n vs n-1 in variance estimation

[brsr]

Received by e-mail in February from Dr. Jerry Lewis:

Variance of m is calculated by noting that within each dose, the number of responders, X, is distributed binomial(n,P), where P is a monotonic function of dose. But, as noted in all editions of Finney (Statistical Method in Biological Assay), while p = X/n is unbiased for P, E[ p_(1-p)/n ] < P_(1-P)/n. You need p_(1-p)/(n-1) for unbiased estimation of P_(1-P)/n, which Finney used in his worked Karber example (omitted from 3rd ed). Without this modification, the Karber intervals seem too narrow when compared to the 50% horizontal slice through intervals from generalized linear models.

The original goal of this package was to replicate a program that used $n$ in the variance estimate, so that's what the current state is. It would be reasonable to add an option to use $n-1$ instead, but I can't determine what needs to be changed in the variance calculation from Dr Lewis's email alone, since there are multiple different $n$s that appear.