Do you know what kinds of keywords are leading to less organic traffic? Are they keywords for your brand name or down in the Long Tail?
My guess is that your organic traffic is, in fact, going down, but not because your rank is going down. Rather, it's because of multi-touch attribution. If you're advertising less (esp. less in the Long Tail), then you'll receive less organic traffic for keyword searches involving your brand name.
This is because a new visitor will often vist your site more than once - the first time, they'll use a PPC click and the second time, they'lls earch on your brand name (because now they know it from the earlier PPC visit). But if you reduce your PPC spend, then you can't get as much multi-session traffic. Make sense?
There's a clear relationship here. See this presentation on Slideshare for an example of how my company tested this relationship in a few different ways. You'll see the kinds of results we ot and then can formulate your own tests.