Generalized Logit Model

For nominal response, a generalized logit model is to fit the ratio of the expected proportion for each response category over the expected proportion of a reference category with a logit link function.

Without loss of generality, let category $D+1$ be the reference category for the response variable Y. Denote the expected proportion for the dth category by $\pi _{hijd}$ as in the section Notation. Then the generalized logit model can be written as

\[  \log \left(\frac{\pi _{hijd}}{\pi _{hij(D+1)}}\right) =\mb {x}_{hij}\bbeta _ d  \]

for $d=1, 2, \ldots , D,$ with the model parameters

\begin{eqnarray*}  \bbeta _ d &  = &  (\beta _{d1}, \beta _{d2}, \ldots , \beta _{dk})’\\ \btheta &  = &  ( \bbeta _1’, \bbeta _2’, \ldots , \bbeta _ D’)’ \end{eqnarray*}