The SURVEYMEANS Procedure

Domain Statistics

When you use a DOMAIN statement to request a domain analysis, the procedure computes the requested statistics for each domain.

For a domain D, let be the corresponding indicator variable:

$I_{D}(h,i,j) = \left\{ \begin{array}{ll} 1 & \mbox{if observation $(h,i,j)$ belongs to \Mathtext{D}} \\ 0 & \mbox{otherwise} \end{array} \right.$

Let

$v_{hij}= w_{hij}I_ D(h,i,j) = \left\{ \begin{array}{ll} w_{hij} & \mbox{if observation $(h,i,j)$ belongs to \Mathtext{D}} \\ 0 & \mbox{otherwise} \end{array} \right.$

The requested statistics for variable y in domain D are computed by using the new weights v.

Domain Mean

The estimated mean of y in the domain D is

$\widehat{\bar{Y}_ D} = \left( \sum _{h=1}^ H\sum _{i=1}^{n_ h} \sum _{j=1}^{m_{hi}} ~ v_{hij} ~ y_{hij} \right) / ~ v_{\cdot \cdot \cdot }$

where

$v_{\cdot \cdot \cdot } = \sum _{h=1}^ H\sum _{i=1}^{n_ h} \sum _{j=1}^{m_{hi}} v_{hij}$

The variance of $\widehat{\bar{Y}_ D}$ is estimated by

$\widehat{V}(\widehat{\bar{Y}_ D}) = \sum _{h=1}^ H \widehat{V_ h}(\widehat{\bar{Y}_ D})$

where, if , then

$\displaystyle \widehat{V_ h}(\widehat{\bar{Y}_ D})$	$\displaystyle =$	$\displaystyle \frac{{n}_ h(1-f_ h)}{{n}_ h-1} ~ \sum _{i=1}^{n_ h} {(r_{hi\cdot }-\bar{r}_{h\cdot \cdot })^2}$
$\displaystyle r_{hi\cdot }$	$\displaystyle =$	$\displaystyle \left( \sum _{j=1}^{m_{hi}}v_{hij}~ (y_{hij}- \widehat{\bar{Y}_ D}) \right) / ~ v_{\cdot \cdot \cdot }$
$\displaystyle \bar{r}_{h\cdot \cdot }$	$\displaystyle =$	$\displaystyle \left( \sum _{i=1}^{n_ h}r_{hi\cdot } \right) / ~ {n}_ h$

and if , then

$\widehat{V_ h}(\widehat{\bar{Y}_ D}) = \left\{ \begin{array}{ll} \mbox{missing} & \mbox{ if } n_{h}=1 \mbox{ for } h’=1, 2, \ldots , H \\ 0 & \mbox{ if } n_{h}>1 \mbox{ for some } 1 \le h’ \le H \end{array} \right.$

Domain Total

The estimated total in domain D is

$\widehat{Y}_ D = \sum _{h=1}^ H\sum _{i=1}^{n_ h} \sum _{j=1}^{m_{hi}} ~ v_{hij} ~ y_{hij}$

and its estimated variance is

$\widehat{V}(\widehat{Y}_ D) = \sum _{h=1}^ H \widehat{V_ h}(\widehat{Y}_ D)$

where, if , then

$\displaystyle \widehat{V_ h}(\widehat{Y}_ D)$	$\displaystyle =$	$\displaystyle { \frac{{n}_ h(1-f_ h)}{{n}_ h-1} \sum _{i=1}^{n_ h} {(z_{hi\cdot }-\bar{z}_{h\cdot \cdot })^2}}$
$\displaystyle z_{hi\cdot }$	$\displaystyle =$	$\displaystyle \sum _{j=1}^{m_{hi}} ~ v_{hij} ~ z_{hij}$
$\displaystyle \bar{z}_{h\cdot \cdot }$	$\displaystyle =$	$\displaystyle \left( \sum _{i=1}^{n_ h}z_{hi\cdot } \right) ~ / ~ {n}_ h$

and if , then

$\widehat{V_ h}(\widehat{Y}_ D) = \left\{ \begin{array}{ll} \mbox{missing} & \mbox{ if } n_{h}=1 \mbox{ for } h’=1, 2, \ldots , H \\ 0 & \mbox{ if } n_{h}>1 \mbox{ for some } 1 \le h’ \le H \end{array} \right.$

Domain Ratio

The estimated ratio of Y to X in domain D is

$\widehat{R}_ D = \frac{ \sum _{h=1}^ H\sum _{i=1}^{n_ h} \sum _{j=1}^{m_{hi}} ~ v_{hij} ~ y_{hij} }{ \sum _{h=1}^ H\sum _{i=1}^{n_ h} \sum _{j=1}^{m_{hi}} ~ v_{hij} ~ x_{hij} }$

and its estimated variance is

$\widehat{V}(\widehat{R}_ D) = \sum _{h=1}^ H \widehat{V_ h}(\widehat{R}_ D)$

where, if , then

$\displaystyle \widehat{V_ h}(\widehat{R}_ D)$	$\displaystyle =$	$\displaystyle \frac{n_ h(1-f_ h)}{n_ h-1} ~ \sum _{i=1}^{n_ h} {(g_{hi\cdot }-\bar{g}_{h\cdot \cdot })^2}$
$\displaystyle g_{hi\cdot }$	$\displaystyle =$	$\displaystyle \frac{\sum _{j=1}^{m_{hi}}v_{hij}~ (y_{hij}- x_{hij}\widehat{R}_ D) }{\sum _{h=1}^ H\sum _{i=1}^{n_ h} \sum _{j=1}^{m_{hi}} ~ v_{hij} ~ x_{hij}}$
$\displaystyle \bar{g}_{h\cdot \cdot }$	$\displaystyle =$	$\displaystyle \left( \sum _{i=1}^{n_ h}g_{hi\cdot } \right) / ~ n_ h$

and if , then

$\widehat{V_ h}(\widehat{R}_ D) = \left\{ \begin{array}{ll} \mbox{missing} & \mbox{ if } n_{h}=1 \mbox{ for } h’=1, 2, \ldots , H \\ 0 & \mbox{ if } n_{h}>1 \mbox{ for some } 1 \le h’ \le H \end{array} \right.$