I have a slightly different approach....
If there weren't any performance issues, I would use in every page more than 1 tracking scripts in parallel to compare they result and to be able to have the most accurate information as possibile about the visitors of my website.
I have already used together (not on a q2a website) google analytics + StatCounter (a service that i like very much and I prefer to google analytics - less features but more simple and immediate to use - but you have to pay in case you have a lot of hits or you want to mantain you historical data).
A server-side only approach (Apache logs analysis) that I have tried is AWStats, but I did not go on with it since it required too many work to setup and mantain (not easy to configure, batch to schedule in production, so not real time results).
But of course everything depends from your own requirements...
About the data that I analyze I give also a lot of importance to the browsers/system statistics, so to have an idea about which type of testing to do, to the visitor path / traffic sources, to have an idea of the goodness of my "marketing" efforts, and of course to the keywords used, to know if my seo efforts are on the right way or not
Speaking about q2a, I've just installed and tested this plugin
that seems to work, but is specific for google analytics and not general.