The PROBIT Procedure

Model Specification

For a two-level response, the probability that the lesser response occurs is modeled by the probit equation as

\[ p = C + (1 - C) F(\mb{x}^{\prime }\mb{b}) \]

The probability of the other (complementary) event is 1 – p.

For a multilevel response with outcomes labeled $l_ i$ for $i=1,2,\ldots ,k$, the probability, $p_ j$, of observing level $l_ j$ is as follows:

\begin{eqnarray*} p_1 & = & C + (1 - C) F(\mb{x}^{\prime }\mb{b}) \\[0.05in] p_2 & = & (1 - C) \left( F(a_2 + {\mb{x}^{\prime }\mb{b}}) - F({\mb{x}^{\prime }\mb{b}}) \right) \\ & \vdots & \\ p_ j & = & (1 - C) \left( F(a_ j + {\mb{x}^{\prime }\mb{b}}) - F(a_{j-1} + {\mb{x}^{\prime }\mb{b}}) \right) \\ & \vdots & \\ p_ k & = & (1 - C) (1 - F(a_{k-1} + {\mb{x}^{\prime }\mb{b}}) ) \end{eqnarray*}

Thus, for a k-level response, there are k – 2 additional parameters, $a_2,a_3,\ldots ,a_{k-1}$, estimated. These parameters are denoted by Interceptj, $j=2,3,\ldots ,k-1$, in the output.

An intercept parameter is always added to the set of independent variables as the first term in the model unless the NOINT option is specified in the MODEL statement. If a classification variable taking on k levels is used as one of the independent variables, a set of k indicator variables is generated to model the effect of this variable. Because of the presence of the intercept term, there are at most k – 1 degrees of freedom for this effect in the model.