This is ranking season. So I did a fun one; many of you will surely disagree. I used cross admit stats from Parchment. I also only focused on HYPSM because I wanted to finish it within a few minutes.
The raw data from Parchment about revealed preference for cross admits:
H 61% Y 39%
H 77% P 23%
H 56% S 44%
H 68% M 32%
Y 70% P 30%
Y 46% S 54%
Y 37% M 63%
P 27% S 73%
P 38% M 62%
S 67% M 33%
So the ranking scores (in terms of popularity) among HYPSM cross admits are:
Thanks, prof2dad. This is quite interesting. I think the Y to M comparison comes out somewhat strangely, because these two colleges best serve rather different student interests, whereas S can offer much the same as M. P is more like H and Y, and not so much like M. H and M really don’t overlap that much in interest served well, though there is cross-enrollment to make up for that.
A college that you did not include is Caltech. This is an interesting one, because the people who really want to go there will choose it over all other options. Hoxby et al threw Caltech out of their study, also based on revealed preferences, because it came out too high in the ranking for their tastes.
To mention another limitation of the revealed preferences methodology: A student who has a very strong preference for Y or P over H may not apply to H at all. (I know a few students in this group.). So the losses by H in this category are not picked up by the method. On the other hand, a student who has a very strong preference for H over Y or P may well apply to Y or P as a back-up choice (not to feed the “safety school” monster, just pointing out common practice). Then if the student gets into H, he/she will almost certainly go there, boosting H’s score.
@profdad Interesting analysis! Shouldn’t you though have a weighting factor to account for the number of kids among the cross admits? The pools with more cross admits should have more statistical relevance than pools with few cross admits. For instance, what if there were 1000 Y & H cross admits but only 10 Y & M cross admits? The former should be weighted more heavily than the latter.
It’s very interesting that P comes in dead last in your analysis (P loses badly to each of HYSM), but P has been at the top of the USNWR ranking for six years. I think that your ranking, which has H and S at the top, better reflects the sense of parents/kids regarding desirability.
@QuantMech @whatisyourquest Admittedly, this is a simple analysis, not sophisticated at all. Of course, it allows for deeper reading into raw data, as you have done.
Among all these raw numbers, I paid most attention to Y 37% M 63%. Yale is overall doing all right, so the loss to M this much (more than H and S) is interesting to me. I think there are two ways to look into this. The obvious one is M is so strong in STEM that Y is far less desirable for STEM cross admits with M. Another one is that Y really wants to nurture its STEM programs and self-selects deeper into this yield-lossing competition with M by admitting more of those traditionally M-STEM style students. Of course, this says something about how simple the ranking being provided it here.
The sample sizes for the cross admits between these five schools are too small.
Hey @prof2dad I’ve been thinking about how to weight with respect to the number of cross admits, and I think that I can do it. Would you please repost your table, and just add a column with the number of cross admits in each row?
For instance, H 61% Y 39% ***
where *** is the number of cross admits between H and Y
Thanks!
The Parchment info is the only public data of this kind available (apart from the Stanford faculty senate minutes that discuss Stanford cross-admits specifically). It seems difficult to draw robust conclusions from it, though, because, if I’m understanding the information on the Parchment site correctly, it looks like the respondents tend to come primarily from California and Michigan, which presumably distorts the results.
I would imagine, for example, that if a plurality of the respondents are from California, as seems to be the case, the comparisons are going to overstate the preferences for Stanford relative to the Ivies, given some assumed amount of home region preference by the respondents.
Also, I’m not sure how Parchment deals with, say, students who are admitted to Harvard, Stanford and Yale and choose Harvard. Presumably, this is a cross-admit win for Harvard over Stanford, but it’s also a Stanford-Yale cross-admit that ended in a tie. I don’t know if this is counted anywhere.
There’s a similar problem inherent in the Stanford faculty senate data; basically, it’s impossible to tell from the outside how much double counting there is among cross-admits because they may be cross-admitted in multiple places with Stanford.
Based on the available info, I’m going to guess that there are at most a couple of hundred cross-admitted students in any pair of the HYPSM schools (and probably considerably fewer in most cases). Each of these schools admits something like 2,000 kids a year, and has a yield roughly between 70%-85% (which means that they’re largely admitting different groups of kids).
It seems unlikely to me that any of these schools wins in cross-admits over any other by more than 80:20. So, really, the difference between the actual cross-admit result of any pair of schools and splitting their cross-admits 50:50 is at most something like 60 kids a year (30% * 200), or roughly 3% of those admitted by any of these schools.
If you had the actual numbers, you might conclude something about revealed preference from them, but I would agree with @hzhao2004 that the sample size is very small, and I believe the potential swing either way isn’t big enough to make a difference in the real world (although I’m sure it matters to these schools’ admissions offices).
I would also note that these numbers presumably don’t account for students who are admitted early to one of the schools, and then don’t pursue applications to other schools.
The situation with M is interesting. However, non-STEM students aren’t going to apply to M at all, so the cross-admit numbers there are probably only relevant to STEM students. I think it’s odd that H would beat M in STEM cross admits, but that Princeton wouldn’t do as well.
What is the time-frame of the Parchment data, i.e. from which year to which year? The website doesn’t seem to disclose.
But what you did is interesting!
@QuantMech Interesting your thoughts on P and M. My D thought P was more like M by far than HYS. She didn’t apply to HYS but MP and Chicago were her top 3. For someone looking to earn a PhD and do research she thought those 3 were better.
Anecdotally she met several HYP cross admits at the local P accepted students event and she said they were all basically interested in Econ/Finance and she was guessing most would chose H.
Here’s another way to look at it:
Let’s say Harvard admits 2,000 kids, all of whom have last names that begin with one of the first 20 letters of the alphabet. The 2,000 kids comprise 20 groups of 100 who share a letter (i.e., there are 100 kids whose last names start with “A”, 100 with “B”, etc.).
Let’s assume all the "A"s and "B"s are also admitted to Princeton, all the "B"s and "C"s to Stanford and all the "C"s and "D"s to Yale.
This means that Harvard has 400 total cross-admits with Princeton, Stanford and Yale. If Harvard wins on average 60% of cross-admits with the other three schools, it gets 240 and loses 160 out of the 400. In other words, the net margin of victory is 80/3, or about 27 students per school on average. That’s less than one entryway in one dorm in Harvard Yard.
But the magnitude of the preference is even smaller than that, because the appropriate way to measure it is against a 50:50 split, or an outcome with no revealed preference. Looked at this way, the preference is a net 40 students - 13 per school on average, or the equivalent of two to three freshman suites.
I think the number of cross-admits is much larger than that, honestly. Every year, there are something like 400 kids admitted to Harvard who don’t enroll. All but a handful of those constitute the other side of Harvard’s percentage of cross-admit success with Stanford, Yale, Princeton, and MIT. The average of those who go to each of the other colleges will be around 100. Given the relative percentages, that’s likely to mean significantly more than 100 at Stanford and Yale.
Why all the guessing about sample size? Does Parchment not disclose the number of cross admits, only the percentages?
What is surprising to me is that someone with a screename of “prof” would even suggest that the self-reported anecdotes on Parchment rise to the level of “raw data”.
It is raw data. It just may not be good raw data 
@prof2dad The funny thing to me is that you are using very questionable data, and still got a better answer than US News!
“@prof2dad The funny thing to me is that you are using very questionable data, and still got a better answer than US News!”
Or just use the yield to admit ratio.
2020 YTAR gives you S slightly ahead of H. Then Y, P, Col, M, Chicago, Penn.
So simple!
Or even simpler – just yield:
S, H, M, Y, P, Penn, Col, Chi.
Could be, @JHS - although the Stanford faculty senate data from a recent year showed around 200 or fewer Stanford cross-admits with each of HYP, as I recall. What I could imagine, though, is that there could be rather more than 200 cross-admits among each pair of the HYP group. A lot of top-tier East Coast kids won’t apply to Stanford, I think, or at least are much more likely to apply to any of HYP than to Stanford, and I’m sure some top-tier West Coast kids don’t apply to any of HYP. So there might be a lot more cross-admits among the HYP group than with any of them and Stanford.
@northwesty - I think comparing yield (or its derivative, YTAR) among schools with ED (particularly UChicago, the most egregious user) and schools with EA is apples and oranges. ED will produce a 95+% yield, EA at HYPS (in my estimation) 85-90%.
Given that there are only about 400 kids anywhere who turn down a Harvard acceptance, and that Stanford and Harvard come very close to splitting their cross-admits, I think 200 Stanford cross-admits is consistent. It would be really interesting to know how many kids who applied to both were accepted by one but not the other.
In my kids’ high school classes, all combined, there were 9 kids admitted to either Harvard or Stanford. Six of the 9 had applied to both, of whom 4 were admitted to both. They split two to Harvard and two to Stanford. One kid accepted at Harvard was turned down by Stanford, and one admitted by Stanford was turned down by Harvard. None of the 6 kids admitted to Harvard went anywhere other than Stanford. One of the 7 kids admitted to Stanford chose Brown instead.