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The same happened to our site. But also Google Analytics shows quite different results : bounce rate, time on site, keywords. All have dramatically changed, so maybe it has more to do with the Google Analytics set up that with real more visitors.
Edit: If like you say, GA metrics do affect serps, removing GA could have a negative impact.
But my return visitors are way, way up. WAY up. As in, over 90% of visits are return now. And the visits are from all over, not just one area. This cannot be accurate. Or if it is, that means...no, it can't be accurate. Yikes!
Google Analytics Blog
Overall, this change may slightly increase the number of visits. Based on our research, most users will see less than a 1% change.