Independence vs. Independency

Personally I would almost always use independence where many historical writers might have used independency. One is example is the "American Declaration of Independency" reported in The Gentleman's Magazine in London in August 1776.

The example where I might use Independency is as a form of church government compared with Episcopacy and Presbyterianism, but even then I might use Congregationalism instead.

I don't think I have such problems with using dependency or interdependency.

This is more worthy of a comment (I don't have enough reputation), but in the math / statistics community, independency gets an inordinate amount of use.

On their own, Google shows that the "Independence" keyword is used ~500x more frequently than its archaic counterpart. However, when you add the keywords "Math" and "Statistics", it is only used ~50x more frequently. In fact, nearly 50% of the references to "Independency" also contain "Math" and "Statistics".

In this context, we often see independence used as a possessed quality, and independency used independently of possession:

• We can verify the linear independence of x and y.
• Every independency contributes to the sparsity of the problem.

Even in this field, they are often used interchangeably.