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, P, of observing a response in a particular subject is the probability that the subjectâ€™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.