jackherbert - 12:17 am on May 12, 2013 (gmt 0)
I was feeling the same way about using google analytics (uncomfortable), since that turns your domain name into a cookie carrier for google. I switched to Piwik [piwik.org] and never looked back. It is more work to setup but the analytic data is yours, not google's.
The switch didn't make any difference to the site's search ranking (at least, I didn't notice any).
Google's line "We wouldn't be so popular if we didn't give fair, un-biased search results" might have been true ten years ago, but I don't think it still holds. A few months ago, when typing the letter 's' in google.co.uk, the first auto-complete suggestion was "Sainsbury's" (a british supermarket chain). I mean, come on! Who on earth would believe that it's the most searched word starting by the letter 's'. Obviously it's 'sax', or maybe 'saxophone' :)