The HPREG Procedure

Diagnostic Statistics

This section gathers the formulas for the statistics available in the OUTPUT statement. All the statistics available in the OUTPUT statement are conditional on the selected model and do not take into account the variability introduced by doing model selection.

The model to be fit is $\mb{Y}=\mb{X}\bbeta + \bepsilon $, and the parameter estimate is denoted by $\mb{b}=(\mb{X'}\mb{X)^{-}}\mb{X'}\mb{Y}$. The subscript i denotes values for the ith observation, and the parenthetical subscript $(i)$ means that the statistic is computed by using all observations except the ith observation.

The ALPHA= option in the PROC HPREG statement is used to set the $\alpha $ value for the confidence limit statistics.

Table 15.6 contains the diagnostic statistics and their formulas. Each statistic is computed for each observation.

Table 15.6: Formulas and Definitions for Diagnostic Statistics

$\begin{array}{c}\mbox{MODEL Option} \\ \mbox{or Statistic}\end{array}$

Formula

PRED ($\displaystyle \widehat{\mb{Y}}_ i$)

$\mb{X}_ i\mb{b}$

RES ($r_ i$)

$\mb{Y}_ i - \widehat{\mb{Y}}_ i$

H ($h_ i$)

$\mb{x}_ i(\mb{X'}\mb{X})^{-}\mb{x}_ i’$

STDP

$\sqrt {h_ i\widehat{\sigma }^2}$

STDI

$\sqrt {(1+h_ i)\widehat{\sigma }^2}$

STDR

$\sqrt {(1-h_ i)\widehat{\sigma }^2}$

LCL

$\displaystyle \widehat{Y}_ i-t_{\frac{\alpha }{2}}$STDI

LCLM

$\displaystyle \widehat{Y}_ i-t_{\frac{\alpha }{2}}$STDP

UCL

$\displaystyle \widehat{Y}_ i+t_{\frac{\alpha }{2}}$STDI

UCLM

$\displaystyle \widehat{Y}_ i+t_{\frac{\alpha }{2}}$STDP

STUDENT

$\displaystyle \frac{r_ i}{\mbox{STDR}_ i}$

RSTUDENT

$\displaystyle \frac{r_ i}{{\hat{\sigma }}_{(i)}\sqrt {1-h_ i}}$

COOKD

$\displaystyle \frac{1}{p}\mbox{STUDENT}^2\frac{\mbox{STDP}^2}{\mbox{STDR}^2}$

COVRATIO

$\displaystyle \frac{\mbox{det}({\hat{\sigma }^2}_{(i)}(\mb{x}_{(i)}'\mb{x}_{(i)})^{-1}}{\mbox{det}({\hat{\sigma }^2}(\mb{X}'\mb{X})^{-1})}$

DFFITS

$\displaystyle \frac{(\widehat{\mb{Y}}_ i-\widehat{\mb{Y}}_{(i)})}{({\hat{\sigma }}_{(i)}\sqrt {h_ i})}$

PRESS($\mbox{predr}_ i$)

$\displaystyle \frac{r_ i}{1-h_ i}$