The SURVEYLOGISTIC Procedure

Complementary Log-Log Model

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

\begin{eqnarray*}  \bbeta &  = &  (\beta _1, \beta _2, \ldots , \beta _ k)’\\ \balpha & =&  (\alpha _1, \alpha _2, \ldots , \alpha _ D)’, \,  \,  \,  \alpha _1<\alpha _2<\cdots <\alpha _ D \\ \btheta &  = &  (\balpha ’,\bbeta ’)’ \end{eqnarray*}