The SURVEYLOGISTIC Procedure

Cumulative Response Models

For a row vector of explanatory variables $\mb {x}$ , the linear predictor

$\eta _ i= g(\mbox{Pr}(Y\leq i~ |~ \mb {x})) = \alpha _ i+\mb {x}\bbeta , \quad 1 \leq i \leq k$

is estimated by

$\hat{\eta }_ i=\hat{\alpha }_ i+\mb {x}\hat{\bbeta }$

where $\hat{\alpha }_ i$ and $\hat{\bbeta }$ are the MLEs of $\alpha _ i$ and $\bbeta$ . The estimated standard error of ${\eta }_ i$ is $\hat{\sigma }({\hat{\eta }}_ i)$ , which can be computed as the square root of the quadratic form $(1, {\mb {x}}^\prime ){\hat{\mb {V}}_\mb {b}}(1, \mb {x}^\prime )^\prime$ , where $\hat{\mb {V}}_\mb {b}$ is the estimated covariance matrix of the parameter estimates. The asymptotic $100(1-\alpha )\%$ confidence interval for ${\eta }_{i}$ is given by

$\hat{\eta }_ i\pm z_{\alpha /2}\hat{\sigma }({\hat{\eta }}_ i)$

where $z_{\alpha /2}$ is the $100(1-\alpha /2)$ percentile point of a standard normal distribution.

The predicted value and the $100(1-\alpha )\%$ confidence limits for Pr $(Y\leq i~ |~ \mb {x})$ are obtained by back-transforming the corresponding measures for the linear predictor.

Link	Predicted Probability	$100(1-\alpha )$ Confidence Limits
LOGIT	$1/(1+e^{-\hat{\eta }_ i})$	$1/(1+e^{-\hat{\eta }_ i \pm z_{\alpha /2}\hat{\sigma }({\hat{\eta }}_ i)})$
PROBIT	$\Phi (\hat{\eta }_ i)$	$\Phi (\hat{\eta }_ i \pm z_{\alpha /2}\hat{\sigma }({\hat{\eta }}_ i))$
CLOGLOG	$1-e^{-e^{\hat{\eta }_ i}}$	$1-e^{-e^{\hat{\eta }_ i\pm z_{\alpha /2}\hat{\sigma }({\hat{\eta }}_ i)}}$