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 $\btheta =\hat{\btheta }$ .

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