It's likely that the two of you are talking across each other because you read slightly different articles. Paul added an assumption to his article, possibly after William read it, which is intended to rule out his posited distribution: "(c) the groups of applicants you're comparing have roughly equal distribution of ability".
This strikes me as a "heroic assumption", but it's true that if you make it most of the flaws in his argument go away. Add in the unspoken assumption that the groups are both are large enough that sampling variation does not matter, and I think he's probably logically correct.
On the other hand, once you make these assumptions, the rest of his argument seems unnecessary, since all you need to know is the ratio of males and females funded. If male and female founders are exchangeable, the process is biased if one group is funded more often than they are represented in the applicants.
You don't even need to look at outcome, since we've already assumed the founders are of equal ability. I think that Paul is aiming at the case where we don't know the ratio of applicants. I think his argument can be useful in this case, but only if you have already accepted his assumptions.
This strikes me as a "heroic assumption", but it's true that if you make it most of the flaws in his argument go away. Add in the unspoken assumption that the groups are both are large enough that sampling variation does not matter, and I think he's probably logically correct.
On the other hand, once you make these assumptions, the rest of his argument seems unnecessary, since all you need to know is the ratio of males and females funded. If male and female founders are exchangeable, the process is biased if one group is funded more often than they are represented in the applicants.
You don't even need to look at outcome, since we've already assumed the founders are of equal ability. I think that Paul is aiming at the case where we don't know the ratio of applicants. I think his argument can be useful in this case, but only if you have already accepted his assumptions.