The SURVEYLOGISTIC Procedure

A probit model uses the probit (or normit) function, which is the inverse of the cumulative standard normal distribution function,

$g(t)=\Phi ^{-1}(t)$

as the link function, where

$\Phi (t)=\frac{1}{\sqrt {2\pi }}\int _{-\infty }^ t e^{-\frac{1}{2} z^2} dz$

Denote the cumulative sum of the expected proportions for the first d categories of variable Y by

$F_{hijd}=\sum _{r=1}^ d \pi _{hijr}$

for $d=1, 2, \ldots , D.$ Then the probit model can be written as

$F_{hijd}=\Phi (\alpha _ d+\mb {x}_{hij}\bbeta )$

with the model parameters

$\displaystyle \bbeta$	$\displaystyle =$	$\displaystyle (\beta _1, \beta _2, \ldots , \beta _ k)’$
$\displaystyle \balpha$	$\displaystyle =$	$\displaystyle (\alpha _1, \alpha _2, \ldots , \alpha _ D)’, \, \, \, \alpha _1<\alpha _2<\cdots <\alpha _ D$
$\displaystyle \btheta$	$\displaystyle =$	$\displaystyle (\balpha ’,\bbeta ’)’$