The jackknife procedure has a number of properties that make it particularly suited for the analysis of NAEP data. When properly applied, a jackknife estimate of the variability of a linear estimator (such as a total) will be the same as the standard textbook variance estimate specified for the sample design (if the first-stage units were sampled with replacement and approximately so otherwise).
Through the creation of student replicate weights, the jackknife procedure allows the measurement of variability attributable to the use of raking and other weight adjustment factors that are dependent upon the observed sample data. Once these replicate weights are derived, it is a straightforward matter to obtain the jackknife variance estimate of any statistic.
The jackknife procedure in this application is based upon the development of a set of jackknife replicate weights for each assessed student, or school depending upon the file involved. The replicate weights are developed in such a way that unbiased estimates of the sampling variance of an estimate result, with an adequate number of degrees of freedom to be useful for purposes of making inferences about the parameter of interest.
The estimated sampling variance of a parameter estimator t is the sum of M squared differences (where M is the number of replicate weights developed):
where ti denotes the estimator of the parameter of interest, obtained using the ith set of replicate weights in place of the original full sample estimate.