Hi.
I ran a PCA with 5 variables, and it seems that I should retain only one PC, which accounts for 70% of the variation. The PC2 eigenvalue is 0.9.
I was wondering:
1- if it makes any sense to use varimax rotation in this particular case retaining only one PC
2- in case I retained two PC, should I rotate the whole loadings matrix (with the five PC) or just those I retain?
Thanks!
David
It is supposed that in PCA usually we have 5-6 eigenvalues adding to unity and our problem is to identify and name the factors that add more, so in your case I cannot understand why you should run PCA at all.
Thanks Demetris,
Well, I just want to get one principal component that may explain most of the variabiliy in the 5 original variables, which are correlated. Does it sound ok?
Thanks!
Thanks Paul, great contribution and references. So I guess rotation is suitable also if only one component is retained, right?
Dear David, it makes no sense to rotate one component. The aim of PCA is to find subspaces explaining the most of original space variability. If you have several components (spanning a multidimensional space), you can find by rotation another set of dimensions spanning the same space and accounting for the same explained variability. If hou have just one dimension spanning a unidimensional space), you already have your varimax solution.
Dear Divide
It seems that you need to more variables for performing to PCA analysis.
Thanks Paul. Could you please provide the reference to the article you mentioned? The link you provided goes directly to your researchgate publications site, and not to a particular paper. Interesting comments!
Thanks!
The fist principal is not necessary the best predictor among all PCs for the outcome, you may not use the 1st PC as the main predictor. The following perform the analysis for Hald’s data (1998), you can see the last PC is most significant, none of other 3 are good predictor:
Inputting the data on p. 279.
data p278;
input u c1 c2 c3 c4 @@;
cards;
.955 1.467 1.903 -.53 .0389 -.746 2.136 .238 -.29 -.03
-2.323 -1.13 .184 -.01 -.094 -.82 .66 1.577 .179 -.033
.471 -.359 .484 -.74 .019 -.299 -.967 .17 .086 -.012
.21 -.931 -2.135 -.173 .008 .558 2.232 -.692 .46 .023
-1.119 .352 -1.432 -.032 -.045 .496 -1.663 1.828 .851 .02
.781 1.641 -1.295 .494 .031 .918 -1.693 -.392 -.02 .037
.918 -1.746 -.438 -.275 .037
run;
Table 10.12-10.13, p. 279.
proc reg data = p278;
model u = C1-C4/ noint;
model u = C1-C3/ noint;
run;
quit;
Parameter Estimates
Variable
DF
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
c1
1
-0.00180
0.00125
-1.44
0.1837
c2
1
-0.00255
0.00149
-1.71
0.1206
c3
1
0.00165
0.00433
0.38
0.7122
c4
1
24.76590
0.04630
534.90
The following are the model fit and estimation by use first 3 PCs
Root MSE
1.09533
R-Square
0.0000
Dependent Mean
5.12411E-17
Adj R-Sq
-0.3000
Coeff Var
2.137605E18
Parameter Estimates
Variable
DF
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
c1
1
-0.00122
0.21144
-0.01
0.9955
c2
1
-0.00012426
0.25186
-0.00
0.9996
c3
1
0.00252
0.73202
0.00
0.9973
Dear David,
I don't exactly know what you are using this analysis for, but using only 1 PC for describing a system might be misleading. For example if it is a calibration using varimax rotation and multilinear fit (PCR) hels you building more robust model than having everything (you errors as well) in 1 PC:
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