Jackknife Method

The jackknife method of variance estimation deletes one PSU at a time from the full sample to create replicates. This method is also known as the delete-1 jackknife method because it deletes exactly one PSU in every replicate. The total number of replicates R is the same as the total number of PSUs. In each replicate, the sampling weights of the remaining PSUs are modified by the jackknife coefficient $\alpha _ r$. The modified weights are called replicate weights.

Let PSU i in stratum $h_ r$ be omitted for the rth replicate; then the jackknife coefficient and replicate weights are computed as

\[  \alpha _ r = \left\{  \begin{array}{ll} \frac{ n_{h_ r} -1 }{n_{h_ r}} &  \text {for a stratified design} \\[0.1in] \frac{R-1}{R} &  \text {for designs without stratification} \end{array} \right.  \]


\[  w_{hij}^{(r)} = \left\{  \begin{array}{ll} w_{hij} &  \text {if observation unit } j \text { is not in donor stratum } h_ r \\ 0 &  \text {if observation unit } j \text { is in PSU } i \text { of donor stratum } h_ r \\ w_{hij} / \alpha _ r &  \text {if observation unit } j \text { is not in PSU } i \text { but in donor stratum } h_ r \end{array} \right.  \]

You can use the VARMETHOD=JACKKNIFE(OUTJKCOEFS=) method-option to store the jackknife coefficients in a SAS data set and use the VARMETHOD=JACKKNIFE(OUTWEIGHTS=) method-option to store the replicate weights in a SAS data set.

If you provide your own replicate weights with a REPWEIGHTS statement, then you can also provide corresponding jackknife coefficients with the JKCOEFS= option. If you provide replicate weights with a REPWEIGHTS statement but do not provide jackknife coefficients, then the procedure uses $(R-1)/R$ as the default jackknife coefficient for every replicate, where R is the total number of replicates.

Let $\hat{\bbeta }$ be the estimated proportional hazards regression coefficients from the full sample, and let ${\hat{\bbeta }_ r}$ be the estimated regression coefficients for the rth replicate. PROC SURVEYPHREG estimates the covariance matrix of $\hat{\bbeta }$ by

\[  \widehat{\mb {V}}(\hat{\bbeta }) = \sum _{r=1}^ R \alpha _ r \left( {\hat{\bbeta }_ r} - \hat{\bbeta } \right) \left( {\hat{\bbeta }_ r} - \hat{\bbeta } \right)’  \]

with $R-H$ degrees of freedom, where R is the number of replicates and H is the number of strata, or R – 1 when there is no stratification.

If you specify the CENTER=REPLICATES method-option, then PROC SURVEYPHREG computes the covariance matrix of $\hat{\bbeta }$ by

\[  \widehat{\mb {V}}(\hat{\bbeta }) = \sum _{r=1}^ R \alpha _ r \left( {\hat{\bbeta }_ r} - \overline{\hat{\bbeta }_ r} \right) \left( {\hat{\bbeta }_ r} - \overline{\hat{\bbeta }_ r} \right)’  \]

where $\overline{\hat{\bbeta }_ r}$ is the average of the replicate estimates as follows:

\[  \overline{\hat{\bbeta }_ r} = \frac{1}{R} \sum _{r=1}^ R \hat{\bbeta _ r}  \]

If one or more components of $\hat\bbeta _ r$ cannot be calculated for some replicates, then the variance estimator uses only the replicates for which the proportional hazards regression coefficients can be estimated. Estimability and nonconvergence are two common reasons why $\hat\bbeta _ r$ might not be available for a replicate sample even if $\hat\bbeta $ is defined for the full sample. Let $R_ a$ be the number of replicates where $\hat\bbeta _ r$ are available, and let $R-R_ a$ be the number of replicates where $\hat\bbeta _ r$ are not available. Without loss of generality, assume that $\hat{\bbeta }_ r$ is available only for the first $R_ a$ replicates; then the jackknife variance estimator is

\[  \widehat{\mb {V}}(\hat{\bbeta }) = \sum _{r=1}^{R_ a} \alpha _ r \left( {\hat{\bbeta }_ r} - \hat{\bbeta } \right) \left( {\hat{\bbeta }_ r} - \hat{\bbeta } \right)’  \]

with $R_ a - H$ degrees of freedom, where H is the number of strata. Alternatively, you can use the VADJUST=AVGREPSS option in the MODEL statement to use the average sum of squares for the invalid replicate samples. See Variance Adjustment Factors for details.