Are we doing something wrong with our Google Analytics installation?
Your tracking code looks OK, providing that you've placed this before the closing
</head> tag as indicated here, which otherwise might result in lower numbers if the page or resources didn't fully load.
they claim our referral numbers for them are lower than theirs by 50%
In addition to the above, make sure that they are not referring traffic from different domains or subdomains that you're not tracking. I'd request all the URL's they're referring from so that you can confirm this, and their tracking reports too.
It might also be helpful to review this: Google Analytics - Troubleshooting the Tracking Code. And use the Debugging Tools listed there to make sure the requests for
__utm.gif are OK.
Also, compare your server logs with your Google Analytics reports to see if there's a variance (and check for errors).
They are tracking clicks in Google Analytics (using events) while we are using visits in Google Analytics.
Comparing events with visits can be like comparing apples with oranges. As indicated in Google Analytics here:
Events are user interactions with content that can be tracked
independently from a web page or a screen load. Downloads, mobile ad
clicks, gadgets, Flash elements, AJAX embedded elements, and video
plays are all examples of actions you might want to track as Events.
Clicks on their site might be higher because they're measuring user interaction, which would suggest that their page, tracking code, and resources were fully loaded. Visits on your site might be lower because these things didn't fully load (as covered above), users didn't wait for your site to load, there were problems reaching your site, etc... So it's best to keep these things in mind when comparing referral numbers between two sites.