Stimulus–Response Studies.

The effects of a drug or other stimulus are often studied by choosing r doses of the drug (levels of the stimulus), say x1,...,xr, and giving the dose xj to each of Nj subjects. The data consist of the number yj who exhibit some predetermined response. Often this response is the death of the subject, but it can be any measure of the effectiveness of the stimulus.

Typically in dose-response studies, interest centers on the median effective dose, the ED(50), or if the response is death, the median lethal dose, the LD(50). The LD(50) is that dose for which the probability is 0.5 that a subject will die. Frequently, a model of the form log pj

(1 − pj )

!

= α + β log(xj )

is fitted to such data. Assume this model holds for any and all doses.

(a) Is the sampling scheme appropriate for a logistic regression?

(b) How could you estimate the LD(50)?

Exercises 11.8.2 and 11.8.3 give additional results on inference for the LD(50).

Suppose that for each individual in the population there is a minimum dose x to which the individual will respond. (The individual is assumed to respond to all doses larger than x.) Let the random variable X be this minimum susceptibility for an individual chosen at random.

(c) Give an estimate of the median of X.

(d) Give an estimate of the 90th percentile of the distribution of X.

(e) What is the distribution of X?