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Dear Brenton,

I got two questions here:

First of all, in tsk::tsk(), you smooth the data (of course only when the data do not show monotonically increasing trend) before doing the Abbott's correction of the control response (when control response is not 0). Shouldn't the Abbott's correction be done before smoothing?

In tsk::tsk(), smoothing using gpava() will sometimes result in a smoothed control p different from the original control p.

# Correct for the control dose.
  data.smoothed$p <- (data.smoothed$p-control)/(1-control)

Then you correct the smoothed data with the original control p based on Abbott's formula. It will not normalize the smoothed corrected control p to 0, likely leading to biases in the results.

Secondly, in the ecotoxicology::TSK(), after Abbott's correction, sample size (n) for all groups is also corrected to n_i_-r[1], where ni is the sample size of each test group, r[1] is the number of original control response. I am not sure if it is correct to do this sample size correction, but would like to hear your opinion. Thank you very much!

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.

According to the correction, the 0,5,10,20% trimmed TSK CI for dataset 1c in Table II is (39.14, 44.49), (39.86, 45.38), (39.51, 46.34) and (40.77, 45.17). However, the tsk package give results as following, (39.19, 44.44), (39.91, 45.32), (39.57, 46.26), and (40.81, 45.12). They are all very close but not identical or with negligible differences. Is there an explanation for this?