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 SURVEYLOGISTIC.

Using the notation described in the section Notation, the estimated covariance matrix of model parameters $\hat{\btheta }$ by the Taylor series method is

\[  \widehat V(\hat{\btheta })= \widehat{\mb {Q}}^{-1} \widehat{\mb {G}} \widehat{\mb {Q}}^{-1}  \]


$\displaystyle  \widehat{\mb {Q}}  $
$\displaystyle  =  $
$\displaystyle  \sum _{h=1}^ H\sum _{i=1}^{n_ h} \sum _{j=1}^{m_{hi}} w_{hij}{ \widehat{\mb {D}}_{hij}} \left(\textrm{diag}({\hat\bpi }_{hij})- {\hat\bpi }_{hij}{{\hat\bpi }_{hij}}’\right)^{-1} \widehat{\mb {D}}_{hij}’ $
$\displaystyle \widehat{\mb {G}}  $
$\displaystyle  =  $
$\displaystyle  \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}_{hi\cdot } $
$\displaystyle = $
$\displaystyle  \sum _{j=1}^{m_{hi}} w_{hij}\widehat{\mb {D}}_{hij} \left(\textrm{diag}({\hat\bpi }_{hij})- {\hat\bpi }_{hij}{{\hat\bpi }_{hij}}’\right)^{-1} (\mb {y}_{hij}-{\hat\bpi }_{hij}) $
$\displaystyle \bar{\mb {e}}_{h\cdot \cdot }  $
$\displaystyle = $
$\displaystyle  \frac1{n_ h}\sum _{i=1}^{n_ h}\mb {e}_{hi\cdot }  $

and $\mb {D}_{hij}$ is the matrix of partial derivatives of the link function $\mb {g}$ with respect to $\btheta $ and $\widehat{\mb {D}}_{hij}$ and the response probabilities ${\hat\bpi }_{hij}$ are evaluated at $\hat{\btheta }$.

If you specify the TECHNIQUE=NEWTON option in the MODEL statement to request the Newton-Raphson algorithm, the matrix $\widehat{\mb {Q}}$ is replaced by the negative (expected) Hessian matrix when the estimated covariance matrix $\widehat V(\hat{\btheta })$ is computed.