Threshold value to determine if the correlation between original variables and given principal component is significant When doing a Principal Component Analysis, how do I choose if the correlation between the original variables and the given principal component is significant? I read that you just look for the highest absolute coefficient values, but at which point are these values considered high or low. I want to justify why I think the correlations are significant when I interpret the PCs in my master thesis. I could not find anything in the literature about a threshold value.
 A: The correlations you refer to are used to determine which variables are important for interpreting a PC. While "significance" is not quite the right concept, "relative importance" is more easily addressed.  You can easily compare which individual variables have more importance in a PC, as well as which pairs of variables (sums or differences) by using a variable pairs heat map.
The following graph shows the variable pairs heatmap for interpreting the first baseline
PC using the epi.bfi data example from the psych package in R. The diagonal shows absolute correlations between the individual variables and the PC; the upper triangle shows absolute correlations between the differences $Z_i − Z_j$ and the PC; the lower triangle shows absolute correlations between the sums $Z_i + Z_j$ and the PC.
In the figure, you can see that even though $PC_1$ is not highly correlated with the variable bdi, which measures depression, it is very highly correlated with the bdi+
traitanx summate (r > 0.90), traitanx measuring anxiety.  Thus, the first PC measures something that is closely related to a "depression with anxiety" scale, where people who are depressed with high anxiety
are at one end of the scale, while not depressed people with
low anxiety are at the opposite end.

Source: Westfall PH, Arias AL, Fulton LV. Teaching Principal Components Using Correlations. Multivariate Behav Res. 2017;52(5):648-660. doi:10.1080/00273171.2017.1340824
