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I have following example of principal component analysis using first 4 variables of iris data set (code in R):

> res = prcomp(iris[1:4])
> res
Standard deviations:
[1] 2.0562689 0.4926162 0.2796596 0.1543862

Rotation:
                     PC1         PC2         PC3        PC4
Sepal.Length  0.36138659 -0.65658877  0.58202985  0.3154872
Sepal.Width  -0.08452251 -0.73016143 -0.59791083 -0.3197231
Petal.Length  0.85667061  0.17337266 -0.07623608 -0.4798390
Petal.Width   0.35828920  0.07548102 -0.54583143  0.7536574

It appears that Sepal.Width has a very small contribution to PC1. How do I know if it is a significant contribution?

Is there any significance test for this? Similarly, I want to determine significance for all values in above matrix.

Also, is there any package in R that does it?

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This is not (yet) and answer, only a comment but too long for the box


I do not really know how to determine such significance; but out of couriosity I did a bootstrap-procedure: from a replication of the original data to a pseudo-population of $N=19200$ I draw $t=1000$ randomsamples of $n=150$ (each row of the dataset could occur at most $128$ times).
From each of this $t=1000$ experiments I computed the pca-solutions and stored the first pc only in a list. From this 1000 instances of first pc's I got the following statistics for their loadings:

PrC[1]:  Mean      Min       Max    Stddev  SE_mean  lb(95%)     mean  ub(95%) 
------------------------------------------------------------------------------
   S.L    0.362    0.314    0.412    0.015    0.000    0.361    0.362    0.362
   S.W   -0.085   -0.131   -0.023    0.017    0.001   -0.086   -0.085   -0.083
   P.L    0.856    0.841    0.869    0.004    0.000    0.856    0.856    0.857
   P.W    0.358    0.334    0.382    0.008    0.000    0.358    0.358    0.359

The 95% confidence interval for the item S.Width was -0.085 .. - 0.083 and this shows that this value seems to be from zero not by the pure random-effect of the sampling. (Similarly narrow appear all 95% confidence intervals for the other loadings)
After that it's clear I need more clarification what it means for a loading to "contribute significantly" - significance derived from what expectance? (But that's what I do not yet understand, I'm competely illiterate yet with the question of significance-estimation for covariances and for loadings in a factormodel, so this all might be of no help at all here)

[Update 2]
Here is a picture which shows the location of the Iris-items in the coordinates of the first 2 principal components, evaluated by the Monte-Carlo-experiment ("population": $N=128 \cdot 150=19200$, "sample": $n=150$, number-of-samples: $s=1000$)

Picture 1: (using covariance-matrix, loadings from eigenvectors as done in the OP's question)
picture1
From the picture I'd say, that the small loading of Sepal.Width of -0.141 on pc1 is a reliable (different from zero, however small) estimate of the loading in the "population" (because the whole cloud is separated from the y-axis)

Using the standard interpretation of PCA (based on correlations, using scaled eigenvectors) the picture looks a bit different, but still with very little disturbances of the loadings of the items.
The statistics are as in the following:

PrC[1]  Mean      Min       Max    Stddev  SE_mean  lb(95%)     mean  ub(95%) 
------------------------------------------------------------------------------
   S.L    0.891    0.840    0.937    0.015    0.000    0.890    0.891    0.892
   S.W   -0.459   -0.705   -0.159    0.081    0.003   -0.465   -0.459   -0.454
   P.L    0.991    0.987    0.994    0.001    0.000    0.991    0.991    0.991
   P.W    0.965    0.946    0.980    0.005    0.000    0.965    0.965    0.965

Picture 2: (using correlation-matrix, principal components taken in the standard method)
picture2

[Update 1] Just for my own couriosity I made a set of plots of the empirical loadings-matrices when samples are drawn from a known population. That's somehow bootstrapping, and I've not yet seen similar images. I took as population a set of 1000 normal random distributed cases with a certain factorial structure. Then I draw 256 random samples from the population with n=40 and did the same components-analysis/rotation for each of that 256 samples. To compare and to see, how the accuracy of the estimation improves I took the same number of samples, but now each sample with n=160. See the comparision at http://go.helms-net.de/stat/sse/StabilityofPC

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  • $\begingroup$ Thanks for your answer. What exact code did you use for above bootstrap procedure? Did you encounter sign issue, since loadings sometimes come with reverse signs on repeated sampling? $\endgroup$ – rnso Jun 26 '15 at 13:30
  • $\begingroup$ I concatened the original dataset several times until N=19200 was reached. Then by a randomgenerator which was restricted to the range 1..19200 (with guaranteed frequency of 1 for each random value if called less than 19200 times) I simply took the first 150 numbers and used them as index into the population-dataset. This made one sample. Evaluate, store the first prcomp and repeat until 1000 samples and thus 1000 first prcomps were collected. Then did simple descriptive statistics on the list. Sign-issue? Hmm, I don't know what you mean? $\endgroup$ – Gottfried Helms Jun 26 '15 at 14:38
  • $\begingroup$ See stackoverflow.com/questions/31057192/… $\endgroup$ – rnso Jun 26 '15 at 15:15
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    $\begingroup$ @rnso: What if not just the signs flip but also the axes of the PCA when choosing different bootstrap samples? (Imagine e.g. the case where the original variables are quite orthogonal). I am asking of pure curiosity. $\endgroup$ – Michael M Apr 18 '16 at 14:12
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    $\begingroup$ The figures you added seem to be very close to the ones I produced answering your follow-up question. That's good. $\endgroup$ – amoeba Apr 18 '16 at 14:17

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