Principal Component Analysis: whether a variable is significantly loaded on a principal component or not? Often, a variable is considered to be significantly loaded on a PC if its loading value in the loading table is above a cut off value (suppose 0.4 or 0.5 in some published cases). Is there any statistical/mathematical method to check whether a variable is significantly loaded on a particular PC or not?
 A: Significance tests are not merely indicators of "how strong" or "how much."  Significance tests serve to contrast a result with the sort of results that typically occur through some known random process.  How large would a group difference be if the groups were merely assigned at random...How large would the correlation between two variables be if in the population those variables were unrelated to one another, and if their relationship in a sample were thus purely the result of random error.  
Now consider your situation:  variables load on some principal component that is formed through a particular procedure -- moreover, one that, as @ttnphns and @jank have said, explicitly "seeks" to form a component that summarizes individual variables.  There is no random process with which to contrast this, no process that might be otherwise occurring that would furnish us with a null hypothesis. This is further reason why it's not meaningful to test the statistical significance of a loading on a PC.
