There is going to be some literal answer to that question that can be determined by looking at the underlying data. I gather another poster is doing that.
To the extent you want a deeper explanation of WHY they have, say, certain relative peer reputations, I think you need to look at a lot of history.
Over really long periods, the relative prominence of different institutions has changed. Indeed, go back far enough, and some institutions didn’t exist yet. And many went through multiple phases.
So like Duke actually started as Brown’s Schoolhouse, and was a combined effort of Methodists and Quakers. After several name changes it became an actual college, and ended up named Trinity. I think it is fair to say it was not necessarily a very prominent institution in the Trinity days.
Starting around 1887, it began to “modernize” itself, which at the time meant adopting a more Germanic model as opposed to the Classical model. “Modern” institutions had already been taking off thanks to the Land Grant movement, so in some sense it was “late” to getting going on this.
The thing is, though, a lot of Gilded Age families not in the Northeast were basically of a mind to create new rivals to colleges like Harvard and Yale in their own backyards. In this case, through a series of gifts, the Duke and Carr families in North Carolina put up the money to move Trinity to Durham, and fund its modernization. Eventually Trinity was actually renamed Duke.
It was basically up and up for Duke from that point forward (more or less), and by, say, 1990, it was firmly established as what was sometimes known as an Ivy+. Meaning they had achieved their goal of creating a rival to the Northeastern colleges.
OK, so Northwestern, Vanderbilt, and Rice actually all have broadly similar stories, and really broadly similar outcomes. Minor rankings differences simply are not that meaningful. Northwestern was founded and developed by Chicago businessmen, Vanderbilt by Cornelius, and Rice–well, by someone named Rice, and if you don’t know that story, you should check it out. It is bonkers.
So why then the (minor) differences in prominence? Well, in a case like Vanderbilt, say, there was sort of this long-standing competition to be the Harvard of the South. And originally, that was actually Princeton, in the sense it was the most favored college of Southern elites for a time. But again by the late 19th century, Southern elites wanted Southern elite colleges actually in the South.
Vanderbilt was actually founded by Cornelius before the Duke and Carr families started transforming Trinity, and I think for a while, it was by no means certain that Vanderbilt was going to lose out to Duke for the Harvard of the South competition. But for a variety of reasons it eventually did, and so Vanderbilt kinda got stuck “behind” Duke.
Interestingly, Northwestern in some sense also got stuck “behind” the University of Chicago. But Chicago was in fact America’s “Second City” for quite a while, and Northwestern leveraged that into a pretty high standing notwithstanding sharing its turf with Chicago.
Rice–actually, Rice in many respects has long just been doing its own thing, really well. Texas is just different to begin with, and I think Texas elites have largely been fine with the role Rice has played. And like all these schools, Rice “nationalized” in appeal, but you still have to be OK with going to Texas (as a student, professor, or so on). And some people are, some people aren’t, but Rice is doing fine being Rice.
That’s actually glossing over a ton of details, but you get the idea. All of these colleges are really huge success stories, and a testament to how much of what leads to a high US News ranking basically comes down to funding. Indeed, you could toss in, say, WashU, which basically is the St Louis version of this story, Johns Hopkins, the Baltimore version, Carnegie-Mellon, the Pittsburgh version, Emory, which has a sort of Duke-like history where it was Coca-Cola money that transformed it . . .
. . . and of course the biggest ultimate success story of them all, Leland Stanford Junior’s university.
Again, you can draw fine distinction between these colleges, but to me the bigger story is how they all succeeded in their own ways in their own areas to provide a worthy alternative to having to send elite kids to the Northeast for college.
. But they should consider groupings of 5 or 10 all of the way down the list.