What test is used to determine communicability of an attribute of God I understand that some attributes of God are said to be communicable, meaning that they can, in some measure, be attributes of men as well. For example, God's holiness is said to be communicable because our holiness is derived from his, as in Lev 11:45 "Be holy, for I am holy." (ESV). God's aseity, on the other hand, could not be attributed to humans, as we are created by him.

Many other attributes, however, seem to be more of a gray area for me. For example, one might mark "knowledge" as communicable, or perhaps omniscience as incommunicable. How does one determine whether an attribute of God is communicable or incommunicable?
The attributes that are speaking of his deity are clearly not communicable.  You mention his aseity.  I also point to his eternal nature. 

However, the attributes that are bound to his personality are communicalbe.  For example, grace and love can be found in us.

We have to be careful though to note that while these attributes of God's can be transferred to us, it is actually God in us that brings about these attributes.

See also:  Attributes of God