Vraag12comp | 0.1917 -0.0191 0.3824 | .4482
Vraag4DTT | 0.0031 0.5954 0.0263 | .2613
Vraag5DTT | 0.0823 0.5963 -0.0402 | .2552
Vraag6DTT | -0.0735 0.5358 0.0365 | .393
----------------------------------------------------------

Component rotation matrix

--------------------------------------------
| Comp1 Comp2 Comp3
-------------+------------------------------
Comp1 | 0.7094 0.1086 0.6365
Comp2 | -0.0922 0.9931 -0.0231
Comp3 | -0.7053 -0.0491 0.7767
--------------------------------------------

. rotate, promax(3) oblique blanks(0.3)

Principal components/correlation Number of obs = 294
Number of comp. = 3
Trace = 9
Rotation: oblique promax (Kaiser off) Rho = 0.6905

--------------------------------------------------------------------------
Component | Variance Proportion Rotated comp. are correlated
-------------+------------------------------------------------------------
Comp1 | 2.09808 0.2331
Comp2 | 2.0778 0.2309
Comp3 | 1.90081 0.2112
--------------------------------------------------------------------------

Rotated components (blanks are abs(loading)<.3)

----------------------------------------------------------
Variable | Comp1 Comp2 Comp3 | Unexplained
-------------+------------------------------+-------------
Vraag4contr | 0.6396 | .2563
Vraag5contr | 0.5813 | .2527
Vraag6contr | 0.4407 | .4252
Vraag10comp | 0.6749 | .2273
Vraag11comp | 0.6055 | .2664
Vraag12comp | 0.3824 | .4482
Vraag4DTT | 0.5954 | .2613
Vraag5DTT | 0.5963 | .2552
Vraag6DTT | 0.5358 | .393
----------------------------------------------------------

Component rotation matrix

--------------------------------------------
| Comp1 Comp2 Comp3
-------------+------------------------------
Comp1 | 0.7094 0.1086 0.6365
Comp2 | -0.0922 0.9931 -0.0231
Comp3 | -0.7053 -0.0491 0.7767
--------------------------------------------

. estat loadings

Principal component loadings
component normalization: sum of squares(column) = 1

--------------------------------------------
| Comp1 Comp2 Comp3
-------------+------------------------------
Vraag4contr | .3808 -.06403 -.5154
Vraag5contr | .4395 -.02825 -.3784
Vraag6contr | .4135 -.07454 -.1991
Vraag10comp | .3706 .01487 .5657
Vraag11comp | .4113 -.04167 .4413
Vraag12comp | .4136 -.05698 .1517
Vraag4DTT | .07295 .5911 -.008317
Vraag5DTT | .08708 .5857 -.1057
Vraag6DTT | .0161 .5402 .04983
--------------------------------------------

. predict pc1 pc2 pc3, score

Scoring coefficients for oblique promax(3) rotation
sum of squares(column-loading) = 1

--------------------------------------------
Variable | Comp1 Comp2 Comp3
-------------+------------------------------
Vraag4contr | 0.6249 -0.0087 -0.0967
Vraag5contr | 0.5792 0.0262 0.0405
Vraag6contr | 0.4514 -0.0297 0.1517
Vraag10comp | -0.0750 0.0234 0.6618
Vraag11comp | 0.0414 -0.0241 0.6043
Vraag12comp | 0.2278 -0.0269 0.4005
Vraag4DTT | -0.0068 0.5951 0.0207
Vraag5DTT | 0.0661 0.5950 -0.0384
Vraag6DTT | -0.0813 0.5370 0.0243
--------------------------------------------

. estat kmo

Kaiser-Meyer-Olkin measure of sampling adequacy

-----------------------
Variable | kmo
-------------+---------
Vraag4contr | 0.7312
Vraag5contr | 0.7639
Vraag6contr | 0.8521
Vraag10comp | 0.7179
Vraag11comp | 0.7477
Vraag12comp | 0.8600
Vraag4DTT | 0.6505
Vraag5DTT | 0.6439
Vraag6DTT | 0.7584
-------------+---------
Overall | 0.7475
-----------------------

As you can see, I reduced the 9 questions I had to 3 components.
However, Stata has standardized my data, while I wanted to retain the scale from 1 to 7.
I wonder what I have to do in order to keep the scale of my original questions.

[Nick Cox]

You aren't reducing them to 3 components; you are producing 9 components and selecting 3, which is not the same.

Scaling to mean zero is what PCA does as a matter of convention. Nothing stops you rescaling PCs to any mean and SD you want, but that is outside the PCA. It is hard to see any real advantage in it.

Note that there might be point in using the covariance option here, as all variables are on the same scale. That won't itself do what you want either.

[Dirk Enzmann]

Your question indicates that you may not be aware of the assumption of quasi-continuous variables when using PCA (or factor analysis) although you actually seem to have a set of 7-point Likert items. This in itself need not be a problem (though I expect that W. Buchanan will dissent; see, for example, http://www.statalist.org/forums/foru...the-scale-mean ).

It may well be that your really want to "reduce the set of variables to three components" (but see the comment of Nick), in that case using PCA is o.k. (although I wonder why you choose mineigen to be 1 - components(3) would have been the more appropriate option - and why you want to allow for correlated components by choosing an oblique rotation).

In many cases, however, PCA is used although the researcher actually wants to employ exploratory factor analysis (EFA) as a method to identify unobservable latent variables. In that case I would recommend to read the article by Preacher and McCallum (2003) (reference below). They recommend

1. using factor analysis (e.g. factor with the option ipf) instead of a PCA;
2. to determine the number of factors by using the parallel analysis criterium instead of a minimum eigenvalue of 1 (in your case a two factor solution would perhaps be more appropriate);
3. followed by an oblique rotation method (e.g. promax) allowing for correlated factors instead of an orthogonal rotation (e.g. varimax).

If you consider using fapara (see findit fapara) to determine the number of factors, I recommend that to run a PCA first to determine the number of factors using fapara followed by a "true" factor analysis to determine factor loadings (and factor scores).

Reference:
Preacher, K. J. & McCallum, R. C. (2003). Repairing Tom Swift's electric factor analysis machine. Understanding Statistics, 2, 13-43.