We currently get web analytics for a WordPress site using WebTrends.
If we use a caching mechanism like Varnish, I would assume WebTrends would suddenly report a dramatic reduction in traffic.
Is this correct and, if so, can you avoid this problem and get the correct statistics reported by WebTrends?
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Just make sure that your varnish config is not removing any webtrends cookies and it should be percetly OK. By default it does not but if you use some ready-made wordpress vcl then it might be you will need to exclude these cookies together with the wordpress-specific ones in the configuration.
In my experience, acceleration caches shouldn't interfere with your Analytics data capture, because the cached content should include all of the on-page data points (such as meta tags) as well as the WT base tag file, which the user's browser will then execute and which will then make the call to the WT data collection server.
By way of a disclaimer, I should add that I haven't got any specific experience with Varnish, but a cache that acts as a barrier to on-page JavaScript executing is basically broken, and I've personally never had a problem with one preventing analytics software from running.
The only conceivable problem I could foresee is if a cache was going to the extent of scanning pages for linked resources (such as the "no javascript" image in the noscript tag), acquiring those resources in advance, and then reconfiguring the page being served to pull those resources from the cache rather than the third party servers. In which case you might end up with spurious "no javascript" records in your data.