As I noted in a different comment here, you can pretty easily fix Graham's test. Compute min(accepted a) and min(accepted B) instead of the means. In your example, the min of the accepted distributions would both work out to be 80%.
Another reason the use of mins here is not helpful, is that adding one equally awful accepted candidate to group A and B would then remove whatever bias there was according to the test, which is not what we want the test to indicate.
The idea by PG is a rough rule of thumb and breaks down trivially - suppose VC fund X were to accept all candidates, but group A was worse than B, the test would falsely imply that the fund was biased.
It's unfortunate the idea was dressed up in statistical persiflage because it isn't rigorous -- it's a rough guideline. To make it rigorous wold be very hard: either the abilities of the candidate populations would have to be measured very closely (unrealistic), or a more scientific experiment conducted (A-B test where candidates from each group are included or excluded opposite to the prior decision, which would need big groups).
Mins will fail if you conspire to cheat the test, it's true. Very few statistical tests stand up to conspiracy theories.
Dude, your comments are normally smarter than this. Yeah, you can easily fix Grahams's test -- all you need are some numbers that do not exist and that we cannot measure.
We're talking about VC's evaluating founders. That does not, and cannot, get reduced to a numerical score. And even if VC's did use some sort of scoring rubric, then we would still not know if there was unfairness in the way they made the scores, or unfairness in the selection process. It would just be punting the problem down a layer. PG's central claim -- that a third-party can detect the bias/unfairness in the funding process just using math -- is false.
You can only know if the process is biased/unfair if you have deep qualitative understanding of the process.
That is fine, that is what he was saying. The point is that his solution is completely impractical for the original goal of finding an objective, statistically valid way of measuring whether bias exists. "Borderline" cannot be measured objectively, only by subjective rubric scoring. And when you only measure the borderline candidates, you have reduced an already way-to-small sample even further.
I made no claims about practicality - right now all I have is a little bit of measure theory showing that pg's algo is, in principle, fixable. I fully agree that the first round capital data he cites is inadequate (and also wrong, due to the unjustified exclusion of uber, which they explicitly note would alter the results).
My concrete claim: PGs idea for a statistical test is solid, I can (and shortly will) prove a toy version works, and given enough work one can probably cook up a practical version for some problems.
"Your idea isn't 100% perfect right out of the gate" is a very unfair criticism. Are we supposed to nurture every idea in complete secrecy until it is perfect?
With statistics on human affairs, 99% of the hard part is not the math, it is applying that math to a complicated, heterogenous, and difficult to measure underlying phenomena. And in most cases, statistics alone will never give you a straight answer, the best they can do is supplement and confirm qualitative observations. Failing to recognize this is how you get all those unending media reports about how X is bad for your health. PG's post was at the level of one of those junk health news articles.
This idea that statistics can only confirm and supplement "qualitative observations" (I.e. my priors) is completely unscientific and anti-intellectual. If that's true, forget stats - lets just write down the one permitted belief on a piece of paper and not waste resources on science. Science is really boring when only one answer is possible.
Since when is investing in startups a science? What is anti-intellectual, what is anti-science is to use the wrong tool for the job. Human affairs are not a science in the way that physics is a science. Statistics are far, far more fraught because there are so many variables in play, phenomena are hard to quantify, each case is so heterogenous, etc. You cannot use statistics in human affairs without also having a very good observational understanding of what is actually going on, otherwise you will end up in all sorts of trouble.
Suppose the cutoff sample is distributed according to f(x)H(x-C). Then the probability of the minima of a sample exceeding C+e by random chance, assuming the null hypothesis, is p = (1-\int_C^{C+e}f(x) dx)^N.
So now you have a frequentist hypothesis test. If you make reasonable assumptions on f(x) (non-vanishing near C, quantified somehow), it's even nice and non-parametric.
I.e., for any d, there is a finite probability of finding an A or a B in [C,C+d]. I don't actually care what the shapes of f or g are at all beyond this - as long as this probability exists and is bounded below (in whatever class of functions f and g might be drawn from), it's all fine.
I am assuming we know exactly one thing about the class the measures f and g come from: for every function in that class, \int_C^{C+d} f(x) dx >= h(d) for some monotonic function h(d).
The p-value is then computed in terms of h(d), since p >= h(d)^N.