In my previous analysis\(^1\), I used the number of undergraduates with a SAT Math score between 700-800 as the school size attribute. This was to correct for the stength of the student body in terms of math.

However, when I fit a binomial regression\(^2\) to the Sloan data using type of school (Liberal Arts, Private Research, Public Research) as a covariate, it was apparent that the number of undergraduates with a SAT Math score between 700-800 along with the school type did not fully explain the number of Sloan Fellows. Some schools produced an unusually high number of Sloan Fellows given their size and the outlier schools did not appear to be random.

1. Baccalaureate Origins of Sloan Fellows: Liberal Arts Colleges Versus Research Universities

2. The binomial regression used number Fellows as the number of ‘successes’ and the number students with SAT Math 700-800 as the number of ‘trials’. Here is the estimated Sloan production in Math, Physics and Econ per 1000 undergrads with SAT Math 700-800. The private research universities are highest, but that is due to the Ivies.

##     TypeL    TypePr    TypePu 
## 0.3231410 0.6749727 0.1361239

Outlier private research universities

This shows the predicted versus observed number of Sloan Fellows. I have labelled those schools that have twice as many fellows as predicted and are in the 95% quantile for number of fellows. So, these are schools with both more than expected fellows and a large number of fellows. The Ivies appear to have more Sloan Fellows than you would expect given the number of students with SAT Math 700-800.

Outlier public research universities

There are also outlier public universities.

Outlier Liberal Arts Colleges

For Liberal Arts colleges there appear to be outliers but actually the samples sizes (number of Sloan Fellows) is small (2-4) so it is hard to make conclusions from this at the school level.

Using fraction of students with SAT Math 700-800 as a covariate

My first idea was to try to use the fraction of students with SAT Math 700-800 as a covariate. The previous analysis\(^1\) showed that for math, physics and economics, schools with higher top SAT 75% quantiles tended to have higher per-capita production of Sloan Fellows. As always ‘capita’ is defined as the number of students with SAT Math 700-800. The fraction of students with SAT Math 700-800 and the number of students with SAT Math 700-800 are not very correlated because Size equals SAT Math 700-800 \(\times\) total undergraduate enrollment. Thus schools can have similar sizes (number students with SAT Math 700-800) and very different fractions of students with SAT Math 700-800.

My idea was that perhaps these outlier schools are simply explained by having a high density of students with SAT Math 700-800. The density of SAT Math 700-800 students might be correlated with something that produces many Sloan Fellows, e.g. the stature of the faculty, or level of mentorship of students, or perhaps extreme talent is a non-linear function of the fraction students in the 700-800 range. The latter is almost certainly true given the nature of statistical distributions.

1. Baccalaureate Origins of Sloan Fellows: Liberal Arts Colleges Versus Research Universities

Effect of SAT Math 700-800 on rate of production of Sloan Fellows

I fit a model with the effect of SAT Math 700-800 varying by type (Liberal Arts, Private, Public) and with the SAT Math 700-800 covariate entered as a 2nd order polynomial. The linear effect didn’t quite explain the pattern and a linear effect was too restrictive. This plot shows the model prediction as a function of the fraction of students with a SAT Math score between 700-800.

Effect of SAT Math 700-800 on rate of production of Sloan Fellows

For both Liberal Arts colleges and public research universities, the fraction of students with SAT Math scores between 700-800 is not predictive of the per-capita Sloan Fellow production. But for private research universities, there is much higher Sloan production in schools that have a high fraction of students with SAT Math 700-800. Nonetheless across most values on the x-axis, Liberal Arts colleges have the highest Sloan production per capita. The exception is in the highest class with SAT Math 700-800 greater than 0.7-0.75.

That’s interesting and we could speculate that perhaps there is something about the ‘density’ of math talent that drives this pattern—which for some reason is mainly seen in the private research universities. The problem with this is that the private school relationship is driven entirely by the Ivies (plus a few other elite schools). Harvard and Princeton, in particular, have a significant impact on the estimate of the effect of ‘SAT Math 700-800’ (as measured by the dfbetas leverage diagnostic).

Predicted SAT Math 700-800 effect versus data

This plot shows the predicted per-capita Sloan Fellow production from the model with ‘SAT Math 200-800’ as a covariate (2nd order polynomial).

Harvard and Princeton are extreme outliers for Math, Physics and Econ. But there generally appears to be an ‘Ivy’ effect. This could be a name-recognition effect. Schools with a world-wide name recognition may be able to draw in international talent to a greater degree than other schools with a similar fraction of students with SAT Math 700-800. Certainly Harvard and Princeton are strong draws for top undergraduate mathematics talent at the international level.

Note, I left out MIT and Caltech from this analysis. MIT is not an outlier on the left plot (x=0.94, y=2.4) but is an outlier for the right plot (x=0.94, y=6.6). Caltech is an outlier for both plots (left x=1, y=12.2; right x=1, y=10.2).

Using ‘reputation’ as a covariate

In progress.

I am going to use the 2016 academic prestige ranking by Times Higher Education\(^1\). This is a ranking of the top 100 schools in terms of the nebulous ‘academic prestige’. I will also try using their ranking for all U.S. schools based on more quantitative metrics.

1. Times Higher Education Prestige Ranking