The SURVEYREG Procedure

Taylor Series (Linearization)

The Taylor series (linearization) method is the most commonly used method to estimate the covariance matrix of the regression coefficients for complex survey data. It is the default variance estimation method used by PROC SURVEYREG.

Use the notation described in the section Notation to denote the residuals from the linear regression as

\[ \mb {r}=\mb {y}-\mb {X}\hat{\bbeta } \]

with $r_{hij}$ as its elements. Let the $p\times p$ matrix $\mb {G}$ be defined as

\[  \mb {G}=\frac{n-1}{n-p} \sum _{h=1}^ H { \frac{n_ h(1-f_ h)}{n_ h-1} \sum _{i=1}^{n_ h} { (\mb {e}_{hi\cdot }-\bar{\mb {e}}_{h\cdot \cdot })’ (\mb {e}_{hi\cdot }-\bar{\mb {e}}_{h\cdot \cdot }) } }  \]


$\displaystyle  \mb {e}_{hij}  $
$\displaystyle = $
$\displaystyle  w_{hij} r_{hij} \mb {x}_{hij}  $
$\displaystyle  \mb {e}_{hi\cdot } $
$\displaystyle = $
$\displaystyle  \sum _{j=1}^{m_{hi}}\mb {e}_{hij}  $
$\displaystyle \bar{\mb {e}}_{h\cdot \cdot }  $
$\displaystyle = $
$\displaystyle  \frac1{n_ h}\sum _{i=1}^{n_ h}\mb {e}_{hi\cdot }  $

The Taylor series estimate of the covariance matrix of $\hat{\bbeta }$ is

\[  \widehat{\mb {V}}(\hat{\bbeta }) = (\mb {X}’\mb {W}\mb {X})^-\mb {G}(\mb {X}’\mb {W}\mb {X})^-  \]

The factor $(n-1)/(n-p)$ in the computation of the matrix $\mb {G}$ reduces the small sample bias associated with using the estimated function to calculate deviations (Hidiroglou, Fuller, and Hickman, 1980). For simple random sampling, this factor contributes to the degrees of freedom correction applied to the residual mean square for ordinary least squares in which p parameters are estimated. By default, the procedure use this adjustment in the variance estimation. If you do not want to use this multiplier in variance estimation, you can specify the VADJUST=NONE option in the MODEL statement to suppress this factor.