The PLM Procedure

Scoring Data Sets for Zero-Inflated Models

The PLM procedure can score new observations for zero-inflated models with the SCORE statement. If you specify the ILINK option, the computed statistics are for estimated counts.

In the following formula, $\mb {x}$ is the design row for covariates that correspond to the the Poisson or negative binomial component, $\hat{\bbeta }$ is the column vector of the fitted regression parameters; $\mb {z}$ is the design row for covariates that correspond to the zero inflation component, $\hat{\bgamma }$ is the column vector of the fitted regression parameters; $g$ and $g^{-1}$ are the link and inverse link functions for the Poisson or negative binomial component, $g_ z$ and $g^{-1}_ z$ are the link and inverse link functions for the zero inflation component; $\Phi$ is the standard normal cumulative distribution function and $\alpha$ is the nominal significance level. Let

$\begin{array}{lclcl} \mu & = & g^{-1}(\eta ) & = & g^{-1}(\mb {x}\hat{\bbeta }) \\ \mu _ z & = & g^{-1}_ z(\eta _ z) & = & g^{-1}_ z(\mb {z}\hat{\bgamma }) \\ v_1 & = & (\frac{\mathrm{d}\mu }{\mathrm{d}\eta })^2 \mb {x}\hat{\mb {V}}_{\bbeta ,\bbeta }\mb {x}’\\ v_2 & = & (\frac{\mathrm{d}\mu _ z}{\mathrm{d}\eta _ z})^2 \mb {z}\hat{\mb {V}}_{\bgamma ,\bgamma }\mb {z}’\\ v_{12} & = & -(\frac{\mathrm{d}\mu }{\mathrm{d}\eta }) (\frac{\mathrm{d}\mu _ z}{\mathrm{d}\eta _ z}) \mb {x}\hat{\mb {V}}_{\bbeta ,\bgamma }\mb {z}’\\ \end{array}$

The formula for statistics in the SCORE statement for zero-inflated models are listed as follows.

$\begin{array}{lclcl} \mr {PZERO} & = & \omega & = & \mu _ z \\ \mr {PRED/ILINK} & = & p_ c & = & \mu (1-\omega )\\ \mr {STD/ILINK} & = & s_ c & = & \sqrt {p_ c^2v_2+(1-\omega )^2v_1+ v_1v_2+2p_ c(1-\omega )v_{12}+v^2_{12}} \\ \mr {UCLM/ILINK} & = & u_ c & = & p_ c\exp (\Phi ^{-1}(1-\alpha /2) s_ c/p_ c) \\ \mr {LCLM/ILINK} & = & l_ c & = & p_ c/\exp (\Phi ^{-1}(1-\alpha /2) s_ c/p_ c) \\ \mr {PRED} & = & p_ l & = & g(p_ c) \\ \mr {STD} & = & s_ l & = & g(s_ c)/(\frac{\mathrm{d}\mu }{\mathrm{d}\eta }) \\ \mr {UCLM} & = & u_ l & = & g(u_ c) \\ \mr {LCLM} & = & l_ l & = & g(l_ c) \\ \end{array}$