Fair warning, this will be quite long, but it is packed with details that will help. I promise.
I sent in my resume and cover letter after hearing about the position on a posting board. I received a short 15 minute phone call with a recruiter from the Illinois offices. After the initial screening, she was my main contact person to set me up for the subsequent interviews. As you move through the interview process, expect a few calls and emails from her as she preps you can schedules your interviews.
I believe my third interview was with my would-be team leader, and it was pretty straightforward. The main thing they wanted to check was that I was interested in the research position and not the statistics position (since there were two positions available at the time and they needed a researcher). After that, there were a few behavioral questions that were asked relating to teamwork and strengths on a team.
[Side note: I had correctly applied to the statistics position, but somehow my application was filtered into the research position entries. They tried to convince me otherwise. The mix-up did work in my favor, but this might also say something about organization.]
My fourth interview was a skype call with two individuals working at the internship location. The first individual mainly covered my previous experiences and asked questions to get a better read on my personality. The second individual asked me more technical questions related to statistics, in case they wanted to use my skills in stats as a follow-up to the research work. The main technical question from that interview is below.
I did note that they made sure to call my listed references. Make sure your references are solid and can really vouch for your work, personality, and work ethic.
If all goes well, you'll be contacted by another recruiter from the main office who will be your new contact person to get you set up to start. This will usually cover salary negotiations, start date, and completion of any remaining paperwork / drug screening. Once this step is done, just get ready for your first day. Prepare for a great learning experience.
The entire process took about... 2 months from initial application to getting all set to start. Applied at the beginning of April. Didn't hear back until beginning of May (but I think I applied really early in their application acceptance period). Once the ball got rolling in May, it was an interview a week until acceptance and paperwork wrap up at the beginning of June.