The SURVEYLOGISTIC Procedure

Generalized Logit Model

For a vector of explanatory variables $\mb {x}$, let $\pi _ i$ denote the probability of obtaining the response value i:

\[  \pi _ i = \left\{  \begin{array}{ll} \pi _{k+1} {e}^{\alpha _ i+\mb {x}\bbeta _ i} &  1\le i\le k \\ \displaystyle \frac{1}{1+\sum _{j=1}^{k} {e}^{\alpha _ j+\mb {x} {\bbeta }_ j}} &  i=k+1 \end{array} \right.  \]

By the delta method,

\[  \sigma ^2({\pi }_ i) = \biggl ( \frac{\partial \pi _ i}{\partial \btheta } \biggr )’ \bV ({\btheta }) \frac{\partial \pi _ i}{\partial \btheta }  \]

A 100(1$-\alpha $)% confidence level for $\pi _ i$ is given by

\[  \hat{\pi }_ i \pm z_{\alpha /2} \hat{\sigma }(\hat{\pi }_ i)  \]

where $\hat{\pi }_ i$ is the estimated expected probability of response i and $\hat{\sigma }(\hat{\pi }_ i)$ is obtained by evaluating $\sigma ({\pi }_ i)$ at $\btheta =\hat{\btheta }$.