I am excited to have been accepted to a few PhD programs in public health. The two top contenders have both offered me full tuition remission, a stipend, and health insurance, but there is a difference in the stipend offers of about 4.5K between the two (with the lower amount being a bit more than $27K). The program with higher stipend offer (which is perhaps a bit less well-known in the field) is also applying on my behalf for an additional university scholarship to augment my stipend. In conversation with my prospective adviser at the program with lower compensation I was told that the stipend amount was limited due to federal regulations, but I am wondering how I have been offered a higher amount at the other program, which is also funded through an NIH grant. I will be speaking with the adviser about whether there's any wiggle room in the compensation, but wanted to get perspectives on 1) what factors might underlie this difference, 2) what I might ask for to offset this difference should I want to accept the lower paying offer, and 3) what discretion the university and adviser may have to help me out.
Don't forget to consider cost-of-living. $27k sounds like the standard NIH PhD stipend. Institutions can supplement NIH stipend levels if they want, but may or may not have the funds to do so. In high-cost areas, they may feel it's necessary to offer more to everyone because it's not possible to live near the institution at the NIH rate. That said, an extra $4500 a year probably isn't going to cover the cost difference between living in, say, NYC versus Champaign, IL, and the smaller number could actually be the more comfortable living.
Another thing to look at is any fees you are required to pay: ordinarily if you're on an NIH-funded stipend you're going to have tuition remission, but there may be separate fees that graduate students still need to pay out of their stipend. When I was a student these were in the range of $500 per semester, but vary by institution.
There's rarely room for negotiation beyond what is offered, though I've known some programs who have a little "slush" money they can use to woo candidates they are especially interested in. Any substantial money available would likely be used in other ways, though: admitting another student, for example, or to fund students near graduation when other funding falls out from under them. I don't think it's bad to ask, especially if it will contribute directly to your decision making process, but it's also not a place where you want to use the negotiation strategies advertised for people taking jobs in industry.
Just to add a bit more detail, my institution sets stipend bands for graduate students, with higher rates for "experienced" students, in our case, ones who are on PhD candidate status. My department negotiated to pay all students the highest rate that the university will allow them.
You might look into the specific university's pay regulations to see if there are university/college rules that will constrain your PI. If they are already maxed out, there's not much your PI can do.
Academia likes to build in layers and layers so everyone has someone else to blame.