For a single independent variable, such as a dosage level, the models for the probabilities can be justified on the basis of a population with mean
and scale parameter
of tolerances for the subjects. Then, given a dose x, the probability,
, of observing a response in a particular subject is the probability that the subject’s tolerance is less than the dose or
Thus, in this case, the intercept parameter,
, and the regression parameter,
, are related to
and
by
Note:The parameter
is not equal to the standard deviation of the population of tolerances for the logistic and extreme value distributions.
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