Selecting genes that contribute the most to the principal components

Question

I have done a PCA analysis on genes expressed in cells under different stimulations, and retrieved the eigenvectors for a number of components.

My question is can I use these to determine which genes (or microarray probes) contribute to the principal components the most? Or is this an inappropriate use of the statistics?

If the answer is yes, is it the magnitude of the eigenvector's element or the sign ($+$ or $-$) of the element that is important?

Below is an example of eigenevectors output for a random selection of probes:

    probe     PC1            PC2            PC3           PC4            PC5
    7894603  -0.00437706    -0.011776456    0.0027118     0.002236574   -0.009450911
    7953873   0.018982512    0.013157616   -0.050475872   0.006077795    0.000618833
    7894231   0.00239363    -0.013645266    0.006057671  -0.007046099    0.006165455
    7895608   0.004984847    0.009435022   -0.00122576    0.009295082   -0.003496827
    7893924   0.004356782    0.002382756    0.002338561  -0.010994646   -0.015574907
    8095680   0.02053544     0.089269843    0.008331465  -0.017682479    0.002557484
    7893452   0.003530932   -0.000148751    0.001704203  -0.0001649     -0.00419296
    7961026   0.018171076    0.011570336   -0.04739463    0.005997931    0.000268158
    8099393  -0.007035288   -0.014426004    0.012909486  -0.024354002   -0.003115809
    7895836   0.007058346   -0.002625799    0.001665055   0.009596538   -0.004969979
    7895884   0.005918773   -0.001403533   -5.63E-06      0.000485206   -0.001941824
    7931681   0.00101047     0.002444639   -0.003501791  -0.007781566    0.003279817
    7896555  -0.001349106   -0.00271741    -0.002717482   0.002233903   -0.007859473
    8117018  -0.00053159     0.004344694    0.001928193  -0.020047103    0.001272397

Answer 1

Normally, an eigenanalysis will provide a set of unit vectors (eigenvectors) and a corresponding vector of values (eigenvalues). The unit vectors define orthogonal directions in the same feature space as before, but optimized to capture the maximum amount of the variance of the input data in the fewest number of vectors. The eigenvalues tell you how much of the original variance is captured in the direction of the corresponding eigenvector. That tells you how important each eigenvector is. Larger eigenvalue means more important. The components of the eigenvector tell you how each of the original dimension directions contribute to the eigenvector. The sign may be significant if the sign of the dimension is significant, but probably not. The absolute value is. In your PC1 above, the second and eighth probes had more influence on that eigenvector than the others.In PC4, dimensions 5,6,9, and 14 were heavier contributors. So the elements of the eigenvectors are weights on the original dimensions. Now you could do lots of things with this information. You could create a new representation of the data in a lower-dimensional space. You could decide some of these probes or some groups of probes are ineffective and throw them out. You could decide that certain groups of probes have key relationships and you should focus on them going forward. Good luck!