The PROBIT Procedure

Distributions

The distributions, upper F left-parenthesis x right-parenthesis, allowed in the PROBIT procedure are specified with the DISTRIBUTION= option in the MODEL statement. The cumulative distribution functions for the available distributions are

Cumulative Distribution Function Distribution
integral Subscript negative normal infinity Superscript x Baseline StartFraction 1 Over StartRoot 2 pi EndRoot EndFraction exp left-parenthesis minus StartFraction z squared Over 2 EndFraction right-parenthesis d z Normal
StartFraction 1 Over 1 plus e Superscript negative x Baseline EndFraction Logistic
1 minus e Superscript minus e Super Superscript x Extreme value or Gompertz

The variances of these three distributions are not all equal to 1, and their means are not all equal to zero. Their means and variances are shown in the following table, where gamma is the Euler constant.

Distribution Mean Variance
Normal 0    1
Logistic 0    pi squared slash 3
Extreme value or Gompertz negative gamma    pi squared slash 6

When comparing parameter estimates by using different distributions, you need to take into account the different scalings and, for the extreme value (or Gompertz) distribution, a possible shift in location. For example, if the fitted probabilities are in the neighborhood of 0.1 to 0.9, then the parameter estimates from the logistic model should be about pi slash StartRoot 3 EndRoot larger than the estimates from the probit model.

Last updated: December 09, 2022