The SURVEYLOGISTIC Procedure

A complementary log-log model uses the complementary log-log function

$g(t)=\log (-\log (1-t))$

as the link function. 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 complementary log-log model can be written as

$\log (-\log (1-F_{hijd}))=\alpha _ d+\mb {x}_{hij}\bbeta$

with the model parameters

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