The PROBIT Procedure

Tolerance Distribution

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 $\mu $ and scale parameter $\sigma $ 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 subject’s tolerance is less than the dose or

\[ P = F \left( \frac{x - \mu }{\sigma } \right) \]

Thus, in this case, the intercept parameter, $b_0$, and the regression parameter, $b_1$, are related to $\mu $ and $\sigma $ by

\[ b_0 = -\frac{\mu }{\sigma }, b_1 = \frac{1}{\sigma } \]

Note: The parameter $\sigma $ is not equal to the standard deviation of the population of tolerances for the logistic and extreme value distributions.