Variance and Standard Error of the Mean

When you specify the keyword STDERR, the procedure computes the standard error of the mean. Also, the procedure computes the standard error by default if you specify the keyword MEAN, or if you do not specify any statistic-keywords in the PROC SURVEYMEANS statement. The keyword VAR requests the variance of the mean.

Taylor Series Method

When you use VARMETHOD=TAYLOR, or by default if you do not specify the VARMETHOD= option, PROC SURVEYMEANS uses the Taylor series method to estimate the variance of the mean $\widehat{\bar{Y}}$. The procedure computes the estimated variance as

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

where, if $n_ h>1$, then

\begin{eqnarray*}  \widehat{V_ h}(\widehat{\bar{Y}}) &  = &  \frac{n_ h(1-f_ h)}{n_ h-1} ~  \sum _{i=1}^{n_ h} {(e_{hi\cdot }-\bar{e}_{h\cdot \cdot })^2} \\ e_{hi\cdot }& =&  \left( \sum _{j=1}^{m_{hi}}w_{hij}~ (y_{hij}- \widehat{\bar{Y}}) \right) / ~  w_{\cdot \cdot \cdot } \\ \bar{e}_{h\cdot \cdot } & =&  \left( \sum _{i=1}^{n_ h}e_{hi\cdot } \right) / ~  n_ h \end{eqnarray*}

and if $n_ h=1$, then

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

When you specify VARMETHOD=BRR or VARMETHOD=JACKKNIFE, the procedure computes the variance $\widehat{V}(\widehat{{\bar{Y}}})$ with replication methods by using the variability among replicate estimates to estimate the overall variance. See the section Replication Methods for Variance Estimation for more details.

Standard Error

The standard error of the mean is the square root of the estimated variance.

\[  \mbox{StdErr}(\widehat{{\bar{Y}}})= \sqrt {\widehat{V}(\widehat{\bar{Y}})}  \]